(Introduction) Linear Optics and Nonlinear Optics

Transcription

1 18. Electro-optics

2 (Introduction) Linear Optics and Nonlinear Optics Linear Optics The optical properties, such as the refractive index and the absorption coefficient are independent of light intensity. The principle of superposition holds. The frequency of light cannot be altered through the medium. Light cannot interact with light; two beams of light in the same region of a linear optical medium can have no effect on each other. Thus light cannot see other lights. Nonlinear optics (NLO) The refractive index, and consequently the speed of light in an optical medium, does change with the light intensity. The principle of superposition is violated. Light can alter its frequency as it passes through a nonlinear optical material (e.g., from red to blue!). Light can interact with light via the medium Thus light cannot see other lights, but, light can control other lights via the nonlinear medium.

3 (Introduction) Nonlinear effects in Optics Polarization : P = εχe 0 Susceptibility : χ = χ + χ E+ χ E D v ε = ε E = ε 0E + ε 0χE ε = ε 0(1 + χ) n = = = 1+ χ c ε εχ 0 1 ε0χ ε0χ3 P = P + P + P + = E+ E + E + Here we will discuss on electro-optic Pockels and Kerr effects

4 (Introduction) Second-order Nonlinear effects P = εχe 0 Second-harmonic generation (SHG) and rectification E = P E ( ) optical ω P ( ω ± = Electro-optic (EO) effect (Pockell s effect) E = 0) P P (0) E P (0), P ( Three-wave mixing P (, P (0) { E (0)}, P ( { 0) }, P ({ } { but, 0) )} + ω, DC optical electrical >> E = E ) ( ω 1) + ω optical P 1 1 E 1 P (ω ) P ( ω + ω ) { 0) } Δn 0) electric, DC optical { E ( ω )}, P (ω ){ E ( ω )}, P ( ω ω ) 1 { ω1) }, { ω ) ω )} 1 Index modulation by DC E-field SHG Frequency doubling Rectification Frequency up-converter Parametric amplifier, parametric oscillator

5 (Introduction) Third-order Nonlinear effects P = εχe Third-harmonic generation (THG) 3 E = P E ( ) optical 3 ω Electro-optic (EO) Kerr effect E = 0) { } 3, P (3ω ){ E ( )} P3 ( 3 ω { but, 0) )} + ω, DC optical electrical >> Frequency tripling P 3 ( ) 0) Δn 0) electric, DC electric, DC ω Index modulation by DC E Optical Kerr effect P3 ( I( Δn I( Index modulation by optical Intensity n = n0 + Δn( I) ϕ = ϕ0 + Δϕ( = k0δnl) { I( x) } Δn{ I( x } n0 n n > 0 + Δn ) Self-phase modulation = Self-focusing, Self-guiding (Spatial solitons) { I ( x) } Δn{ I ( x } n0 n n < = Self-defocusing 0 + Δn )

6 (Introduction) Third-order Nonlinear effects P = εχe Four-wave mixing E = E ) ( ω 1) + ω ) optical + ω optical 3 optical P 3 3 ( ± ω, ± ω, ± ω ) 6 16 terms 3 3 E 1 3 = One example : P3 ( ω1 + ω + ω3 ω4) ω1) ω3) If ω = ω = ω ω 3ω = THG Frequency up-converter Another * example : P3 ( ω1 + ω -ω3 ω4) ω1) E ( ω3) ω + If 1 + ω = ω3 ω4 ω = 1 = ω = ω3 ω4 Degenerate four-wave mixing If we assume two waves among them are plane waves traveling in opposite directions * P3 ( ω4 = E ( Optical phase conjugation

7 18.1 Principles of of Electro-optic effects The electro-optic effect is the change in the refractive index resulting from the application of a DC or low-frequency electric field. Linear electro-optic effect or Pockels effect : The refractive index changes in proportion to the applied electric field. Quadratic electro-optic effect or Kerr effect : The refractive index changes in proportion to the square of the applied electric field.

8 Pockels effect and Kerr effect Polarization : P = εχe Susceptibility : η( E) = η + re + RE 0 0 χ = χ + χ + χ + n = ( 1+ χ) 1 E 3E ne ( ) = rn 0 Rn 0 n E E Pokels Effect Kerr Effect

9 Pockels effect (Linear electro-optic effect) 1 η( E) = η+ re ( η = ) n dη( E) 1 dn 1 3 = = r dn= rn ( de) 3 de n de 1 3 ne ( ) = n+ dn= n rne Pockels coefficient (linear electro-optic coefficient)

10 Kerr effect (Quadratic electro-optic effect) R

11 Electro-optic modulators and switches 1 Phase modulators ( Pockels cell) 3 ne ( ) = n+ dn= n rne

12 Phase modulators ( Pockels cell)

13 Dynamic wave retarders L Pockels cell SA (n1) FA (n) V

14 Intensity modulators : Use of an interferometer

15 Intensity modulators : Use of crossed polarizers

16 Scanners : electro-optic prisms Position switch

17 Directional couplers

18 Spatial light modulators (SLM)

19 Q-switching lasers

20 18. Electro-optics of anisotropic media where η = η η = ; η = ; η = ij n1 n n3 ji

21 Pockels and Kerr coefficients Impermeability at E = 0 ( 3 = 9 elements ) : Linear E-O (Pockels) coefficients ( 3 3 = 7 elements ) : Quardratic E-O (Kerr) coefficients ( 3 4 = 81 elements )

22 Symmetry in Pockels and Kerr coefficients 6 independent elements (6 x 3) independent elements (6 x 6) independent elements It is conventional to rename the pair of indices (i, j), i, j = 1,,3 as a single index I = 1,,..., 6. (k, l), k, l = 1,,3 as a single index K = 1,,..., 6.

23 Pockels effect The index ellipsoid is modified as a result of applying a steady electric field. To determine the optical properties of an anisotropic material exhibiting the Pockels effect, (that is, to find modified principal refractive indices)

24 Example Find the index change of uniaxial crystal by E = E z r113 = r13; r13 = r63 = 0; r133 = r53 = 0 r3 = r3 = r13; r13 = r63 = 0; r33 = r43 = 0 r = r = r ; r = r = 0; r = r = Only r ( ) 0 ij3 E for i = j E η ( E) = η (0) + r E ij ij ij3 η ( Ex ) + η ( Ex ) + η ( Ex ) = re x = + re x ( η (0) ) no ( η (0) ) no ( η (0) ) 1 + re x = + re x 1 + re x = + re x ne

25 Example E When an electric field is applied along the optic axis of this uniaxial crystal, it remains uniaxial with the same principal axes, but its refractive indices are modified.

26 Homework Derive their final principal refractive indices, in DETAIL step-by-step.