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Complete Gradient Refractive Index Lens Schematic Human Eye Model by Jesús Emmanuel Gómez Correa A dissertation submitted in partial fulfillment of the requirements for the PhD Degree in Optics at the Instituto Nacional de Astrofísica, Óptica y Electrónica Thesis Advisor: Dr. Sabino Chavez Cerda Researcher at INAOE Mayo 2015 Tonantzintla, Puebla c INAOE 2015 "Empieza haciendo lo necesario, continúa haciendo lo posible; y de repente te encontrarás haciendo lo imposible". San Francisco de Asís Abstract A new theoretical schematic model of the human eye is introduced. The new model considers that the human eye crystalline is a gradient index lens composed by two oblate half spheroids of different heights. By modifying the spherically symmetric Luneburg model for a gradient index lens, we created a model for the anterior and posterior half spheroids matching the corresponding geometric and gradient index boundary conditions at the plane of fusion of the spheroids. We tested the imaging capabilities of our model and found that it is more realistic compared with those reported in the literature concluding that the gradient index is dynamic and cannot be modeled by one single explicit equation. Resumen Un nuevo modelo teórico del ojo humano es introducido. El nuevo modelo considera que el cristalino del ojo humano es una lente con índice de refracción gradiente, la cual está compuesta por dos esferoides achatadas con diferente radio. Al modificar el modelo de la lente simétrica de Luneburg esférica, creamos un modelo para las esferoides anterior y posterior, las cuales se ajustan en el eje de fusión con las condiciones de frontera correspondientes a la geometría y al índice de refracción. Hemos probado las capacidad de formación de imágenes en el ojo completo y encontramos que es más realista en comparación con los reportados en la literatura, en conclusión el índice del gradiente es dinámico y no puede ser modelado por una sola ecuación explícita. To my dad, my mom, and my son. Para mi papá, mi mamá y mi hijo. Acknowledgments/ Agradecimientos • Macario Gómez: Gracias por todo el apoyo que me has dado durante toda mi vida pero sobretodo por ser el mejor papá de todo el universo y enseñarme a ser el hombre que soy hasta el día de hoy. Porque gracias a sus consejos de todos los días, cariño, dedicación, caricias y sobre todo por su amor he logrado cumplir mis metas y sueños que me he propuesto. Espero algún día llegar hacer un papá como tu lo eres conmigo. Papá Te Amo como no tienes idea. • Victoria Correa: Gracias por buscar siempre que esté bien en todos los aspectos y por ser la mejor mamá de todo el universo y llevarme de la mano día a día sin importar que sucediera y por comprenderme durante toda mi vida. Gracias a sus caricias, por cuidarme, por aconsejarme y por buscar mi bienestar, pero sobre todo por su gran amor. Espero siempre darle alegrías sin importar lo que suceda. Mamá Te Amo como no tienes idea. • Iker Emmanuel: Por darme la alegría para poder seguir adelante y ser mi fuente de inspiración. Te amo demasiado. • Anel: Gracias por todo lo bueno y lo malo que me has dado a lo largo de estos años, pero sobretodo muchas gracias por darme lo mejor que tengo en mi vida. Te quiero mucho. • Ian y Lars: Por ser como son conmigo, por quererme y por compartir horas jugando Beisbol y por todas las alegrías y momentos bonitos a mi lado. Los quiero mucho. • A mis hermanas: Conchi y Blanca: Por hacerme la vida más feliz y estar conmigo cuando más las he necesitado. Por apoyarme cada día que pasa y darme ánimos x Chapter 0. Acknowledgments/ Agradecimientos para seguir adelante. Por quererme y amarme tanto. Porque sin ustedes mi vida sería complicada en todos los aspectos. Gracias por demostrarme su amor día a día. Las Adoro y las Amo hermanitas. • A mis hermanos: Sergio y Arcenio: Gracias por ser los hermanos que nunca tuve, sin embargo en ustedes los he encontrado. Por considerarme su hermano aunque sea su cuñado, por divertirnos y disfrutar de la vida con muchas sonrisas y sobre todo por quererme tanto. Los quiero demasiado hermanitos. • A mis sobrinos: Cheito, Katy, Karol, Arcenito y Ale: Por ser mi alegría de todos los días y compartir hermosos momentos a mi lado y por demostrarme cuanto me quieren cada día que pasa. Los adoro como no tienen idea. • Dr. Adrián Carbajal: Gracias por sus enseñanzas y por la gran calidad que tienen sus clases, porque gracias a estas clases yo me motivé a seguir adelante en esta área. Muchas Gracias, por todo el conocimiento que me ha compartido a lo largo de estos años de conocernos. • Dr. Jesús Rogel-Salazar: Muchas gracias por apoyarme en el artículo de guías de ondas y por ayudarme en los detalles de Latex para que esta tesis fuera posible. • Dr. David Sánchez de la Llave: Muchas gracias por apoyarme durante todo el doctorado, gracias por los comentarios a mi trabajo y gracias por buscar lo mejor para mí en cuestiones académicas. Muchas gracias Doctor. • Barbara Pierscionek: Thank you very much for your help to improve the papers of this thesis and to believe that this work is important in the visual optics. • A los doctores: De la Llave, Cornejo, Arrizón, Balderas y xi Malacara: Por tomarse el tiempo de revisar esta tesis, de hacerme comentarios para mejorar este trabajo y por aceptar ser mis sinodales. • A mis amigos: Julio Ramirez San Juan: Gracias Julio por esas largas horas que nos pasamos hablando de box y de fútbol, gracias por tus comentarios tan alentadores que me ayudaron ha seguir adelante en gran camino que fue desde la maestría hasta el doctorado. Muchas gracias y sabes que aquí tienes un amigo. Sandra: Por reírnos de cualquier cosa, por ayudarme, aconsejarme y por ser una de mis mejores amigas de una buena parte de mi vida. Gracias Sandrita y aquí sabes muy bien que tienes a un muy buen amigo. José Adán: Por tu amistad dentro y fuera del INAOE, la cual, día a día se va haciendo más Grande y por la gran recomendación que me diste para salir a tiempo del doctorado. Julio García y Karla Sánchez: Gracias por dejarme compartir mis ocurrencias, alegrías y tristeza con ustedes. Muchas Gracias a los dos y por tantas salidas a comer, cenar y al cine. Los quiero mucho. Marco Canchola: Por darme tu apoyo cada día y por compartir muy buenos momentos conmigo y hacerme sentir que tengo un muy buen amigo de verdad. Gracias viejo sabes que te quiero mucho. Juan Pablo (JP): Gracias por tantas horas compartidas en el cubículo, porque sin tu amistad hubiera sido complicado estar tantas horas en el INAOE. Gracias mi buen JP. Jorge Ugalde: Gracias por tu amistad y por compartir tantas horas en el mismo cubo, un gran futuro te espera. Juan Carlos: Gracias por las largas horas que hemos pasado riéndonos y disfrutando de la vida. Gracias mi estimado Juan Carlos. Sergio Mejia: Por divertirnos tanto y pasar varias horas riéndonos, por Tu xii Chapter 0. Acknowledgments/ Agradecimientos apoyo y sinceridad, por la gran compañía que me has dado y por tu gran amistad Eicela: Por ser mi amiga y por todos los consejos que me ha dado en estos 6 años de conocernos y por facilitarme la vida en el INAOE. Paty: Por el gran apoyo que me dio a lo largo de estos 6 años en y por facilitarme la vida en el INAOE. • CONACYT: Por otorgarme la beca de doctorado 235164, para que fuera posible obtener el grado. • INAOE: Por darme las comodidades para poder trabajar a gusto y darme el privilegio de estudiar un doctorado en esta reconocida institución. Special Acknowledgment Dr. Sabino Chávez-Cerda: Le quiero agradecer por las tres tesis que me ha dirigido en estos 8 años. Pero sobretodo le quiero agradecer por todos los consejos que me ha dado tanto personales como académicos. Gracias por enseñarme que el conocimiento no se mide con una regla de 30 centímetros si no que el conocimiento se mide con la regla más grande que se tiene. Después de estos 8 años de conocernos yo no lo considero mi asesor, yo lo considero un muy buen amigo en mi vida. Muchas Gracias por todo, espero que sigamos colaborando como hasta el día de hoy lo hemos hecho. Muchas Gracias Dr. Sabino. Contents i Abstract iii Resumen v vii Acknowledgments/ Agradecimientos ix Special Acknowledgment xiii 1 Preface 1 2 Gradient Index Media 5 2.1 The Ray Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The Linear Gradient Index Medium . . . . . . . . . . . . . . . . . 10 2.3 The Radial Cylindrical Gradient Index Medium . . . . . . . . . . 14 2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 The Spherical Luneburg Lens 19 3.1 The Spherical Gradient Index Medium . . . . . . . . . . . . . . . 19 3.2 The Ray Integral Equation . . . . . . . . . . . . . . . . . . . . . . 22 3.2.1 26 3.3 Generalized Snell Law for Inhomogeneous Media with Spherical Symmetry . .q. . . . . . . . . . . . . . . . . . . Rays in a medium with n (r) = . . . . . . . . . . . . . . 29 3.4 Maxwell’s Fisheye . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.5 The Luneburg Lens . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.6 The Generalized Luneburg Lens . . . . . . . . . . . . . . . . . . . 49 3.7 The Elliptical Luneburg Lens . . . . . . . . . . . . . . . . . . . . 52 C+ 1 r xvi Contents . . . . 53 55 57 60 4 The Human Eye As An Optical System 4.1 The Human Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Refracting components of the human eye: Cornea and Lens 4.1.2 Pupil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Retina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Schematics Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 62 63 67 68 70 73 5 Schematic Eye with Composite Luneburg Crystalline 5.1 Composite Modified Luneburg Lens . . . . . . . . . . . . . . . . . 5.2 Schematic Luneburg Eye . . . . . . . . . . . . . . . . . . . . . . . 5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 76 88 94 6 Conclusions and Future Work 95 List of Figures 99 3.8 3.9 3.7.1 Linear Transformation of The Spherical 3.7.2 GRIN of The Elliptical Luneburg Lens Ray Tracing . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . Bibliography Luneburg . . . . . . . . . . . . . . . . . . Lens . . . . . . . . . . . . . 103 Chapter 1 Preface Some laws of optics have been known for the centuries, for example, the Greeks were aware with some of the properties of light, they understood the law of reflection. However, they were unable to understand the nature of the eye because the eye has an extraordinary complexity. They believed that light was emitted by the eye and only produced a visual response when the emitted rays struck an object. Many centuries passed before it was realized that light passes from the object to the eye and not from the eye to the object [1]. Nowadays, the human eye is considered an optical system and its properties are studied by the Visual Optics. Many studies from the Visual Optics establish that the eye is very complex due to the fact that its refractive surfaces are not strictly spherical and its lens has a gradient refractive index (GRIN). At the present time, one of the challenges in Visual Optics is creating a schematic model of the human eye. Since the GRIN profile may change from one individual to another, the human lens has been the most challenging task in order to have a complete and realistic schematic model. Many generic expressions for the refractive index based in biometric data of animal and human lens have been proposed over the years that provide a good estimation of the actual GRIN distribution [2, 3, 4]. In some models the anterior and posterior faces are considered to be symmetric [2] while in more recent models a realistic asymmetry of the faces is taken into account [3, 4]. In the latter, the GRIN is described by two different equations with respect to a plane or a curved surface that intersects the human lens at its equator [3, 4, 5]. A drawback is that a ray (or its derivative) travelling in the proposed GRIN distribution may undergo a discontinuity at any of such surfaces. 2 Chapter 1. Preface The most famous lens with a gradient refractive index is the Spherical Luneburg Lens. This lens was introduced in 1944 by Rudolf K. Luneburg in the book "Mathematical Theory of Optics" [6]. A Luneburg lens is a GRIN lens with spherical geometry with normalized unitary radius and stigmatic property that focuses a sphere into a sphere [7]. As a particular example, Luneburg solved the problem for incident rays at the anterior surface coming from infinity (infinite sphere) focused at the opposite side surface of the spherical lens (sphere with radius r = 1). The importance of this lens in visual optics is due that four decades ago it was proposed that the human lens can be studied as a Luneburg lens [8]. It is very important to say that the Maxwell’s Fisheye Lens also has been proposed for represent the GRIN of the human lens [9]. However, the Maxwell’s Fisheye has the drawback that only points within or on the surface of the lens are sharply imaged while that the lens proposed by Luneburg images sharply every parallel bundle of rays incident on the outer surface of the lens [10]. In this work we propose a more realistic theoretical schematic eye using the idea that the human lens can be represented as a composite asymmetric Luneburg lens calculating a continuos GRIN function from the anterior to the posterior conicoid faces. Our model takes into account the obliquity of the rays coming from the cornea illuminated by a source at infinity. We have used the biometric parameters provided by A. M. Rosen, et al. [11] but our method allows obtaining custom made GRIN distributions given the corresponding biometric parameters. This work is divided in six chapters: The Chapter 1 is this introduction. In the Chapter 2 we make a analysis for the gradient mediums with planar and cylindrical geometry. Right after, the spherical gradient index medium is explained and we study fully the classical Luneburg lens and the Elliptical Luneburg Lens in the Chapter 3. The Chapter 4 explains the human eye as an optical system, where the optical components are described. Also, in this Chapter the most importants schematic human eye models are presented and its properties and its drawbacks are studied. In the Chapter 5 we introduce a new model of the human lens and a new schematic human eye model using this new lens. Finally, in the last 3 Chapter we sum up the material and project future work and perspectives of our investigations. Chapter 2 Gradient Index Media Gradient index media have many applications in telecommunications to model slim antennas and in Visual Optics to model the human eye lens. Because of this, it is necessary to start by defining: what is a gradient index medium?. The gradient index medium is an inhomogeneous medium in which the refractive index varies from point to point [12]. The gradient index type depends on the geometry of the medium is the gradient index type. The three most important types of gradient index are: The linear gradient, The radial gradient and The spherical gradient. We know that rays propagate as straight lines in a homogeneous medium, i e, in mediums that have a constant refractive index. For a medium with Gradient Refractive index, the rays are represented as curves. The shape of these curves are given by the solution of the Ray Equation. For this reason, we will begin this section by describing Fermat’s Principle in order to define the ray equation so that we will be able to analyze the different types GRIN types. 