Innovation and Development of Study Field. nano.tul.cz

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1 Innovation and Development of Study Field Nanomaterials at the Technical University of Liberec nano.tul.cz These materials have been developed within the ESF project: Innovation and development of study field Nanomaterials at the Technical University of Liberec

2 Miroslav Šulc

3 Outline Incident, transmitted, and reflected beams at interfaces Reflection and transmission coefficients The Fresnel Equations Brewster's Angle Total internal reflection Power reflectance and transmittance Phase shifts in reflection The mysterious evanescent wave 3

4 Definitions: Planes of Incidence and the Interface and the polarizations Perpendicular ( S ) polarization sticks out of or into the plane of incidence. ki Incident medium kr Plane of incidence (here the xy plane) is the plane that contains the incident and reflected k- vectors. Parallel ( P ) polarization lies parallel to the plane of incidence. E i q i q r Interface Plane of the interface (here the yz plane) y (perpendicular to page) z x q t E r E t Transmitting medium k t n i n t 4

5 Shorthand notation for the polarizations Perpendicular (S) polarization sticks up out of the plane of incidence. Parallel (P) polarization lies parallel to the plane of incidence. 5

6 Fresnel Equations E i E r We would like to compute the fraction of a light wave reflected and transmitted by a flat interface between two media with different refractive indices. E t r E / E 0r 0i 0r 0i perpendicular 0t / 0i polarization t E0t / E0i t E E for the r E / E for the parallel polarization where E 0i, E 0r, and E 0t are the field complex amplitudes. We consider the boundary conditions at the interface for the electric and magnetic fields of the light waves. We ll do the perpendicular polarization first. 6

7 Boundary Condition for the Electric Field at an Interface In other words: The total E-field in the plane of the interface is continuous. Here, all E-fields are in the z-direction, which is in the plane of the interface (xz), so: The Tangential Electric Field is Continuous E i (x, y = 0, z, t) + E r (x, y = 0, z, t) = E t (x, y = 0, z, t) ki B i Interface E i q i q r q t B t E r E t kr k t B r z y x n i n t 7

8 Boundary Condition for the Magnetic Field at an Interface Reflection and refraction of light The Tangential Magnetic Field* is Continuous In other words: The total B-field in the plane of the interface is continuous. ki q i B i q i E i q i q r E r kr B r y n i Here, all B-fields are in the xy-plane, so we take the x-components: Interface B i (x, y=0, z, t) cos(q i ) + B r (x, y=0, z, t) cos(q r ) = B t (x, y=0, z, t) cos(q t ) q t B t E t k t z x n t 8 *It's really the tangential B/m, but we're using m m 0

9 Reflection and Transmission for Perpendicularly (S) Polarized Light Canceling the rapidly varying parts of the light wave and keeping only the complex amplitudes: E E E 0i 0r 0t B cos( q ) B cos( q ) B cos( q ) 0i i 0r r 0t t But B E /( c / n) ne / c and q q : 0 0 r i n ( E E )cos( q ) n E cos( q ) i 0r 0i i t 0t t Substituting for E E E E : 0t using 0i 0r 0 t n ( E E )cos( q ) n ( E E )cos( q ) i 0r 0i i t 0r 0i t 9

10 Reflection & Transmission Coefficients for Perpendicularly Polarized Light Rearranging n ( E E )cos( q ) n ( E E )cos( q ) yields: i 0r 0i i t 0r 0i t cos( q ) cos( q ) cos( q ) cos( q ) E n n E n n 0r i i t t 0i i i t t Solving for E / E yields the reflection coefficient : 0r 0i q q q q r E0r / E 0i n cos( ) i i nt cos( t ) / ni cos( i) nt cos( t ) Analogously, the transmission coefficient, E / E 0t t E / E n cos( q ) / n cos( q ) n cos( q ) t i i i i i t t i, is 10

11 Simpler expressions for r and t Recall the magnification at an interface, m: wt cos( qt) m w cos( q ) i i wi n i n t q i q t w t Also let r be the ratio of the refractive indices, n t / n i. Dividing numerator and denominator of r and t by n i cos(q i ): i cos( qi) t cos( qt ) / i cos( qi) t cos( qt ) 1 r / 1 r t n cos( q ) / n cos( q ) n cos( q ) / 1 rm r n n n n m m i i i i t t r 1 rm t 1 rm 1 rm 11

12 Fresnel Equations Parallel electric field ki B i E i q i q r B r kr E r This B-field points into the page. n i z y x Interface Beam geometry for light with its electric field parallel to the plane of incidence (i.e., in the page) q t B t E t k t n t Note that Hecht uses a different notation for the reflected field, which is confusing! Ours is better! Note that the reflected magnetic field must point into the screen to achieve E B. kthe x means into the screen. 1

