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Nonlinear Optics (PHYC/ECE 568) Fall 2015 Instructor: M. Sheik-Bahae Midterm Exam, Open Text-Book, Closed Notes, Time: 2 hours 5:30pm-7:30pm NAME ...................................... ........................................ last first Score: Please staple and return this page with your exam. Happy Thanksgiving! 1. Self-Phase Modulation (SPM) (20 points) An ultrashort laser pulse with an electric field E (t ) E0 e t / 2t cos( 0 t ) is incident on a nonlinear medium of length L with a cumulative nonlinear refraction (i.e. noninstantaneous with long decay time) described by 2 n(t ) t I (t ')dt ' , Where I(t) in the instantaneous intensity, and ( cm2/J ) is the nonlinear coefficient. Show (derive) how the spectrum of the laser pulse changes (i.e. gets chirped) upon exiting the material. Qualitatively draw and compare the incident and transmitted E(t) and |E()| assuming <0. (20 points) 𝜔 𝑡 𝜔 ΔΦ(𝑡) = 0 Δ𝑛(𝑡)𝐿 = 0 γ𝐿 ∫−∞ 𝐼(𝑡 ′ )𝑑𝑡′ 𝑐 𝑐 ∂ΔΦ(𝑡) 𝜔0 δω = − =− γ𝐿𝐼(𝑡) ∂t 𝑐 <0, therefore >0 for all t. 5 4 out E2( t)+ 3 2 E( t) in 0 1.5 2 3 0 Time t 2 3 E(t) 2 p 2. Optical Rectification (25 points) An ultrashort laser pulse with instantaneous electric field E (t ) E0 e t / 2t cos( 0 t ) is incident on a 2nd order NLO material with known (2). 2 E (t ) 2 p (2) (a) Write down, and plot the second-order nonlinear polarization term 𝑃̃(2) (𝑡) for optical rectification. 𝑃̃(2) (𝑡) = 𝑡2 − 2 𝜖0 𝜒 (2) 𝐸02 𝑒 𝑡𝑝 𝜕2 𝑃̃ (2) (b) As you well know, the driving term in the Maxwell’s equation involves 𝜕𝑡 2 . Qualitatively plot this term, and describe the generated waveform (temporal and spectral) assuming tp=100 femtoseconds (femto=10-15 ) The generated wave form will look like the above (i.e. a single-cycle wave form) Spectrally it has the contents of the incident pulse but at zero freq. Note there is no DC term as the area under the curve is zero. The width of its spectrum is nearly 1/tp=10 THz. (c) Show the photon-picture (i.e. the energy conservation arrows) in describing the new frequencies that are generated. 3. Spontaneous Parametric Down Conversion (30 points): In a quantum optics experiment, photon pairs (signal and idler) at s and i (<s) are generated in a (2) material by a pump beam at p polarized along the c-axis as shown below. Using appropriate polarizers, we only detect photons generated in a type-I phase matching. The normal dispersion no() and ne() are known. (a) Draw the wave-vector matching diagram. What is the polarization of generated photon pairs (show graphically)? (8 pts) (b) What type of birefringence (positive, negative, none or both) will work in this geometry? Explain. (7 pts) As shown in (a), we must have |kp|<|ks|+|ki|. Since p=s+i, then ns, ni> np or no>ne (negative birefringence only). (c) Write down the equation(s) that will give s and i. (15 pts) From the above triangle, we get the following two equations: no[icos(i)+ scos(s)]=nep , and isin(i) - ssin(s)=0 (d) (Bonus: 10 pts) Repeat (c) but assume type-II phase matching where idler (signal) is extra ordinary In this case, assuming signal wave is e-ray, making an angle of /2+s with the c-axis. The new equations are like in part c, but ne(s,s) is obtained from index ellipsoid equation: 1 𝑛𝑒2 (𝜃𝑠 ) = 𝜋 2 𝑛𝑒2 sin2 ( +𝜃𝑠 ) + 𝜋 2 2 𝑛𝑜 cos2 ( +𝜃𝑠 ) noicos(i)+ s ne(s)cos(s)=nep , thus and noisin(i)- s ne(s)sin(s)=0 4. NLO susceptibilities: resonances and selection rules (25 points) A fictional molecule has the following 3 energy levels. Draw the spectrum (for 0< <10 eV) for the (a) linear absorption coefficient α(), (b) two-photon absorption (TPA) coefficient (), (c) SHG: |(2)(2;,)| and (d) THG: |(3)(3;,,)|. (25 points) Be quantitative in your x-axis. Assume a finite broadening in your drawings. Point out the resonances (diagrammatically) on your graph for each case and show the relative strengths if obvious. Note: no calculations needed for this problem. E (eV) 8 |3> s & p-type (mixed parity hybrid) 6 |2> p-type 0 |1> p-type (ground state)