Derivation of the General Propagation Equation Phys 477/577: Ultrafast and Nonlinear Optics, F. Ö. Ilday, Bilkent University February 25, 26 1
1 Derivation of the Wave Equation from Maxwell s Equations Much of ultrafast and nonlinear optics is based on an understanding and a quantitative description of the propagation of laser light in an optical medium. In this technical note, we will derive the general propagation equation for laser light in a passive optical medium. This description includes the major effects that act on continuous beams as well as short pulses with high intensities, but amplification and loss are not included. Diffraction is included to properly describe evolution of real beams, i.e., we do not necessarily make the plane wave approximation. Further limitations and assumptions are explained below among the steps of the derivation. We start from Maxwell s equations given below, (i). D = ρ f, (1) (ii). B =, (2) (iii) E = B t, (3) (iii) H = J f + D t, (4) where in an optical medium, D = ɛ E + P and B = µ H + M. Here, we will only consider non-magnetic media with no free charges, which reduces the Maxwell equations to, (i). D =, (5) (ii). B =, (6) (iii) E = B t, (7) (iii) B = µ D t, (8) 2
These coupled equations describe the time evolution of the electric and the magnetic fields. It is possible to eliminate B and extract an equation for E only. Using equation (iv) and the relation D = ɛ E + P, we obtain, B E = µ ɛ t + µ P t. (9) By taking the curl of both sides of equation (iii) given above and using the mathematical identity, E = (. E) 2 E, we obtain (. E) 2 E = t ( B). (1) Using equation 9, we get an equation containing E only: (. E) 2 E 2 E = µ ɛ t 2 µ 2 P t 2. (11) For a homogenous media, we have. E = and recalling ɛ µ = 1/c 2, the equation above reduces to the well-known wave equation, 2 E 1 2 E c 2 t 2 = µ 2 P t 2. (12) This concludes the derivation of the wave equation. Keep in mind that so far we only assumed that the material is non-magnetic and it is homogenous. 2 Derivation of the Equation Governing Propagation from the Wave Equation Now we will work with this equation and begin to write out what P is. As a first step, we will assume that the polarization response of the material is instantaneous, i.e., P is determined only by the present conditions, there is no delayed response or memory effect in the system. This is valid for a nonlinear response that is electronic in origin, since the reconfiguration time 3
of the electron cloud is in order of.1 fs. This can be seen to correspond roughly to the travel time of the electron around the nucleus in the Bohr atom model. For almost all pulses, this is as good as instantaneous. Other material responses, for example, deriving from the nucleus can be much slower, but they are also typically weaker and we ignore them for now. The second important assumption we will make is that although P is in general nonlinear with respect to the electric field, the nonlinearity is weak enough that it can be treated perturbatively. In other words, P = ɛ (χ (1) E + χ (2) E 2 + χ (3) E 3 +...) = P linear + P nonlin. (13) For the linear part of the polarization, the precise value depends on the frequency. Expressed in the frequency domain, P linear (ω) = ɛ χ (1) (ω) E(ω), the wave equation can be written in the f-domain as follows (recall t iω), ( 2 + ω2 c 2 (1 + χ(1) )) E(ω) = µ ω 2 Pnonlin. (14) You might have noticed that for sake of simplicity we are being a little sloppy with the nonlinear polarization term on the right side. This amounts to assuming that the nonlinear contribution is dispersionless, i.e., does not depend on ω, among other mathematical complications. However, this is sufficient for our purposes. Note that everything that will be considered here is non-dissipative (energy is constant, no amplification, no loss). It should be recognized that the electric field has a time structure that has a slow and a fast varying component. The fast time scale corresponds to the optical cycle, which is order of λ/c 3 fs. The slow time scale corresponds to the width of the pulse, which is proportional to E(t) 2, which is typically 1 fs or much longer. Rarely do we have to deal with pulses less than 1 fs, where this separation of the time scales is not so clean anymore. 4
