Phys 531 Lecture 25 17 November 2005 Retarders and the Jones Calculus Last time, discussed how work: - transmit one - blocks orthogonal polarization Described ways to implement Discussed

Today: Discuss retarders Develop computational tools

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Outline: Review polarizers • Retarders • Matrix technique • Partially polarized light • Finish polarization

Next time: Eat turkey After that: Quantum

2 Polarizers Ideal has transmission axis a transmits light with ˆ = a blocks light with ˆ a ⊥ If a complex, transmit circ or ellip polarization no true “axis” If ˆ neither a nor a k ⊥ use T = ˆ a 2 | ∗ · | Output light is polarized along a

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Retarders (Hecht 8.7) Use polarizers to make linear polarized light What about circular or elliptical? Use retarder

Most common retarder = wave plate Based on birefringence Light polarized along optic axis: index ne Light polarized axis: index no ⊥

4 Suppose linear polarized light, angle α Incident on uniaxial crystal with axis vertical Crystal thickness d

E y

x α axis

z

d

Coordinates as shown: z = 0 at front of crystal

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Incident wave has ˆ = cos α ˆx + sin α yˆ i(k r ωt) Einc = E0ˆe · −

In crystal k nk = nω/c → 0 different for Ex and Ey components So

inok0z inek0z iωt Ecrystal = E0 cos α e xˆ + sin α e yˆ e− At output z = d: 

inok0d inek0d iωt Ed = E0 cos α e ˆx + sin α e yˆ e−   inok d i(ne no)k d iωt = E0e 0 cos α ˆx + sin α e − 0 yˆ e− h i 6 i(ne no)k d Define ˆout = cos α xˆ + sin α e − 0 yˆ Then after crystal, have

inok0d i[k(z d) ωt] E(z, t) = E0e ˆout e − −

i(no 1)k0d i(kz ωt) = E0e − ˆout e − i(kz ωt) = E0 ˆout e −

Get plane wave out with

ˆ = cos α ˆx + sin α eiε yˆ and ε = (ne no)k d retardance − 0 ≡ Set d to achieve desired ε value 7

By adjusting α and ε, make arbitrary polarization state

Example: kd(ne no) = π/2 − Then d(ne no) = λ/4: call quarter-wave plate − iπ/2 Get ˆout = cos α ˆx + e sin α yˆ

= cos α ˆx + i sin α yˆ

xˆ iyˆ For α = 45◦, ˆout =   √2 Make left and right circular polarizations Main use of quarter-wave plate: convert linear to

8 Other common configuration: ε = π

Then d(ne no) = λ/2: half-wave plate − Have eiπ = 1 so − ˆ = cos α xˆ sin α yˆ out − Changes polarization angle α α → −

If α = 45 , then α = 45 in ◦ out − ◦ Rotated by 90◦: orthogonal to input

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Generally, use half-wave plate at arb angle θ how to calculate effect? need to express input in crystal basis ⇒ Define x0 = axis of x'

E y θ x α

z

y'

10 Then ˆx0 = cos θ ˆx + sin θ yˆ yˆ = cos θ yˆ sin θ ˆx 0 − and ˆ = cos α xˆ + sin α yˆ Calculate j = xˆ ˆ x0 0 · = cos α cos θ sin α sin θ = cos(α θ) − − and j = yˆ ˆ y0 0 · = cos α sin θ + sin α cos θ = sin(α θ) − − So: ˆ = cos α0 xˆ0 + sin α0 yˆ0 for α = α θ 0 − (Or just use x0-y0 coords from beginning) 11

Effect of half wave plate is α α 0 → − 0 So α θ = (α θ) out − − in − α = 2θ α out − in Change θ by ∆θ, output polarization rotates by 2∆θ

Main use of half-wave plate: rotate linear polarization by arbitrary angle

Question: What happens when θ = αin? What is physically happening in this situation? 12 One problem: very thin

Quarter wave plate: d = λ/4(ne no) λ − ∼ Often make ε = π/4 + 2πm for integer m: get same effect Called multiple order waveplate typical m 10 ≈

Most common waveplate material: cost $200 for 1-cm diameter plate Cheaper: plastic ($5 for 5 cm square) - ε less accurate - distorts laser beams 13

Terminology: Fast axis: axis of wave plate with lower n • Slow axis: axis of wave plate with higher n • Doesn’t really matter which is optical axis depends on whether no > ne (“negative” crystal) or no > ne (“positive” crystal)

