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Download Lecture Note No. 3 Reflection and Refraction (Sections 4.3 to 4.8)
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Engineering Optics and Optical Techniques Lecture Note No. 3 Spring 2007, Prof. Kenneth D. Kihm Engineering Optics and Optical Techniques Lecture Note No. 3 Reflection and Refraction (Sections 4.3 to 4.8) [Reading Assignment: Sections 4.9 to 4.12] TRANSMISSION: In a conventionally dense medium, scattered E-M wavelets (secondary wave) cancelled each other in all but the forward direction, i.e., the sustained ongoing beam. *Most ordinary air environment below 100 miles altitude, liquid, glass, crystals etc., for example, ~ 500 nm and l ~ 3 nm for STP air. (Reminder) /2 REFLECTION: Back-scattered E-M wavelets based on the non-resonant ground level vibration by a thin layer (~/2) of unpaired atomic oscillators near the surface, i.e., bounced portion of the ongoing beam (see Fig. 4-15). nti nt ni ni: refractive index of the medium where the primary E-M wave is incident. nt: refractive index of the medium where the primary E-M wave is transmitted. Reflection layer thickness 1 Engineering Optics and Optical Techniques Lecture Note No. 3 Spring 2007, Prof. Kenneth D. Kihm For ni < nt, the reflection is called “external” For ni > nt, the reflection is called “internal” REFLECTION has no apparent color preference: 1 2 Reflection layer thickness No. of reflectors (atoms) Rayleigh scattering dependency Reflection efficiency *Ray *Wavefront *Plane-of-incidence *Specular reflection *Diffuse reflection 2 Engineering Optics and Optical Techniques Lecture Note No. 3 Spring 2007, Prof. Kenneth D. Kihm REFRACTION Ferma’s Principle … The beam path between two points takes one that needs the least travel time, or shortest optical path length (OPL ns ds n j s j ). The resulting travel time is b a given as n j j s j / c with c = speed light in vacuum or free space. j *OPL is different from the shortest path. OPL min n j s j j t min n j s j /c Snell’s Law … ni sin i nt sin t SO O P t Vi Vt From b 2 a x h2 x2 Vi Vt 2 d t 0 for least travel time condition: dx sin i sin t Vi Vt ni sin i nt sin t 3 c j Engineering Optics and Optical Techniques Lecture Note No. 3 Spring 2007, Prof. Kenneth D. Kihm QUICK REVIEW ON WAVE NATURE OF E-M RADIATION 1-D Wave Equation and Plane Waves (Revisit) § Def’n of plane: k r ro 0 or k r const (k is the wave propagation direction.) x, t A sin kx t A Ime i kxt k 2 V …wave frequency 2 …angular frequency 1 1 k 2 …wave number …wave period 2 2 for the case of constant V V / V V 4 Engineering Optics and Optical Techniques Lecture Note No. 3 Spring 2007, Prof. Kenneth D. Kihm REFLECTION/REFRACTION BY E-M WAVE APPROACH Revisit of the Snell’s Law Plane-of-Incidence view of Fig. 4.38 E i E oi cosk i r i t E r E or cosk r r r t r E t E ot cosk t r t t t *The tangential [perpendicular to the page] component of the electric oscillation vector must remain the same before and after the interface, i.e., Etangential on one side of the surface Etangential on the other side. This is required to satisfy no arbitrary slip of atoms at the interface: un E i un E r un E t This leads to k i r i t y b k r r r t r y b k t r t t t y b which requires an identity i r t *Set t = 0 for the moment of incident of the beam on the surface: ki r kr r r kt r t k i k r r r … The interface vector r sweeps out a plane perpendicular to a vector k i k r , i.e., k i k r is perpendicular to the interface and parallel to u n , or equivalently, k i k r un 0 . This last equation gives, k i sin i k r sin r , and i r since k i k r ( k 2 / / V ). 