2.1 The Ray Equation If we take any two points P1 and P2 , these points can be connected with an infinite number of curves. If each curve travels a distance ds with a velocity v = c/n (x, y, z), where n (x, y, z) is the space dependent refractive index in a given point and c represents the speed of light in free space. The time taken to transverse the geometrical path ds in a medium of refractive index n (x, y, z) is 6 Chapter 2. Gradient Index Media given by n (x, y, z) ds (2.1) c where the distance ds is a small part of a curve between P1 and P2 , as show in the Fig. 2.1. dt = Now, if t represents the total time taken by the ray to transverse the path from P1 to P2 along the curve, we have t= 1X ni (x, y, z) dsi c i (2.2) dsi represents the ith ds along of a curve, to each dsi corresponds only one ni (x, y, z). We can observe that this equation represents the time for a discrete refractive index, i.e., we have the sum for all segments dsi . Figure 2.1: Fermat’s Principle. If we have a continuously refractive index, where the refractive index is changing by each point, the equation can be rewritten as 1 t= c Z P2 n (x, y, z) ds P1 (2.3) 2.1. The Ray Equation 7 from this equation is very easy see that the path length of ray is P2 Z n (x, y, z) ds. L= (2.4) P1 Once obtained this result we can define Fermat’s Principle: ”The actual ray path between two points is the one for which the optical path length is stationary with respect to variations of the path". This represents that Z P2 n (x, y, z) ds = 0. δ (2.5) P1 where δ variation of the integral means that it is a variation of the path of the integral such that the endpoints P1 and P2 are fixed. The equation (2.5) represents that the ray path is an extremum and it may be maxima, minimum or stationary. For example, if n (x, y, z) is a constant at each point, i e, a homogenous medium, the rays are straight lines that correspond to a minimum value of the optical path connecting two points in the medium. The trajectory of rays can be compared with the trajectory of the particles in classical mechanics. The Hamilton’s principle in classical mechanics establishes that the trajectory of a particle between times t1 and t2 is such that Z t2 Ldt = 0. δ (2.6) t1 where L is called the Lagrangian. The difference between Eq. (2.5) and Eq. (2.6) is that the integration is over time in the first equation and in the second one is over the space. This difference can be eliminated if the infinitesimal arclenght ds is rewritten as s 2 q 2 dx dy 2 2 2 ds = (dx) + (dy) + (dz) = dz + +1 (2.7) dz dz 8 Chapter 2. Gradient Index Media if ẋ = dx/dz and ẏ = dy/dz, then ds = dz p ẋ2 + ẏ 2 + 1 (2.8) where dz is equivalent to dt in the Eq. (2.6) and in this case, Eq. (2.5) Z P2 n (x, y, z) δ p 1 + ẋ2 + ẏ 2 dz = 0. (2.9) P1 From this equation we can define an optical Lagrangian as L (x, y, ẋ, ẏ, z) = n (x, y, z) p 1 + ẋ2 + ẏ 2 . (2.10) As in classical mechanics, the solution of this problem must satisfy the EulerLagrange equations, in this case these equations are given by and d dz d dz ∂L ∂ ẋ ∂L ∂ ẏ = ∂L ∂x (2.11) = ∂L . ∂y (2.12) If we substitute Eq. (2.10) into Eq (2.11), we obtain " d n (x, y, z) ẋ p dz 1 + ẋ2 + ẏ 2 # = p ∂n (x, y, z) 1 + ẋ2 + ẏ 2 ∂x (2.13) from Eq. (2.7) is easy to see that Eq. (2.13) can be reduced to because d dx ∂n (x, y, z) n (x, y, z) = ds ds ∂x (2.14) d 1 d =p . ds 1 + ẋ2 + ẏ 2 dz (2.15) 2.1. The Ray Equation 9 Similarly for y and z components, we have and d dy ∂n (x, y, z) n (x, y, z) = ds ds ∂y (2.16) dz ∂n (x, y, z) d n (x, y, z) = ds ds ∂z (2.17) If we define ~r as a position vector, we can obtain a vector equation which contains equations (2.14), (2.16) and (2.17), i e, d d~r n (x, y, z) = ∇n (x, y, z) ds ds (2.18) this equation is known as the Ray Equation. It is very important to say that Eq. (2.18) can also be obtained from the Eikonal equation. The Eikonal equation is important in geometrical optics because it describes the phasefront of a wave. In order to obtain the Ray Equation from the Eikonal equation, we start with the latter one as [14] [∇S (x, y, z)]2 = n2 (x, y, z) (2.19) where S (x, y, z) = Constant and S (x, y, z) can be interpreted as the function describing the phasefront of the wave. Now, if we defined a unit vector normal to the phase fronts and tangent to the light ray ŝ, i.e., along the ray, then ∇S (x, y, z) = n (x, y, z) ŝ (2.20) d d d~r [∇S (x, y, z)] = n (x, y, z) ds ds ds (2.21) where ŝ = d~r/ds, hence 10 Chapter 2. Gradient Index Media since as d/ds = ŝ · ∇, this equation can be represented as d d~r n (x, y, z) = (ŝ · ∇) ∇S ds ds (2.22) We want to know the direction of the rays, hence ∇ [∇S (x, y, z)]2 = ∇n2 (x, y, z) (2.23) ∇ [∇S (x, y, z)]2 = 2n (x, y, z) (ŝ · ∇) ∇S (2.24) ∇n2 (x, y, z) = 2n (x, y, z) ∇n (x, y, z) (2.25) (ŝ · ∇) ∇S = ∇n (x, y, z) (2.26) d~r d n (x, y, z) = ∇n (x, y, z) ds ds (2.27) using and we obtain from the Eq. (2.22) we end up with the Ray equation, which was obtained from the Eikonal equation. Using the Eq. (2.27), we will study the axial and the radial gradients index medium in sections 2.2 and 2.43, respectively. 2.2 The Linear Gradient Index Medium In the linear gradient index medium, the refractive index varies in a continuous way along the optical axis of the inhomogeneous medium. The isoindicial surfaces (surfaces of constant index) are planes that are parallel to the optical axis, as is show in Fig. 2.2. 2.2. The Linear Gradient Index Medium 11 Figure 2.2: The linear gradient index medium. A linear refractive index in the x-axis is given by ( 2 n (x) = n20 − αx x > 0 n20 x < 0 (2.28) and in the y-axis by ( 2 n (y) = n20 − αy y > 0 n20 y < 0 (2.29) where α is any small number and n0 is the refractive index in x = 0 or y = 0. These refractive indixes are represented in the Fig. 2.2. To learn how rays propagate within a gradient index medium is necessary to solve the ray equation, which is d~r d n (x, y, z) = ∇n (x, y, z) ds ds (2.30) this equation can be rewritten as a set of three equations, shown in equation 12 Chapter 2. Gradient Index Media (2.31). ∂n(x,y,z) d n (x, y, z) dx = ∂x ds ds dy d n (x, y, z) ds = ∂n(x,y,z) ds ∂y ∂n(x,y,z) d dz n (x, y, z) = . ds ds ∂z (2.31) If the refractive index does not depend on z, we have d dz ∂n (x, y) n (x, y) = = 0. ds ds ∂z (2.32) implying that dz =β (2.33) ds where β is an invariant of the ray path. From Fig. 2.3 we can observe that n (x, y) Figure 2.3: An arc lenght along the ray path. dz = cos θ (x, y) ds (2.34) ⇒ β (x, y) = n (x, y) cos θ (x, y) . (2.35) where θ (x, y) is the angle make the optical axis (z-axis) and the tangent line to the curve at one point (x, y). Using Eqs. (2.8) and (2.35), we can rewrite (2.31) 2.2. The Linear Gradient Index Medium as 13 ∂n2 (x, y) d2 x 1 = 2 dz 2 2β (x, y) ∂x (2.36) d2 y ∂n2 (x, y) 1 = dz 2 2β 2 (x, y) ∂y (2.37) dβ (x, y) 1 ∂n2 (x, y) = = 0. dz 2β (x, y) ∂z (2.38) If the linear refractive index is given by Eq. (2.28), then Eq. (2.36), for x > 0, becomes α d2 x =− 2 (2.39) 2 dz 2β and the general solution is x (z) = − α 2 z +C 4β 2 (2.40) where C = Constant, β = n0 cos θ0 and θ0 = θ (0, 0). In this case, θ0 is the incident angle of the ray at the point x = 0 and y = 0. Thus for a ray that passes through the points x = 0 and z = 0, the constant C, takes the value of C = z tan θ0 (2.41) therefore x (z) = − α 2 z + z tan θ0 . 4β 2 (2.42) We can observe in the Fig. 2.4 that if x < 0 the trajectory of the ray is a straight line and if x > 0 the trajectory is a parabola. The solution for a linear refractive index in the y-axis is the same that for the x-axis, for this reason, it is not necessary solve it. 14 Chapter 2. Gradient Index Media Figure 2.4: Solution for the linear gradient index medium. 2.3 The Radial Cylindrical Gradient Index Medium In the radial cylindrical gradient index medium, the index profile has a maximum at the cylinder axis and decreases continuously from the axis to the periphery along the transverse direction in such a way that the isoindicial surfaces are concentric cylinders about the optical axis [12], as is show in Fig. 2.5. Figure 2.5: The radial gradient index medium. The radial cylindrical gradient index is represented by 2.3. The Radial Cylindrical Gradient Index Medium 2 2 2 r x + y 2 2 n1 1 − 2∆ = n1 1 − 2∆ 0<r<a a2 a2 n2 (r) = n1 [1 − 2∆] r>a n22 = 15 (2.43) where ∆ is any small number, n1 is the refractive index in r = 0, n2 is a constant, a is the maximum radius of the refractive gradient index (r = a). This equation is known as parabolic refractive index and it has no dependency on z. To find the solution of the rays in radial cylindrical gradient index medium, we substitute Eq. (2.43) into Eqs. (2.36) and (2.37), then d2 x dz 2 d2 y dz 2 where Γ = √ n1 2∆ aβ + Γ2 x = 0 + Γ2 y = 0 (2.44) and the solutions are given by x = A sin Γz + B cos Γz y = C sin Γz + D cos Γz. (2.45) The easiest solution is when we only work in a sagittal plane, therefore, it is necessary to impose initial launching conditions [13], like these x (z = 0) = 0 y (z = 0) = 0 dx | =0 dz z=0 dy | = 0. dz z=0 (2.46) These launching conditions make that the meridional rays are confined in the sagittal plane; we may mention that the meridional rays are defined such that they are confined in a plane and intersect the sagittal plane. Using these conditions 16 Chapter 2. Gradient Index Media the solution is of the form x= aβ √ n1 2∆ tan θ0 sin h √ 2∆ z a cos θ0 i = a√ sin θ0 2∆ sin h √ 2∆ z a cos θ0 i (2.47) y=0 where θ0 is the incident angle of the ray at the point x = 0 and y = 0. The solution for two different values of β are shown in the Fig. 2.6. Figure 2.6: Solution for the Sagittal plane. The difference between the gradient in the sagittal plane of the radial cylindrical gradient index and the linear gradient in the same plane is that in the first the gradient is given in an interval of [−a, a] while in the second one the gradient is in an interval of [0, a], where a is any number, for this reason the solutions are different. The most complete solution is when we impose the initial launching conditions as x (z = 0) = a0 y (z = 0) = 0 (2.48) dx | = 0 z=0 dz dy | = tan θ0 dz z=0 where a0 is the distance on the x axis, which was launched the ray in the y−z plane and θ0 is the angle that the ray makes with the z axis. With these conditions, one obtains x = a0 cos Γz (2.49) y = a0 sin Γz 2.4. Conclusions 17 Equations (2.49) are represented in the Fig. 2.7. Figure 2.7: Solution for the radial gradient index medium. These solutions describe which are know as skew rays which do not remain confined to a plane and represent the solution for the radial cylindrical gradient index medium, as show in Fig. 2.7. 2.4 Conclusions The importance of this Chapter is that we have reviewed the theory of propagation of the rays in gradient media and with this we have presented the ray equation which we will be used from now on. In the next Chapter we will present the extension to a spherical gradient index. This study must be carried out carefully, due to the fact that the crystalline is a inhomogenous lens with a gradient index that can be represented as a spherical gradient index. Chapter 3 The Spherical Luneburg Lens This chapter provides a description of light propagation through the spherical gradient medium. We will analyse the spherical Luneburg lens and Maxwell’s Fish Eye who are the most famous examples of lenses with spherical gradient index. The difference between the spherical Luneburg lens and the Maxwell’s Fisheye is the representation of the mathematical approach of the refractive index and that the Maxwell fish eye lens has a drawback; only points within or on the surface of the lens are sharply imaged. The lens proposed by Luneburg images sharply every parallel bundle of rays incident on the outer surface of the lens [10]. This analysis will be done with the integral ray equation because we want to know the form of the ray in an interval, i.e., from point P0 to point P1 . The importance of this analysis is due to the fact that both lenses have been proposed to represent the refractive gradient index of the crystalline. It is very important to mention that it has been proposed, but for the case of the Luneburg lens does not exist a complete analysis. For this reason, in this chapter we will focus our attention to study the spherical Luneburg lens. 3.1 The Spherical Gradient Index Medium Lets suppose that we have a refractive index that is constant on concentric spheres, i.e., each sphere has a thickness of dr and in this thickness, the refractive index is constant. The smaller sphere has the highest refractive index and it decreases from the inner sphere to a sphere of radius a, then we can say that we have a spherical gradient index medium, see Fig. 3.1. 20 Chapter 3. The Spherical Luneburg Lens Figure 3.1: The spherical gradient. The spherical gradient index can be represented by different mathematical expressions. An interesting case is when we define the spherical gradient index as C + 1 0 < r < a 2 r n (r) = n22 r>a (3.1) p where r = x2 + y 2 + z 2 , C is a constant, n2 = C − a1 , where a is the maximum radius of the refractive gradient index. This medium is interesting because the light rays in this medium are identical with the paths of particles which move in a Coulomb field of potencial φ = −1/2r, and with the energy C/2. The variation of the spherical gradient index given by Eq. (3.1) is represented in the Fig. 3.2 for different values of C. Notice that the spherical gradient index has a singularity at r = 0, for this reason, the radius is r > 0 in Eq. (3.1). The singularity at r = 0 is not a problem for the ray tracing in this medium, because we will find a ray integral equation that depend on a new function ρ (r). This function will be define in the section 3.3. At this point, it is important to answer: which is the the ray integral equation?, so we can solve any gradient index with spherical symmetry. In the next section we will answer this question and later we will solve the refractive gradient index 3.1. The Spherical Gradient Index Medium Figure 3.2: The Index Variation. 21 22 Chapter 3. The Spherical Luneburg Lens given by the Eq. (3.1). 