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 (ω) = 0Ei (ω) + Pi (ω) (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 (ω) = 0χij (ω : ω1)Ej (ω1) (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(ω) = χ(1)(ω) + 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(ω0t) (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(ω0t) (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(ω t) Linear Optics out 0 0 0 1 Nonlinear = χ(2)E 2 [1 + cos(2ω 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: 2ω0 and 0. Now consider the cubic term Parametric Processes (3) (3) 3 3 Applications Pout (t) = 0χ E0 cos (ω0t) 1 = χ(3)E 3 cos(ω t) [1 + cos(2ω t)] 0 0 2 0 0 1 = χ(3)E 3 [cos(ω t) + cos(ω t)cos(2ω t)] 0 0 2 0 0 0 3 1 = χ(3)E 3 cos(ω t) + cos(3ω t) 0 0 4 0 4 0 Example: New Frequencies
ECE 455 Problem: Two lasers with frequencies ω and ω 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(ω1t) + E0cos(ω2t)) (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:
ω ∈ {ω1, 3ω1, ω2, 3ω2, |2ω1 ± ω2|, |ω1 ± 2ω2|} (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 ωout = |ωi ± ωj ... ± ωq| (10)
where ωi , ωj , ...ωq ∈ {ω1, ω2, ..., ωn}.
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 (ω) = 0χ¯ (ω)E(ω) (11) Optics χ(2) Processes which is a compact way of writing χ(3) Processes Non- P (ω) χ (ω) χ (ω) χ (ω) E (ω) Parametric x xx xy xz x Processes Py (ω) = 0 χyx (ω) χyy (ω) χyz (ω) Ey (ω) Applications Pz (ω) χzx (ω) χzy (ω) χzz (ω) Ez (ω) (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 (ω) = 0χij (ω : ω1)Ej (ω1) (13) χ(3) Processes Non- For example, when considering the y coordinate, this becomes Parametric Processes h (1) (1) Applications Py (ω) = 0 χyx (ω : ω1)Ex (ω1) + χyy (ω : ω1)Ey (ω1)
(1) i +χyz (ω : ω1)Ez (ω1) (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 (ω : ω1, ω2) (15) χ(3) Processes
Non- should be interpreted in the following manner: Parametric Processes The coefficient of the material response at frequency ω and Applications polarization i due to fields with frequencies ω1 and ω2 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 ∇ × E = − (16) Optics ∂t χ(2) Processes ∂D~ ∇ × H~ = (17) χ(3) Processes ∂t Non- ~ Parametric ∇ · B = 0 (18) Processes ∇ · 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~ ∇ × ∇ × 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 ∇ × ∇ × 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 ∇ × ∇ × E = ∇ · ∇E − ∇ E. (2) χ Processes However, unlike linear optics, the equation ∇ · ∇E~ = 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 ω3 = ω1 + ω2 (27)
Non- Parametric They must also obey the conservation of momentum: Processes
Applications k~3 = k~1 + k~2 (28)
By convention ω3 > ω2 ≥ ω1 (29)
In the context of frequency mixing, ω3, ω2 and ω1 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 ~ω1 + ~ω1 → ~(2ω1) Second Harmonic Generation Math I
ECE 455 Let us examine the mathematics behind SHG. Let ω be the Lecture 6 1 ω1n1 fundamental optical frequency, and define k1 ≡ k(ω1) = c 2ω1n2 Linear Optics and k2 ≡ k(2ω1) = c . The electric field in the medium will Nonlinear consist of two parts: the fundamental frequency and its second Optics harmonic. χ(2) Processes
χ(3) Processes 1 h i ı(ω1t−k1z) ı(2ω1t−k2z) E(z, t) = A1(z)e + A2(z)e + c.c (30) Non- 2 Parametric Processes The nonlinear polarization will then be: Applications 2 PNL(z, t) = 20deff E (z, t) 0d h i = eff A2eı(2ω1t−2k1z) + 2A A∗eı(ω1t−(k2−k1)z) + c.c. 2 1 2 1 Note some terms of the nonlinear polarization (such as 2 ı(4ω1t−2k2z) A2e ) have been discarded. We’ll see why in a few slides Second Harmonic Generation Math II
