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Nonlinear Optics Nonlinear Optics Χ(2) Processes ECE 455 Optical Electronics Χ(3) Processes ECE 455 Lecture 6 Linear Optics Nonlinear Optics Nonlinear Optics χ(2) Processes ECE 455 Optical Electronics χ(3) Processes Non- Parametric Processes Tom Galvin Applications Gary Eden ECE Illinois Introduction ECE 455 Lecture 6 Linear Optics Nonlinear Optics χ(2) Processes In this section, the following subjects will be covered: χ(3) Processes What is meant by 'nonlinear optics' Non- Parametric Second and third order optical nonlinear effects Processes Applications Applications of nonlinear phenomenon Linear Optics I ECE 455 Lecture 6 To understand what is meant by nonlinear, it is first necessary to understand what is meant by linear Linear Optics The electric displacement field is most generally written as: Nonlinear Optics Di (!) = 0Ei (!) + Pi (!) (1) χ(2) Processes χ(3) Processes In linear optics, it is assumed that the dipole induced by Non- an electric field is linearly proportional the the strength of Parametric Processes that field: (1) Applications Pi (!) = 0χij (! : !1)Ej (!1) (2) This results from assuming the effect of the optical electric field on the medium is perturbative The net effect of the induced polarization is to change the effective speed of light in the material as: q n(!) = χ(1)(!) + 1 (3) Linear Optics II ECE 455 Lecture 6 Linear optical phenomenon are characterized by Linear Optics interactions of materials with a single photon Nonlinear Optics All of the phenomenon discussed in this class so far are (2) χ Processes linear optic effects χ(3) Processes Non- Pulses sent through linear materials may be amplified and Parametric Processes phase distorted, but the output frequency is equal to the Applications input frequency Linear optical phenomenon are independent of the intensity of the field Lenses, mirrors, prisms, gratings, optical fibers, and cavities can all be described by the theory of linear optics. Nonlinear Optics ECE 455 Lecture 6 Linear Optics Nonlinear Nonlinear optical phenomenon are characterized by Optics interactions of materials with multiple photons χ(2) Processes χ(3) Processes Nonlinear optical phenomenon are in general a strong Non- function of the intensity Parametric Processes When any system is driven hard enough, it will exhibit Applications nonlinear behavior. With ordinary light intensities, nonlinear optic effects are too small to be noticed Linear and Nonlinear Behavior ECE 455 Lecture 6 Linear Optics P P Nonlinear Optics χ(2) Processes χ(3) Processes Non- E E Parametric Processes Applications Linear material response Nonlinear material response Generation of New Frequencies I ECE 455 Lecture 6 The polarization response can be written as a power series: Linear Optics h (1) (2) 2 (3) 3 i Nonlinear P = 0 χ E + χ E + χ E + ::: (4) Optics (2) χ Processes Suppose the input field is: χ(3) Processes Non- Parametric Ein(t) = E0cos(!0t) (5) Processes Applications We know the χ(1)E term is the linear part of the polarization and is responsible for the index of refraction. The polarization from this linear term is: (1) (1) Pout (t) = 0χ E0cos(!0t) (6) Generation of New Frequencies II ECE 455 The polarization from the χ(2) term is: Lecture 6 P(2)(t) = χ(2)E 2cos2(! t) Linear Optics out 0 0 0 1 Nonlinear = χ(2)E 2 [1 + cos(2! t)] (7) Optics 0 0 2 0 χ(2) Processes χ(3) Processes There are two new frequencies which weren't present in the Non- original signal: 2!0 and 0. Now consider the cubic term Parametric Processes (3) (3) 3 3 Applications Pout (t) = 0χ E0 cos (!0t) 1 = χ(3)E 3 cos(! t) [1 + cos(2! t)] 0 0 2 0 0 1 = χ(3)E 3 [cos(! t) + cos(! t)cos(2! t)] 0 0 2 0 0 0 3 1 = χ(3)E 3 cos(! t) + cos(3! t) 0 0 4 0 4 0 Example: New Frequencies ECE 455 Problem: Two lasers with frequencies ! and ! pass through Lecture 6 1 2 a material with a χ(3) coefficient. Find all frequencies at which Linear Optics there is a polarization response. Nonlinear Solution: If we were to continue as we have been doing, we Optics would expand χ(2) Processes χ(3) Processes (3) (3) 3 Pout = 0χ (E0cos(!1t) + E0cos(!2t)) (8) Non- Parametric Processes and simplify it with sine and cosine identities. Applications However, we can recognize that when sines and cosines are mutliplied, they generate frequencies with the sum and difference of the arguments. The polarization response frequencies will be: ! 