3.2 The Ray Integral Equation We know from the literature about media with radial symmetry that the rays are confined to a single plane [13]. In this work we are interested in knowing the solutions the a x − y plane, for this reason it is possible to reduce this problem of three variables into a two variables problem. Now, if we consider a continuous medium with a refractive index that has radial symmetry n (r), as shown in the Fig. 3.3, we can easily know the ray integral equation. Figure 3.3: Medium with radial symmetry The ray integral equation is obtained from the Eikonal equation, then we can rewrite the Eq. (2.19) as ∂ψ ∂x 2 + ∂ψ ∂y 2 = n2 (x, y) (3.2) as we are working in a medium with radial symmetry, we make a change of 3.2. The Ray Integral Equation 23 variables of the form x = r cos θ y = r sin θ (3.3) and p r= we obtained ∂ψ ∂r 2 1 + 2 r x2 + y 2 ∂ψ ∂θ 2 (3.4) = n2 (r) (3.5) This equation can be solved with the method of Separation of Variables, then the Eq. (3.5) can be rewritten as ∂ψ =r ∂θ s n2 (r) − ∂ψ ∂r 2 (3.6) we can observe that the left side of the equation does not depend on r, then dψ(θ) dθ =K dψ(θ) = Kdθ R R dψ(θ) = Kdθ R R dψ(θ) = K dθ ψ(θ) = Kθ (3.7) 24 Chapter 3. The Spherical Luneburg Lens and the right of the equation does not depend on θ, we have r n2 (r) − K=r 2 K =r 2 2 n (r) − dψ(r) dr dψ(r) dr 2 2 2 − n2 (r) = − dψ(r) dr q 2 dψ(r) = n2 (r) − Kr2 dr q 2 dψ(r) = n2 (r) − Kr2 dr q R Rr 2 dψ(r) = r0 n2 (r) − Kr2 dr Rr q 2 ψ(r) = r0 n2 (r) − Kr2 dr K2 r2 (3.8) if the solution is given by therefore ψ = ψ(θ) ± ψ(r) (3.9) Z rr K2 n2 (r) − 2 dr ψ = Kθ ± r r0 (3.10) where K is for the moment an arbitrary constant. We know that the Eikonal equation describes the phase front of a wave, and the rays are lines that are normal to these wavefronts, showing the direction of energy flow at one particular point, for this reason it is necessary to apply the Jacobi Theorem to obtain the integral ray equation. This theorem is represented by ∂ψ =α (3.11) ∂K 3.2. The Ray Integral Equation 25 applying the Jacobi theorem, we have ∂ψ ∂K Rr q K2 2 = Kθ ± r0 n (r) − r2 dr Rr q 2 ∂ψ ∂ ∂ = ∂K [Kθ] ± ∂K r0 n2 (r) − Kr2 dr ∂K Rr ∂ q 2 ∂ψ ∂K = θ ± n2 (r) − Kr2 dr ∂K ∂K ∂K r 0 1/2 h Rr ∂ 2 K2 ∂ = θ ± r0 ∂K n (r) − r2 n2 (r) − ∂K −1/2 Rr 2 ∂ψ K2 2K n (r) − = θ ± dr ∂K r2 r2 r0 2K R r ∂ψ = θ ± r0 q 2 r2 K 2 dr ∂K n (r)− 2 r Rr ∂ψ q 2K = θ ± dr ∂K r0 r2 n2 (r)− K 2 r2 Rr ∂ψ = θ ± 2K r0 r2 √ 2dr 2 2 ∂K n (r)r −K Rr r ∂ψ = θ ± 2K r √ 2 dr 2 2 ∂K ∂ψ ∂K ∂ ∂K 0 α = θ ± 2K − θ 2 = ±K i dr (3.12) n (r)r −K Rr √ dr r0 r n2 (r)r2 −K 2 Rr √ dr r0 r n2 (r)r2 −K 2 α − θ = ±2K α 2 r K2 r2 Rr r0 r √ (3.13) dr n2 (r)r2 −K 2 we defined α =θ 2 θ = θ0 2 (3.14) the rays of light in the x − y plane are obtained by the following equation Z r θ − θ0 = ±K r0 dr p r n2 (r)r2 − K 2 (3.15) The latter equation is known as the integral ray equation. The integration constants θ0 and r0 are the initial coordinates of the ray on the spherical refractive gradient index. In this analysis, the constant K is very important because it represents the ray direction at the point (r0 , θ0 ), then we need to know the value of K. In the 26 Chapter 3. The Spherical Luneburg Lens next section, we will found this value. 3.2.1 Generalized Snell Law for Inhomogeneous Media with Spherical Symmetry From the literature [14], it is known that the path of rays in a medium with gradient index n = n (r) are planar curves in a plane containing the origin. We − can consider a position vector → r along the path of the curve that represents a light − ray inside the spherical refractive gradient index, and a unit vector → s tangent to the ray, as shown in the Fig. 3.4. Figure 3.4: The path of rays in a medium with gradient index − − If we consider the vector variation → r × [n (r) → s ] along the ray (curve) and we derive with respect to s, we obtain − d → d→ r d − − − − [− r × [n (r) → s ]] = × [n (r) → s]+→ r × [n (r) → s] ds ds ds (3.16) − d→ r − =→ s ds (3.17) but and from Eq. (2.27) d ds → → −− d− r n = 5→ n ds (3.18) 3.2. The Ray Integral Equation 27 therefore → − −−→ d → − − − − [− r × [n (r) → s ]] = → s × [n (r) → s]+→ r × 5 n (r) ds it is very easy to see that the first term is 0 and the equation reduces to → − −−→ d → − − [− r × [n (r) → s ]] = → r × 5 n (r) ds (3.19) (3.20) as n (r) is a function that only depend on r − → − −−→ dn (r) → r 5 n (r) = dr r (3.21) from this equation it is clear to see d → − [− r × [n (r) → s ]] = 0 ds (3.22) → − − r × [n (r) → s ] = cte (3.23) this leads us to a vector product that can be expressed as → − − − − r × [n (r) → s ] =k → r kk n (r) → s k sin ϕ (3.24) − − − where ϕ is the angle between the vectors → r and → s , and as the vector → s is a unit vector, we have − k→ r k= r (3.25) − s k= n (r) k n (r) → if we substitute these equations into Eq. (3.25) → − − r × [n (r) → s ] = rn (r) sin ϕ (3.26) rn (r) sin ϕ = cte (3.27) hence 28 Chapter 3. The Spherical Luneburg Lens and this constant is K, i.e., K = rn (r) sin ϕ (3.28) The Eq. (3.28) is known as the generalized Snell law for inhomogeneous media whit spherical symmetry. This demonstrates that the path of rays in a medium with gradient index n = n (r) are planar curves because K is always a constant with positive or negative sign and there is not sign change along the ray. Figure 3.5: Rays on Horizontal axis It is easy to calculate the variation range of K, due the fact that the maximun point of incidence can be obtained when r = 1, n = 1 and ϕi = π2 . From the generalized Snell’s law, we have that K = 1 and the minimum point is when we are on the axis of propagation, which leads us to have ϕi = 0 and for consequence K = 0, then the variation range of K is 0 ≤ K ≤ 1. (3.29) If we considered an angle α0 , which is measured relative to the normal and it is on the negative side on the same axis. We observe that the incident angle relative to the normal is given by θ − π, i.e., θ0 = π, ϕ = α0 , and K = n0 r0 sin α0 , by 3.3. Rays in a medium with n (r) = q C+ 1 r 29 substituting these values in Eq. (3.15), we obtain r Z θ − π = ±n0 r0 sin α0 r0 dr p r n2 (r)r2 − n20 r02 sin2 α0 (3.30) dr p r n2 (r)r2 − n20 r02 sin2 α0 (3.31) dr p r n2 (r)r2 − n20 r02 sin2 α0 (3.32) for α0 < π2 , we have r Z θ − π = +n0 r0 sin α0 r0 and for α0 > π2 , we have r Z θ − π = −n0 r0 sin α0 r0 If we considered that α0 is given from 0 to π2 , we obtain that 0 ≤ K ≤ n0 r0 3.3 (3.33) q Rays in a medium with n (r) = C + 1r We have that the integral ray equation is Z r θ − θ0 = ±K r0 dr p r n2 (r)r2 − K 2 (3.34) we can define a function ρ (r), that has the form ρ(r) = n(r)r. (3.35) 1 r (3.36) if n2 (r) = C + then ρ2 (r) = Cr2 + r (3.37) 30 Chapter 3. The Spherical Luneburg Lens by substituting Eq. (3.37) into Eq. (3.34) and we obtain Z r θ − π = −K r0 dr √ 2 r Cr + r − K 2 (3.38) in this case θ0 = π, because we are considering that origin of ray is on the horizontal axis. We can find the solution of the integral when we rewrite the lower term as v" r # u K 1 2 u √ − 1 2K t 1− r (3.39) Cr2 + r − K 2 = r C + 4K 2 C + 4K1 2 and by making a change of variable K r z=q r2 1 2K − C+ q C+ dr = − z θ−π =− z0 1 4K 2 K we observe that Z (3.40) 1 4K 2 Kr2 q C+ dz 1 4K 2 dz K q √ r2 C + 4K1 2 1 − z 2 therefore Z z θ−π =− z0 (3.41) dz √ 1 − z2 (3.42) (3.43) the solution is very easy if we assume then Z dz √ = 1 − z2 Z z = sin u (3.44) dz = cos udu (3.45) cos udu p = 1 − sin2 u Z du = u = arcsin z (3.46) 3.3. Rays in a medium with n (r) = hence Z z √ θ−π =− z0 Now, if arcsin z0 = β − π 2 q C+ 1 r dz = arcsin z0 − arcsin z 1 − z2 31 (3.47) is constant, we have z = sin π 2 − [θ − β] = cos (θ − β) (3.48) we substitute Eq. (3.48) into Eq. (3.40) and obtain r= 1+ √ 2K 2 4CK 2 + 1 cos (θ − β) (3.49) the latter equation determines the type of curves and these curves can be easily seen if us assume that β=0 x = r cos (θ) (3.50) y = r sin (θ) p r = x2 + y 2 with equations (3.50) we find from the Eq. (3.49) 2 C 1 √ 1 + 4CK 2 − 2 y 2 = 1 4C x − 2C K (3.51) h i √ y 2 = K 2 4Cx2 − 4x 1 + 4K 2 C + 4K 2 . (3.52) 2 or Equation 3.51 tell us that the curves are conic sections and that a given C all these conics have the same principal axes A = 1/2C. From this equation we can observe that the eccentricity is [6] 1 √ e= 1 + 4K 2 C = 2C r 1 K2 + . 4C 2 C The type of these conics is determined by the value of C, i.e.: (3.53) 32 Chapter 3. The Spherical Luneburg Lens If C > 0: Equation 3.51 represents hyperbolas with the same principal axis A = 12 C, with the point x = y = 0 as common focal point, as shown in Fig. 3.6. Figure 3.6: The rays are hyperbolas when C > 0. If C = 0: Equation 3.51 represents parabolas with x = y = 0 as common focal point, as shown in Fig. 3.7. If C < 0: Equation 3.51 represents ellipses with the same principal axis A = 1 and x = y = 0 as common focal point, as shown in Fig. 3.8. In this case, the 2|C| 1 . light rays cannot penetrate into the region r > − 2C In this section our aim was to find the shape of the rays, but it is important to see that the refractive index given by Eq. (3.36) can be transformed into another vital refractive index. By recalling the Eikonal equation ψx2 + ψy2 + ψz2 = n2 (r) (3.54) and substituting the Eq. (3.36) in cartesian coordinates into the Eq. (3.54), we obtain 1 ψx2 + ψy2 + ψz2 = C + p (3.55) x2 + y 2 + z 2 3.3. Rays in a medium with n (r) = q C+ 1 r Figure 3.7: The rays are parabolas when C = 0. Figure 3.8: The rays are ellipses when C < 0. 33 34 Chapter 3. The Spherical Luneburg Lens using Legendre transformations ψ = ψ (x, y, z) ω = ω (ξ, η, ς) ψ + ω = xξ + yη + zς (3.56) ξ = ψx η = ψy ς = ψz x = ωξ y = ωη z = ως (3.57) where it is very easy to see that the Eq. (3.55) is transformed into 1 ξ 2 + η2 + ς 2 = C + q ωξ2 + ωη2 + ως2 (3.58) and this equation can be rewritten as ωξ2 + ωη2 + ως2 = 1 2 −C + ξ + η 2 + ς 2 2 (3.59) if C = −1, so this equation is reduced to the Eikonal equation with a refractive index given by 1 n (ξ, η, ς) = (3.60) 2 1 + ξ + η2 + ς 2 or 1 n (r) = . (3.61) 1 + r2 This optical medium is know as Maxwell’s Fisheye. A medium with this refractive index will be studied in next section. 3.4. Maxwell’s Fisheye 3.4 35 Maxwell’s Fisheye The Maxwell’s Fisheye has been proposed to represent the refractive gradient index of the crystalline [9], for this reason is important to study it. The gradient refractive index of Maxwell’s fisheye is generally represented as n= (b2 a + r2 ) (3.62) if a and b are constants, and a = b = 1, we have n= 1 . (1 + r2 ) (3.63) We want to know the form of the rays, in order to do so we use the integral ray equation Z dr √ (3.64) θ−π =K 2 r n r2 − K 2 substituting the gradient refractive index into Eq. (3.64) and by doing some algebra, we obtain Z K (1 + r2 ) q θ−π = dr (3.65) r r2 − K 2 (1 + r2 )2 this integral can be solve by using f (r) = arcsin where r2 − 1 K √ r 1 − 4K 2 (3.66) K (1 + r2 ) df (r) = q dr 2 2 2 2 r r − K (r + 1) (3.67) we can observe that K (1 + r2 ) Z θ−π = r q r2 − K2 (1 + Z dr = r2 )2 df (r) (3.68) 36 Chapter 3. The Spherical Luneburg Lens and the solution is θ − π = arcsin K r2 − 1 √ r 1 − 4K 2 +C (3.69) where C is the integration constant. Equation (3.69) is rewritten as √ 1 − 4K 2 r2 − r sin (θ − π − C) − 1 = 0 K C can be defined as C=β− 3π 2 substituting the value of C and using the trigonometric identities sin cos θ and, cos θ = cos (−θ) into Eq. (3.70), we have √ 2 r −r 1 − 4K 2 cos (θ − β) − 1 = 0 K if β = 0, then 1 − 4K 2 cos θ − 1 = 0 K if we introduce cartesian coordinates into this equation, we have √ x− 1 , 2K π 2 (3.71) −θ = (3.72) √ r2 − r and if R = (3.70) then 1 − 4K 2 2K 2 + y2 = 1 4K 2 2 √ 2 x − R − 1 + y 2 = R2 . (3.73) (3.74) (3.75) The form of the rays is given by Eq. (3.75). We can observe that this equation √ represents a set of displaced circles on the x axis by a factor of R2 − 1 and all the circles intersect at y = ±1. We can say that all rays have a circular path, as shown in the Fig 3.9. So far, we have not said where the rays were originated, we only know that the rays have a circular path. Now, we will consider that all rays are originated in a 3.4. Maxwell’s Fisheye 37 Figure 3.9: Maxwell Solution point P0 (x, y) on the x-axis, i.e., x = x0 and y = 0. From Eq. (3.72) in cartesian coordinates, we have where R = 1 2K x− √ R2 − 1 cos β 2 2 √ + y − R2 − 1 sin β = R2 (3.76) and this equation is a generalization of Eq. (3.75). If y = 0 in Eq. (3.76) represents the interception points on the x axis, and this leads us to the equation √ x2 − 2x R2 − 1cosβ − 1 = 0. It can be solved if we use x0,1 = b± (3.77) √ b2 − 4ac 2a (3.78) p (R2 − 1) cos2 +1 (3.79) √ if a = 1, b = −2 R2 − 1 cos β and c = −1, then x0,1 = √ R2 − 1 cos β ± 38 Chapter 3. The Spherical Luneburg Lens Figure 3.10: Maxwell Solution and the solutions are p √ x0 = R2 − 1 cos β + (R2 − 1) cos2 +1 p √ x1 = R2 − 1 cos β − (R2 − 1) cos2 +1 (3.80) the relationship between x0 and x1 is given by x0 x1 = −1 (3.81) we can observe that this relationship is independent of the parameters R and β and this tells us that all rays that originate at x0 are intersected at a point x1 , where 1 x1 = − . (3.82) x0 the solution is sketched in Fig. 3.10. This result can be generalized if we have a source that is outside the x axis, i.e., if all rays are originated in a point P0 = (x0 , y0 ), there is an intersected point 3.4. Maxwell’s Fisheye 39 Figure 3.11: Maxwell Solution Off-Axis P1 = (x1 , y1 ), where x1 = − xr20 0 y1 = − yr20 (3.83) 0 the solution is sketched in Fig. 3.11. From the latter solution we can observe that all the rays are focusing in one only point, i.e., it is an optical instrument free of the aberrations. For this reason, the Maxwell fish eye is considered a perfect optical Instrument in the x − y plane, the image produced is inverted and it has a magnification of M = − rr10 . These solutions are interesting, because the curves that represent the rays are inside the medium, i.e., the Maxwell’s Fisheye is not a lens, for this reason, it cannot be proposed to represent the refractive gradient index of the human lens. The well known perfect optical Instrument is the Luneburg lens which is our main interest in this work and it will study in the next section. 