study of self-refractive effects in nonlinear wave propagation. A. Korpel*. Institute ... Our method is closely related to the so-called split-step. Fourier technique of ...
Vol. 3, No. 6/June 1986/J. Opt. Soc. Am. B
Korpelet al.
885
Split-step-type angularplane-wave spectrum method for the study of self-refractive effects in nonlinear wave propagation A. Korpel* Institute for Applied Physics, University of Erlangen, Nuremberg, Federal Republic of Germany K. E. Lonngren, P. P. Banerjeet,
H. K. Sim, and M. R. Chatterjee
Department of Electrical and Computer Engineering, The University of Iowa, Iowa City, Iowa 52242 Received April 26, 1985; accepted February 18, 1986
We introduce an efficient algorithm to simulate nonlinear self-induced waverefraction effects. The algorithm is nnlied to the case of self-focusing. defocusina. and predicted steady-state
behavior.
The results show reasonably
good agreement with available approximate theory and excellent agreement when exact analytic solutions are available for comparison.
1.
INTRODUCTION
The effect of a cubic nonlinearity
on wave propagation has
been theoretically and experimentally studied over the past few years.' This paper reports on a novel algorithm that simulates the effect by combining angular plane-wave spec-
trum techniques in the Fourier domain with self-induced thin-lens techniques in the space domain. (For the linear case, both techniques are adequately explained in Ref. 2.) The essence of our method is as follows. If the complex field distribution across any plane is Fourier analyzed, the
various spatial Fourier components can be identified as plane waves traveling in different directions. The field amplitude at any other point can then be calculated by adding the contributions of all plane waves, taking into account the phase shifts that they have undergone during propagation. We apply this concept to a thin slice perpendicular to the normal propagation direction in order to obtain the field amplitude behind the slice. We then correct for the nonlinear effect by passing through the slice again as if it were a
thin lens, self-induced by the field just calculated. Our method is closely related to the so-called split-step Fourier technique of Hardin and Tappert,3 used by Hasegawa and Tappert to calculate nonlinear pulse propagation in dispersive
fibers 4 5 and by Fisher and Bischel to 6
investigate one-dimensional self-phase modulation. As far as we are aware, the first complete description of the method was given by Fisher and Bischel, who apparently arrived at it independently. The analogies between Fisher and Bischel (F/B) and ourselves are as follows:
time t (F/B) versus
transverse position x; temporal-frequency spectrum (F/B) versus angular plane-wave spectrum; and frequency disper-
through the use of alternate Fourier and inverse Fourier transforms, thereby speeding up the calculations considerably.
Other investigators, too, have used fast-Fourier-trans-
form (FFT) methods to calculate propagation in three-dimensional linear media8 and one-dimensional nonlinear media. 9 We restrict ourselves to the case of a cubic nonlinearity.
In principle, however, our method can be used for any odd nonlinearity because, to first order (i.e., neglecting higher harmonics), the refractive index of the medium can then be represented to have an even-order dependence on the modulus of the complex amplitude. Such a dependence generally leads to a self-focusing or a defocusing of a hump-type beam.
In addition, we treat only the two-dimensional case in order to avoid difficulties with amplitude singularities at the focus.' 0 The organization of our paper is as follows: In Section 2
we briefly derive the amplitude-dependent propagation constant from the scalar wave equation with cubic nonlinearity. In Section 3 we use the angular spectrum method to derive an expression for the emerging wave in the presence of cubic
nonlinearity, including the nonlinearly induced thin-lens effect. In order to check our algorithm, we first calculate in Section 4 some well-known linear cases, viz., the propagation
of a Gaussian beam and the diffraction from a rectangular aperture. In Section 5 the nonlinear evolution of these cases is presented. Finally, in Section 6 we demonstrate the existence of a particular analytic steady-state solution and test it using our algorithm. 2. THE EFFECT OF CUBIC NONLINEARITY
sion (F/B) versus diffraction.
