Phys 531 Lecture 25 17 November 2005 Retarders and the Jones Calculus Last time, discussed how polarizers work: - transmit one polarization - blocks orthogonal polarization Described ways to implement Discussed birefringence Today: Discuss retarders Develop computational tools 1 Outline: Review polarizers • Retarders • Matrix technique • Partially polarized light • Finish polarization Next time: Eat turkey After that: Quantum Optics 2 Polarizers Ideal polarizer has transmission axis a transmits light with ^| = a blocks light with ^| a ? If a complex, transmit circ or ellip polarization no true \axis" If ^| neither a nor a k ? use T = ^| a 2 j ∗ · j Output light is polarized along a 3 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 5 Incident wave has ^| = cos α ^x + sin α y^ i(k r !t) Einc = E0^|e · − In crystal k nk = n!=c ! 0 different for Ex and Ey components So inok0z inek0z i!t Ecrystal = E0 cos α e x^ + sin α e y^ e− At output z = d: inok0d inek0d i!t Ed = E0 cos α e ^x + sin α e y^ e− inok d i(ne no)k d i!t = E0e 0 cos α ^x + sin α e − 0 y^ e− h i 6 i(ne no)k d Define ^|out = cos α x^ + sin α e − 0 y^ Then after crystal, have inok0d i[k(z d) !t] E(z; t) = E0e ^|out e − − i(no 1)k0d i(kz !t) = E0e − ^|out e − i(kz !t) = E0 ^|out e − Get plane wave out with ^| = cos α ^x + sin α ei" y^ and " = (ne no)k d retardance − 0 ≡ Set d to achieve desired " value 7 By adjusting α and ", make arbitrary polarization state Example: kd(ne no) = π=2 − Then d(ne no) = λ/4: call quarter-wave plate − iπ=2 Get ^|out = cos α ^x + e sin α y^ = cos α ^x + i sin α y^ x^ iy^ For α = 45◦, ^|out = p2 Make left and right circular polarizations Main use of quarter-wave plate: convert linear to circular polarization 8 Other common configuration: " = π Then d(ne no) = λ/2: half-wave plate − Have eiπ = 1 so − ^| = cos α x^ sin α y^ out − Changes polarization angle α α ! − If α = 45 , then α = 45 in ◦ out − ◦ Rotated by 90◦: orthogonal to input 9 Generally, use half-wave plate at arb angle θ how to calculate effect? need to express input in crystal basis ) Define x0 = axis of waveplate x' E y θ x α z y' 10 Then ^x0 = cos θ ^x + sin θ y^ y^ = cos θ y^ sin θ ^x 0 − and ^| = cos α x^ + sin α y^ Calculate j = x^ ^| x0 0 · = cos α cos θ sin α sin θ = cos(α θ) − − and j = y^ ^| y0 0 · = cos α sin θ + sin α cos θ = sin(α θ) − − So: ^| = cos α0 x^0 + sin α0 y^0 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: waveplates very thin Quarter wave plate: d = λ/4(ne no) λ − ∼ Often make " = π=4 + 2πm for integer m: get same effect Called multiple order waveplate typical m 10 ≈ Most common waveplate material: quartz cost $200 for 1-cm diameter plate Cheaper: plastic ($5 for 5 cm square) - " 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 15 Basic idea: ^| is a vector j write ^| = x " jy # Polarizers and retarders are linear elements write as matrices ^| = ^| out U in = 2 2 matrix U × Hecht writes matrix as A I'll use U 16 Examples: Matrix for polarizer with a = x^ is 1 0 = U " 0 0 # Arbitrary incident polarization: cos α ^|in = i" " e sin α # 1 0 cos α So ^|out = i" " 0 0 # " e sin α # cos α = " 0 # 17 See ^| a and ^| 2 = ^| a 2 out k j outj j in∗ · j as required Not really any different than previous calcs Just written differently: Matrix is taking ^| a for you · Question: What would be for polarizer with transmission U axis along y^? 18 One issue: ^| is supposed to be unit vector 1 Really have t ^|out = ^|in = cos α U " 0 # where t = amplitude transmission coeff E0out = tE0in Then intensity transmittance T = t 2 j j Easier to write ^| = ^| out U in use T = ^| 2 j outj Just remember to make ^| 2 = 1 j inj 19 More examples: Wave plate with optic axis along y in k d in k d Saw ^| jxe o 0 x^ + jye e 0 y^ ! 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 − − 21 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 ^| in x0-y0 coords If x0 axis at angle θ to x, then ^x0 = cos θ ^x + sin θ y^ y^ = cos θ y^ sin θ ^x 0 − as before So j = x^ ^| = cos θjx + sin θjy x0 0 · j = y^ ^| = 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 ^| = (θ)^| 0 R In primed coordinates, is easy U For polarizer 1 0 0 = U " 0 0 # and ^| = ^| = (θ) ^| out0 U0 in0 U0 R 24 Still need to get ^|out in lab coords 1 Generally ^| = R(θ)− ^|0 (θ) 1 = matrix inverse of (θ) R − R 1 cos θ sin θ Find (θ)− = − = ( θ) R " sin θ cos θ # R − So ^| = ( θ) (θ)^| 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 ^|in = = ^x " 0 # cos2 θ cos θ get ^|out = = cos θ " sin θ cos θ # " sin θ # Get T = ^| 2 = cos2 θ as expected j outj 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) " − # 27 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 ^| = so ^| = in 1 out 0:25 + 0:433i So T = ^| 2 = 2 0:25 + 0:433 2 = 0:5 j outj × j j Question: What is the polarization state of the output? 29 Often output state is \disguised" by extra phase factors Example: Light polarized along y^ 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 ^| = out 2 1 i " − # Doesn't look like circular polarization! 30 Note 1 + i = p2eiπ=4 1 i = p2e iπ=4 − − 1 eiπ=4 eiπ=4 1 So ^|out = iπ=4 = iπ=2 p2 " e− # p2 " e− # eiπ=4 1 = p2 i " − # = right circular polarization Good approach: always factor out phase to make jx real Gives standard form ^| = cos βx^ + ei" sin βy^ 31 Can generalize and work in any basis (^e1; ^e2) j Write ^| = j ^e + j ^e = 1 1 1 2 2 j " 2 #12 j ^e ^x ^e y^ jx then 1 = 1∗ · 1∗ · j ^e ^x ^e y^ jy " 2 #12 " 2∗ · 2∗ · # " #xy jx (xy 12) j ≡ S ! " y #xy Then in 1-2 basis, have = 12 (xy 12) xy (yxy 12) U S ! U S ! where = conjugate transposition of Sy 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: ^| 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 33 For polarized light, G and ^| 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 Uy 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 # 35 2 2 Total irradiance = jx + jy = Trace G j j j j = sum of diagonal elements Here write 0:55 0 Gout = 0:81 " 0 0:45 # Total