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Spatial-temporal characteristics of ultrashort optical pulses in a resonantly absorbing Bragg reflector are numerically evaluated. The moving and stationary spatial-temporal gap solitons are shown to exist in the photonic structure for a finite spatially-distributed light field in a finite thickness sample. A practical method of trapping light pulses in the photonic structure is presented.
©2005 Optical Society of America
1. Introduction
Electromagnetic wave propagation and localization in nonlinear periodic dielectric structures has been of great research interest in recent years [1]. It has been shown that these nonlinear periodic structures have many different applications, such as optical bistability [2], compression and shaping of laser pulses [3, 4], gap soliton generation [5, 6], storage of ultrashort optical pulses [7] etc., The structure of resonantly absorbing Bragg reflectors (RABR), i. e., a periodic array of thin layers of resonant two-level systems separated by half wavelength nonabsorbing dielectric layers, was introduced by Kurizki et al. [8, 9, 10]. Gap solitons (GS) in these structures have been predicted at much lower input intensities (10 MW/cm2 or less) than GS in a Bragg mirror [10, 11], which is significant for practical applications.
When the spatial effect is taken into account, there exist spatiotemporally localized soliton solutions, i.e., “light bullets.” [12, 13] These stable “light bullets” suggest new possibilities of signal transmission control and self-trapping of light. In ref. [9], Blaauboer et al. studied the probability and stability of approximate soliton solutions in an infinite RABR. In practice, the structure of RABR has a finite length and spatially-distributed light pulse is applied to excite the sample. Whether the spatial-temporal propagation of the input wave can evolve into the “light bullets” predicted in [9] is worthy of research for practical experimental system. In this Letter, we numerically demonstrate complicated propagation behavior of laser pulses in a finite RABR, and examine moving and stationary spatial-temporal solitons in the structure.
It was shown that multiple GS can be simultaneously spatially localized, resulting in efficient optical energy storage in an RABR as atomically coherent states [7]. In practice, pulses are spatially localized, and will diffract in the transverse direction. We also demonstrate in the Letter that pulses with finite beam size can be efficiently stored in an RABR.
2. Computation model and method
The theory used here for the interaction of light pulses with RABR is based on the two-wave Maxwell-Bloch (TWMB) equations [8] in the slowly varying envelope approximation of the forward and backward propagating electric fields E ± (r, t). With reference to [9, 14], we introduce diffraction in the transverse direction. Then a set of TWMB equations can be obtained:
where Ω ± (r, t) = (2τ0 μ/ħ)E ± (r, t); E ± (r, t) are the smooth field-amplitude envelopes of the forward and backward Bloch waves; denotes ∂xx + ∂yy; P and w are the polarization and population inversion; In these equations, dimensionless space z = z '/cτ 0, r⃑ = r⃑'/c τ 0 and time t = t '/τ 0 are used; here τ 0 is the cooperative time; The parameter η = lc /lr is the ratio of the Arrechi-Courtens cooperativity length (lc =cτ 0 = 2c/ω, where ωp is an effective plasma frequency) to the reflection length (lr = 4dε 0/πΔε, where d is the period and Δε the variation of the dielectric index of the periodic structure with average dielectric index ε 0) [8]; δ = (ω 0 - ωc )τ 0 is the detuning of two-level atoms in the periodic structures; the Fresnel number F is a diffraction parameter.
We assume that no initial polarization and population inversion exists, i.e. P(x, z, t = 0) = 0 and w(x, z, t = 0) = -1 , in the RABR and the initial pulse is displaced by z 0 to a position outside of the RABR. Only one transverse direction is considered without lose of generality but for the sake of numerical evaluation convenience. The initial conditions are given by:
Where, (2a) and (2b) are initial conditions obtained from boundary conditions using the method in [13]; Ω is the amplitude of the incident pulse; χ is the frequency detuning of the incident pulse in the RABR; σz and σx are parameters which denote the longitudinal and transverse width respectively.
A split-step numerical method [14] and a predict-correct method are adopted respectively to numerically solve the equations (1a–1b) and equations (1c–1d) with the initial and boundary conditions described in equations (2).
The parameters used are as follows: The initial displacement, z 0, of the pulse is chosen to be several times greater than the pulse spread in space and σz = σx = 1. This ensures no light field in the medium at the initial time. The z-axis spans 12π and the x-axis span is 3π. To do fast Fourier transforms (FFT), we use 1024 points in the longitudinal direction and 256 points in the transverse direction. In our computations, F=100, which means weak transverse coupling. The accuracy of the numerical calculation and hence the validity of the numerical evaluation was verified by changing the grid size from 512*128 points to 2048*512 points. No diversion of the results was observed and conservation of |P|2 + |w|2 at each time step was warranted.
