Chapter 7. Polarization and Modulation of Light. 7.1 Polarization. A. State of Polarization

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1 Chapter 7 olariation and Modulation of Light 7.1 olariation. State of olariation 1 lane of polariation ^ (a) (b) (c) (a) linearl polaried wave has its electric field oscillations defined along a line perpendicular to the direction of propagation,. The field vector and define a plane of polariation. (b) The -field oscillations are contained in the plane of polariation. (c) linearl polaried light at an instant can be represented b the superposition of two fields and with the right magnitude and phase. ˆ ˆ 0 = o ^ ^ o ^ 3 olariation Linearl polaried lane polaried Unpolaried olarier = cos( ω t k) o = cos( ω t k + φ ) o dd electric field components vectoriall (1) () = ˆ + ˆ = ˆ cos( ωt k) ˆ cos( ωt k) o o = cos( ω t k) ˆ ˆ 0 = o o o 4

2 = cos( ωt k) = sin( ωt k) (5a) (5b) + = (6) (a) (b) (c) (d) Δ = kδ right circularl polaried light. The field vector is alwas at right angles to, rotates clockwise around with time, and traces out a full circle over one wavelength of distance propagated. 5 o = 0 o φ = 0 o o φ = 0 o o φ = π/ o o φ = π/ amples of linearl, (a) and (b), and circularl polaried light (c) and (d); (c) is right circularl and (d) is left circularl polaried light (as seen when the wave directl approaches a viewer) 6 (a) (b) (c) o o = φ = 0 o o = φ = π/4 o o = φ = π/ (a) Linearl polaried light with o = o and φ = 0. (b) When φ = π/4 (45 ), the light is right ellipticall polaried with a tilted major ais. (c) When φ = π/ (90 ), the light is right ellipticall polaried. If o and o were equal, this would be right circularl polaried light. 7 8

3 . Malus s Law I( ) = I(0)cos Linearl polaried light T cos Light detector Liquid Crstal T 1 olarier = naler 平行於光軸 olarier 1 Unpolaried light Randoml polaried light is incident on a olarier 1 with a transmission ais T 1. Ligh emerging from olarier 1 is linearl polaried with along T 1, and becomes incident on olarier (called "analer") with a transmission ais T at an angle to T 1. detector measures the intensit of the incident light. T 1 and T are normal to the light direction. 9 垂直於光軸 10 Liquid Crstal LCD There are man modes operation TN 90 deg twist STN 70 deg twist IS in plane switching atterned vertical alignment MV multi-domain vertical alignment 11 1

4 7. Light ropagation in an nisotropic Medium: irefringence. Optical nisotrop Crstals are generall anisotropic Man properties depend on the crstal direction lectronic polariation depends on the crstal direction The refractive inde, or dielectric constant, of a crstal depends on the direction of the electric field in the propagation beam. Noncrstalline materials and cubic crstals are opticall isotropic. For all classes of crstals, ecluding cubic structures, the refractive inde depends on the propagation direction and the state of polariation. irefringence Doubl refracted 15 line viewed through a cubic sodium chloride (halite) crstal (opticall isotropic) and a calcite crstal (opticall anisotropic). rincipal refractive indices: n 1, n and n 3 rincipal aes:, and iaial crstals - Two optic aes Uniaial crstals - One optic ais (n 1 =n ) ositive, n 3 >n 1 Negative, n 3 <n 1 Optic ais: direction that two waves with the same velocit 16

5 Flurite Uniaial - positive Quart Rutile Uniaial - negative Calcite Tourmaline biaial 17. Uniaial Crstals and Fresnel s Optical Indicatri Two orthogonal linearl polaried waves Ordinar (o) wave Same velocit in all directions etraordinar (e) wave Velocit depends on the polariation direction lectric field phase propagation direction => not necessar Optic ais Direction: two waves with the same velocit 18 Optical indicatri -- Fresnel s refractive inde ellipsoid Two orthogonall polaried M waves Minor O n 1 =n =n o Major O 1 cos sin = + n ( ) n n e o e rotates with -ais in - plane k n 1 Two polaroid analers are placed with their transmission aes, along the long edges, at right angles to each other. The ordinar ra, undeflected, goes through the left polarier whereas the etraordinar wave, deflected, goes through the right polarier. The two waves therefore have orthogonal polariations. n O n 3 n 1 =n O Optic ais 19 (a) Fresnel's ellipsoid (b) n M wave propagating along O at an 0 angle to optic ais.

