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קורס 336533 Fundamentals of biomedical optics and photonics יסודות אופטיקה ופוטוניקה ביו-רפואית מרצה: ד"ר דביר ילין, שעות קבלה :יום ג' 14:00-15:00 מתרגל: ליאור גולן, שעות קבלה יקבעו בהמשך. ספר: Saleh & Teich, “Fundamentals of Photonics”, 2nd edition היקף: 3נקודות זכות (שעתיים שיעור ,שעתיים תרגול). מבנה ציון: 80%בחינה 20% ,תרגילים. * ינתנו כחמישה תרגילים בסך הכל .כל התרגילים יבדקו ,יוחזרו עם הערות ,וינתנו ציונים בהתאם. תרגיל שלא יוגש במועד יקבל ציון אפס .לא יפורסמו פתרונות לתרגילים ולבחינות קודמות. * הבחינה (כשלוש שעות) עם חומר פתוח מלבד ספרים ומחשבים ניידים ,תכלול כארבע שאלות פתוחות ,מתוכן אחת המבוססת על שאלה מתוך אחד התרגילים. “… the market for biomedical optics doubled from 1985 to 1995, and then tripled from 1995 to 2005 to reach just over $6 billion… 5-fold increase is expected over the next five years. … One example is the PillCam developed by Given Technology in Israel, which optics.org reported on a few months ago. The 11×31 mm capsule contains automatic lighting control, as well as tiny cameras at each end that capture four images per second for up to 10 hours. Although the technology is nothing new, the device enables clinicians to see parts of the body that cannot be reached by an endoscope. “… to succeed in this market companies must be sure that their optics-based technology addresses a specific medical need. Indeed, analysis by medical device maker Johnson & Johnson reveals that some 87% of all innovations originate from the clinicians working in hospitals… companies must therefore attempt to focus on highvalue solutions that address real-life problems…” David Benaron, CEO of Spectros Corp. 2007 Photonics West conference in San Jose, CA (BIOS, LASE, MOEMS-MEMS, OPTO) BIOS (biomedical optics) conference 2006: 65/260 pages 2007: 81/284 pages 2008: 81/308 pages 2009: 95/324 pages 2010: moving to a larger convention center in San Francisco. Laser invention Optical technologies CCD technologies Better light handling Improved detection, imaging Biomedical applications מטרת הקורס להעניק יסודות רחבים באופטיקה ,בשיטות מדידה אופטיות, ובהדמיה ,על מנת לתת כלים לסטודנט • להבין אופטיקה • להשתמש בשיטות מדידה והדמיה אופטיות בביו-רפואה • לשנות אמצעים אופטיים קיימים • לפתח טכנולוגיות חדשות למחקר ולפיתוח Course outline 1. Maxwell equations, wave equation 2. Electromagnetic waves, Gaussian beams 3. Fourier optics, the lens, resolution 4. Light-matter interaction: scattering, absorption 5. Fluorescence, photo dynamic therapy משוואות גלים, משואות מקסוול.1 קרניים גאוסיניות, גלים אלקטרומגנטים.2 הפרדה, העדשה, אופטיקת פורייה.3 בליעה, האטום, פיזור:רקמה- אינטראקציה אור.4 דינמי- טיפול פוטו, פלואורסנציה.5 6. Fundamentals of lasers עקרונות לייזרים.6 7. Lasers in medicine לייזרים ברפואה.7 8. Basics of light detection, cameras 9. Microscopy, contrast mechanism 10. Confocal microscopy, laser scanning microscopy 11. Nanoparticles in biomedical optics 12. Optical fibers and waveguides 13. Endoscopy 14. Advanced microscopy techniques, super resolution מצלמות, עקרונות גילוי אור.8 ניגודיות, מיקרוסקופיה.9 מיקרוסקופית לייזר סורק, מיקרוסקופיה קונפוקלית.10 רפואית-חלקיקים באופטיקה ביו- ננו.11 סיבים אופטים.12 אנדוסקופיה.13 רזולוציה- סופר, מיקרוסקופיה מתקדמת.14 Lectures 1-2 1. Maxwell equations 2. The wave equation 3. Maxwell equations in medium 4. Helmholtz equation 5. Electromagnetic waves: plane, spherical, Gaussian beams 6. Properties of Gaussian beams Maxwell equations An electromagnetic field is described by two related vector fields that are functions of position and time: Electric field: E r , t Magnetic field: H r , t In free space: E H 0 t H 2. E 0 t 1. 