5 Engineering Optics and Optical Techniques Lecture Note No. 3 Spring 2007, Prof. Kenneth D. Kihm k i k t r t , or k i k t is normal to the interface, or parallel to un . Now, k i k t u n 0 and k i sin i k t sin t 2 i sin i 2 t sin t , and by c i or c t gives ni sin i nt sin t … Snell’s Law AMPLITUDE OF REFLECTION (r) and TRANSMISSION (t) I. E the Plane-of-Incidence (Refer to Fig. 4.39) (1) Continuity of tangential E (electric)-field at the interface: E oi E or E ot (2) Continuity of tangential H (magnetic) B/-field at the interface: Bi i cos i Br r cos r Bt t cos t with v i v r , i r t o (for most transparent or optically thin materials), and i r , combining the related equations [ V j B j E j , Eqs. (4.12), (4.13) & (4.14)] gives the amplitude reflection coefficient and the amplitude transmission coefficient respectively as, E n cos i nt cos t r or i E oi ni cos i nt cos t E 2ni cos i t ot E oi ni cos i nt cos t 6 Engineering Optics and Optical Techniques Lecture Note No. 3 Spring 2007, Prof. Kenneth D. Kihm Substituting the Snell’s law, ni sin i nt sin t r t sin( i t ) sin( i t ) 2 sin t cos i sin( i t ) II. E the Plane-of-Incidence (Refer to Fig. 4.40) E r or E oi n cos i ni cos t t ni cos t nt cos i E 2ni cos i t ot E oi ni cos t nt cos i Substituting the Snell’s law gives, r tan( i t ) tan( i t ) t 2 sin t cos i sin( i t ) cos( i t ) For graphical results, refer to Figs. 4.41 and 4.42. For normal incidence ( i 0 ), r r t t nt ni nt ni 2ni nt ni 7 Engineering Optics and Optical Techniques Lecture Note No. 3 Spring 2007, Prof. Kenneth D. Kihm By examining the sign conversion of r, the phase shift of E-field for internal and external reflections (refer to Fig. 4.44). ni nt for External Reflection Figs. 4.44-a & b (air >> glass) ni nt for Internal Reflection: Fig. 4.44-c, d & e (glass >> air) E n cos i nt cos t By examining r or i E oi ni cos i nt cos t For example, for external reflection “-“ for all i … (Fig. 4.44-a) E n cos i ni cos t By examining r or t … (Fig. 4.44-b) E oi ni cos t nt cos i For external reflection, “+” then “-“ beyond p (Critical angle for total reflection) No phase shift is occurred for transmission as t is always “+”. E 2ni cos i t ot E oi ni cos i nt cos t E 2ni cos i t ot E oi ni cos t nt cos i . 8 Engineering Optics and Optical Techniques Lecture Note No. 3 Spring 2007, Prof. Kenneth D. Kihm REFLECTANCE (R) AND TRANSMITTANCE (T) E2 I c o o 2 Reflectance reflected power/incident power: 2 I A cos r I r E or r2 R r I i A cos i I i E oi for both and cases. Transmittance transmitted power/incident power I A cos t nt cos t T t I i A cos i ni cos i E or E oi For graphical results, refer to Fig. 4.48. And R + T =1 2 n cos t t ni cos i for both and cases. 9 2 t for both and cases. Engineering Optics and Optical Techniques Lecture Note No. 3 Spring 2007, Prof. Kenneth D. Kihm REFLECTANCE (R) AND TRANSMITTANCE (T) When i 0 (Normal incidence), n ni R R R t n n t i T T T Dielectric substances (TransmittedIncident) Diamond-air (Hardest and most dense: highest refractive index: most sparkling) 2 4n t n i nt ni 2 nt ni R T 2.41 1 0.17 0.83 Glass-air 1.5 1 0.04 0.96 Water-air 1.33 1 0.02 0.98 Diamond-water (not as glittering as in air) 2.41 1.33 0.083 0.917 10 Engineering Optics and Optical Techniques Lecture Note No. 3 Spring 2007, Prof. Kenneth D. Kihm TOTAL INTERNAL REFLECTION From Snell’s law ni sin i nt sin t , i c when t 90 , i.e., c sin 1 nti [ nti nt / ni ] Visible light wave in glass (ni = 1.5) on air (nt = 1.0), c 41.8 Condition for total internal reflection: sin i nti When total internal reflection is occurred, Ir = Ii and It = 0, R = 1.0 and T = 0. EVANESCENT (short-lived) TRANSMITTED WAVES [Existing under total internal reflection] 11 Engineering Optics and Optical Techniques Lecture Note No. 3 Spring 2007, Prof. Kenneth D. Kihm The E-M wave expression of the transmitted electric field is E t E ot