6 Optical indicatri -- Fresnel s refractive inde ellipsoid Two orthogonall polaried M waves Minor O n 1 =n =n o Major O n O n 1 (a) Fresnel's ellipsoid 1 cos sin = + n ( ) n n e o e n 3 n 1 =n O Optic ais (b) n M wave propagating along O at an 1 angle to optic ais. k (a) n 1 =n =n o n o = n 1 o e n e (0 ) = n = n 1 = Optic ais (b) 1 cos sin = + n ( ) n n e o e n o = n 1 o = o-wave and e = e-wave (a) Wave propagation along the optic ais. (b) Wave propagation normal to optic ais o e n e (90 ) = n 3 = Optic ais = Optic ais Wavevector Surface Wavevector Surface k Optic ais S e = ower flow k Optic ais S e = ower flow e k e e k e e-wave o-wave e k e Q k e e-wave o-wave e k e Q k e o k o O k o k o O k (a) oscillations to paper oscillations // to paper Wavefronts (constant phase fronts) (b) (a) oscillations to paper oscillations // to paper Wavefronts (constant phase fronts) (b) (a) Wavevector surface cuts in the plane for o- and e-waves. (b) n etraordinar wave in an anisotropic crstal with a k e at an angle to the optic ais. The electric field is not normal to k e. The energ flow (group velocit) is along S e which is different 3 than k e. (a) Wavevector surface cuts in the plane for o- and e-waves. (b) n etraordinar wave in an anisotropic crstal with a k e at an angle to the optic ais. The electric field is not normal to k e. The energ flow (group velocit) is along S e which is different 4 than k e.

7 C. irefringence of Calcite (CaCO 3 ) Cleaved form: rhombohedron: calcite rhomb rincipal selection: plane contains optical ais and is normal to a pair of opposite crstal surfaces Incident ra Optic ais rincipal section rincipal section // e-wave, n e Optic ais e-wave, n o e-wave calcite rhomb e-ra o-ra Incident wave Optic ais (in plane of paper) o-wave n M wave that is off the optic ais of a calcite crstal splits into two waves called ordinar and etraordinar waves. These waves have orthogonal polariations and travel with different velocities. The o-wave has a polariation that is alwas perpendicular to the optical ais. 5 (a), n o o-wave (b), n o o-wave Optic ais (a) birefringent crstal plate with the optic ais parallel to the plate surfaces. (b) birefringent crstal plate with the optic ais perpendicular to the plate surfaces. 6 D. Dichroism Optical absorption in a substance depends on the direction of propagation and state of polariation of the light beam Dichroic crstals.g. tourmaline (aluminum borosilicate) o-wave is much more heavil absorbed 7.3 irefringent Optical Devices. Retarding lates 7 8

8 ositive uniaial: Half wavelength plate: φ = Quarter wavelength plate: φ =? Optic ais = Slow ais Optic ais Input α α 45 // α L L = Fast ais // φ n e = n 3 n o Output α = arbitrar α 0 < α < 45 α = 45 retarder plate. The optic ais is parallel to the plate face. The o- and e-waves travel in the same direction but at different speeds. φ (a) (b) π = ( ne no ) L 9 Input and output polariations of light through (a) a half-wavelength 30 λ plate and (b) through a quarter-wavelength plate. Quart Rutile Calcite Tourmaline 31 3