3. E 0 4. H 0 where E E x E y E z x y z E E y E x E z E y E x E z , , y z z x x y 0 1 36 109 F m Electrical permittivity of free space in MKS units: 0 4 107 H m Magnetic permeability of free space E t H 2. E 0 t 1. H 0 The wave equation 3. E 0 H E 0 t H 2 E E 0 t 4. H 0 3 1 E 0 2 E 0 Speed of light in free space: c0 1 0 0 3 108 2 H E E 2E t 0 E t t 2E E 0 0 2 0 t 2 m s Similar procedure is followed for H For each component: 2 1 E 2E 2 2 0 c0 t 1 2Ei Ei 2 0 2 c0 t 2 i x, y , z Maxwell equations in medium D 0E P Assuming no free electric charges or currents. Electric flux density: Magnetic flux density: D r ,t B r ,t B 0H 0M E In source-free media: B t D 2. H t 1. D E 3. D 0 4. B 0 In free space: P 0 M0 P + D 0E B 0H Nucleus Electron cloud Electromagnetic waves in dielectric media definitions 1. A dielectric medium is said to be linear if the vector field P(r,t) is linearly related to the vector field E(r,t). The principle of superposition then applies. 2. The medium is said to be nondispersive if its response is instantaneous, i.e., if P at time t is determined by E at the same time t and not by prior values of E. Nondispersiveness is clearly an idealization since all physical systems, no matter how rapidly they may respond, do have a response time that is finite. 3. The medium is said to be homogeneous if the relation between P and E is independent of the position r. 4. The medium is said to be isotropic if the relation between the vectors P and E is independent of the direction of the vector E, so that the medium exhibits the same behavior from all directions. The vectors P and E must then be parallel. Induced polarization E P 0 E D - P + D 0E P 0E 0 E Or: where: D E 0 1 Electric permittivity Electric susceptibility We will assume that P is linear with E, which is valid for low field intensities. + Ep Eint ! - In isotropic media: Eint E - E P The refractive index (n) The wave equation (in a medium): 1 2E E 2 2 0 c t 2 Speed of light in free space: 1 m c0 3 108 s 0 0 where the speed of light in the medium is denoted c: c 1 The ratio of the speed of light in free space to that in the medium, c0/c, is defined as the refractive index n: c0 n c 0 0 For a nonmagnetic material, =0 and: n 1 0 0 1 Boundary conditions In a homogeneous medium, all components of the fields E, D, H, B are continuous functions of position. At the boundary between two dielectric media, in the absence of free electric charges and currents, the tangential components of the electric E and magnetic H fields, and the normal components of the electric D and magnetic B flux densities must be continuous. E=0 Poynting vector The flow of electromagnetic power is governed by the Poynting vector: Which is orthogonal to both E and H. S E H H E S The optical intensity I (power flow across a unit area normal to the vector S) is the magnitude of the time-averaged Poynting vector S. I r ,t S The average is taken over times that are long in comparison with an optical cycle. Monochromatic EM waves For the case of monochromatic electromagnetic waves in an optical medium, all components of the fields are harmonic functions of time with the