ON WAVE PROPAGATION
The advantage of the technique is that diffraction (dispersion) is calculated in the Fourier domain and nonlinearity in
We assume a cw situation (e.g., starting with a Gaussian
the space (time) domain, both effects requiring only simple
multiplications in their own domain. In this respect our 7 method is analogous to that of Bojarski, who calculates
scattering by an external (not self-induced) potential 0740-3224/86/060885-06$02.00
profile) in two dimensions and use the conventional nondispersive wave equation with cubic nonlinear term as a2 U /at
2
-
3 C02 (a 2 U/ax 2 + 02 U/0z 2 ) = +A 8 U .
© 1986 Optical Society of America
(1)
886
/ J. Opt. Soc. Am. B/Vol. 3, No. 6/June 1986
Korpel et al.
x Y
tude values' being larger by a factor of 2. This formula approximately agrees with our results, as we show in Section 5.
Y (x;z)
Conversely, when A3 < 0, the medium behaves as a nega-
propagation direction
tive lens for an amplitude profile that is maximum on the @
-
-
-
-
t
-
-
-
propagation axis, resulting in self-defocusing.
z Fig. 1.
3. ANGULAR SPECTRUM METHOD
Definition of coordinate axes.
We now introduce the two-dimensional U= / 2 [Yexp(jwt) + Y* exp(jct)]
phasor Y as follows: = Re[Y exp(jcot)], Y = Y(x; z),
a transverse coordinate, as shown in Fig. 1.
Substituting Eq. (2) into Eq. (1) and neglecting contributions from the 3wcomponent, we find that -
Co2(a 2y/0X 2 +
2y/az 2 )
+3/4A3 1 1 2y.
(3)
The amplitude-dependent dispersion relation may now be readily obtained from expression (3), on substituting Y = exp(-jkx - jkz) as
W2/CO2 = k
2
A k2-(
3
/)(A3/C
2
)j j
2
spectrum in order to study the effect of cubic nonlinearity on wave propagation.
(2)
where z denotes the normal direction of propagation and x is
_(2y
As mentioned in Section 1, we use the angular plane-wave
The field Y(x; z) can be thought of as consisting of an angular plane-wave spectrum A(hx; z): Y(x; z) = (1/27r)
J
A(k; z)exp(-jhkxx)dkx.
(7)
(Note that evanescent and reflected waves are neglected in this treatment.) The angular plane-wave spectrum A(k; z) may be found from Y(x; z) as follows2 :
lM
(4a)
Consider a slice dz at z, as shown in Fig.
2. Immediately to the left of the slice, the field is Y(x; z).
A(k ; z) =
J
Y(x; z)exp(jkx)dx.
(8)
(Note that Ref. 2 followsthe optics convention in defining Y. Hence, to compare equations, j should be replaced by j.) We use the following symbolic notation:
or 2
k2 = ko + ( 3/4 )(A 3/CO2 ) 112,
this, however, bears relevance only if I Y I is constant.]
The physical implications of Eqs. (4) may be summarized as follows: When A3 > 0, kl increases with l yl2, i.e., a beam with a maximum in the center tends to self-focus, whereas a beam with a minimum in the center will defocus. From another point of view, for the assumption of weak nonlinearity, k 2 >>I (1 4)(A3 /Co2 )I1 21, Eq. (4a) can be used to define a large-signal refractive index as follows: nj
1 + ( 3 /8 )[A3 /(k 0 2 CO2 )]I Y12 ,
(5)
where n, = kl/ko. Note that, in expression (5), the refractive index increases quadratically with field amplitude for A3 > 0. The medium will thus behave as a positive lens for a Gaussian input amplitude profile or any other profile with a maximum in the center. This causes self-focusing to occur periodically for a paraxial assumption, k k used here. The theoretical distance to the first focus for a Gaussian amplitude profile in the absence of diffraction is given by12
zf - (1.4w01a0)V,1-(aw1,9a 2) i 12,
Y(x; z) = F[A(kx; z)]
(9)
A(k,; z) = IF[Y(x; z)]
(10)
(4b)
where k = (k 2 + k 2)1/2 may be termed the "large-signal amplitude propagation constant." [The often-used parameter da/0a 2 = w/l YVI (Ref. 11) may be calculated to be approximately equal to -(%)A 3/W;
and
where F and IF represent direct and inverse Fourier transform in the x and k domains, respectively. In what follows,a paraxial approximation is assumed, i.e., kx