Let us incorporate these two scales into our mathematical formulation, because we will make use of this separation later: E( r, t) = A( r, t)exp(ik z iω t) + c.c., (15) P ( r, t) = P ( r, t)exp(ik z iω t) + c.c., (16) the exponential term is a plane wave which propagates in the +z direction and has a definitive frequency, ω and an associated wavevector k = k k. c.c. stands for complex conjugate. For brevity, we will not explicitly write c.c. anymore, but it should be understood to be there. Indeed, we are doing something rather simple here: in the time domain, we are writing the fields as an envelope ( A( r, t) and P ( r, t)) times a field that oscillates rapidly at the frequency ω. We choose ω to be the center frequency of the optical spectrum of the pulse. In the frequency domain, this corresponds to shifting the spectrum centered at ω = ω altogether to ω =, multiplied by a plane wave with ω = ω. In mathematical terms, in the frequency domain (and similarly for P ), this is equivalent to E( r, ω) = + A( r, t)exp(ik z iω t)exp(iωt) (17) = A( r, ω ω )exp(ik z). (18) Let us continue from equation (14), recall from linear optics that n 2 (ω) = 1 + χ (1) (ω) and k(ω) = ω c n(ω), so χ(1) (ω) = c2 ω 2 k 2 (ω) 1. Then, we obtain, ( 2 + k 2 (ω)) E(ω) = µ ω 2 Pnonlin. (19) We now switch to the amplitude times the oscillation field description from equation (15) and (16). Only the 2 z 2 term brings down new terms. Since each term has exp(ik z iω t), they all cancel out. We further drop 5
the vector signs from now on, assuming that the field is polarized in a fixed direction in the plane perpendicular to the direction of propagation. ( 2 z 2 + 2 x 2 + 2 y 2 + 2ik z k2 )A(x, y, t) + k 2 (ω)a(x, y, t) (2) = µ ω 2 P nonlin (21) The first parenthesis groups all the space terms, the next term incorporates dispersion and the term on the right describes the nonlinear corrections to this otherwise linear equation. We make the usually very well justified approximation that k(ω) is a reasonably slowly varying function of ω, such that it can expanded in powers of ω, k(ω) = k + k 1 (ω ω ) + 1 2! k 2(ω ω ) 2 + 1 3! k 3(ω ω ) 3 +... (22) Here, k = n ω /c is related to the phase velocity light in the medium, k 1 = dk dω ω = 1 v g is the inverse of the group velocity. k 2 = d2 k dω 2 ω is the 2 nd order dispersion and k n gives the n th order dispersion. Now, let us go back to the time and in doing so, use ω = ω ω, this gives, ( 2 z 2 + 2 x 2 + 2 y 2 + 2ik z k2 )A(x, y, t) (23) 2 +(k + ik 1 t 1 2 k 2 +...)A(x, y, t) (24) t2 = µ (ω + i t )2 P nonlin. (25) As a next step, we will make a reference frame transformation to that of the pulse, which propagates nominally at the group velocity, (z, t) (z, τ = t z/v g ). We now describe the optical pulse by the function A (x, y, τ) = 6
A(x, y, t). The chain rule of differentiation yields z = z + τ and similarly, we have, 2 z 2 t = z τ z = z k 1 τ, (26) z t + τ τ t = τ, (27) = 2 2k z 2 1 z τ + k2 1 2. We collect all second τ 2 and higher order dispersion terms under D for notational simplicity. We rewrite the equation in the new coordinates and simplify, ( 2 z 2 + 2 x 2 + 2 y 2 )A (x, y, τ) (28) +i2k (1 k 1 k τ ) z A (x, y, τ) + (2k (1 + i k 1 k τ )D + D2 )A (x, y, τ) (29) = µ ω 2 (1 + i 1 ω τ )2 P nonlin. (3) We will make several important approximations, (i) the slowly varying envelope approximation (SVEA) in time and (ii) SVEA in space and (iii) ignore D 2, since the leading term is 4 τ 4, which is already too weak to be of interest to us. The SVEA in time relates to k 1 /k = c/(n ω v g ) = 1 ω k 1 k τ T optical T pulse v p v g 1 ω. Therefore, 1. Physically speaking, this approximation is valid when the pulse is long enough to contain more than just a few optical cycles within the envelope. The SVEA in space is very similar in its nature, namely it is valid when the beam is wider than just a few wavelengths: 2 A z 2 k A z. Then, we are left with the much simplified equation below, A i2k z + ( 2 x 2 + 2 y 2 )A + 2k DA = µ ωp 2 nonlin, or (31) A z i 1 ( 2 2k x 2 + 2 y 2 )A + i 1 2 k 2 A 2 τ 2 = iµ ω 2 P nonlin 2k. (32) 7