ECE 455 Substituting the above into the nonlinear wave equation (26), Lecture 6 we obtain: 2 Linear Optics 1 d A dA ω n 2 1 1 2 1 1 ı(ω1t−k1z) − + ı2k1 + k A1 − A1 · e + Nonlinear 2 dz2 dz 1 c Optics 2 χ(2) Processes 1 d A dA ω n 2 2 1 2 2 2 ı(2ω1t−k2z) − + ı2k2 + k2 A2 − A2 · e + χ(3) Processes 2 dz2 dz c Non- d h i Parametric +c.c. = eff 2ω2A2eı(2ω1t−2k1z) + ω2A A∗eı(ω1t−(k2−k1)z) + c.c. (31) Processes c2 1 1 1 2 1 Applications The above equation looks horrible, but fortunately, it can be simplified: ωi ni First note that since ki ≡ c , the last two terms in the brackets both cancel. 2 d Ai dAi It will also be assumed that dz2 ki dz . This is commonly known as the slowly varying envelope approximation Second Harmonic Generation Math III
ECE 455 Lecture 6 From phase matching considerations, the equation can be Linear Optics broken into two equations: the parts that vary as eıω1t and Nonlinear ı2ω1t Optics those which vary as e
χ(2) Processes (3) χ Processes dA1 −ıω1deff ∗ −ı∆kz Non- = A2A1e (32) Parametric dz n1c Processes dA2 −ıω1deff 2 ı∆kz Applications = A1e (33) dz n2c
where ∆k ≡ k2 − 2k1
Equations 41 and 42 are coupled nonlinear differential equations. The Undepleted Pump Approximation
ECE 455 Lecture 6
In the limit of low conversion efficiency A1 can be considered Linear Optics constant. In this limit, Equation 42 can be integrated simply: Nonlinear Optics Z L −ıω1d χ(2) Processes eff 2 ı∆kz A2(L) = A1 e dz χ(3) Processes n2c 0 Non- −ı2ω1deff 2 ı∆kL ∆kL Parametric = A1e 2 sin (34) Processes n2c∆k 2 Applications The conversion efficiency is then
∗ 2 I2ω1 A2A2 2ω1deff 2 2 ∆kL = ∗ = |A1| sin (35) Iω1 A1A1 n2c∆k 2 Second Harmonic Generation Picture
ECE 455 Lecture 6
Linear Optics
Nonlinear Optics
χ(2) Processes
χ(3) Processes
Non- Parametric Processes
Applications Final Thoughts on SHG
ECE 455 Lecture 6
Linear Optics
Nonlinear Small index differences severely lower conversion efficiency: Optics The ∆n = 0 curve on the previous slide is far off the scale. χ(2) Processes ∆n = 0 is known at the phase matching condition χ(3) Processes Non- Processes which are not phase matched are not Parametric Processes macroscopically observed because their efficiency is low Applications With careful design, conversion efficiencies approaching 100% can be achieved For extremely short crystals, ∆n has no effect! Visual Image of Phase Matching Condition
ECE 455 Lecture 6
Linear Optics
Nonlinear Optics
χ(2) Processes
χ(3) Processes
Non- Parametric Processes
Applications Birefringent Media I
ECE 455 Lecture 6 Obeying the conservation of momentum is non-trivial
Linear Optics because in general n is a function of ω
Nonlinear Optics ωn(ω) k(ω) = (36) χ(2) Processes c χ(3) Processes
Non- Therefore in general k(ω1 + ω2) 6= k(ω1) + k(ω2) Parametric Processes In order to overcome this difficulty, the properties of Applications birefringent crystals are used Uniaxial media have indicies of refraction of the form: no 0 0 n¯ = 0 no 0 (37) 0 0 ne Birefringent Media II
ECE 455 Lecture 6 For a plane wave whose ~k vector Linear Optics makes an angle θ with thez ˆ axis, Nonlinear there will be two indices of n Optics e (2) refraction, depending on the χ Processes polarization of the wave: χ(3) Processes The ordinary index of Non- Parametric refraction is no Processes The extraordinary index of θ Applications refraction can be found with the following equation:
2 2 n 1 sin (θ) cos (θ) o 2 = 2 + 2 n (θ) ne no (38) Birefringent Media III
ECE 455 Lecture 6
Linear Optics
Nonlinear Optics Type I Phasematching involves the following two types χ(2) Processes of interacations χ(3) Processes o + o → e Non- e + e → o Parametric Processes Type II Phasematching Applications o + e → o o + e → e Second Harmonic Generation Example I
ECE 455 Lecture 6 Problem: BBO is a negative uniaxial crystal (ne < no). The
Linear Optics indicies of refraction for the ordinary and extraordinary axes are
Nonlinear given by: Optics χ(2) Processes 0.01878 n2(λ) = 2.7359 + − 0.01354λ2 χ(3) Processes o λ2 − 0.01822 Non- Parametric Processes 2 0.01224 2 ne (λ) = 2.3753 + 2 − 0.01516λ Applications λ − 0.01667 where λ is expressed in µm. Find the phasematching angle to create the second harmonic of 780 nm light. Solution: Because BBO is negative uniaxial, the 780 nm must be input as the ordinary wave. The frequency doubled output will be an extraordinary wave. Second Harmonic Generation Example II
