Reflection and refraction

Ivan Bazarov Cornell Physics Department / CLASSE

Outline • Reflection & refraction in geometric optics • Fresnel equations • Applications

1 Reflection and refraction P3330 Exp Optics Fermat’s principle: reflection

Fermat’s principle: optical path is stationary (has an extremum with Optical path = path × index of refraction respect to small changes)

Law of reflection ⇥i =⇥r

2 Reflection and refraction P3330 Exp Optics Fermat’s principle: refraction

Snell’s law n sin ⇥= n sin ⇥ ⇢ 0 0

⇢0

SP = n⇢ + n0⇢0

SP + dSP = n(⇢ dx sin ⇥) + n0(⇢0 + dx sin ⇥0) f dSP f =0=f n( sin ⇥) + n0 sin ⇥0 dx f

3 Reflection and refraction P3330 Exp Optics Look at the problem as E&M wave at the interface

• Geometric optics can provide no information about wave amplitudes • Need E&M (vector!) theory of light

Perpendicular (“S”) polarization “sticks out” of or into the plane of ! Incident medium ! incidence. ki kr Ei Er n Plane of incidence i (here the xy plane) is qi qr the plane that contains InterfacePlane of the interface (here the yz the incident and plane)y (perpendicular to page) reflected k-vectors. qt x nt z E ! Parallel (“P”) t k polarization lies parallel Transmitting medium t to the plane of incidence. B-field not shown here, but it’s always perpendicular to E-field 4 Reflection and refraction P3330 Exp Optics Fresnel equations

• Goal: relate amplitudes of transmitted and reflected waves • Fresnel was the first to obtain the expressions in early 1800’s.

• A complication: must consider two polarizations separately S- and P- or ⊥ and || w.r.t. E-field orientation relative to the incidence plane

rr ==EE0r0/Er/E0i0,ti,tt ===EEE0r00/Ert/E0t0it forfor S-polarization S-polarization ?? ??? rr ==EE0r0/Er/E0i0,ti,tt===EEE0r00/Ert/E/E0t00ti forfor P-polarization P-polarization k k k kk

• These equations emerge from boundary conditions at the interface, namely a requirement for the tangential E- and B- fields to be continuous (assuming non-magnetic material)

5 Reflection and refraction P3330 Exp Optics S-polarization

! ! ki kr Er Ei ni B B i qi qr r y Interface z x qt Et n ! t Bt kt

Ei (x, y = 0, z, t) + Er (x, y = 0, z, t) = Et (x, y = 0, z, t) –Bi (x, y=0, z, t) cosqi + Br (x, y=0, z, t) cosqr = –Bt (x, y=0, z, t) cosqt or in terms of amplitudes:

E0i + E0r = E0t

− B0i cosθi + B0r cosθr = −B0t cosθt

6 Reflection and refraction P3330 Exp Optics S-polarization Fresnel ratios

Using B = nE/c and qr = qi one finds E n cos ✓ n cos ✓ r = 0r = i i t t ? E0i ni cos ✓i + nt cos ✓t E 2n cos ✓ t = 0t = i i ? E0i ni cos ✓i + nt cos ✓t

w cosθ n Nicer form if we define m = t = t , n = t wi cosθi ni

q w i 1 nm i n r = i ? 1+nm nt w 2 q t t = t ? 1+nm

7 Reflection and refraction P3330 Exp Optics P-polarization Fresnel ratios ! ! k This B- ki r field points E into the i Br ! ! ! page Esince×B∝k Bi E qi qr r ni Interface Beam geometry for light with its qt electric field E parallel to the t nt plane of incidence Bt ! kt Or more compactly From boundary conditions: n cos ✓ n cos ✓ m n r = i t t i r = k B0i – B0r = B0t k ni cos ✓t + nt cos ✓i m + n E0icosqi + E0rcosqr = E0tcosqt 2 2ni cos ✓i t = t = k m + n k ni cos ✓t + nt cos ✓i

8 Reflection and refraction P3330 Exp Optics Reflection from air-to-glass interface

1.0 nair » 1 < nglass » 1.5

Note that: r Brewster’s angle .5 r =0! r Total reflection at q = 90° || || for both polarizations 0 Zero reflection for parallel polarization at Brewster's angle (56.3° for these