2 f!1; 3!1;!2; 3!2; j2!1 ± !2j; j!1 ± 2!2jg (9) General Rule for finding new frequencies ECE 455 Lecture 6 Our insight from the last example can be formalized with the Linear Optics aid of fourier transform theory: Multiplication in the time Nonlinear Optics domain is convolution in the frequency domain. χ(2) Processes χ(3) Processes In general, if n frequencies are inicident upon a medium with a th Non- k order nonlinearity, the frequencies at which the material will Parametric Processes respond are: Applications !out = j!i ± !j ::: ± !qj (10) where !i ;!j ; :::!q 2 f!1;!2; :::; !ng. In practice, most of these frequencies will not be observed because they will destructively interfere macroscopically. Einstein Notation I ECE 455 Lecture 6 The linear response of a material to an optical field may in general be represented with the vector equation Linear Optics ~ (1) (1) ~ Nonlinear P (!) = 0χ¯ (!)E(!) (11) Optics χ(2) Processes which is a compact way of writing χ(3) Processes Non- 2 P (!) 3 2 χ (!) χ (!) χ (!) 3 2 E (!) 3 Parametric x xx xy xz x Processes 4 Py (!) 5 = 0 4 χyx (!) χyy (!) χyz (!) 5 4 Ey (!) 5 Applications Pz (!) χzx (!) χzy (!) χzz (!) Ez (!) (12) The above matrix equation doesn't seem so bad; The tensor has 9 terms (not all independent). However the tensor for the second-order nonlinear response contains 27 terms! We need a compact way of representing these equations. Einstein Notation II ECE 455 Lecture 6 Repeated subscript inidices imply summation over spatial coordinates x, y and z. The following equation represents the Linear Optics Einstein notation for the linear polarization. Nonlinear Optics (2) (1) χ Processes Pi (!) = 0χij (! : !1)Ej (!1) (13) χ(3) Processes Non- For example, when considering the y coordinate, this becomes Parametric Processes h (1) (1) Applications Py (!) = 0 χyx (! : !1)Ex (!1) + χyy (! : !1)Ey (!1) (1) i +χyz (! : !1)Ez (!1) (14) Equations 11, 12, and 13 are all equivalent ways of expressing the same idea. The χ(n) Tensors ECE 455 Lecture 6 Linear Optics Nonlinear Optics The expression (2) χ(2) Processes χijk (! : !1;!2) (15) χ(3) Processes Non- should be interpreted in the following manner: Parametric Processes The coefficient of the material response at frequency ! and Applications polarization i due to fields with frequencies !1 and !2 and polarizations j and k respectively. The Nonlinear Wave Equation ECE 455 Lecture 6 In a sourceless material medium, Maxwell's four equations are Linear Optics ~ ~ @B Nonlinear r × E = − (16) Optics @t χ(2) Processes @D~ r × H~ = (17) χ(3) Processes @t Non- ~ Parametric r · B = 0 (18) Processes r · D~ = 0 (19) Applications If we take the curl of Equation 16 and assume a non-magnetic material (B~ = µ0H~), then we may substitute Equation 17 into Equation 16 @2D~ r × r × E~ = −µ (20) 0 @t2 The Nonlinear Wave Equation ECE 455 Lecture 6 In linear optics, the next step would be to substitute D~ = ¯E~ and go on our merry way, but the relationship between D~ and Linear Optics E~ is more complicated in nonlinear optics. Nonlinear Optics ~ ~ ~ χ(2) Processes D = 0E + P (21) (3) (1) NL χ Processes = 0E~ + P~ + P~ (22) Non- (1) NL Parametric = 0 1 +χ ¯ E~ + P~ (23) Processes Applications (1) NL = 0¯ E~ + P~ (24) substituting this back into Equation 20 (1) 2 ~ 2 ~ NL ~ ¯ @ E 1 @ P r × r × E + 2 2 = − 2 2 (25) c @t 0c @t The Nonlinear Wave Equation ECE 455 Lecture 6 Linear Optics The next step in deriving the nonlinear wave equation is to Nonlinear ~ ~ 2 ~ Optics apply the vector identity r × r × E = r · rE − r E. (2) χ Processes However, unlike linear optics, the equation r · rE~ = 0 is χ(3) Processes Non- only approximate. However it is a good approximation as long Parametric as the pulses are not too 'short.' The resulting wave equation is Processes Applications (1) 2 ~ 2 ~ NL 2 ~ ¯ @ E 1 @ P −∇ E + 2 2 = − 2 2 (26) c @t 0c @t What would the corresponding linear wave equation look like? χ(2) Processes ECE 455 Lecture 6 In order to posess even-ordered nonlinear coefficients, the Linear Optics material must lack inversion symmetry Nonlinear All nonlinear processes must obey the conservation of Optics energy: χ(2) Processes χ(3) Processes !3 = !1 + !2 (27) Non- Parametric They must also obey the conservation of momentum: Processes Applications k~3 = k~1 + k~2 (28) By convention !3 > !2 ≥ !1 (29) In the context of frequency mixing, !3, !2 and !1 are referred to as the pump, signal, and idler respectively Second Harmonic Generation ECE 455 Lecture 6 Linear Optics Nonlinear Optics (2) χ(2) Processes The simplest χ interaction is second harmonic χ(3) Processes generation (SHG) Non- Two photons from the source are combined into a single Parametric Processes photon Applications ~!1 + ~!1 ! ~(2!1) Second Harmonic Generation Math I ECE 455 Let us examine the mathematics behind SHG. Let ! be the Lecture 6 1 !1n1 fundamental optical frequency, and define k1 ≡ k(!1) = c 2!1n2 Linear Optics and k2 ≡ k(2!1) = c .
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