40 Chapter 3. The Spherical Luneburg Lens 3.5 The Luneburg Lens The Luneburg lens is the answer to the problem posed by R. K. Luneburg in 1944, this problem is stated as follows: If we have a refractive index with radial symmetry, where the maximum radius is 1 and it is submerged in a refractive index n = 1 (air), What should be the refractive index of the medium where all incident rays focus at a single point?. In order to solve this problem, we begin by considering a sphere with maximum radius equal to 1 and where the refractive index is n = n (r) , r < 1 n = 1, r≥1 n (1) = 1, r = 1. (3.84) At this point, we do not know as is the refractive index n (r), but we are considering some boundary conditions, i.e., the maximum radius (rm ) is 1 and also we have n (rm ) = 1. The first boundary condition (rm = 1) does not generate any loss of generality of this problem solution, because r is converted into a normalized variable. The second boundary condition (n (rm ) = 1) does not mean that this medium is air or vacuum, this means that the refractive index of the element considered is normalized with respect to the refractive index where the sphere is submerged [15]. Figure 3.12 has the parameters that we use for solving the Luneburg problem. From this figure we can observe that the incident ray is originated at the point r00 and the point r10 is the point where all rays are focused after passing through the sphere. The point P0 indicates the point where the incident ray enters at the sphere and the point P1 is the point at which the ray exits the same sphere. The radius r∗ = r (θ) is the minimum radius at the angle θ∗ . The rays outside the sphere propagates in straight lines, i.e., the ray from r00 to P0 and the ray P1 to r10 , because they are in a medium with constant refractive index. Now, our interest is to know the propagation of the rays inside of the sphere. 3.5. The Luneburg Lens 41 Figure 3.12: Luneburg Sphere Parameters For this reason, it is necessary to know the variation from the angle θi to the angle θs . In the point P0 we have that π = α0 + Ψi + π − θi θi = α0 + Ψi + π − π θi = α0 + Ψi (3.85) and K in this point is K = n (r0 ) r0 sin Ψ (3.86) from the boundary conditions is very easy to see that n (r0 ) = 1 and r0 = 1, then sin Ψ = K Ψ = arcsin K (3.87) Ψi + Ψ = π Ψi + arcsin K = π Ψi = π − arcsin K (3.88) we observe that 42 Chapter 3. The Spherical Luneburg Lens substituting the value of Ψi into Eq. (3.85) Ψi + α0 + π − θi = π Ψi + α0 − θi = 0 π − arcsin K + α0 − θi = 0 (3.89) θi = π − arcsin K + α0 . (3.90) therefore In the outter point P1 , we have θs + α1 + π − Ψs = π θs = Ψs − α1 + π − π θs = Ψs − α1 (3.91) but from the Generalized Snell’s Law in this point Ψs = arcsin K (3.92) and substituting the value of Ψs into Eq. (3.91) θs = arcsin K − α1 . (3.93) The angular variation inside the sphere is given by θs − θi = arcsin K − α1 − [π − arcsin K + α0 ] θs − θi = arcsin K − α1 − π + arcsin K − α0 θs − θi = 2 arcsin K − π − [α0 + α1 ] (3.94) using the trigonometric identity arcsin k + arccos k = π 2 (3.95) 3.5. The Luneburg Lens 43 we have θs − θi = − [2 arccos K − (α0 + α1 )] . (3.96) The sum of the angles α0 + α1 can be defined as the deflexion function given by αT (K) = α0 (K) + α1 (K) (3.97) this function represents the direction change of the ray passing through the medium. With this equation, we can rewrite Eq. (3.96) as θs − θi = − [2 arccos K + αT (K)] . (3.98) From Eq. (3.33) is easy to find the K value range, because we know the value of n0 and r0 , then 0 ≤ K ≤ 1. (3.99) It is possible to obtain from the ray integral equation, the variation from θi to θs , i.e., Z r1 dr p (3.100) θs − θi = K ρ2 − K 2 r0 r to evaluate the integral is necessary to divide the variation of the angle into two variations. The first variation is from r0 to r∗ and the second is r∗ to r1 , this leads to Z r∗ Z r1 dr dr p p θs − θi = K −K (3.101) ρ2 − K 2 ρ2 − K 2 r0 r r∗ r but r0 and r1 are equal to 1, then Z 1 θs − θi = −2K r∗ dr p . r ρ2 − K 2 (3.102) Since Eq. (3.96) is equal to Eq. (3.102) we have Z 1 K r∗ dr 1 p = arccos K + αT (K) 2 r ρ2 − K 2 (3.103) 44 Chapter 3. The Spherical Luneburg Lens where 0 ≤ K ≤ 1. Since the Luneburg lens is a stigmatic system is possible to find the deflexion function [7]. From the triangle given by the points r00 , P0 , and the origin, we can obtain the value α0 r0 1 = sin0Ψi sin α0 sin α0 = sinr0Ψi h0 i (3.104) α0 = arcsin sinr0Ψi h 0i α0 = arcsin rK0 0 and from the triangle given by the points r10 , P1 , and the origin; we have that the α1 value is 1 r10 (3.105) = sin α1 sin (π − Ψs ) but sin (π ± x) = ∓ sin x (3.106) sin (π − Ψs ) = sin Ψs (3.107) then we obtain r0 1 sin α1 = sin 1Ψs sin α1 = sinr0Ψs h1 i α1 = arcsin sinr0Ψs h 1i α1 = arcsin rK0 (3.108) 1 α0 and α1 only depend on K, then α0 (K) = arcsin h i K ; r0 0 α1 (K) = arcsin h i K r10 (3.109) therefore, the deflexion function is given by K K αT (K) = arcsin 0 + arcsin 0 . r0 r1 (3.110) 3.5. The Luneburg Lens 45 Substituting this equation into Eq. (3.103), Z 1 K r∗ if K 1 K p + arcsin = arccos K + arcsin 0 2 r0 r10 r ρ2 − K 2 (3.111) 1 K K f (K) = arccos K + arcsin + arcsin 0 2 r0 r10 (3.112) dr we have Z 1 K r∗ dr p = f (K) . r ρ2 − K 2 (3.113) Since we want to know the refractive index of the medium, we will consider a change of variable of the form τ = log r (3.114) where dr (3.115) r we know that ρ is a function of r, but with the change of variable, ρ becomes a function of τ , i.e., ρ = ρ (τ ). With this change of variable, we have dτ = Z 0 K −∞ dτ p = f (K) . ρ2 (τ ) − K 2 (3.116) the minimum value that r∗ can have is 0, then, the lower limit of integration is −∞, and when ρ is a function of r the upper limit of integration is 1, and with the change of variable it is 0. Now, we assume that ρ (r) and ρ (τ ) are invertible functions, i.e., r = r (ρ) τ = τ (ρ) (3.117) τ (ρ) = log r (ρ) (3.118) this leads us to 46 Chapter 3. The Spherical Luneburg Lens if and Ω (ρ) = τ (ρ) = log r (ρ) (3.119) dτ (ρ) 1 dr (ρ) dΩ (ρ) = = dρ dρ r (ρ) dρ (3.120) we can rewrite this derivative as Ω0 (ρ) dρ = where Ω0 (ρ) = dr (ρ) r (ρ) dΩ (ρ) dρ (3.121) (3.122) Equation (3.116) is transformed with these equations in Z 1 K K Ω0 (ρ) dρ p = f (K) ρ2 − K 2 (3.123) from the denominator we can see that the highest value of ρ is 1 and the smallest value is K, because 0 ≤ K ≤ 1. This is an of Abel’s type integral where the solution is known [16], but we will find its solution using the Luneburg theorem which tells us [15]: If the function f (K) is defined by the integral Z λ K K Ω0 (ρ) dρ p = f (K) ρ2 − K 2 (3.124) in the interval 0 ≤ K ≤ λ, then Ω (ρ) is determined by the integral 2 Ω (λ) − Ω (ρ) = π in the interval 0 ≤ ρ ≤ λ. Z ρ 1 f (K) p dK K 2 − ρ2 (3.125) 3.5. The Luneburg Lens 47 Using the Luneburg theorem we can solve the integral of Eq. (3.123), then log (1)−log (r) = 2 π 1 Z ρ arccos K 1 p dK + π K 2 − ρ2 Z Z 1 1 ρ h i arcsin rK0 + arcsin rK1 p dK (3.126) K 2 − ρ2 but as log (1) = 0 2 π − log (r) == Z ρ 1 arccos K 1 p dK + π K 2 − ρ2 ρ h i arcsin rK0 + arcsin rK1 p dK. K 2 − ρ2 (3.127) The first integral is easy to solve when we define f (K) = arccos K (3.128) where Z 1 f (K) = K K Z dρ p =K ρ K 2 − ρ2 d dρ 1 K p log ρdρ K 2 − ρ2 Z 1 =K K d log ρ p K 2 − ρ2 (3.129) and applying the Luneburg theorem, we have 2 − log ρ = π 1 Z ρ arccos K p K 2 − ρ2 (3.130) therefore − log (r) = − log ρ + 1 π Z ρ 1 h i arcsin rK0 + arcsin rK0 1 p0 dK. 2 2 K −ρ (3.131) We introduce a new function given by 1 ω (ρ, a) = π Z ρ 1 arcsin at p dt t2 − ρ2 (3.132) 48 Chapter 3. The Spherical Luneburg Lens we obtain ρ = ω (ρ, r00 ) + ω (ρ, r10 ) r but knowing that ρ = nr, then we will have that log (3.133) 0 0 n = e[ω(ρ,r0 )+ω(ρ,r1 )] (3.134) r = ρe−[ω(ρ,r0 )+ω(ρ,r1 )] (3.135) and 0 0 Finally, if the solution for the ω (ρ, a) function is known, it is possible to solve the system of equations given by Eqs. (3.134) and (3.135), and we will be able to know the refractive index n (r). For example, if we have a system where r00 = ∞ and r10 = 1, the solutions for the ω (ρ, a) function when a = r00 and a = r10 are arcsin ∞t p dt = 0 t2 − ρ2 (3.136) h i p arcsin t 1 p dt = log 1 + 1 − ρ2 . 2 t2 − ρ2 (3.137) 1 ω (ρ, ∞) = π and 1 ω (ρ, 1) = π Z ρ 1 Z 1 ρ Substituting these expressions into the Eqs. (3.134) and (3.135), we have that q p n = 1 + 1 − ρ2 r= ρ n (3.138) If we solve this system of equations, we obtain the refractive index n (r) given by n (r) = where r = √ 2 − r2 (3.139) p x2 + y 2 . A sphere with this refractive index makes that all rays from infinity focused on the point r10 = 1, i.e., all the rays focused on the horizontal axis and on the 3.6. The Generalized Luneburg Lens 49 surface of the sphere, as shown in the Fig. 3.13. This sphere is known as The Luneburg Lens. Figure 3.13: The Luneburg Lens The Luneburg lens can be generalized to focus the rays outside or inside the lens because Luneburg’s original lens is a limiting case in which r00 is at infinity and r10 is on the surface of the lens. We are interested in the first case because in the eye all rays focus on the retina rather than inside the crystalline. Recalling that our goal is to build a model of the crystalline based on the Luneburg lens. 3.6 The Generalized Luneburg Lens In 1958, Morgan extended Luneburg’s analysis in his paper entitled "General Solutions of the Luneburg Lens Problem", where he demonstrates the possibility of a spherical gradient lens that images one finite sphere sharply onto another [17]. In some cases both; the object and image surfaces lie outside the lens itself [10]. The principle of the generalized Luneburg lens is shown in Fig. 3.14. But Morgan was not the only person who studied this problem. In 1957, Toraldo di Francia published his paper entitled "Il Problema matematico del sis- 50 Chapter 3. The Spherical Luneburg Lens Figure 3.14: The Generalized Luneburg Lens tema ottico concentrico stigmatico", where he does the analysis of Luneburg lens, but his method can be applied easily to the most general case [18]. In this section, we rely on the Morgan paper because this work provides a more complete and explicit analysis of the generalized Luneburg’s problem [17]. The initial conditions to solve this problem are: a refractive index n (r) must exist in the interval 0 ≤ r ≤ 1, and the external foci are given by r00 > 1 and r10 > 1. To find the refractive index n (r) we start from Eq. (3.113), Z 1 K r∗ dr p = f (K) . r ρ2 − K 2 (3.140) Before solving this equation we can note that, if the index of the lens at r = a, where a ≤ 1, is less than unity, some rays from r00 will be totally reflected. Then, the full aperture of the lens will not be used [10]. To avoid this problem, let us introduce a new refractive index in the interval a ≤ r ≤ 1, which is given by n (r) = P (r) 1 ≥ r r (3.141) if P (r) is known and a ≤ 1, the solution of the Eq. (3.140) is Z a K r∗ dr p = f (K) − F (K) r ρ2 − K 2 (3.142) 3.6. The Generalized Luneburg Lens where 1 Z 51 dr p r P 2 (r) − K 2 F (K) = K a (3.143) and f (K) is given by Eq. (3.112). If we consider that ρ (r) is a invertible function then we can introduce the Ω (ρ) function (Eq. (3.119)) to solve Eq. (3.142), i.e., the integral reduces to Z 1 K K Ω0 (ρ) dρ p = f (K) − F (K) ρ2 − K 2 (3.144) using the Luneburg theorem, the solution of this integral is 2 a − log = r π 1 Z ρ f (K) − F (K) p K 2 − ρ2 (3.145) it can rewrite as 2 a − log = r π Z ρ 1 2 p − K 2 − ρ2 π f (K) Z 1 ρ F (K) p K 2 − ρ2 (3.146) from Eqs. (3.130) and (3.132) is very easy to see that 2 π Z ρ 1 f (K) p = ω (ρ, r00 ) + ω (ρ, r10 ) − log ρ K 2 − ρ2 (3.147) and we define a new Υ (ρ) function given by 2 Υ (ρ) = π Z ρ 1 F (K) p K 2 − ρ2 (3.148) where Υ (ρ) can be known if P (r) is known. Substituting Eqs. (3.147) and (3.148) into Eq. (3.146), we have 0 0 n = e[ω(ρ,r0 )+ω(ρ,r1 )−Υ(ρ)] (3.149) 52 Chapter 3. The Spherical Luneburg Lens and ρ (3.150) n with these two equations we can find a refractive index for any value of r00 and r10 . For example, if a = 1 is very easy to see that Υ (ρ) = 0 and leading to Luneburg solution, i.e., when r00 = ∞ and r10 = 1. r= Morgan established a condition for the existence of a solution, that is, in order for a solution to exist for any value of r00 and r10 it is necessary that the following condition is satisfied Z 1 1 1 dr p arcsin 0 + arcsin 0 ≥ 2 . (3.151) r0 r1 P 2 (r) − 1 a r Nowadays, Luneburg lens and the generalized Luneburg lens have great interest in various areas. For example; in telecommunications, the Luneburg lens is studied because it eliminates the aberrations caused by the antennas and the scanners [19]. Also, this lens provides excellent steering capabilities for incident wide angles [20, 21]. The Luneburg lens have a spherical geometry, this geometry can be changed to a ellipsoidal geometry when we make a linear transformation. The new lens is named as the elliptical Luneburg lens and it has the same property of the Luneburg lens, i.e., all rays are focused on one point after the lens. In the next section, a detailed analysis of the Luneburg lens with an elliptical geometry will be made. 3.7 The Elliptical Luneburg Lens We can define the Luneburg lens as a marvellous optical lens because it is a aberrations free lens. In telecommunications is extremely difficult to be applied in any practical antenna system due to its large spherical shape. Currently, it has been proposed a transformation that reduces the profile of the original Luneburg lens without affecting its unique properties. The new trans- 3.7. The Elliptical Luneburg Lens 53 formed slim lens is then discretized and simplified for a practical antenna application [21]. The Luneburg lens and The ellipsoidal Luneburg lens have been designed experimentally using Polymeric nanolayered. The first lens is presented as a developing application of the nanolayered polymer technology and the second lens is used to model a human crystalline lens using the anterior and posterior shape of the crystalline for a age=5 years. The importance of this work is that the anterior and posterior GRIN lenses were assembled into a bio-inspired GRIN human eye lens through which a clear imaging was possible [22, 23]. Due of these experimental advances is necessary to analyze the Elliptical Luneburg lens. 3.7.1 Linear Transformation of The Spherical Luneburg Lens We will demonstrate that from a linear transformation of a circle we can generate an ellipse. If we have a vector of the form " C= xc yc # where this vector represents the components of a unit circle and it has the property of C t C = x2c + yc2 = 1 (3.152) and also has a imaging vector given by " C0 = xc0 yc0 # where C 0 = AC (3.153) 54 Chapter 3. The Spherical Luneburg Lens then C = A−1 C 0 t C t = (A−1 C 0 ) (3.154) from Eq. (3.152) t C t C = (A−1 C 0 ) (A−1 C 0 ) t (A−1 C 0 ) (A−1 C 0 ) = 1 if " A= a b c d (3.155) # " A−1 # a c At = b d " # " # d −b d0 −b0 1 = = det A −c a −c0 a0 # " 0 0 d −c t A−1 = −b0 a0 now, we have " A−1 C 0 = d0 −c0 −b0 a0 #" xc0 yc0 # " = d0 xc0 − c0 yc0 −b0 xc0 + a0 yc0 # i d0 xc0 − c0 yc0 −b0 xc0 + a0 yc0 " # i 0 0 0 0 h 0 d x − c y c c −1 0 t −1 0 A C A C = d xc0 − c0 yc0 −b0 xc0 + a0 yc0 −b0 xc0 + a0 yc0 i t h A−1 C 0 A−1 C 0 = (d0 xc0 − c0 yc0 )2 + (−b0 xc0 + a0 yc0 )2 A−1 C 0 t = h 2 2 (d0 xc0 − c0 yc0 ) + (−b0 xc0 + a0 yc0 ) = 1. (3.156) Taking the appropriate values of the matrix A, Eq. (3.156) becomes to the 3.7. The Elliptical Luneburg Lens 55 equation of an ellipse, given by " A= then (A)−1 1 = α " α 0 0 1 # # 1 0 0 α " = 1 α 0 0 1 # is clear to see that if a0 = 1, b0 = 0, c0 = 0 y d0 = α1 , then Eq. (3.156) becomes 1 xc0 α 2 and if " A= + yc20 = 1 1 0 0 α (3.157) # Equation (3.156) becomes x2c0 + 1 y c0 α 2 = 1. (3.158) The parameter α is very important because it represents a compression or a expansion of the x-axis or y-axis, as it is shown in the Eqs. (3.157) and (3.158). With these two equations it is possible to generate the elliptical Luneburg lens and we can to know the refractive index shape. 