13 Reflection & Transmission Coefficients for Parallel (P) Polarized Light For parallel polarized light, B 0i - B 0r = B 0t and E 0i cos(q i ) + E 0r cos(q r ) = E 0t cos(q t ) Solving for E 0r / E 0i yields the reflection coefficient, r : q q q q r E / E n cos( ) n cos( ) / n cos( ) n cos( ) 0r 0i i t t i i t t i Analogously, the transmission coefficient, t = E 0t / E 0i, is t E / E n cos( q ) / n cos( q ) n cos( q ) 0t 0i i i i t t i These equations are called the Fresnel Equations for parallel polarized light. 13

14 Simpler expressions for r and t q q q q r E / E n cos( ) n cos( ) / n cos( ) n cos( ) 0r 0i i t t i i t t i t E / E n cos( q ) / n cos( q ) n cos( q ) 0t 0i i i i t t i Again, use the magnification, m, and the refractive-index ratio, r. And again dividing numerator and denominator of r and t by n i cos(q i ): r/ r r m m t /mr r m r m r t m r 14

15 Reflection coefficient, r Reflection and refraction of light Reflection Coefficients for an Air-to-Glass Interface n air 1 < n glass Note that: Total reflection at q = 90 for both polarizations.5 Brewster s angle r =0! r Zero reflection for parallel polarization at Brewster's angle (56.3 for these values of n i and n t ) r (We ll delay a derivation of a formula for Brewster s angle until we do dipole emission and polarization.) Incidence angle, q i 15

16 Reflection coefficient, r Reflection and refraction of light Reflection Coefficients for a Glass-to-Air Interface n glass 1.5 > n air Critical angle Note that:.5 r Total internal reflection above the critical angle 0 Total internal reflection q crit arcsin(n t /n i ) (The sine in Snell's Law can't be > 1!): -.5 Brewster s angle Critical angle r sin(q crit ) n t /n i sin(90) Incidence angle, q i

17 Transmittance (T) T Transmitted Power / Incident Power IA t t IA i i c 0 0 I n E0 A = Area Compute the ratio of the beam areas: wi 1D beam expansion n i n t q i q t w t A A w cos( q ) w cos( q ) t t t i i i m T The beam expands in one dimension on refraction. c I A w n I A w n 0 0 nt 0 E t t t nt E t 0t wt t t t 0c 0 i i i ni E0i wi i i ni E 0i nt cos qt) ) n cos q ) T t r mt i i ) E E cos( q ) cos( q ) 0t 0i The Transmittance is also called the Transmissivity. t 17

18 Reflectance (R) R Reflected Power / Incident Power IA r IA i 0 0 I n E0 r i c A = Area w i ni n t q i q r w i Because the angle of incidence = the angle of reflection, the beam area doesn t change on reflection. Also, n is the same for both incident and reflected beams. So: R r The Reflectance is also called the Reflectivity. 18

19 Reflectance and Transmittance for an Air-to-Glass Interface Perpendicular polarization Parallel polarization 1.0 T 1.0 T R R Incidence angle, q i Incidence angle, q i Note that R + T = 1 19

20 Reflectance and Transmittance for a Glass-to-Air Interface Perpendicular polarization Parallel polarization 1.0 R 1.0 R.5.5 T T Incidence angle, q i Incidence angle, q i Note that R + T = 1 0

21 Reflection at normal incidence When q i = 0, R n n t i nt ni and T 4nn t i n n ) t i For an air-glass interface (n i = 1 and n t = 1.5), R = 4% and T = 96% The values are the same, whichever direction the light travels, from air to glass or from glass to air. The 4% has big implications for photography lenses. 1

22 Practical Applications of Fresnel s Equations Windows look like mirrors at night (when you re in the brightly lit room) One-way mirrors (used by police to interrogate bad guys) are just partial reflectors (actually, aluminum-coated). Disneyland puts ghouls next to you in the haunted house using partial reflectors (also aluminum-coated). Lasers use Brewster s angle components to avoid reflective losses: R = 100% 0% reflection! Laser medium R = 90% 0% reflection! Optical fibers use total internal reflection. Hollow fibers use highincidence-angle near-unity reflections.

23 Reflection coefficient, r Phase shifts in reflection (air to glass) p 180 phase shift for all angles Brewster s angle r 0 p Incidence angle -1.0 r Incidence angle, q i Incidence angle 180 phase shift for angles below Brewster's angle; 0 for larger angles 5

24 Reflection coefficient, r Phase shifts in reflection (glass to air) p 0 p Incidence angle Interesting phase above the critical angle Critical angle Brewster s angle Critical angle r r Total internal reflection Incidence angle, q i Incidence angle 180 phase shift for angles below Brewster's angle; 0 for larger angles 6

25 Phase shifts vs. incidence angle and n i Note the general behavior above and below the various interesting angles /n t n i /n t q i Li Li, OPN, vol. 14, #9, pp. 4-30, Sept. 003 n i /n t q i 7

26 If you slowly turn up a laser intensity incident on a piece of glass, where does damage happen first, the front or the back? The obvious answer is the front of the object, which sees the higher intensity first. But constructive interference happens at the back surface between the incident light and the reflected wave. This yields an irradiance that is 44% higher just inside the back surface! 8 (1 0.) 1.44