9 . Soleil-abinet Compensator Optical Compensator: control the retardation S.-. Compensator: control and anale the polariation state π φ1 = ( nd e + nd o ) λ π φ = ( nd o + nd e ) λ d 1 Wedges can slide Optic ais hase difference φ = φ φ 1 π = ( n n )( D d ) λ D late Optic ais e o Soleil-abinet Compensator C. irefringent rism Optic ais e-ra o-ra Optic ais 1 1 e-ra 7.4 Optical ctivit and Circular irefringence 1 Optic ais Optic ais o-ra The Wollaston prism is a beam polariation splitter. the paper and also to the optic ais of the first prism. and orthogonal to 1.?1999 S O K O l i ( i H ll) 1 is orthogonal to the plane of is in the plane of the paper 35 Commercial products: tpical beam splitting angle 5º-45º 36

10 Optical activit the rotation of the plane of polariation b a substance spiraling or helical motion of electrons magnets Clockwise (detrorotator) or counterclockwise (levorotator) retardation Observer receiving the wave Quart L Optic ais Levo Detro n opticall active material such as quart rotates the plane of polariation of the incident wave: The optical field rotated to. If we reflect the wave back into the material, rotates back to.?1999 S O K O l i ( i H ll) (left) (right) clockwise 37 Quart: right-handed or left handed (atomic rrangement) Man biological substances, liquid solution (e.g. corn srup) Specific rotator power (/L) Wavelength dependent.g. quart: 400nm, 650nm Diff. speeds for left and right circularl polaried waves, i.e., diff. n linearl polaried wave Sum of left- and right-handed circularl polaried waves (fig. 7.18) π = ( nr nl) L λ Circular birefringence If the direction of the light wave is reversed, the ra simpl retraces itself. 38 Input L α α = β β R 7.5 lectro-optic ffects Output L α α β β R Slow Fast Verticall polaried wave at the input can be thought of as two right and left handed circularl polaried waves that are smmetrical, i.e. at an instant α = β. If these travel at different velocities through a medium then at the output the are no longer smmetric with respect to, α β., and the result is a vector at an angle to

11 . Definitions Changes in refractive inde of a material induced b application of an eternal field, which modulates the optical properties. ternal field Changes in electron motions or in crstal structure Changes in optical properties O effects ' n = n+ a1+ a + ockels effect (linear effect) Δ n= a1 Kerr effect (second effect) ( ). ockels ffect smmetr in the structure (-Δn, -) and (+Δn, +) Not in noncrstalline materials Centrosmmetric materials, e.g. NaCl Onl in noncentrosmmetric crstals, e.g. Gas Depends on the directions of applied field, propagation and polariation..g. Gas isotropic => uniaial Uniaial => biaial: e.g. KD and LiNbO3 ropagation in -direction (same in, directions) KD a in (Fig. 7.19(b)) LiNbO3 a in (Fig. 7.19(c)) Induce birefringence 41 Δ n= a = λk K: Kerr coefficient 4 n = n n o n 1 45 ' 1 3 n1 n1+ n1ro ' 1 3 n n nro r : ockels coeffcient n a ' 1 3 ' 1 3 n1 n1+ n1ro and n n nr φ π n π L 1 V ( ) λ λ d ' 3 1 L = n + 0 n0r π 3 L Δ φ = φ1 φ = nr 0 V Δ φ = π <=> V = V λ / λ d half-wave voltage o n 1 = n o a n 1 KD, LiNbO 3 KD LiNbO 3 (a) (b ) (c) (a) Cross section of the optical indicatri with no applied field, n 1 = n = n o (b) The applied eternal field modifies the optical indicatri. In a KD crstal, it rotates the principal aes b 45 to and and n 1 and n change to n 1 and n. (c) pplied field along in LiNbO modifies the indicatri and changes n 1 and n change 43to n and n. Input light 45 V d a olariation modulator Δφ Outpu light Tranverse ockels cell phase modulator. linearl polaried input light 44 into an electro-optic crstal emerges as a circularl polaried light.