3.7.2 GRIN of The Elliptical Luneburg Lens We know that the Luneburg lens has a refractive index given by n (r) = where r = √ 2 − r2 (3.159) p x2 + y 2 is the the radius on one point inside the lens and 0 ≤ r ≤ 1. 56 Chapter 3. The Spherical Luneburg Lens Now, if we defined a new radius given by s re = 1 x α 2 + y2 (3.160) and we substitute it into Eq. (3.159), we have v # " u 2 u 1 x + y2 . n (re ) = t2 − α (3.161) This equation represents the refractive index of the elliptical Luneburg lens. If we consider that the α parameter is α < 1, we have an elliptical refractive index with semi-major axis in the vertical axis, as shown in the Fig. 3.15. Figure 3.15: The Elliptical Luneburg Lens We have been designed theoretically the ellipsoidal Luneburg lens using a linear transformation. Now, we need to know how the rays propagate inside this lens. 3.8. Ray Tracing 3.8 57 Ray Tracing If we know the refractive index n (r), it is possible to obtain the rays inside of this refractive index from Eq. (3.15). But in this section, we will use the theory of J. A. Grzesik published on the paper entitled "Focusing properties of a threeparameter class of oblate, Luneburg-like inhomogeneous lenses", because it is a lot easier to understand and also because this theory can apply to both spherical and elliptical Luneburg Lenses [24]. The refractive index of spherical or elliptical Luneburg lenses can be represented as p (3.162) n (x, y) = nc − (nc − ns ) (µ2 x2 + y 2 ) √ √ where nc and ns is the central and surface refractive index of the lens, respectively, and µ is a compression factor along x-axis. If µ−1 < 1 we obtain an oblate ellipsoidal lens with |x| < µ−1 , and if µ = 1 we have a sphere of unit radius. We can observe that when we have nc = 2 and ns = 1, Eq. (3.163) generates the same refractive index that in Eq. (3.161) if α = µ1 . At this point, this is all what we need to consider. Figure 3.16: Parameters As the refractive index depends only on x and y then is possible to use the ray equation given by Eq. (2.14) in order to see how the rays propagate in the 58 Chapter 3. The Spherical Luneburg Lens medium. Equation (2.14) with some minor manipulation, adopts the form ∂n2 (x, y, z) d n2 (x, y) = dx 1 + ẏ 2 ∂x (3.163) free from any square root entanglement. To solve this equation, we will consider only two parameters α0 and P0 . The parameter α0 is the angle that the ray makes with respect to the direction of compression and P0 = (x0 , y0 ) is the point where the ray enters the lens, as shown in Fig. 3.16. Now, if the rays are given by y (x) function, the solution of this equation is y (x) = Λ2 sin sin−1 y0 Λ−1 ± µ−1 sin−1 xΛ−1 − sin−1 x0 Λ−1 2 1 1 where Λ21 = and Λ22 ns cos2 (α0 ) + µ2 (nc − ns ) x20 µ2 (nc − ns ) ns sin2 (α0 ) + (nc − ns ) y02 . = (nc − ns ) (3.164) (3.165) (3.166) The ± sign in Eq. (3.164) provides the possibility to have an ascendent or a descent ray, as dictated by the sign of injection angle α0 . The ray tracing with this solution is very easy to obtain. For example, if we have nc = 2, ns = 1, µ = 1 and the rays come from infinity, i.e. α0 = 0, the Λ1 and Λ2 parameters are q Λ1 = 1 + x20 (3.167) and Λ2 = y0 . (3.168) With these parameters and if x goes from x0 to 1 for each ray, we obtain the ray tracing of the Luneburg lens when we substituted these values into Eq. (3.164), as shown in the Fig (3.17). For the elliptical Luneburg lens, we need nc = 2.5, ns = 1, µ > 1 (in this case µ = 2), α0 = 0 and x goes from x0 to x1 for each ray. With these values we have 3.8. Ray Tracing 59 Figure 3.17: Ray tracing of The Luneburg Lens that r Λ1 = 1 + x20 6 (3.169) and Λ2 = y 0 . (3.170) then we can find the ray tracing of the elliptical Luneburg lens, as shown in Fig. 3.18. Figure 3.18: Ray tracing of The Elliptical Luneburg Lens 60 Chapter 3. The Spherical Luneburg Lens Figure 3.19: Ray tracing of The Elliptical Luneburg Lens with nc = 2, ns = 1, µ = 2 and α0 = 0 We can note that the central refractive index is different for both examples. This should not worry us, because the ray propagation depends on three parameters (nc , ns and µ) and we must choose very carefully the values of these parameters so that the rays are focused. For example, if nc = 2, ns = 1, µ = 2 and α0 = 0, we obtain a lens which only focuses rays entering in the center of the lens, as shown in Fig. 3.19. 3.9 Conclusions Throughout this chapter we have studied the properties of a spherical gradient index medium; its refractive index and how the rays propagate inside that medium. We have given several examples using the ray equation, the ray integral equation and the Grzesik’s theory to have a better understanding of the ray propagation in a spherical gradient index medium. We have mainly studied the classical Luneburg lens (Spherical Luneburg lens) and the elliptical Luneburg lens. With the latter, we finally have the fundament to build our proposed crystalline. But, before we move on, we need to know the anatomy of the eye and the geometrical parameters (measurements) in order to use them in our model. Chapter 4 The Human Eye As An Optical System The human visual system is composed, in one hand, by the optics of the eye, and in the other one, by a signal processing system wired to the brain. In this work we are interested in the optics of the eye, for this reason, the signal processing system will not be treated here. However, there is a considerable number of information about this area in the literature that can be consulted [25, 26, 27]. The eye is an optical system with an extraordinary complexity. The description or modeling of the eye is done by organizing average measurements and properties into simplified models called schematic eyes [25, 27]. These models are useful to systematically study the properties and performance of individual components of the eye. The complexity of the eye is due to the fact that its refractive surfaces are not strictly spherical and its lens has a gradient refractive index. Also, the eye is complex because it can be seen as an optical imaging and detection instrument. The complex structure of its components make the eye to be difficult to study. The most complex component is the lens, for this reason, almost all schematic models do not include the lens or the gradient refractive index of it, and it is changed by a constant refractive index. In this chapter, the components of the human eye are studied and also we describe some interesting schematic models are described. 62 Chapter 4. The Human Eye As An Optical System 4.1 The Human Eye Physiologically the eye has many elements, as shown in Fig. 4.1. The inside of the eye is divided into three compartments [28, 29]: • The anterior chamber, between the cornea and the iris, which contains the aqueous fluid. • The posterior chamber, between the iris, the ciliary body and the lens, which contains the aqueous fluid. • The vitreous chamber, between the lens and the retina, which contains a transparent colourless and gelatinous mass called the vitreous humour or vitreous body. Figure 4.1: Human eye and its optical elements [28]. The number of elements can be minimized if we consider the eye as an optical system, i.e., the elements to be considered in this section are cornea, pupil, lens, retina, aqueous humor and vitreous humor. 4.1. The Human Eye 63 The elements of the eye not considered in this chapter can be consulted with more detailed in an anatomical description sense in many specialized books in this area [30, 29]. 4.1.1 Refracting components of the human eye: Cornea and Lens In the human eye, there are two refracting optical elements; the cornea and the lens. These elements are very different in their geometrical shape and in their refractive index, but in order to provide a good quality retinal image, these elements must be transparent and have appropriate curvatures and refractive indices [28]. Given the importance of these two elements in the human eye, the cornea and the lens are described in this section. 4.1.1.1 Cornea The cornea is the curved transparent front surface of the eye. It has approximately spherical shape with a radius of curvature of about 8 mm. The anterior surface of the cornea is protected by the sclera which is the outermost structure of the human eye. It is a dense, white, opaque, fibrous tissue that is mainly protective in function and is approximately spherical with a radius of curvature of about 12 mm. The centres of curvature of the sclera and cornea are separated by about 5 mm [28, 29]. The posterior surface of the cornea is in contact with the aqueous humour that is a colorless liquid and is composed by a 98% of water. The aqueous humour is a liquid with constant refractive index. The value of its refractive index is well defined and is of 1.336. The cornea in optical terms is one of the two refracting elements of the eye. This element contributes with one third of the optical power of the human vision system. The refractive index of the cornea is usually taken as 1.376 and this value is considered constant over the entire cornea. However, the anterior corneal surface 64 Chapter 4. The Human Eye As An Optical System is not a smooth optical surface due to its cellular structure, although an optically smooth surface is provided by the very thin tear film which covers it. While this has an index less than 1.376, for most optical calculations this tear film can be regarded as a very thin optical element consisting of two concentric surfaces of almost equal radii of curvature and therefore has negligible power [1]. In some works, the refractive index of the cornea is considered a gradient refractive index [31, 32], because the cornea is composed by many layers, as shown in Fig. 4.2, and each corneal layer has its own refractive index. Figure 4.2: The structure of the cornea [28]. From Fig. 4.2, we can observe that the stroma is by far the thickest layer, i.e., its refractive index dominates. For this reason, the value of refractive index of the cornea is usually taken as 1.376. Since the cornea is the first and most accessible optical element in the human eye and by its important optical power makes it the object of several investigations and treatments [33]. And it is probably also the most measured. 4.1. The Human Eye 4.1.1.2 65 Lens The other refracting element is the crystalline lens and it is commonly called lens. The anterior surface of the lens is in contact with the posterior surface of the iris and the aqueous humour. The posterior surface is in contac with the vitreous humour that is a clear gel that occupies the posterior segment of the eye and its refractive index can be considered equal to aqueous humour, 1.336. The geometric shape of the lens and its refractive index are very difficult to measure in vivo, because the lens is a dynamic lens responsible for adjusting the focusing distance. This changed of focusing distances is called accommodation. Also, as the crystalline lens ages, its geometric shape changes, and it becomes less flexible, and consequently a person’s ability to accommodate is lost slowly with time [25]. It is very important to say that also its refractive index changes with age. The lens is supported by tiny muscular fibers, called ciliary muscles that pull the lens. During accommodation, when the eye needs to change focus from distant to closer objects, the ciliary muscle contracts and causes the suspensory ligaments, which support the lens, to relax. This allows the lens to become more rounded, thickening at the centre and increasing the surface curvatures. The front surface moves slightly forward. These changes result in an increase in the equivalent power of the eye. When the eye has to focus from close to more distant objects, the reverse process occurs [25, 28, 29]. However, measures of the geometric shape and refractive index of the lens in vitro, exist [11]. These show that the lens is not axially symmetric, as can be seen in Fig. 4.3, and the lens due to its biological nature and protein distributions is an optically inhomogeneous lens with a gradient refractive index. Since the lens does not has an axial symmetry, it is represented by two lenses: one anterior and one posterior, which intersect at the equator. The most important characteristic of the lens is that the refractive index is represented by a gradient refractive index where the iso-indicial surfaces are ellipsoidal curves. Throught the years, the gradient index of the lens has been represented by 66 Chapter 4. The Human Eye As An Optical System Figure 4.3: The geometric shape of the lens. za and zp are differents. two mathematical equations, one equation for anterior lens and one equation for the posterior lens [3, 4]. These models present drawbacks because they are discontinuous at the equator. These discontinuities can be classified into two main groups: • In the first group the gradient index variation in the equator is discontinuaous between the anterior lens and posterior lens [3]. • In the second group the discontinuities are between the derivatives of first and second order with respect to the iso-indical surface geometry [34, 4]. These problems are generated by the accommodation of the lens, because it reshapes is required to change its internal gradient index distribution imposing a constraint to stablish a definitive expression for modeling its gradient refractive index [35]. In other words, there cannot exists an unique mathematical expression to simulate a living biological crystalline. The paper of A. M. Rosen, et al. entitled "In vitro dimensions and curvatures 4.1. The Human Eye 67 of human lenses" provides excellent experimental data for the geometric shape and refractive index of the lens [11]. In this paper, the refractive index ranges from nc = 1.4181 ± 0.075 in the core to ns = 1.3709 ± 0.0039 at the boundary surface of the lens. And also, it provides the equations for the parameters of the lens with age dependence given by R = [0.0138 (±0.002) ∗ Age + 8.7] /2 za = 0.0049 (±0.001) ∗ Age + 1.65 zp = 0.0074 (±0.002) ∗ Age + 2.33 (4.1) where the ratio of anterior thickness (za ) to posterior thickness (zp ) is constant at 0.70. Measurements were made on 37 human lenses ranging in age from 20 to 99 years. Numerous studies and measurements of the defects of the cornea and crystalline have given the basis to the hypothesis that the crystalline compensates the defects of the cornea by minimizing or balancing them. A discussion on this can be found in a review by Artal [33]. These and other important characteristics of the lens will be studied with more detail in the Chapter 5. Also, we will studied the light propagating inside lens. 4.1.2 Pupil The iris forms the aperture stop of the eye. Its aperture or opening is known as the pupil. The pupil size is determined by two antagonistic muscles, which are under autonomic (reflex) control [28]: • The sphincter pupillae, which is a smooth muscle forming a ring around the pupillary margin of the iris. When it contracts, the pupil constricts. It is innervated by the parasympathetic fibres from the oculomotor (3rd cranial) nerve by the way of the ciliary ganglion and the short ciliary nerves. • The dilator pupillae, which is more primitive and consists of myo-epithelial cells that extend radially from the sphincter into the ciliary body. It dilates 68 Chapter 4. The Human Eye As An Optical System the pupil and is innervated by sympathetic nerve fibres, which synapse in the superior cervical ganglion and enter the eye by way of the short and long ciliary. The diameter of the pupil may vary from about 2-3 mm at high illumination to about 8 mm in darkness. If we considered identical illumination conditions is possible to observe that the diameter is different for each person. This is due to pupil size decreases with increasing in age and pupils react less to changes to light levels. For example, for a 10 years old person can consider a typical diameters of 4.8 mm, while at 45 will be of 4.0 mm, and 3.4 mm at 80 years. For an eye in total darkness the most common diameters are 7.6 mm at 10 years, 6.2 mm at 45, and 5.2 mm at 80 years [29]. 