3. Results
In this computation, both of the amounts of detuning δ and parameter η are initially set equal to 0, which means a structure of resonant photonic crystal (RPC) [15, 16, 17]. The dispersion relation of an infinite RPC without diffraction in the transverse direction was discussed in [8], and there is a frequency band gap -√2 < χ < 2 where soliton solution exists.
Figure 1 is an animation showing a low-power pulse amplitude propagating through the structure (Ω0 = 0.5). The absolute amplitude of the field at each pixel is represented by a color. Z is the propagation direction and X is the transverse coordinate. The location of the RPC is denoted by the green boxes in the subfigures. The initial pulse is displaced outside of the structure and propagates to the right. Here, the input pulse is resonant with the band gap, i.e. χ = 0 . The forward wave is shown to break up into a set of sub-pulses. Every sub-pulse is smooth but has small amplitude. The backward pulse has a major peak accompanied by several satellite pulses, with the major peak amplitude greater than the forward wave amplitude. These phenomena can be explained with the spectral analysis where the input pulse spectrum embraces the forbidden band, which is reflected strongly for weak amplitude, and which forms a comparative smooth backward pulse. The spectrum outside the band gap transmits through and gives rise to a set of forward sub-pulses as a result of frequency beating.
In addition, because of the weak electromagnetic field, the nonlinearity in the transverse direction is small, which has little effect in the change of the transverse shape.
In all of the numerically simulations of the weak incident pulse (Ω0 < 1), the pulse break up in the longitudinal direction is observed and the change of the transverse shape is small. When the intensity of the incident pulse is increased to Ω0 = 5.5, the propagation of the pulse exhibits complicated spatial-temporal behavior.
Fig. 1. Animation of the pulse evolution through an RPC with detuning χ = 0. The amplitude of the initial pulse is Ω0 = 0.5. The top panel is the forward-propagating pulse and the bottom panel is the back ward propagating pulse. The numbers in brackets denote the maximum value of the intensity in that plot (727 KB gif animation).
Figure 2 is an animation of a strong pulse detuned at the upper band gap edge (χ = 1.4) propagating through the sample. As the pulse propagates through Z axis, the forward field self focuses and then propagates with a stable “bullet-like” profile. The backward pulse breaks up both in the longitudinal and transverse direction, which contributes to a very complicated distribution of the backward field. For this input pulse, most of the pulse energy enters into the structure and propagates in the forward field.
We show the full width of the half maximum (FWHM) of the forward pulse in the longitude and transverse direction in figure 3. After the pulse focuses to a small dimension, the FWHM of the forward pulse in both directions keeps a stable value for some time, which contributes to a property of moving spatial-temporal solitons.
For η = 0.1, the structure becomes an RABR. According to [8], there will be two soliton zones: -1.47 < χ < -0.1and 0.1 < χ < 1.37. We numerically simulate the spatial-temporal propagation of ultrashort pulses in an RABR. The spatial-temporal solitons obtained in an RPC can also be observed in an RABR.
Fig. 2. Animation of Gaussian pulse evolution through the RPC. Parameters are the same as Fig. 1 except for χ =1.4 and Ω0 = 5.5 (710 KB gif animation).
Fig. 3. Full width of the half maximum of the forward pulse. Solid line: FWHM in the longitude direction; dotted line: FWHM in the transverse direction; space bracketed by the dash line: RPC structure.
A novel result is that stationary solitons can be generated and most of the pulse energy can be trapped inside in either an RPC or an RABR when a hyperbolic secant shape pulse of suitable pulse area is incident into the structures with a certain detuning. Figure 4 is a similar animation of pulses propagating in the RABR. Here, the initial and boundary conditions are the same as equations (2) except for a different spatial profile of the incident pulse:
Here, Gaussian profile of the pulse in the transverse direction is employed. The parameters used in figure 4 are as follows: Ω0 = 3.5, σz = 0.5, σx = 1, χ = -0.15.
It is necessary to emphasize that the pulse incident into the RABR and is trapped at a position near the input boundary, indicating that the required sample length is small (<6lc ).
Fig. 4. The evolution of hyperbolic secant pulse through an RABR with detuning χ = -0.15 and initial amplitude Ω0 = 3.5 (856 KB gif animation).
4. Conclusions
The spatial-temporal propagation of ultrashort pulses in a finite length RABR is investigated. The propagation behavior is mainly determined by the temporal spectra for weak incident pulses. For strong input pulses, moving spatial-temporal GS can be generated inside the photonic structure. Stationary solitons can be obtained in an RABR using appropriate pulse as input, resulting in optical pulse trapping in a finite length sample.
Acknowledgments
The supports for the project by the National Key Basic Research Special Foundation (NKBRSF) under Grant No. G2004CB719805, Chinese National Natural Science Foundation (90201027, 10374120) and Guangdong Natural Science foundation (036593) are acknowledged. The authors thank critical reading of the manuscript by J. T. Li and M. Li.
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