3 The Nonlinear Schrödinger Equation (NLSE) In this section, we will write out the nonlinear polarization term but we will keep only the χ (3) term, derive the NLSE. In general, P nonlin = ɛ (χ (2) E 2 + χ (3) E 3 +...). All higher order contributions are extremely weak and need not be considered in the great majority of the time. χ (2) can be significant, however it requires proper phase-matching and it is also zero for centrosymmetric and most amorphous materials. It will be considered later on in parametric processes, second-harmonic generation, etc. For now, we concentrate on χ (3). Thus, we have P nonlin = 3 4 ɛ χ (3) a 2, where the units are such that a 2 is the optical intensity. Note that, I = 2n Z E 2, so A = a Z 2n. Rewriting in terms of a, with n 2 3 8n χ (3). a z i 1 ( 2 2k x 2 + 2 y 2 )a + i1 2 k 2 a 2 τ 2 = iµ ɛ ω 2 3 k 8 χ(3) a 2 a. (33) = i 3ω 8n c χ(3) a 2 a = i 2πn 2 λ a 2 a. (34) This brings us to the NLSE in space and time. This is a ubiquitous equation, which appears in many branches of physics and engineering, describing extremely different phenomena, from laser beams to Bose-Einstein condensates (see Özgür Öktel of the Physics Department for more information). a z i 1 ( 2 2k x 2 + 2 y 2 )a + i1 2 k 2 a 2 τ 2 = i2πn 2 a 2 a. (35) λ 4 NLSE in Time Only Of special relevance to ultrafast optics is the NLSE in time only. Often times, the spatial dynamics is either trivial (as in propagation in optical fiber, or a large enough beam for which diffraction is negligible) or it is not coupled strongly to the temporal dynamics, so it can be analyzed separately. In fact, 8
either of these two cases is overwhelmingly more common. Then, ignoring diffraction, we have only dispersion and nonlinearity. This corresponds to making the plane wave approximation. A more common way to write the equation is such that the absolute square of the field gives power, instead of intensity. Let us give it a new name, u, to avoid confusion, if we have a = u/ A eff, with A eff being the effective area of the beam, also introducing γ = 2πn 2 /(λ A eff ), we have the following rather simple looking equation, u z + i1 2 k 2 u 2 τ 2 = iγ u 2 u. (36) It is extremely useful to define characteristic length scales, which describe the strength of the effects, because by simply comparing them to each other and the total propagation distance, one can immediately perceive which effects are strong, if one dominates over the other, and if any significant variations on the pulse are expected. The characteristic dispersion length is L ds = τ 2/k 2, where τ is the width of the pulse. Precisely how this width is defined is not important (although there is a convention we should stick to), the point is that the shorter the pulse, the stronger is dispersion and it goes quadratically. In fact, a more general way to define it is to use the inverse bandwidth squared of the pulse, since then L ds applies even for chirped pulses. However, for now, let us stay with this definition. The characteristic nonlinear length can be defined simply as L nl = 1/(γP p ), where P p is the original peak power of the pulse. In other words, P p = max( u(, τ) 2 ). We usually choose the initial pulse shape such that it is centered at τ =, so P p = u(, ) 2. In fact, we can simplify equation (33) and express everything in terms of L, L ds, L nl, here L is the length of the medium. To this end, we make the 9
following transformations, τ ττ, and u u P p and finally z z/l, u z + i1 L 2 u 2 L ds τ 2 = i L u 2 u. (37) L nl All we have done is to choose our time, amplitude and length units such that now z goes from to 1, the initial peak power is 1 and the pulse width is also 1. This normalization is completely arbitrary, there is an infinitely many other ways to do it. However, it is intuitive and simple to keep in mind. Importantly, now all the quantities are order of 1, this makes it convenient to see which effect dominates and it is especially convenient for numerical simulations on the computer. In fact, this is what I do in my simulations. 1