4.1.3 Retina The retina is the ocular surface where images are projected and is the lightsensitive tissue of the eye. It is composed of a number of cellular and pigmented layers, and a nerve fibre layer, i.e., the retina consists of ten layers, as shown in Fig. 4.4. The study by direct observation of this membrane is known as Retinoscopy. This procedure has undergone significant evolution since the adaptive optics systems were first implemented [36]. The retina is especially interesting since it is directly connected to the central nervous system and some experts even consider it to be part of the brain since neural cells can be found in it [37]. Here is where specialized cells that can convert light into nervous impulses can be found. There are two main types of corneal cells which are responsible for light conversion. The cones are the ones responsible for color discrimination, they come in three types, each one capable of detecting a specific frequency band associated to color blue, green, and red, respectively. The rods are responsible for night vision, since they are very sensitive to light. This high sensitivity is somehow related to the wiring between rods which combines 4.1. The Human Eye Figure 4.4: The layers at the back of the human eye [28]. 69 70 Chapter 4. The Human Eye As An Optical System the signals into the brain, giving them as a consequence a poor spatial resolution [33, 37]. There are around seven millions cones, a hundred and twenty five millions of rods and as much as a million nervous fibers. The fovea is the region of the retina where a higher cone density is found. 4.2 Schematics Eye During the last decades different schematic eye models have been proposed. The paraxial schematic models are the most relevant [28, 1]. However, more complicated schematic eye models containing a Gradient Refractive Index lens have been published in recent years [3, 38]. The paraxial schematic eye models have a problem; they use a homogeneous index lens and a limited numbers of surfaces. In the Emsley reduced model, the eye is represented only by a single refractive surface; being the anterior surface of the cornea [28, 1, 39], as shown in Fig. 4.5. Figure 4.5: The Emsley reduced schematic eye. The Gullstrand-Emsley model has three refracting surfaces, as shown in Fig. 4.6; one for the cornea and two for the lens, moreover the aqueous and vitreous refractive index were modified up to 1.416 when it is well known that the real values are 1.336 [28]. The Le Grand full theoretical model is represented by four surfaces; two for the cornea and two for the lens, as shown in Fig. 4.7. Both models provide the 4.2. Schematics Eye 71 Figure 4.6: The Gullstrand-Emsley schematic eye. accommodation of the lens. Figure 4.7: The Le Grand full theoretical schematic eye. Gullstrand was the first that proposed a model with an inhomogeneous lens, but the lens only have four refracting surfaces [40]. This lens has a core and a cladding, the core has a high refractive index limited by the interior surfaces and the cladding has lower refractive index limited by the exterior surfaces. The lower refractive index is surrounding the core, i. e., Gullstrand represented the lens as a lens inside other lens, but it is very important to say that it does not represent a GRIN lens, because both lenses, independently, have a homogeneous refractive index. In 1974, R. G. Zainullin and et al. in the paper entitled "The crystalline lens as a Luneburg lens" proposed to use the Luneburg lens to represent the crystalline 72 Chapter 4. The Human Eye As An Optical System lens of the eye of vertebrates [8]. We say that they proposed, because they do not make a schematic eye in that paper. Using the same idea of R. G. Zainullin and et al., Campbell and Hughes published a paper entitled "An analytic, gradient index schematic lens and eye for the rat which predicts aberrations for finite pupils", where the lens is represented as a Luneburg lens [2]. But, there is a problem in this paper, it can not be used to represent a schematic human eye since the anterior and posterior faces of the lens are considered to be symmetric, as shown in Fig. 4.8. Figure 4.8: The schematic eye is analytically derived from the refractive index profile of the crystalline lens and anatomical measurements of a rat eye. [2]. Also, the Maxwell fish eye has been considered to build a schematic human eye, however, its design is a spherically symmetric lens, and the index of refraction varies symmetrically about a point [9]. For this reason, the schematic eye have the same problem that the schematic eye of Campbell and it is not realistic. This Schematic eye has many conceptual mistakes, because it does not exist a generalized Maxwell’s Fisheye, and the Maxwell’s Fisheye is not a lens (see Section 3.4), then It is not possible to make a ray tracing outside the environment of the 4.3. Conclusions 73 Maxwell’s Fisheye. This design was proposed by Yun Wu and et al. in 2010 and its schematic human eye is shown in Fig. 4.9. Figure 4.9: Schematic eye model with Maxwell fish-eye spherical lens [9]. In this thesis we propose a more realistic theoretical schematic eye using the idea that the lens can be represented as a composite modified Luneburg lens. 4.3 Conclusions In this chapter we have analysed the anatomy of the human eye and we have presented the optical characteristics of each element of the human eye. Also, we have studied the different schematic eyes, in which we have observed their problems to represent a realistic schematic eye model. With the characteristics of the human eye given in this chapter, we may be able to build a schematic human eye using a new lens that we will call the composite modified Luneburg lens, which will be discussed in the next chapter. This lens is based on the elliptical Luneburg lens and the new model can be modified for 74 Chapter 4. The Human Eye As An Optical System different eye conditions, as we will show in Chapter 5. Chapter 5 Schematic Eye with Composite Luneburg Crystalline The original Luneburg lens is a sphere with gradient index that has the property of being spherical aberrations free [6, 17]. For every pair of object and image conjugate points the internal refractive index distribution is different presenting a similitude with the human lens [8]. Now, it has been demonstrated that it is possible to perform a geometric transformation in such a way that the original spherical Luneburg lens can be transformed into an oblate spheroidal shape maintaining its aberration free property [24, 21]. This can be used as a first approximation to the human crystalline. However, the human crystalline is not symmetrical; to take this into account an alternative model will be proposed having an asymmetric bi-spheroidal shape but assuming that the gradient refractive index can be represented with a given analytical expression [41, 42]. As discussed above, the crystalline shape and refractive index are being modified continuously as the vision point changes, so a realistic model of the crystalline must consider the dynamics of both of them. In this chapter we propose a dynamic model of the crystalline constructed with two separate spheroidal hemispheres with variable curvatures for the anterior and posterior sections and considering a varying Gradient Refractive Index (GRIN). Its imaging properties are investigated based on the Luneburg lens theory modified accordingly to the proposed composite lens. We impose the condition in our model that the isoindical lines be continuous as well as their first order derivatives at the equatorial plane. This condition creates a perfect match of the gradient refractive index of both hemispheres guaranteeing the smooth continuity of the rays. This 76 Chapter 5. Schematic Eye with Composite Luneburg Crystalline model will be referred to as the composite modified Luneburg lens. Also, in this chapter we will propose a more realistic theoretical schematic eye using the composite modified Luneburg lens. This schematic eye is based in biometric parameters reported in Ref. [11]. 5.1 Composite Modified Luneburg Lens As mentioned above, the problem of the spherical Luneburg lens after a simple transformation can be reformulated into that of a spheroid maintaining its aberration free property [17, 21]. The index in the spherical Luneburg lens is described p by a function n (r) where r = x2 + y 2 + z 2 and x, y and z are the Cartesian p coordinates. We set ρ = x2 + y 2 and by symmetry around the z-axis we will work only with y. As r defines a spherical shape, the transformation to obtain the desired spheroidal shape can be either 1 0 y s 2 + z 02 = r2 or 02 y + 1 0 z s 2 = r2 . (5.1) (5.2) In these equations s is a constant parameter. By properly choosing the value of s we can compress or expand the sphere along the y or z axis as needed. The model of crystalline will be approximated by two spheroidal hemispheres [21, 24, 41, 42]. We now proceed to construct the model by imposing the condition that the gradient indices and the axial derivatives of the anterior and posterior hemispheres match at every point on the equatorial plane. For this purpose the refractive indices, na (r) for the anterior and np (r) for the posterior hemisphere, must be of the form 5.1. Composite Modified Luneburg Lens s 0 0 n2c − na (y , z ) = s n(y 0 , z 0 ) = np (y 0 , z 0 ) = n2c − y0 2 R y0 2 R 77 + 1 z0 Rsa + 1 z0 Rsp 2 2 z0 ≤ 0 (5.3) z0 ≥ 0 where nc is the refractive index at the center of the lens, R is the radius of the lens p measured on the equatorial plane. The scaling factor is s(a,p) = z(a,p) /R n2c − n2s where za is the anterior vertex and zp the posterior vertex. For na (r) the variable z 0 is negative and for np (r) the variable z 0 is positive. Figure 5.1 displays the projection on a meridional plane of the constructed crystalline lens with the bielliptical iso-indical lines showing continuity at the equatorial plane. Figure 5.1: Geometry of the bi-spherical model showing the continuity of the isoindical lines at the equator plane. 78 Chapter 5. Schematic Eye with Composite Luneburg Crystalline For the numerical example under study we used as a reference the biometric parameters reported in Ref. [11]. The refractive index ranges from nc = 1.4181 ± 0.075 at the core to ns = 1.3709 ± 0.0039 at the boundary surface of the lens. The age dependence of geometric parameters of their lens are estimated by the equations given by Eq. (4.1). For a lens aged 35 years the corresponding parameters are R = 4.4005mm za = 1.8215mm zp = 2.5890mm. (5.4) The curvatures of the anterior and posterior hemispheres are such that the ratio between za and zp is 0.7035. The lens aged 35 years is shown in Fig. 5.1. Now, to investigate the imaging properties of our model we follow a procedure similar to that described in Ref. [24]. this is, to modified it accordingly to fit the new geometry and the imposed continuity conditions. We assumed that the lens is embedded in a medium with refractive index equals to 1.336 and by keeping constant the surface refractive index ns . The first case that will be studied simulates a relaxed crystalline assuming an object at infinity so that the rays impigne parallel to its anterior surface. The chosen geometrical parameters and refractive indices produce practically a perfect focus at the image plane placed at a distance of 63.05mm from the posterior surface of the lens as shown in Fig. 5.2 a). For the next two cases we modify the vertex ratio za /zp and calculate the refractive index that is necessary to form a point image at the same plane of a point object placed at 250 mm from the anterior surface. In Fig. 5.2 b) we see that for a reduced ratio the Composite Modified Luneburg (CML) lens is thinner resulting in a GRIN distribution such that the central refractive index has been increased. The opposite case of making thicker the CML lens with a larger ratio za /zp = 0.7185 results in a GRIN redistribution with a reduced refractive index at its center, as is shown in Fig 5.2 c). The results of our simulations confirm the prediction discussed in Ref. [35]. By changing the shape of the crystalline implies a gradient refractive index dis- 5.1. Composite Modified Luneburg Lens 79 Figure 5.2: Ray tracing through the proposed CML lens embedded in a medium with refractive index of 1.336. a) Rays incident from an infinite distance. Rays incidents from a finite distance of 250 mm with b) za /zp = 0.6872 and c) za /zp = 0.7185. 