12 Total e-field at the analer o o =? cos( ω t) + cos( ωt + φ) Field pass through 1 1 = o sin Δ φ sin ωt+ Δφ Detected I = Io sin Δφ 1 I sin π V = Io V λ / QW Input light 45 V Crstal Transmission intensit I o Detector Q 0 V λ/ V Left: tranverse ockels cell intensit modulator. The polarier and analer have their transmission ais at right angles and polaries at an angle 45 to -ais. Right: Transmission intensit vs. applied voltage characteristics. If a quarter-wave plate ( 45 QW) is inserted after, the characteristic is shifted to the dashed curve. 46 C. Kerr ffect pplied field distorts the electron motions n o => n e Δ n=λk a ll materials Induced birefringence Second order effect => smaller than ockels effect Short response time (a) a (b) n o n e n o Input light a Δφ Output light (a) n applied electric field, via the Kerr effect, induces birefringences in an otherwise opticall istropic material. (b) Kerr cell phase modulator

13 7.6 Integrated Optical Modulators Integrated optics: various optical devices and components on a single substrate e.g. LiNbO 3 Miniaturiation etter performance etter usabilit hase and olariation Modulation π 3 L Δ φ = φ φ =Γ nr 0 V LV λ d Spatial overlap efficienc: Γ=0.5~0.7. Mach-Zehnder Modulator Interferometer: a device that interferes two waves of the same frequenc but diff. phase out cos( ωt+ φ) + cos( ωt φ) = cos( φ)cos( ωt) out out ( φ) = cos φ (0) olaried input light LiNbO 3 Coplanar strip electrodes L O Substrate V(t) Waveguide Thin buffer laer d a Cross-section LiNbO 3 Integrated tranverse ockels cell phase modulator in which a waveguide is diffused into an electro-optic (O) substrate. Coplanar strip electrodes appl a transverse field a through the waveguide. The substrate is an -cut LiNbO 3 and tpicall there is a thin dielectric buffer laer (e.g. ~00 nm thick SiO ) between the surface 51 electrodes and the substrate to separate the electrodes awa from the waveguide. LiNbO 3 In C V(t) Waveguide O Substrate lectrode n integrated Mach-Zender optical intensit modulator. The input light is split into two coherent waves and, which are phase shifted b the applied voltage, and then the two are combined again at the output. D Out 5

14 C. Coupled Waveguide Modulators Cross-section Coupled waveguides n d (a) n n s Input (0) Top view light been transferred to. eond this point, light begins to be transferred back 53 to (b) () () L o (L o ) (L o ) (a) Cross section of two closel spaced waveguides and (separated b d) embedded in a substrate. The evanescent field from etends into and vice versa. Note: n and n > n s (= substrate inde). (b) Top view of the two guides and that are coupled along the -direction. Light is fed into at = 0, and it is graduall transferred to along. t = L o, all the in the same wa. (L o )/ (0) 100% 0 (π 3)/L o π 1 3 V π Δ β =Δn n r λ d λ V(t) Fibers L o In lectrode LiNbO 3 Transmission power ratio from guide to guide over the transmission length L o as a function of mismatch Δβ. Δβ V Waveguides ( Lo) = f ( Δβ ) (0) Cross-section V(t) d π 3 3 λd Δ β = π => V0 = 3 L nrl o a Coupled waveguides LiNbO 3 54 o 7.7 cousto-optic Modulator hotoelastic effect Induced strain => refractive inde change 1 Δ = ps n Depends on the directions (Cf. lements of hotonics) ieoelectric effect Generation of strain b appling an eternal field (Fig. 7.8) Surface acoustic wave (SW) b modulating voltage at RF eriodic Δn due to periodic S due to photoelastic effect 55 56