ECE 455 Evaluating the previous two equations, we find the indices of Lecture 6 refraction to be:
Linear Optics no(780 nm) = 2.7595 Nonlinear Optics
χ(2) Processes no(390 nm) = 2.8741
χ(3) Processes ne (390 nm) = 2.4634 Non- Parametric Processes Setting the ordinary index at 780 nm to be equal to the
Applications extraordinary index at 390 nm yields the following equation:
2 2 1 1 sin (θpm) cos (θpm) 2 = 2 = 2 + 2 no(780 nm) n (θpm) ne (390 nm) no(390 nm)
Solving this equation for θpm yields:
◦ θpm = 30 Non-critical Phase Matching
ECE 455 Lecture 6 Angle tuning a crystal to for phasematching purposes is known as critical phase matching Linear Optics
Nonlinear Critical phase matching is sensitive to misalignment, and Optics suffers from problems such as walk off. χ(2) Processes
χ(3) Processes In special cases, the phasematching can be tuned by
Non- varying the temperature and composition of the crystal Parametric Processes Periodically Poled Crystals
Applications 2 2 2ω0deff 2 2 ∆kL I2ω = 3 3 Iωsinc (39) 0c n 2 Efficiency is not as good as angle phase matching, but this process is more versatile ADD PICTURE OF PPLN crystal and its response Frequency Mixing
ECE 455 Lecture 6 When ω1 6= ω2 The propagation of
Linear Optics dA1 −ıω1deff ∗ −ı∆kz Nonlinear = A3A2e (40) Optics dz n1c (2) dA −ıω d χ Processes 2 = 2 eff A A∗e−ı∆kz (41) (3) 3 1 χ Processes dz n2c Non- dA2 −ıω3deff ı∆kz Parametric = A1A2e (42) Processes dz n3c Applications where ∆k = k3 − k1 − k2 In sum frequency generation (SFG), two lower frequency photons combine to make a higher frequency photon In difference frequency generation (DFG), a high frequency photon is split into two lower frequency photons SHG and DFG can be used to amplify weak signals or convert a signal to from one center wavelength to another χ(3) Processes
ECE 455 Lecture 6
Linear Optics Nonlinear The general form of the χ(3) tensor is: Optics
χ(2) Processes P(3)(ω) = χ(3)(ω : ω , ω , ω )E (ω )E (ω )E (ω ) χ(3) Processes i 0 ijkl 1 2 3 j 1 k 2 l 3
Non- (43) Parametric (2) Processes χ effects are only observed in materials without Applications inversion symmetry. Can be shown that all materials have nonzero χ(3) coefficients Intensity Dependent Index of Refraction
ECE 455 Lecture 6
Linear Optics The χ(3) coefficient gives rise to an intensity dependent Nonlinear Optics index of refraction χ(2) Processes (1) (3) 3 χ(3) Processes P = χ E + χ E Non- (1) (3) 2 Parametric = χ + χ |E| E (44) Processes
Applications This can also be witten as:
n = n0 + n2I (45) Kerr Lensing
ECE 455 Lecture 6
Linear Optics Nonlinear In gaussian laser beams, the intensity is highest in the Optics center of the beam. Hence the index of refraction is χ(2) Processes highest in the center. χ(3) Processes
Non- The field turns the medium into a lens. The effect is Parametric Processes known as self-focusing Applications Because the only frequency present in these interactions is ω, dispersion is irrelevant and this interaction is always phase matched! Self Phase Modulation (SPM)
ECE 455 Lecture 6 Temporal result of Kerr effect is known as self phase modulation Linear Optics A pulse can be written as: Nonlinear Optics ı(ω0t+φ(t)) Ein(t) = E(t)e (46) χ(2) Processes χ(3) Processes where E(t) is real. Non- Parametric If this pulse is passed through a non-dispersive Kerr Processes medium, the output is Applications ωn0L ωn2I (t)L ı ω0t+φ(t)+ c + c Eout (t) = E(t)e (47) where I (t) is the instantaneous intensity of the pulse Last term in exponential spectrally broadens pulse Because multiple frequencies are involved, dispersion and phase matching are important Frequency Tripling
ECE 455 Lecture 6