exp i k t r t E ot exp i k tx x k ty y t (1) E ot exp ikty y exp i k tx x t Using the Snell’s law [ ni sin i nt sin t ] and the condition for total internal reflection [ sin i nti ], we have k sin i k tx k t sin t t nti and sin 2 i k ty k t cos t k t 1 nti2 (2) Combining (1) and (2) gives, 1/ 2 E t E ot e y e i k t x sin i / nti t sin2 i with t , i , nt , ni kt 1 2 nti The transmitted wave amplitude drops off exponentially as it penetrates the less dense medium in the y-direction (the first exponential function). The amplitude diminishes within a few -distance. The transmitted wave advances in the x-direction as a surface wave or evanescent wave (the second exponential function). FTIR (Frustrated Total Internal Reflection) – energy flow creeping the gap (of an order of -thickness) or tunneling … a principle of beam-splitters. *Example applications: Total Internal Reflection Fluorescence Microscopy (TIRFM) 12 Engineering Optics and Optical Techniques Lecture Note No. 3 Spring 2007, Prof. Kenneth D. Kihm Substance Categories in view of Fluid Dynamics: Fluid (gas, liquid), Solid Substance Categories in view of Optics: Free space (vacuum, air): Di-electric medium (Non-conducting): Metal (Conducting): Metal: E (E-M) V I i2 R No electrons All bound electrons Free electrons + Bound electrons Thermal diffusion … significant absorption E-M WAVES ON METAL SURFACES Since metal is conducting (abundant free electrons), i.e., 0 , the Maxwell equation is given as, 2 E 2E 2E 2E 2E E 2 2 2 2 t x y z t dx ] in the oscillator model. dt We then need to substitute the complex index to account for the absorption, i.e., The conduction term acts like a damping term [ n nR inI Substituting n into E Eo cost ky Eo cos t ky / Eo cos t y / Eo cos t ny / c gives: y E E o e nI y / c cos t c / nR 13 Engineering Optics and Optical Techniques Lecture Note No. 3 Spring 2007, Prof. Kenneth D. Kihm This depicts that the transmitted E-field amplitude is given as E o e nI y / c , whereas the incident E-field Amplitude is specified as E o . 2 I E Amplitude tranmitted e 2nI y / c e y And thus, I o E Amplitude incident where the attenuation coefficient, 2n I c y =1/ is called the skin depth through which the light intensity drops to 1/3 (~1/e = 1/2.7) of the incident beam intensity (extremely thin in the order of nm for most metals) *Since metal has abundant free electrons ( o k e / m e 0 with k e 0 and 0 ), the expression of refractive index based on the oscillator model, Eq. (3.72), should be extended to account for the additional oscillation of free electrons: Eq. (3.72) n 2 nr ini 1 2 1 Nqe2 o me Nqe2 o me fj j 2 2 i j oj fj fe j 2 2 i 2 i e j oj Assuming the contribution of bound electrons are negligibly small compared with that of free electrons, i.e., fe = 1.0, e = 0, and fj = 0: p Nqe2 n 1 1 2 o m e 2 2 with the plasma frequency p Nqe2 / o me For incident wave frequency p , n-real and the metal is transparent to , and for p , ncomplex and the metal is opaque to . This is the reason why X-rays are penetrating most materials including metals (exception: lead). >>> Table 4.3 14 Engineering Optics and Optical Techniques Lecture Note No. 3 Spring 2007, Prof. Kenneth D. Kihm Reflectance from a Metal: The Amplitude of Reflection for i 0 is given as (Eq. 4.47): ro r r nt ni n t ni ~ n in ), For normal incidence in air (ni = 1.0) on a metal surface (nt = n R I R ro 2 ro ro* nR 12 nI2 nR 12 nI2 R becomes Rdielectric for nI ~ 0 (no free electron). R ~ 1 for nI >> 0 (100% reflection). * For o 589.3 nm , Metals nR nI R Sodium (Na) 0.04 2.4 0.9 Tin (Sn) 1.5 5.3 0.8 Galium (GA) 3.7 5.4 0.7 Glass (dielectric) 1.5 0 0.04 Homework Assignment #3 E-o-C Problems: 4.19, 4.24, 4.40, 4.44, 4.57, 4.61, 4.77, 4.80 Due by 1 p.m. on Tuesday of February 6, 2006 15