15 coustic absorber Incident light Induced diffraction grating Diffracted light Λ sin = λ /n : ragg angle Incident optical beam Diffracted optical beam ' ' coustic wave coustic wave fronts Through light ieoelectric crstal n min n ma n min n ma Λ Λsi n O Q Λsi n O' v acoustic coustic wave fronts Modulating RF voltage Interdigitall electroded transducer Traveling acoustic waves create a harmonic variation in the refractive inde and thereb create a diffraction grating that diffracts the incident beam through 57 an angle. ω ' = ω ±Ω due to Doppler effect Simplified ctual Ω : acoustic wave freq. Consider two coherent optical waves and being "reflected" (strictl, scattered) from two adjacent acoustic wavefronts to become ' and '. These reflected waves can onl constitute the diffracted beam if the are in phase. 58The angle is eaggerated (tpicall this is a few degrees). n min n ma n n min n ma n 7.8 Magneto-Optic ffects Farada effect Opticall inactive material, e.g. glass, is placed in a strong magnetic field (e.g. solenoid) Light polariation propagating in the direction of magnetic field is rotated = ϑl ϑ : Verdet constant.g. glass L=0 mm in ~ 0.1T, =1º Comparison Optical activit Farada effect 59 60

16 Source = ϑl ϑ : Verdet constant = 45 => isolator avoid returned light to interfere the source olarier Light Reflected light Farada medium Reflector 7.9 Nonlinear Optics and Second Harmonic Generation The sense of rotation of the optical field depends onl on the direction of the magnetic field for a given medium (given Verdet constant). If light is reflected back into the Farada medium, the field rotates a further in the same sense to come out as with a rotation with respect to Linear = ε χ χ : electric susceptibilit o 3 Nonlinear = εχ o 1+ εχ o + εχ o 3 χ 1 : linear susceptibilit χ : second order susceptibilit, non-centrosmmetric e.g. quart (also pieoelectric) χ 3 : third order susceptibilit Second Harmonic Generation (SHG) = ε χ + ε χ and = sin( ωt) o 1 1 = εχ sin( ωt) εχ cos( ωt) + εχ + o 1 o 0 o 1 o 0 o ο o o - o t o + t - sinωt -cosωt DC t (a) (b) (c) 63 (a) Induced polariation vs. optical field for a nonlinear medium. (b) Sinusoidal optical field oscillations between ± o result in polariation oscillations between + and -. (c) The polariation oscillation can be represented b sinusoidal oscillations at angular 64 frequencies ω (fundamental), ω (second harmonic) and a small DC component.

17 Constructive interference => same velocit => n( ω) = n( ω) Inde matching n ( ω) = n ( ω) e o at phase matching angle Second harmonics Fundamental v 1 S k 1 S v S 3 k 1 Crstal s the fundamental wave propagates, it periodicall generates second harmonic waves ( S 1, S, S 3,...) and if these are in phase then 65 the amplitude of the second harmonic light builds up. Laser Nd:YG λ 064 nm KD Optic ais IM λ 064 nm λ = 53 nm Filter λ = 53 nm simplified schematic illustration of optical frequenc doubling using a KD (potassium dihdrogen phosphate) crstal. IM is the inde matched direction at an angle (about 35 ) to the optic ais along which n e (ω) = n o (ω). The focusing of the laser beam onto the KD crstal and the collimation of the light emerging from the crstal are not shown. 66 Conservation of momentum Conservation of energ v k k1 k1 Second harmonic photon, k Fundamental photon, k 1 ω 1 hotonic interpretation of second ω harmonic generation (SHG) 1 ω Fundamental photon, k 1 ω ω ω = = = = v Dipole moment-photon interaction region k + k = k 1 1 ω1+ ω1 = ω SHG is onl effective over a coherence length π lc = Δ k = k k1 Δk 眼睛在不同的亮度下, 其對不同波長的 sensitivit 也不同, 在正常 亮度下, 對黃 - 綠光最敏感 (0.55 um) 在昏暗下, 對綠光 (0.51 um ) 較敏感 The relative sensitivit of the ee to different wavelengths for normal levels of illumination (photopic vision )and under conditions of dark adaption (scotopic vision) CFLIN 68 NTU&O

18 The relative sensitivit of rods and cones as a function of wavelength. NTU&O CFLIN 69