same frequency . Angular frequency H r , t Re H r e E r , t Re E r eit 2 frequency Similarly: it D r , t Re D r e M r , t Re M r e B r , t Re B r e P r , t Re P r eit it it it “complex-amplitude” vectors Maxwell equations in medium (Complex amplitude) D t Re D r eit t H Re H r e it The Maxwell equations become (source-free medium, monochromatic): If the operations on the complex fields are linear, one may drop the symbol Re and operate directly with the complex functions. The real part of the final expression will represent the physical quantity in question. it H r e D r eit t eit H r D r eit t H r i D r Also, D 0E P B 0 H 0 M 1. H i D 2. E i B 3. D 0 4. B 0 Intensity and power (Complex amplitude) S E H Re Eeit Re Heit 1 1 Eeit E *e it Heit H *e it 2 2 1 E H * E * H e i 2 t E H e i 2 t E * H * 4 The last two terms on the right oscillate at optical frequencies and are therefore will be washed out by the averaging process, which is slow in comparison with an optical cycle: 1 1 * * S E H E H S S * Re S 4 2 1 Where the new vector S E H* 2 may be regarded as a complex Poynting vector. The optical intensity is the magnitude of the vector S: I S Linear, nondispersive, homogenous, and isotropic media 1. H i D D E 2. E i B B H 3. D 0 4. B 0 If we use the “material equations” for monochromatic waves: We obtain the Maxwell's equations which depend solely on the complex-amplitude vectors E and H: 1. H i E 2. E i H 3. E 0 4. H 0 * linear, non-dispersive, homogenous, isotropic, source-free medium, monochromatic. Helmholtz equation E r , t Re E r eit Substitute the complex amplitude notation into the wave equation yields: 2 1 E 2E 2 2 0 c t i 2 E 2 c2 E0 c 2 E 2 E 0 U k U 0 2 1 k nk0 2 E k 2 E 0 or: 2 Helmholtz equation where U U r represents the complex amplitude of any of the components of the electric and magnetic fields: U Ex , E y , Ez , H x , H y , H z Elementary electromagnetic waves Assumptions: Medium: linear, homogenous, non-dispersive, isotropic. Light: monochromatic - Plane waves - Spherical waves - Gaussian waves Plane waves “wavelength” E r E0eik r |k | 2 Solutions for the wave equation: H r H 0eik r The real electric field: E r , t Re E r eit Re E0 eik r eit E0 cos k r t k r: E r , t0 cos k r t0 k r: E r , t0 constant k Plane waves Proof: Plane waves satisfies the Helmholtz equation: 2 E k 2 E 0 E r E0eik r 2 E0 eik r k 2 E0eik r 0 ik 2 E0 eik r k 2 E0eik r 0 k 2 k 2 0 k k nk0 Implying that the length of the wave vector parameter k in Helmholtz equation: Also: k of the plane wave must be equal to the k nk0 k n 2 0 n c n c0 2 2 n n c0 c0 c0 Plane waves H r H 0e E r E0e 1. H i E ik r 2. E i H ik r 3. E 0 Substitute in Maxwell equations 1 & 2 yields (exercise): 4. H 0 k H 0 E0 E is perpendicular to both k and H k E0 H 0 H is perpendicular to both k and E Transverse electromagnetic (TEM) wave: Intensity of TEM waves The ratio between the amplitudes of the electric and magnetic fields is known as the impedance of the medium: E0 H 0 the impedance of free space: The complex Poynting vector: 0 0 0 377 120 0 n 1 S E H* 2 E0 1 * I S E0 H 0 2 2 2 Example: an intensity of 10 W/cm2 in free space corresponds to an electric field of 87 V/cm: V E0 2 I 2 377 10 87 cm Spherical waves Another simple solution (proof in exercise) of the Helmholtz equation is the scalar spherical wave: 0 ikr A U r e r U is spherically symmetric: U r U r Reminder: Laplacian in