Linear Optics Nonlinear (3) Optics The χ (3ω : ω, ω, ω) coefficient is typically very small. χ(2) Processes Intensities required to excite a third-harmonic response χ(3) Processes may be near or in excess of the damage threshold of the Non- Parametric material Processes Direct tripling is inefficient and rarely performed Applications To obtain the third harmonic of a laser, first frequency double the beam and mix the output with the fundmental Four Wave Mixing (FWM)
ECE 455 Lecture 6
Linear Optics
Nonlinear Optics
χ(2) Processes
χ(3) Processes
Non- Parametric Processes
Applications Common Nonlinear Crystals
ECE 455 Lecture 6
Linear Optics
Nonlinear Optics Crystal Transparency
χ(2) Processes BBO Beta Barium Borate 198-2600 nm
χ(3) Processes BiBO Bismuth Triborate 286-2700 nm Non- KDP Potassium Dihydrogen Phosphate 176-1550 nm Parametric Processes KTP Potassium Titanyl Phosphate 352-4500 nm Applications LBO Lithium Triborate 160-2300 nm LN Lithium Niobate 400-5500 nm Non-Parametric Processes
ECE 455 Lecture 6
Linear Optics
Nonlinear Optics
χ(2) Processes χ(3) Processes In non-parametric nonlinear processes, energy can be Non- transferred to the medium in which the light is propagating Parametric Processes
Applications Two Photon Absorption
ECE 455 Lecture 6
E2 E2 Linear Optics Nonlinear Virtual Optics States (2) χ Processes E1 χ(3) Processes
Non- Parametric Processes Eo Eo Applications
Two photons absorbed simultaneously to bring atom to high-lying energy level Absorption enhanced by enhanced by allowed transitions near single photon resonance Can limit the performance of high-powered lasers Stimulated Raman Scattering (SRS)
ECE 455 Lecture 6
Linear Optics
Nonlinear Optics
χ(2) Processes χ(3) Processes Interaction of light with an acoustic phonon in the material Non- Parametric Processes
Applications Stimuated Brillioun Scattering (SBS)
ECE 455 Lecture 6
Linear Optics
Nonlinear Optics
χ(2) Processes
χ(3) Processes Interaction of light with an acoustic phonon in the material Non- The scattered wave is always backwards propagating Parametric Processes
Applications Frequecy Conversion
ECE 455 Lecture 6
Linear Optics
Nonlinear Optics
χ(2) Processes
χ(3) Processes
Non- Parametric Processes
Applications Optical Parametric Oscillator (OPO)
ECE 455 Lecture 6
Linear Optics
Nonlinear Optics I Ip sig χ(2) Processes Iidler χ(3) Processes
Non- Parametric NL Crystal Processes
Applications
Place nonlinear crystal inside cavity When pumped with a high-intensity laser, crystal acts a gain medium Resulting device behaves very similar to a laser OPOs can angle or temperature tuned Continuum Generation
ECE 455 Lecture 6
Linear Optics
Nonlinear Optics χ(2) Processes Focus a femtosecond laser into a fiber or a sapphire plate χ(3) Processes Exact spectrum structure is chaotic Non- Parametric Processes Useful as a high intensity, broadband spectroscopic light
Applications source Solitons
ECE 455 Lecture 6
Linear Optics
Nonlinear Optics
χ(2) Processes Linear dispersion and self phase modulation balance each
χ(3) Processes other Non- Pulse progates without changing shape Parametric Processes Nonlinear propagation equation solutions have discrete Applications energies and stable pulse profiles Two Photon Photoresist
ECE 455 Lecture 6
Linear Optics
Nonlinear Optics
χ(2) Processes
χ(3) Processes
Non- Parametric Processes
Applications
Two photon absorption permits high contrast 3D patterning Courtesy Sidartha Gupta Conclusions
ECE 455 Lecture 6 Nonlinear processes are those in which the output may contain frequencies of light not present in the input. Linear Optics Nonlinear Lasers are essential for observing nonlinear optical Optics properties. χ(2) Processes
χ(3) Processes Nonlinear optical interactions require a material medium
Non- (solid, liquid, gas, plasma) Parametric Processes Photon energy must be conserved in parametric nonlinear Applications processes Some photon energy is transferred to the medium in non-parametric nonlinear processes Momentum must be conserved in nonlinear processes (phase-matching) in order for them to be observable on a macroscopic scale