80 Chapter 5. Schematic Eye with Composite Luneburg Crystalline tribution change. Although in the present model we have used perfect geometric surfaces, implementing minor modifications to our model, by using real biometric data it is possible to simulate imaging of a real human lens imaging [38]. The GRIN profiles and the GRIN distribution for each lens are shown in Fig. 5.3. From this figure, we can observe the refractive index changes when we change the ratio za /zp . Figure 5.3: The GRIN distribution and the GRIN profiles for each lens with different ratio za /zp . We have said the central refractive index increases when za /zp decreases and the central refractive index decreases when za /zp decreases, this is much easier to see when all the profiles are plotted in the same figure, as is shown in Fig 5.4. If we observe the rays inside the CML lens, we can say that its bending due to the GRIN is almost imperceptible appearing as if the refractive index was homogeneous. However, from Fig. 5.5 we observe that the rays are planar curves 5.1. Composite Modified Luneburg Lens 81 Figure 5.4: The GRIN profiles for each lens with different ratio za /zp . as is predicted from the Luneburg theory. These rays are different for each CML lens; in its length and its trajectory, because the propagation distance is different for each ratio za /zp , and the change on the refractive index causes a different trajectory for the rays are different, i.e., a higher refractive index causes the ray to have a greater curvature, while a lower refractive index causes the ray to have a smaller curvature, as shown in Fig 5.5. The second case that is investigated simulates different CML lenses keeping the surface refractive index ns constant and the ratio za /zp of a lens aged 35 years. Also, we assumed that the lens is embedded in a medium with a refractive index equals to 1.336. In these computational simulations, the only parameter of the lens that can be modified is the central refractive index nc , because we decided that the parameters ns and za /zp to be constants. A parameter that also can be modified, but this is not a parameter of the lens, is the plane of a point object, these planes were placed at different distances from the anterior surface, i.e., the distances were: infinity, 2000 mm, 1000 mm, 500 mm, and 250 mm. We calculate the central refractive index nc that is necessary to produce prac- 82 Chapter 5. Schematic Eye with Composite Luneburg Crystalline Figure 5.5: Trajectory of the inner rays for each CML lens with different ratio za /zp . tically a perfect focus at the image plane placed at a distance of 63.05 mm from the posterior surface of the lens. From Fig. 5.6, we can observe that the central refractive index decreases when we placed the point object plane near the anterior surface of the lens. This means that the refractive power of the lens is lower when the objects are closer to the lens. These results give us a better idea of the changes in the refractive index of the lens, because we are not altering the geometry of the lens and due to this fact, we do not make assumptions of changes in the ratio za /zp for different object point distances. The change of the refractive index is much easier to see when all GRIN profiles are plotted in the same figure, as is shown in Fig 5.7. If we knew experimentally, how the ratio za /zp changes for specific distances, we could find how the refractive index varies in the real life. Unfortunately, it is very difficult to measure this change in vivo, because the lens is dynamic and 5.1. Composite Modified Luneburg Lens 83 Figure 5.6: The GRIN distribution and the GRIN profiles for each lens with the same ratio za /zp and different central refractive index. 84 Chapter 5. Schematic Eye with Composite Luneburg Crystalline Figure 5.7: The GRIN profiles for each lens with the same ratio za /zp and different central refractive index. there are no measurements of this change. The propagation of the rays are shown in Fig. 5.8, in which can be observed that the rays are focused on an image plane placed at 63.05 mm from the posterior surface of the lens. The incident rays have been cut, because the distances of where the rays come from are large compared to the size of the CML lens. From Fig. 5.8 a) to Fig. 5.8 e), the rays and the gradient refractive index appear to be equal to each other, however, the rays and the gradient refractive index are different in each case, as shown in Fig. 5.9 and Fig. 5.6, respectively. From Fig. 5.9, the rays inside the lens appear to have larger curvature when we have a lower gradient refractive index, but this is not true, because we must remember that we are changing the incidence point on the lens. From Fig. 3.16 is easy to see that the angle α0 is larger when the point of incidence approaches to the lens. This implies that the constant K given by generalized Snell law for inhomogeneous media with spherical symmetry is changing when the angle α0 is changing, i.e., the K values depende on the values of α0 , as we see from Eq. 3.86. The two cases have been studied with a human lens of 35 years old. However, 5.1. Composite Modified Luneburg Lens 85 Figure 5.8: Ray tracing through the proposed CML lens embedded in a medium with a refractive index of 1.336 and a constant ratio za /zp = 0.7036. Rays incidents from a finite distance of a) Infinity, b) 2000 mm, c)1000 mm, d) 500 mm, and e)250 mm. 86 Chapter 5. Schematic Eye with Composite Luneburg Crystalline Figure 5.9: Trajectory of the inner rays for each CMLL with constant ratio za /zp = 0.7036. we have equations where the geometrical parameters of the human lens depend on age. Then, it is very interesting to make an analysis of second case when we have a human lens of a different age. For example, the Fig. 5.10 represents the GRIN profiles of a human lens that has 20 years old for different object distances. The different with a human lens of 35 years old is the increment of 0.0002 in the central refractive index and a image distance of 62.35 mm. The increment in the central refractive index is due to the lens dimensions of 20 years old is lower compared a lens of 35 years old. Now, the question is: Why the image distance changed?. The response is: It is always possible to find a distance where practically all the rays are focused, but this distance depend on the geometrical parameters and the refractive index of the lens. In the case where the lens has 20 years old, it is impossible to find a refractive index where all the rays are focused in a distance of 63.05 mm. Rosen in his paper say: "The ratio of anterior thickness to posterior thickness is constant at 0.70 for all age". It is very interesting because we have the same 5.1. Composite Modified Luneburg Lens 87 Figure 5.10: The GRIN profiles for each lens with the same ratio za /zp and different central refractive index. The human lens is 20 years old. 88 Chapter 5. Schematic Eye with Composite Luneburg Crystalline ratio for all age and different Za and Zp , for this reason, we have different image distances for all age. The ratio Za /Zp changes in the accommodation process of the human lens. This change allows that we can focus all rays at retina for different object distances. We can observe the Fig. 5.2 and the question arises: why in the Fig. 5.2 was possible to find the image distance of 63.05 mm when we have different Za /Zp ?. In this case, we have fixed the image distance and the central refractive index (nc) and we have found a ratio Za /Zp where all rays are focused at 63.05 mm. Observe that the ratio Za /Zp is different to 0.70 as we said in previous paragraphs. To find this ratio is very difficult because we have a big number of possibilities. We must understand that we have two cases very different when we fixed the geometrical parameters or when we fixed the image distance and the central refractive index. With the carried out analysis of the CML lens is possible to construct a new Schematic Model of the Human Eye using the CML lens as a human lens. This model is studied in next section. 5.2 Schematic Luneburg Eye In the previous section, we constructed a CML lens where its geometrical parameters and refractive indices were chosen to produce practically a perfect focus. It is woeth to note that, the human lens does not have those central refractive indices used in Section 5.1. In this section, we are interested in understanding the behavior of the light inside the complete eye, but first we want to know about the behavior of the light when passing through the lens inside the eye. The lens to be used is 35 years old with a ratio of za /zp = 0.7036, R = 4.7, nc = 1.4181, and ns = 1.3709, as is shown in Fig. 5.1. With these parameters and refractive indices of the eye, we can generate the ray tracing along of this new CML lens. It is very important to say that the refractive indices and parameters 5.2. Schematic Luneburg Eye 89 will be modified along the simulations to study the function of accommodation in the eye. We can see in Fig. 5.11 that rays inside the lens are planar curves and also we can see that the lens produce a negative spherical aberration when rays are impigne from an infinite distance. Figure 5.11: Negative spherical aberration from the human lens. This aberration is important when we considered the cornea as it is perfectly spherical, because the cornea has a positive spherical aberration as it is shown in Fig. 5.12. The positive aberration from the cornea can be suppressed by the negative aberration from the crystalline [43] when we choose an appropriate refractive index. For example, if we have nc = 1.41262, ns = 1.3709, za /zp = 0.7036, and rays that impigne from infinity distance, then the aberration from the cornea is suppressed by the lens aberration as is shown in Fig. 5.13. In this figure, we can see that the rays focused on the retina. With this model, by knowing the curvature it is possible to know the refractive index or if knowing the refractive index it is possible to know the curvatures of the lens. Also, we can change the distance from which the rays are coming from and we can construct a new lens that can suppress the cornea positive spherical 90 Chapter 5. Schematic Eye with Composite Luneburg Crystalline Figure 5.12: Positive spherical aberration due to the cornea. Rays incident from infinity distance, the anterior and posterior surfaces of the cornea are immersed in a refractive index of 1 and 1.336, respectively. Figure 5.13: Schematic eye model with CML lens. 5.2. Schematic Luneburg Eye 91 aberration. The parameters of the human lens for different distances are given in Table 5.1 and the propagations are presented in Fig. 5.14. Infinity Ratio za /zp nc 0.7036 1.41262 250 mm Ratio za /zp nc 0.7185 1.41477 0.7323 1.41249 0.7450 1.41088 0.7568 1.40897 0.7677 1.40803 500 mm Ratio za /zp nc 0.7185 1.41255 0.7323 1.41092 0.7450 1.40939 0.7568 1.40801 0.7677 1.40669 ns 1.3709 ns 1.3709 1.3709 1.3709 1.3709 1.3709 ns 1.3709 1.3709 1.3709 1.3709 1.3709 Table 5.1: Parameters of the crystalline for different rays impigned distances. Simulations in Fig. 5.14 are made having the refractive index of the surface fixed, then the ratio between vertex, za /zp , is change, and a suitable refractive index is search for the centre so the light rays focused on the retina. We can observe from Table 5.1 that the accommodation of the surface does affect the refractive index of the crystalline. When the vertex ratio is increased, the refractive index decreases. In particular, when the crystalline has a vertex ratio of 0.7185 and rays coming from a distance of 250 mm, we can observe that the refractive index is bigger than when the ratio is of 0.7036 and rays come from infinity. This means that for closer distances, the accommodation must be greater than for larger distances. It is clear, from Fig. 5.13 that the marginal rays do not focused on the retina. This can be seen as a problem, but due to the fact that these rays never enter 92 Chapter 5. Schematic Eye with Composite Luneburg Crystalline Figure 5.14: Examples of Schematic model eyes with CML lens: a) Incident rays coming from a 250 mm distance and b) Incident rays coming from a 500 mm distance. Both Schematic eyes have a relation of 0.7450. 5.2. Schematic Luneburg Eye 93 into the eye, because they are blocked by the eye’s pupil. Figure 5.15: Schematic eye when rays are off-axis. This analysis is completed when an off-axis analysis is made. When rays are off-axis, the rays reach the retina with a coma aberration. This aberration is larger when the rays are farther from the axis as shown in Fig. 5.15. This schematic model can be modified for different object distances, different refractive indices and different curvatures of the crystalline; this is to make it as versatile as possible. Also, it is possible to change the corneal topography to get a more realistic scenario, since every single eye is different. The lens accommodation is obtained by reshaping the lens, resulting in the necessity of a modification in its gradient refractive index. As can be seen from the simulations, the rays inside the crystalline are planar curves. The analysis of the off-axis case shows that when the rays enter with a certain angle with respect to the optical- or z-axis into the eye, a coma aberration is presented. As this angle increase the resultant coma aberration increase. 