Other ways to make retarders (Hecht pg 356-357): - Fresnel rhomb: use phase shift from TIR - Babinet-Soliel compensator: waveplate with variable thickness

14 Jones Calculus (Hecht 8.13.2-3) Polarizers and retarders simple in proper basis Polarizer: pass one component, block other Retarder: add phase shift to one component Always good to choose basis carefully

But with multiple elements, no one basis is simple Half-wave plate example illustrated how to deal: change basis for each element Easy way to implement: use matrix notation

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Basic idea: ˆ is a vector j write ˆ = x " jy #

Polarizers and retarders are linear elements write as matrices ˆ = ˆ out U in = 2 2 matrix U ×

Hecht writes matrix as A I’ll use U

16 Examples: Matrix for polarizer with a = xˆ is 1 0 = U " 0 0 # Arbitrary incident polarization: cos α ˆin = iε " e sin α #

1 0 cos α So ˆout = iε " 0 0 # " e sin α # cos α = " 0 #

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See ˆ a and ˆ 2 = ˆ a 2 out k | out| | in∗ · | as required Not really any different than previous calcs Just written differently: Matrix is taking ˆ a for you ·

Question: What would be for polarizer with transmission U axis along yˆ?

18 One issue: ˆ is supposed to be unit vector 1 Really have t ˆout = ˆin = cos α U " 0 # where t = amplitude transmission coeff E0out = tE0in Then intensity transmittance T = t 2 | |

Easier to write ˆ = ˆ out U in use T = ˆ 2 | out| Just remember to make ˆ 2 = 1 | in|

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More examples: Wave plate with optic axis along y

in k d in k d Saw ˆ jxe o 0 xˆ + jye e 0 yˆ → einok0d 0 1 0 So = = einok0d inek d i(ne no)k d U " 0 e 0 # " 0 e − 0 #

Say quarter wave plate, ne > no - then slow axis along y 1 0 = U " 0 i # ignoring overall phase

20 Or, quarter wave plate with fast axis along y: 1 0 = U 0 i " − # slow axis along x ≡ (Doesn’t matter which is optic axis)

Half wave plate, either axis along y: 1 0 = U 0 1 " − # since eiπ = e iπ = 1 − −

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Generally called Jones matrices Easy when element axes aligned to x or y

What if axis at angle θ? Define two coordinates: x-y = “lab” coordinates x0-y0 = “element” coordinates

For instance, x0 = transmission axis of polarizer

22 Express ˆ in x0-y0 coords

If x0 axis at angle θ to x, then ˆx0 = cos θ ˆx + sin θ yˆ yˆ = cos θ yˆ sin θ ˆx 0 − as before So j = xˆ ˆ = cos θjx + sin θjy x0 0 · j = yˆ ˆ = sin θjx + cos θjy y0 0 · − Write

j cos θ sin θ jx x0 = j sin θ cos θ jy " y0 # " − # " # 23

Define rotation matrix cos θ sin θ (θ) = R sin θ cos θ " − # Then ˆ = (θ)ˆ 0 R In primed coordinates, is easy U For polarizer 1 0 0 = U " 0 0 # and ˆ = ˆ = (θ) ˆ out0 U0 in0 U0 R

24 Still need to get ˆout in lab coords 1 Generally ˆ = R(θ)− ˆ0 (θ) 1 = matrix inverse of (θ) R − R 1 cos θ sin θ Find (θ)− = − = ( θ) R " sin θ cos θ # R −

So ˆ = ( θ) (θ)ˆ out R − U0 R in

Or, without specifying coordinates: (θ) = ( θ) (θ) U R − U0 R for (θ = 0) U0 ≡ U 25

So polarizer with axis at angle θ: cos θ sin θ 1 0 cos θ sin θ (θ) = − U sin θ cos θ 0 0 sin θ cos θ " # " # " − # cos2 θ sin θ cos θ = 2 " sin θ cos θ sin θ #

1 If ˆin = = ˆx " 0 #

cos2 θ cos θ get ˆout = = cos θ " sin θ cos θ # " sin θ # Get T = ˆ 2 = cos2 θ as expected | out|

26 Other examples: QWP fast axis at angle θ (measured from x axis)

cos θ sin θ 1 0 cos θ sin θ (θ) = − U sin θ cos θ 0 i sin θ cos θ " # " # " − # cos2 θ + i sin2 θ (1 i) sin θ cos θ = − (1 i) sin θ cos θ sin2 θ + i cos2 θ " − #

HWP at angle θ cos(2θ) sin(2θ) (θ) = U sin(2θ) cos(2θ) " − #

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Get matrices for all elements in lab coordinates For multiple elements, multiply matrices

Example: Suppose light linearly polarized along y passes through a quarter-wave plate with slow axis at θ1 = 30◦ from x, then a half-wave plate with axis at θ2 = 60◦, and

finally a polarizer with axis at θ3 = 45◦. What is the total transmission?