radial coordinates: 2 1 2 r 2 r r r An oscillating dipole radiates a wave with features that resemble the scalar solution. For points at distances from the origin much greater than a wavelength (r»λ or kr»2π), the complexamplitude vectors may be approximated by: E r E0 sin U r ˆ H r H 0 sin U r ˆ Gaussian beams – the paraxial wave A0 ikr e Spherical wave: U r r ik r Plane wave: E r E0 e A paraxial wave is a plane wave traveling along the z direction (e-ikz, with k=2π/λ), modulated by a complex envelope that is a slowly varying function of position, so that its complex amplitude is: ikz Paraxial waves: U r A r e ‘Carrier’ plane wave Slowly varying complex amplitude (in space) The paraxial Helmholtz equation 2U k 2U 0 Substitute the paraxial wave into the Helmholtz equation: 2U 2U 2U 2 k U r 0 2 2 2 x y z 2 ik z 2 A r ik z 2 A r ik z A r e 2 ik z e e k A r e 0 2 2 2 x y z Paraxial wave U r A r eik z 2 A r ik z 2 A r ik z 2 A r ik z A r ik z 2 ik z 2 ik z e e e 2 ik e k A r e k A r e 0 x 2 y 2 z 2 z 2 A 2 A 2 A A 2 ik 0 2 2 2 x y z z A A A A 2 ik 0 2 2 2 x y z z 2 2 2 Paraxial wave U r A r eik z We now assume that the variation of A(r) with z is slow enough, so that: 2 A 2 k A z 2 2 A k A z 2 z These assumptions are equivalent to assuming that sin and tan 2 A 2 A A 2 2ik 0 2 x y z Paraxial Helmholtz equation: Transverse Laplacian: A 2 T A 2ik 0 z 2 2 A A T2 2 2 x y Gaussian beams A A 2ik 0 z 2 T Paraxial Helmholtz equation Substitute the paraxial wave U r A r eik z into the paraxial Helmholtz equation 2 yields a solution of the form: ik A1 Ar e q z Where q z z iz0 and 2 q z z0: Rayleigh range 2 x2 y 2 q(z) can be separated into its real and imaginary parts: 1 1 i q z R z W 2 z Where W(z): beam width R(z): wavefront radius of curvature Gaussian beams The full Gaussian beam: 2 2 W0 W 2 z ikz ik 2 R z i z U r A0 e e W z With beam parameters: z W z W0 1 z0 2 A0 A1 iz0 z0 2 R z z 1 z z z tan 1 z0 W0 z0 A0 and z0 are two independent parameters which are determined from the boundary conditions. All other parameters are related to z0 and by these equations. Gaussian beams - properties Intensity I r U r I 0 A0 2 2 W0 U r A0 e W z 2 W0 I , z I0 e W z 22 W 2 2 W 2 z e z W z W0 1 z0 z ikz ik 2 z0 2 R z z 1 z At any z, I is a Gaussian function of . On the beam axis: z tan 1 2 W0 I0 I 0, z I 0 2 W z 1 z z 0 W0 z z0 z0 - Maximum at z=0 - Half peak value at z=±z0 z=0 z=z0 1 1 z=2z0 1 2 2 R z i z Gaussian beams - properties Beam width W0 U r A0 e W z 2 W 2 z e z W z W0 1 z0 2 z0 2 R z z 1 z z W z W0 1 z0 2W0 2 z tan 1 W0 z0 z z0 ikz ik 2 2 R z i z Gaussian beams - properties Beam divergence z z0 z W z W0 1 z0 W W z 0 z z0 2 W0 U r A0 e W z 2 W 2 z e z W z W0 1 z0 ikz ik 2 2 R z 2 z0 2 R z z 1 z z tan 1 W0 W0 0 2 z0 W0 W0 Thus the total angle is given by 4 2 0 2W0 W0 z z0 z0 2W0 2 0 4 i z Gaussian beams - properties Depth of focus W0 U r A0 e W z z0 W0 W02 2 z0 2 z0: Rayleigh range 2 W 2 z e z W z W0 1 z0 2 z0 2 R z z 1 z z tan 1 W0 z0 z z0 ikz ik 2 2 R z i z Gaussian beams - properties Phase W0 U r A0 e W z W0 U r A0 e W z 2 W 2 z e ikz ik 2 2 R z i z 0, z kz tan 1 2 W 2 z e z W z W0 1 z0 z z0 ikz ik 2 2 R z 2 z0 2 R z z 1 z z tan 1 W0 z z0 z0 A. Ruffin et al., PRL (1999) The total accumulated excess retardation as the wave travels from - to is . This phenomenon is known as the Gouy effect. i z Gaussian beams - properties Wavefront , z kz tan 1 z k2 z0 2 