94 5.3 Chapter 5. Schematic Eye with Composite Luneburg Crystalline Conclusions In conclusion, a schematic model eye based on a composite modified Luneburg lens acting as the eye lens has been presented. Chapter 6 Conclusions and Future Work Along this work, we have presented a new model of the human lens and a new schematic human eye using a composite modified Luneburg lens. This lens has been proposed as a bi-spheroidal model of the human lens based on the gradient index Luneburg lens. Using biometrical data reported in the literature our model accurately predicts the imaging properties of a human lens, where we have observed that the composite modified Luneburg lens has the same physiological properties of the human lens. It was also obtained that is necessary a modification in its gradient refractive index when simulating accommodation by reshaping the lens it is necessary a modification in its gradient refractive index implying that it is not possible to establish a definitive mathematical expression for the gradient refractive index of a biological crystalline. Also, with this lens model it was possible to create a schematic GRIN eye that emulates the human lens behavior. With this model, we did an analysis that had never been done, i.e., this model allows the evaluation of different imaging situations; i.e., the effects of the accommodation in the human lens due to the changed in distance from which rays impigne. The analysis predicts that the accommodation of the human lens is accompanied by a change in the gradient refractive index of this lens, which depends on the ratio za /zp . This analysis was completed when an off-axis analysis was made. This demonstrated that the rays off-axis reach the retina with a coma aberration, due to the inclination of the rays. In the specialized literature, the coma aberration is obtained when the lens has a inclination on the optical axis, however, in our schematic model, the rays off-axis act in a similar manner as the lens inclination, 96 Chapter 6. Conclusions and Future Work for this reason, the coma aberration is present. It is very important to say that our schematic model can be modified for different object distances, different refractive indices and different curvatures of the human lens; also, our model can be modified for different topographies of the cornea, i.e., it can be modified fully to emulate any experimental data. The work presented in this thesis has opened various research topics in the optical area. For example, in Visual Optics , it will be of interest to study the ray propagation in a three-dimensional elliptical Luneburg lens, the lens is shown in Fig. 6.1. Figure 6.1: Three-dimensional elliptical Luneburg lens. The importance of considering the study of the elliptical Luneburg lens is because we can do a study of a three-dimensional human lens, as shown in Fig. 6.2, with which we can generate a three-dimensional schematic human eye, in which we will be able study the effect that causes the fovea to be on the visual axis. Also, this study is important in the communications area because the elliptical Luneburg lens can be used to replace conventional antenna systems. In the fiber integrated optics, it is possible to produce a loss-free waveguide using the Luneburg lens [44]. The problem with this waveguide will be that the 97 Figure 6.2: Three-dimensional human lens. incident rays must always be parallels to its optical axis. However, we can do an analysis using the off-axis rays to obtain a loss-free waveguide that does not depend on the angle of incidence of the rays. Note that this idea generates a critical angle of π. In general, we have introduced a model of the human lens built as a composite Luneburg lens whose gradient index function and its derivatives are continuous at the equatorial plane. This allowes the construction of a more realistic schematic model of the eye. We demonstrated its imaging capabilities of the whole schematic eye for sources at infinity. In our model we have used conicoid surfaces but it can be generalized to any surface with the condition of having continuos derivatives at the equator plane. Our method allows us to obtain custom made GRIN distributions once the biometric parameters are given. List of Figures 2.1 Fermat’s Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 The linear gradient index medium. . . . . . . . . . . . . . . . . . 11 2.3 An arc lenght along the ray path. . . . . . . . . . . . . . . . . . . 12 2.4 Solution for the linear gradient index medium. . . . . . . . . . . . 14 2.5 The radial gradient index medium. . . . . . . . . . . . . . . . . . 14 2.6 Solution for the Sagittal plane. . . . . . . . . . . . . . . . . . . . . 16 2.7 Solution for the radial gradient index medium. . . . . . . . . . . . 17 3.1 The spherical gradient. . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 The Index Variation. . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 Medium with radial symmetry . . . . . . . . . . . . . . . . . . . . 22 3.4 The path of rays in a medium with gradient index . . . . . . . . . 26 3.5 Rays on Horizontal axis . . . . . . . . . . . . . . . . . . . . . . . 28 3.6 The rays are hyperbolas when C > 0. . . . . . . . . . . . . . . . . 32 3.7 The rays are parabolas when C = 0. . . . . . . . . . . . . . . . . . 33 3.8 The rays are ellipses when C < 0. . . . . . . . . . . . . . . . . . . 33 3.9 Maxwell Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.10 Maxwell Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.11 Maxwell Solution Off-Axis . . . . . . . . . . . . . . . . . . . . . . 39 3.12 Luneburg Sphere Parameters . . . . . . . . . . . . . . . . . . . . . 41 3.13 The Luneburg Lens . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.14 The Generalized Luneburg Lens . . . . . . . . . . . . . . . . . . . 50 3.15 The Elliptical Luneburg Lens . . . . . . . . . . . . . . . . . . . . 56 3.16 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.17 Ray tracing of The Luneburg Lens . . . . . . . . . . . . . . . . . 59 3.18 Ray tracing of The Elliptical Luneburg Lens . . . . . . . . . . . . 59 3.19 Ray tracing of The Elliptical Luneburg Lens with nc = 2, ns = 1, µ = 2 and α0 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 100 List of Figures 4.1 Human eye and its optical elements [28]. . . . . . . . . . . . . . . 62 4.2 The structure of the cornea [28]. . . . . . . . . . . . . . . . . . . . 64 4.3 The geometric shape of the lens. za and zp are differents. . . . . . 66 4.4 The layers at the back of the human eye [28]. . . . . . . . . . . . 69 4.5 The Emsley reduced schematic eye. . . . . . . . . . . . . . . . . . 70 4.6 The Gullstrand-Emsley schematic eye. . . . . . . . . . . . . . . . 71 4.7 The Le Grand full theoretical schematic eye. . . . . . . . . . . . . 71 4.8 The schematic eye is analytically derived from the refractive index profile of the crystalline lens and anatomical measurements of a rat eye. [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.9 Schematic eye model with Maxwell fish-eye spherical lens [9]. . . . 73 5.1 Geometry of the bi-spherical model showing the continuity of the isoindical lines at the equator plane. . . . . . . . . . . . . . . . . . 77 Ray tracing through the proposed CML lens embedded in a medium with refractive index of 1.336. a) Rays incident from an infinite distance. Rays incidents from a finite distance of 250 mm with b) za /zp = 0.6872 and c) za /zp = 0.7185. . . . . . . . . . . . . . . . . 79 The GRIN distribution and the GRIN profiles for each lens with different ratio za /zp . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.4 The GRIN profiles for each lens with different ratio za /zp . . . . . 81 5.5 Trajectory of the inner rays for each CML lens with different ratio za /zp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 The GRIN distribution and the GRIN profiles for each lens with the same ratio za /zp and different central refractive index. . . . . 83 The GRIN profiles for each lens with the same ratio za /zp and different central refractive index. . . . . . . . . . . . . . . . . . . . 84 Ray tracing through the proposed CML lens embedded in a medium with a refractive index of 1.336 and a constant ratio za /zp = 0.7036. Rays incidents from a finite distance of a) Infinity, b) 2000 mm, c)1000 mm, d) 500 mm, and e)250 mm. . . . . . . . . . . . . . . . 85 5.2 5.3 5.6 5.7 5.8 List of Figures 5.9 5.15 Trajectory of the inner rays for each CMLL with constant ratio za /zp = 0.7036. . . . . . . . . . . . . . . . . . . . . . . . . . . . . The GRIN profiles for each lens with the same ratio za /zp and different central refractive index. The human lens is 20 years old. Negative spherical aberration from the human lens. . . . . . . . . Positive spherical aberration due to the cornea. Rays incident from infinity distance, the anterior and posterior surfaces of the cornea are immersed in a refractive index of 1 and 1.336, respectively. . . Schematic eye model with CML lens. . . . . . . . . . . . . . . . . Examples of Schematic model eyes with CML lens: a) Incident rays coming from a 250 mm distance and b) Incident rays coming from a 500 mm distance. Both Schematic eyes have a relation of 0.7450. Schematic eye when rays are off-axis. . . . . . . . . . . . . . . . . 6.1 6.2 Three-dimensional elliptical Luneburg lens. . . . . . . . . . . . . . Three-dimensional human lens. . . . . . . . . . . . . . . . . . . . 5.10 5.11 5.12 5.13 5.14 101 86 87 89 90 90 92 93 96 97 Bibliography [1] David A. Atchison and George Smith. The Eye and Visual Optical Instruments. Cambridge University Press, USA, 1997. (Cited on pages 1, 64 and 70.) [2] M.C.W. Campbell and A. Hughes. An analytic, gradient index schematic lens and eye for the rat which predicts aberrations for finite pupils. Vision Res., 21(7):1129–1148, 1981. (Cited on pages 1, 72 and 100.) [3] Alexander V. Goncharov and Chris Dainty. Wide-field schematic eye models with gradient-index lens. J. Opt. Soc. Am. A, 24(8):2157–2174, 2007. (Cited on pages 1, 66 and 70.) [4] Fernando Palos Rafael Navarro and Luis González. Adaptive model of the gradient index of the human lens. i. formulation and model of aging ex vivo lenses. J. Opt. Soc. Am. A, 24(8):2175–2185, 2007. (Cited on pages 1 and 66.) [5] R. Meder C. E. Jones, D. A. Atchison and J. M. Pope. Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (mri). Vision Res., 45:2352–2366, 2005. (Cited on page 1.) [6] R. K. Luneburg. Mathematical Theory of Optics. University of California Press, Los Angeles California, 1964. (Cited on pages 2, 31 and 75.) [7] Rudolf Kingslake and R. Barry Johnson. Lens Design Fundamentals. Academic Press (SPIE Press), USA, 2010. (Cited on pages 2 and 44.) [8] A. B. Kravtsov R. G. Zainullin and E. P. Shaitor. The crystalline lens as a luneburg lens. Biofizica, 19(5):913–915, 1974. (Cited on pages 2, 72 and 75.) [9] Hao Lv Yun Wu, Aimei Liu and et al. Finite schematic eye model with maxwell fish-eye spherical lens. In Symposium on Photonics and Optoelectronic (SOPO) 2010, pages 1–4, 2010. (Cited on pages 2, 35, 72, 73 and 100.) 104 Bibliography [10] Erich W. Marchand. Gradient Index Optics. Academic Press, New York, 1978. (Cited on pages 2, 19, 49 and 50.) [11] V. Fernandez A. M. Rosen, D. B. Denham and et al. In vitro dimensions and curvatures of human lenses. Vision Res., 46(6):1002–1009, 2006. (Cited on pages 2, 65, 67, 76 and 78.) [12] María Victoria Pérez Carlos Gómez-Reino and Carmen Bao. Gradient-Index Optics Fundamentals and Applications. Springer, New York, 2002. (Cited on pages 5 and 14.) [13] Ajoy K. Ghatak Vasudevan Lakshminarayanan and K. Thyagarajan. Lagrangian Optics. Kluwer Academid Publishers, Massachusetts USA, 2002. (Cited on pages 15 and 22.) [14] Max Born and Emil Wolf. Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light. Cambridge University Press, UK, 7th edition, 1978. (Cited on pages 9 and 26.) [15] J. R. Flores. Estudio de Elementos Ópticos de Gradiente de Índice de Simetría Esférica. PhD thesis, Universidade de Santiago, España, 1992. (Cited on pages 40 and 46.) [16] A. I. Kiselev M. L. Krasnov and G. I. Makarenko. A book of problems in ordinary differential equations. MIR, URSS, 1981. (Cited on page 46.) [17] Samuel P. Morgan. General solution of the luneburg lens problem. J. Appl. Phys., 29(9):1358–1368, 1958. (Cited on pages 49, 50, 75 and 76.) [18] G. Toraldo di Francia. Il problema matematico del sistema ottico concentrico stigmatico. Ann. Mat. Pura Appl., 44:35–44, 1957. (Cited on page 50.) [19] Ning Wang Yoke Leng Loo, Yarong Yang and et al. Broadband microwave luneburg lens made of gradient index metamaterials. J. Opt. Soc. Am. A, 29(4):426–430, 2012. (Cited on page 52.) Bibliography 105 [20] Nathan Kundtz and David R. Smith. Extreme-angle broadband metamaterial lens. Nature Materials, 9:129–132, 2010. (Cited on page 52.) [21] Angela Demetriadou and Yang Hao. Slim luneburg lens for antenna applications. Opt. Express, 19(21):19925–19934, 2011. (Cited on pages 52, 53, 75 and 76.) [22] Matthew Mackey Shanzuo Ji, Kezhen Yin and et al. Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses. Opt. Eng., 52(11):112105–1– 112105–13, 2013. (Cited on page 53.) [23] Richard S. Lepkowicz Shanzuo Ji, Michael Ponting and et al. A bio-inspired polymeric gradient refractive index (grin) human eye lens. Opt. Express, 20(24):26746–26754, 2012. (Cited on page 53.) [24] J. A. Grzesik. Focusing properties of a three-parameter class of oblate, luneburg-like inhomogeneous lenses. J. of Electromagn. Waves and Appl., 19(8):1005–1019, 2005. (Cited on pages 57, 75, 76 and 78.) [25] Michael P. Keating. Geometric, Physical and Visual Optics. ButterworthHeinemann, Boston, 2ed edition, 2002. (Cited on pages 61 and 65.) [26] E Dalimier and Science Faculty. Adaptive Optics Correction of Ocular HigherOrder Aberrations and the Effects on Functional Vision. PhD thesis, Universidade de Santiago, España, 2007. (Cited on page 61.) [27] Steven H. Schwartz. Geometrical and Visual Optics, a clinical approach. McGraw-Hill, 2002. (Cited on page 61.) [28] David A. Atchison and George Smith. Optics of the Human Eye. Butterworth-Heinemann, Oxford, 2000. (Cited on pages 62, 63, 64, 65, 67, 69, 70 and 100.) 106 Bibliography [29] María Cinta Puell Marín. Óptica Fisiológica: El sistema óptico del ojo y la visión binocular. E-Prints Complutense (Universidad Complutense Madrid), Madrid, 2006. (Cited on pages 62, 63, 65 and 68.) [30] J. A. Alvarado M. J. Hogan and J. E. Weddell. Histology of the Human Eye. Saunders and Co., 1971. (Cited on page 63.) [31] Sergio Barbero. Refractive power of a multilayer rotationally symmetric model of the human cornea and tear film. J. Opt. Soc. Am. A, 23(7):1578– 1585, 2006. (Cited on page 64.) [32] A. Díaz del Rio M. T. Flores-Arias and et al. Description of gradient-index human eye by a first-order optical system. J. Opt. A: Pure Appl. Opt., 11(12), 2009. (Cited on page 64.) [33] Pablo Artal and Juan Tabernero. The eye’s aplanatic answer. Nature Photonics, 2:586–589, 2008. (Cited on pages 64, 67 and 70.) [34] Enrique Gambra Alberto de Castro, Sergio Ortiz and et al. Three-dimensional reconstruction of the crystalline lens gradient index distribution from oct imaging. Opt. Express, 18(21):21905–21917, 2010. (Cited on page 66.) [35] Barbara K. Pierscionek and Justyn W. Regini. The gradient index lens of the eye: An opto-biological synchrony. Progress in Retinal and Eye Research, 31(4):332–349, 2012. (Cited on pages 66 and 78.) [36] David R. Williams Junzhong Liang and Donald T. Miller. Supernormal vision and high-resolution retinal imaging through adaptive optics. J. Opt. Soc. Am. A, 14(11):2884–2892, 1997. (Cited on page 68.) [37] G. J. Augustine D. Purves and D. Fitzpatrick (Editors). Neurociencia. Editorial Médica Panamericana, Madrid, 3rd edition, 2008. (Cited on pages 68 and 70.) Bibliography 107 [38] Barbara Pierscionek Mehdi Bahrami, Masato Hoshino and et al. Optical properties of the lens: An explanation for the zones of discontinuity. Experimental Eye Research, 124:93–99, 2014. (Cited on pages 70 and 80.) [39] H. H. Emsley. Visual Optics. Butterworth-Heinemann, Oxford, 1952. (Cited on page 70.) [40] J. P. C. Southall. Helmholtz?s Treatise on Physiological Optics. Optical Society of America, New. York, 1924. (Cited on page 71.) [41] B. K. Pierscionek G. Smith and D. A. Atchison. The optical modelling of the human lens. Ophthal. Physiol. Opt., 11(4):359–369, 1991. (Cited on pages 75 and 76.) [42] C. Bao M. A. Rama, M. V. Pérez and et al. Gradient-index crystalline lens model: A new method for determining the paraxial properties by the axial and field rays. Optics Communications, 249(4):595–609, 2005. (Cited on pages 75 and 76.) [43] Antonio Benito Pablo Artal and Juan Tabernero. The human eye is an example of robust optical design. Journal of Vision, 6(1):1–7, 2006. (Cited on page 89.) [44] G. P. Tsironis M. M. Mattheakis and V. I. Kovanis. Luneburg lens waveguide networks. J. Opt., 14(11):1–8, 2012. (Cited on page 96.)