28 Solution: Get matrix for each element. 0.75 + 0.25i 0.433 0.433i 1 = − U 0.433 0.433i 0.25 + 0.75i  −  0.5 0.866 0.5 0.5 2 = − 3 = U 0.866 0.5 U 0.5 0.5     0.433 0.25i 0.25 + 0.433i = 3 2 1 = − Utot U U U 0.433 0.25i 0.25 + 0.433i  −  0 0.25 + 0.433i Given ˆ = so ˆ = in 1 out 0.25 + 0.433i     So T = ˆ 2 = 2 0.25 + 0.433 2 = 0.5 | out| × | | Question: What is the polarization state of the output?

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Often output state is “disguised” by extra phase factors Example: Light polarized along yˆ into QWP at 45◦ Expect to get circular polarized light out 1 1 + i 1 i Have = − U 2 1 i 1 + i " − # 1 1 + i so ˆ = out 2 1 i " − # Doesn’t look like circular polarization!

30 Note 1 + i = √2eiπ/4 1 i = √2e iπ/4 − − 1 eiπ/4 eiπ/4 1 So ˆout = iπ/4 = iπ/2 √2 " e− # √2 " e− # eiπ/4 1 = √2 i " − # = right circular polarization Good approach: always factor out phase to make jx real Gives standard form ˆ = cos βxˆ + eiε sin βyˆ

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Can generalize and work in any basis (ˆe1, ˆe2) j Write ˆ = j ˆe + j ˆe = 1 1 1 2 2 j " 2 #12

j ˆe ˆx ˆe yˆ jx then 1 = 1∗ · 1∗ · j ˆe ˆx ˆe yˆ jy " 2 #12 " 2∗ · 2∗ · # " #xy jx (xy 12) j ≡ S → " y #xy Then in 1-2 basis, have = 12 (xy 12) xy (†xy 12) U S → U S → where = conjugate transposition of S† S 32 Partially Polarized Light Sometimes light is not perfectly polarized also not unpolarized Example: pass natural light through poor polarizer No Jones vector in this case: ˆ is fluctuating Instead define coherency matrix

jxjx∗ jxjy∗ jx G = h i h i = jx∗ jy∗ " jyjx∗ jyjy∗ # *" jy # + h i h i h i

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For polarized light, G and ˆ have same information Unpolarized light = mixture of x and y polarized light 1 1 0 1 0 0 Then Gunpolarized = + 2 " 0 0 # 2 " 0 1 # 1/2 0 = " 0 1/2 #

Get G in other cases by propagation If system has Jones matrix , then have U G = G out U in U† 34 Example: Suppose glass surface has t = 0.95, t = 0.85 ⊥ from Fresnel relations k 0.95 0 Then = in - basis U " 0 0.85 # k ⊥

Illuminate with natural light: 0.95 0 0.5 0 0.95 0 Gout = " 0 0.85 # " 0 0.5 # " 0 0.85 # 0.45 0 = " 0 0.36 #

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2 2 Total irradiance = jx + jy = Trace G | | | | = sum of diagonal elements Here write 0.55 0 Gout = 0.81 " 0 0.45 # Total transmission is 0.81

Transmitted light is partially polarized: 55% and 45% k ⊥

36 Could then pass through a QWP at 45◦ eiπ/4 1 i = − U √2 i 1 " − #

1 1 i 0.55 0 1 i Get G = − out 2 i 1 0 0.45 i 1 " − # " # " # 0.5 0.05i = 0.05i 0.5 " − # Describes partial circular polarization

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Coherency matrix is generally nice way to describe polarization Takes care of phase factors • Has correct symmetry property: • 180◦ rotation doesn’t change state

Older way to describe partially polarized light: (Hecht 8.13.1) Coherency matrix is better Reference Klein and Furtak §9.3

38 Summary: Retarders based on birefringence • - Quarter-wave plate: make circ polarization - Half-wave plate: rotate linear polarization Use Jones matrices for multiple elements • Partially polarized light • coherency matrix → - Also uses Jones matrices

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