R z W0 U r A0 e W z 2 W 2 z e z W z W0 1 z0 2 z0 2 R z z 1 z z tan 1 W0 z0 ~ spherical wave z z0 ikz ik 2 2 R z i z Gaussian beams - properties Wavefront W0 U r A0 e W z 2 W 2 z e z W z W0 1 z0 Plane wave z0 2 R z z 1 z z tan 1 W0 Spherical wave Gaussian beam 2 z0 z z0 ikz ik 2 2 R z i z Gaussian beams - properties Propagation W0 U r A0 e W z 2 W 2 z e z W z W0 1 z0 ikz ik 2 2 R z i z 2 z0 2 R z z 1 z z tan 1 W0 z z0 z0 Consider a Gaussian beam whose width W and radius of curvature R are known at a particular point on the beam axis. The beam waist radius is given by W0 W 1 W R 2 located to the left at a distance z R 1 R W 2 2 2 Gaussian beams - properties Propagating through lens W0 U r A0 e W z 2 W 2 z e ikz ik 2 2 R z i z The complex amplitude induced by a thin lens of focal length f is proportional to exp(ik2/2f). When a Gaussian beam passes through such a component, its complex amplitude is multiplied by this phase factor. As a result, the beam width does not altered (W'=W), but the wavefront does. Consider a Gaussian beam centered at z=0, with waist radius W0, transmitted through a thin lens located at position z. The phase of the emerging wave therefore becomes: kz k 2 2R k 2 2f kz k 2 2R ' Where 1 1 1 R' R f The transmitted wave is a Gaussian beam with width W'=W and radius of curvature R'. The sign of R is positive since the wavefront of the incident beam is diverging whereas the opposite is true of R'. Gaussian beams - properties Propagating through lens The magnification factor M evidently plays an important role. The waist radius is magnified by M, the depth of focus is magnified by M2, and the divergence angle is minified by M. Gaussian beams - properties Beam focusing For a lens placed at the waist of a Gaussian beam (z=0), the transmitted beam is then focused to a waist radius W0’ at a distance z' given by: W0 ' z' M W0 z 1 z0 f 2 f z 1 f z0 1 1 z0 f f f W0 z0 0 f 2 z 2 f 0 f 0 f z0 Gaussian beams - properties The ABCD low Reminder: A A r 1 e q z ik 2 2 q z , where q z z iz0 1 1 i or: q z R z W 2 z The ABCD Law The q-parameters, q1 and q2, of the incident and transmitted Gaussian beams at the input and output planes of a paraxial optical system described by the (A,B,C,D) matrix are related by: Aq1 B q2 Cq1 D Example: transmission Through Free Space When the optical system is a distance d of free space (or of any homogeneous medium), the ray-transfer matrix components are (A,B,C,D)=(1,d,0,1) so q2 = q2 + d. *Generality of the ABCD law The ABCD law applies to thin optical components as well as to propagation in a homogeneous medium. All of the paraxial optical systems of interest are combinations of propagation in homogeneous media and thin optical components such as thin lenses and mirrors. It is therefore apparent that the ABCD law is applicable to all of these systems. Furthermore, since an inhomogeneous continuously varying medium may be regarded as a cascade of incremental thin elements, the ABCD law applies to these systems as well, provided that all rays (wavefront normals) remain paraxial. Summary Maxwell equation: linear, non-dispersive, homogenous, isotropic, source-free medium, monochromatic light H i E E i H k E 0 H H 0 Helmholtz equation 2U k 2U 0 Plane waves: E r E0eik r Spherical waves: U r A0 ik r e r Paraxial Helmholtz equation: T2 A 2ik The Gaussian beam: E A 0 z W0 U r A0 e W z 2 W 2 z e ikz ik 2 2 R z i z