27 Phase shifts with coated optics Reflections with different magnitudes can be generated using partial metallization or coatings. We ll see these later. But the phase shifts on reflection are the same! For near-normal incidence: 180 if low-index-to-high and 0 if high-index-to-low. Example: Laser Mirror Highly reflecting coating on this surface Phase shift of 180 9

28 Total Internal Reflection occurs when sin(q t ) > 1, and no transmitted beam can occur. Note that the irradiance of the transmitted beam goes to zero (i.e., TIR occurs) as it grazes the surface. Brewster s angle Total Internal Reflection Total internal reflection is 100% efficient, that is, all the light is reflected. 30

29 Applications of Total Internal Reflection Beam steerers Beam steerers used to compress the path inside binoculars 31

30 Three bounces: The Corner Cube Corner cubes involve three reflections and also displace the return beam in space. Even better, they always yield a parallel return beam: If the beam propagates in the z direction, it emerges in the z direction, with each point in the beam (x,y) reflected to the (-x,-y) position. Hollow corner cubes avoid propagation through glass and don t use TIR. 3

31 Fiber Optics Optical fibers use TIR to transmit light long distances. They play an ever-increasing role in our lives! 33

32 Design of optical fibers Core: Thin glass center of the fiber that carries the light Cladding: Surrounds the core and reflects the light back into the core Buffer coating: Plastic protective coating n core > n cladding 34

33 Propagation of light in an optical fiber Light travels through the core bouncing from the reflective walls. The walls absorb very little light from the core allowing the light wave to travel large distances. Some signal degradation occurs due to imperfectly constructed glass used in the cable. The best optical fibers show very little light loss -- less than 10%/km at 1,550 nm. Maximum light loss occurs at the points of maximum curvature.

34 Microstructure fiber In microstructure fiber, air holes act as the cladding surrounding a glass core. Such fibers have different dispersion properties. Air holes Core Such fiber has many applications, from medical imaging to optical clocks. 36

35 Frustrated Total Internal Reflection By placing another surface in contact with a totally internally reflecting one, total internal reflection can be frustrated. Total internal reflection Frustrated total internal reflection n=1 n=1 n n n n How close do the prisms have to be before TIR is frustrated? This effect provides evidence for evanescent fields fields that leak through the TIR surface and is the basis for a variety of spectroscopic techniques. 37

36 FTIR and fingerprinting See TIR from a fingerprint valley and FTIR from a ridge. 38

37 The Evanescent Wave The evanescent wave is the "transmitted wave" when total internal reflection occurs. A mystical quantity! So we'll do a mystical derivation: Since sin( q ) 1, q doesn't exist, so computing r is impossible. t r t E n cos( ) cos( ) 0 i qi n r t qt E n cos( q ) n cos( q ) 0i i i t t Let's check the reflectivity, R, anyway. Use Snell's Law to eliminate q : n i cos( qt ) 1 sin ( qt ) 1 sin ( qi) Neg. Number nt Substituting this expression into the above one for r and * a bi a bi redefining R yields: R rr 1 a bi a bi So all power is reflected; the Reflection evanescent and refraction wave of light contains no 39 power. k i y x q i q t kr k t n i n t t

38 The Evanescent-Wave k-vector The evanescent wave k-vector must have x and y components: Along surface: k tx = k t sin(q t ) Perpendicular to it: k ty = k t cos(q t ) k i y x q i q t kr k t n i n t Using Snell's Law, sin(q t ) = (n i /n t ) sin(q i ), so k tx is meaningful. And again: cos(q t ) = [1 sin (q t )] 1/ = [1 (n i /n t ) sin (q i )] 1/ = ± ib Neglecting the unphysical -ib solution, we have: E t (x,y,t) = E 0 exp[ kb y] exp i [ k (n i /n t ) sin(q i ) x w t ] 40 The evanescent wave decays exponentially in the transverse direction.

39 Optical Properties of Metals A simple model of a metal is a gas of free electrons (the Drude model). These free electrons and their accompanying positive nuclei can undergo "plasma oscillations" at frequency, w p. where: w p Ne m ) 0 e The refractive index for a metal is : n w p 1 w When n 0, nis imaginary, and absorption is strong. So for w w metals absorb strongly. For w w meta ls are transparent. p p 41

40 Reflection from metals At normal incidence in air: R ( n 1) ( n 1) Generalizing to complex refractive indices: R * ( n1)( n 1) * ( n1)( n 1) 4

41 On-line measurement of thickness of foil for safety windows Important polarization of beams n 1 = 1 n = 1,51 n 3 = 1,48 n 1 = 1,51 43

42 Elipsometry 44

43 elipsometer 45

44 References/Further reading HECHT E., ZAJAC A.,Optics. Addison-Wesley Publishing Company, 1979 SALEH, B. E. A., TEICH, M.C. Fundamentals of Photonics, John Wiley & Sons, 1991 On the Web Source text Polarization of light, reflection and refraction, Fresnel law Reflection and refraction, Huygens principle refraction.htm 46

45 THANK YOU FOR YOUR ATTENTION 47