Reflection coefficient, coefficient, Reflection -.5 r values of ni and nt). ┴

-1.0 0° 30° 60° 90° Incidence angle, qi

9 Reflection and refraction P3330 Exp Optics Reflection from glass-to-air interface

1.0 nglass » 1.5 > nair » 1 Critical angle r Note that: .5 r ┴ Total internal reflection Total internal above the critical angle reflection 0

qcrit º arcsin(nt /ni) Brewster’s angle (The sine in Snell's Law coefficient, Reflection -.5 can't be > 1!): Critical r|| angle

sin(qcrit) = nt /ni sin(90°) -1.0 0° 30° 60° 90° Incidence angle, qi

10 Reflection and refraction P3330 Exp Optics Transmittance (T)

IA T º Transmitted Power / Incident Power = tt A = Area IAii Compute the qi wi ratio of the ni beam areas: nt 1D beam wt expansion qt 2 The beam expands in one dimension on refraction. E0t 2 2 = t æöe� c 2 00 E0i ç÷nEtt0 2 IAttèø2 éù w tnEtt0 w t n t2 cos(q� t ) Tt==êú =2 = IAæöe�00c 2 wnE w n cos(q� ) iinEëû iii0 i i i ç÷ii2 0 n èø n = t n ! n cos θ $ i ( t ( t )) 2 2 Þ T = # &t = nmt The Transmittance is also # n cos θ & called the Transmissivity. "( i ( i ))%

11 Reflection and refraction P3330 Exp Optics Reflectance (R)

IA R º Reflected Power / Incident Power = rr A = Area IAii

qi qr wi wi ni nt

Because the angle of incidence = the angle of reflection, the beam area doesn’t change on reflection.

Also, ni = nr for both incident and reflected beams.

So: Rr= 2 The Reflectance is also called the Reflectivity.

12 Reflection and refraction P3330 Exp Optics Reflectance and transmittance for an air- to-glass interface

Perpendicular polarization Parallel polarization 1.0 1.0 T T

.5 .5

R R 0 0 0° 30° 60° 0° 30° 60° 90° 90° Incidence angle, qi Incidence angle, qi

Note that R + T = 1

13 Reflection and refraction P3330 Exp Optics Reflectance and transmittance for a glass- to-air interface

Perpendicular polarization Parallel polarization 1.0 1.0 R R

.5 .5 T T 0 0 0° 30° 60° 0° 30° 60° 90° 90° Incidence angle, qi Incidence angle, qi

Note that R + T = 1

14 Reflection and refraction P3330 Exp Optics Reflection at normal incidence

When q = 0, 2 i " % nt − ni R = $ ' # nt + ni &

and 4n n T t i = 2 n n ( t + i )

For an air-glass interface (ni = 1 and nt = 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.

15 Reflection and refraction P3330 Exp Optics Practical implications of Fresnel 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 Mansion 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 high- incidence-angle near-unity reflections.

16 Reflection and refraction P3330 Exp Optics Polarized sunglasses

Sunglasses are made to transmit only vertically polarized light to cut glare due to reflections from water

17 Reflection and refraction P3330 Exp Optics Photography with polarizers

(no polarizer) (with polarizer)

Same principle to ‘see below the surface’ or remove an unwanted reflection/glare

18 Reflection and refraction P3330 Exp Optics Phase shifts for reflections

r has a sign, i.e. when negative it means 180º phase shift 180° if low-index-to-high and 0 if high-index-to-low.

E.g. if you slowly turn up a laser intensity incident on a piece of glass, where does damage happen first, the front or the back?

Actually, at the back surface because of constructive interference, which gives about 44% higher intensity than at the front: (1+r)2 = (1+0.2)2 = 1.44

Important application of phase shifts: anti-reflection coating

19 Reflection and refraction P3330 Exp Optics Total Internal Reflection occurs when sinqt > 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.

20 Reflection and refraction P3330 Exp Optics Applications of total internal reflection

Beam steerers

Beam steerers used to compress the path inside binoculars

21 Reflection and refraction P3330 Exp Optics Back-reflectors: corner cubes and cat’s eyes

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.

Cat’s eye reflectors use a spherical geometry to achieve the same goal.

22 Reflection and refraction P3330 Exp Optics Optical fibers

Optical fibers use TIR to transmit light long distances.

They play an ever-increasing role in our lives!

23 Reflection and refraction P3330 Exp Optics Design of an optical fiber

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

ncore > ncladding

24 Reflection and refraction P3330 Exp Optics Propagation of light in 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.

25 Reflection and refraction P3330 Exp Optics Frustrated total internal reflectance

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

n0=1 n0=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.

26 Reflection and refraction P3330 Exp Optics FTIR for fingerprinting

See TIR from a fingerprint valley and FTIR from a ridge.

27 Reflection and refraction P3330 Exp Optics Links/References

http://www.teknik.uu.se/ftf/education/ftf2/Optics_FresnelsEqns.pdf

www.physics.rutgers.edu/ugrad/389/FresnelsEqns.ppt

http://optics.hanyang.ac.kr/~shsong/23-Fresnel%20equations.pdf

Some figures taken from the lecture notes by Dr. Agladze and Wikipedia

28 Reflection and refraction P3330 Exp Optics