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xin, qin The Ray Vector xout, qout A light ray can be defined by two co-ordinates: its position, x q its slope, q x Optical axis These parameters define a ray vector, which will change with distance and as the ray propagates through optics. x q Ray Matrices For many optical components, we can define 2 x 2 ray matrices. An element’s effect on a ray is found by multiplying its ray vector. Ray matrices can describe simple and complex systems. Optical system ↔ 2 x 2 Ray matrix xin q in A C B D xout q out These matrices are often called ABCD Matrices. Ray matrices as derivatives Since the displacements and angles are assumed to be small, we can think in terms of partial derivatives. spatial magnification xout xin xout xout xout xin qin xin qin q out q out q out xin qin xin qin xout q in xout A B xin q C D q in out q out xin q out q in angular magnification We can write these equations in matrix form. Ray matrix for free space or a medium If xin and qin are the position and slope upon entering, let xout and qout be the position and slope after propagating from z = 0 to z. xout qout xin, qin xout xin z qin qout qin Rewriting these expressions in matrix notation: z z=0 Ospace 1 z = 0 1 xout 1 z xin q 0 1 q in out Ray matrix for a lens Olens 1/ f (n 1)(1/ R1 1/ R2 ) 1 = -1/f 0 1 The quantity, f, is the focal length of the lens. It’s the single most important parameter of a lens. It can be positive or negative. In a homework problem, you’ll extend the Lens Maker’s Formula to lenses of greater thickness. R1 > 0 R2 < 0 f>0 If f > 0, the lens deflects rays toward the axis. R1 < 0 R2 > 0 f<0 If f < 0, the lens deflects rays away from the axis. A lens focuses parallel rays to a point one focal length away. For all rays A lens followed by propagation by one focal length: xout = 0! f xin 0 xout 1 f 1 0 xin 0 q 0 1 1/ f 1 0 1/ f 1 0 x / f in out f f Assume all input rays have qin = 0 At the focal plane, all rays converge to the z axis (xout = 0) independent of input position. Parallel rays at a different angle focus at a different xout. Looking from right to left, rays diverging from a point are made parallel. Consecutive lenses Suppose we have two lenses right next to each other (with no space in between). 1 Otot = -1/f 2 f1 f2 0 1 0 1 = 1 -1/f1 1 -1/f1 1/ f 2 0 1 1/f tot = 1/f1 + 1/f 2 So two consecutive lenses act as one whose focal length is computed by the resistive sum. As a result, we define a measure of inverse lens focal length, the diopter. 1 diopter = 1 m-1 A system images an object when B = 0. When B = 0, all rays from a point xin arrive at a point xout, independent of angle. xout = A xin xout A 0 xin A xin q C D q C x D q in in in out When B = 0, A is the magnification. The Lens Law From the object to the image, we have: 1) A distance do 2) A lens of focal length f 3) A distance di 0 1 d o 1 d i 1 O 1/ f 1 0 1 0 1 do 1 d i 1 1/ f 1 d / f 0 1 o 1 di / f d o di d o di / f 1/ f 1 d / f o B d o di d o di / f do di 1/ do 1/ di 1/ f 0 if 1 1 1 d o di f This is the Lens Law. Imaging Magnification If the imaging condition, 1 1 1 d o di f is satisfied, then: 1 di / f O 1/ f 0 1 do / f So: M O 1/ f 0 1/ M 1 1 A 1 di / f 1 di d o di di M do 1 1 D 1 do / f 1 do d o di do 1/ M di magnification • Linear or transverse magnification — For real images, such as images projected on a screen, size means a linear dimension • Angular magnification — For optical instruments with an eyepiece, the linear dimension of the image seen in the eyepiece (virtual image in infinite distance) cannot be given, thus size means the angle subtended by the object at the focal point (angular size). Strictly speaking, one should take the tangent of that angle (in practice, this makes a difference only if the angle is larger than a few degrees). Thus, angular magnification is defined as Depth of Field Only one plane is imaged (i.e., is in focus) at a time. But we’d like objects near this plane to at least be almost in focus. The range of distances in acceptable focus is called the depth of field. It depends on how much of the lens is used, that is, the aperture. Out-of-focus plane Image Object f Aperture Focal plane The smaller the aperture, the more the depth of field. Size of blur in out-of-focus plane Gaussian Beams F-number The F-number, “f / #”, of a lens is the ratio of its focal length and its diameter. f/# = f/d f d1 f f/# =1 f d2 f f/# =2 Large f-number lenses collect more light but are harder to engineer. Numerical Aperture Another measure of a lens size is the numerical aperture. It’s the product of the medium refractive index and the marginal ray angle. NA = n sin(a) a f High-numerical-aperture lenses are bigger. Why this definition? Because the magnification can be shown to be the ratio of the NA on the two sides of the lens. Numerical Aperture • The maximum angle at which the incident light is collected depends on NA Remember diffraction Numerical Aperture • The maximum angle at which the incident light is collected depends on NA Remember diffraction Scattering of Light • What happens when particle size becomes less than the wavelength? • Rayleigh scattering : – The intensity I of light scattered by a single small particle from a beam of unpolarized light of wavelength λ and intensity I0 is given by: – – where R is the distance to the particle, θ is the scattering angle, n is the refractive index of the particle, and d is the diameter of the particle. Scattering of Light • What happens when particle size becomes less than the wavelength? • MIE scattering : http://omlc.ogi.edu/calc/mie_calc.html – Use a series sum to calculate scattered intensity at arbitrary particle diameter Sphere Diameter 0.10 microns Refractive Index of Medium 1.0 Real Refractive Index of Sphere 1.5 Imaginary Refractive Index of Sphere 0 Wavelength in Vacuum 0.6328 microns Concentration 0.1 spheres/micron3 MIE scattering Sphere Diameter 0.2 microns Refractive Index of Medium 1.0 Real Refractive Index of Sphere 1.5 Imaginary Refractive Index of Sphere 0 Wavelength in Vacuum 0.6328 microns Concentration 0.1 spheres/micron3 Telescopes Keplerian telescope Image plane #1 Image plane #2 M1 M2 A telescope should image an object, but, because the object will have a very small solid angle, it should also increase its solid angle significantly, so it looks bigger. So we’d like D to be large. And use two lenses to square the effect. Oimaging Otelescope M 1/ f 0 1/ M where M = - di / do 0 M1 0 M2 1/ f 1/ M 1/ f 1/ M 2 2 1 1 M 1M 2 0 M / f M / f 1/ M M 1 2 2 1 1 2 Note that this is easy for the first lens, as the object is really far away! So use di << do for both lenses. Telescope Terminology Microscopes Image plane #1 Objective M1 Eyepiece Image plane #2 M2 Microscopes work on the same principle as telescopes, except that the object is really close and we wish to magnify it. When two lenses are used, it’s called a compound microscope. Standard distances are s = 250 mm for the eyepiece and s = 160 mm for the objective, where s is the image distance beyond one focal length. In terms of s, the magnification of each lens is given by: |M| = di / do = (f + s) [1/f – 1/(f+s)] = (f + s) / f – 1 = s / f Many creative designs exist for microscope objectives. Example: the Burch reflecting microscope objective: Object To eyepiece Example: Magnifying Glass