Computational design of one-dimensional nonlinear photonic crystals with material dispersion for efficient second-harmonic generation

Sangbum Kim; Kihong Kim; Fabian Rotermund; Hanjo Lim

doi:10.1364/OE.17.019075

1. Introduction

The recent development in microfabrication technology has incurred renewed interest in the study of electromagnetic wave propagation through inhomogeneous nonlinear dielectric structures, which provide a possibility for the realization of ultra-compact devices, toward all-optical signal processing. Dielectric photonic crystals have unique physical and technological properties. One of the most promising fields of application of photonic crystals is the one that utilizes nonlinear optical phenomena, especially second-order phenomena, that occur in them. Efficient second-harmonic (SH) generation using materials with large second-order optical nonlinearities such as GaAs has been theoretically and experimentally demonstrated [1–16]. Such technology may be employed to create a monolithic active SH generation device by integrating the passive SH generation region into a semiconductor laser. It can also be a way to achieve direct modulation of SH light in compact SH generation devices.

When an electromagnetic wave propagates in a photonic crystal, its dispersion relation is strongly influenced by the periodicity of the structure. In the case of second-order nonlinear optical processes, one may use this fact to accomplish phase matching between the waves of different wavenumbers. For this purpose, it is common to include in the SH generation devices a quasi-phase-matching (QPM) structure which consists of GaAs/AlAs multilayers [17]. However, the SH light power from such QPM structure saturates within a small limiting thickness, since GaAs has a large absorption coefficient in a visible and shorter wavelength region. Another way to increase the conversion efficiency is to increase the intensity of fundamental field inside the SH generation region since the intensity of SH field is proportional to the square of the fundamental field intensity. Another ramification of the periodicity of the structure is an efficient confinement of the light within the structure. These features result in an enhancement of the interacting fields’ amplitudes and an increase of the interaction time between the waves to improve the efficiency of the SH generation process.

It is well-known that photonic crystals display a range of frequencies for which electromagnetic wave propagation is forbidden, which is called a photonic bandgap. In previous studies, it has been discussed that the enhancement of SH generation in photonic crystals is attributed to the high density of modes near the band edges under phase-matching conditions [1, 3]. By the combination of high density of modes and exact phase-matching conditions in a mixed half-quarter-wave periodic photonic bandgap structure, a scheme was proposed to get doubly resonant SH generation near the band edge. In the absence of material dispersion, it can be shown that an optimal SH conversion efficiency is obtained when the pump field is tuned to the band edge of the lower bandgap and the density of modes for the SH field is at the second peak away from the band edge of the higher bandgap. When considerable material dispersion is present, it has been suggested that phase-matching can be achieved by manipulating the geometry of the structure such as the layer thicknesses [3]. Later, an enhancement in SH generation by ‘true’ double resonance was obtained by keeping the FF and SH fields tuned to the band edge frequencies of the two consecutive bandgaps where the center frequency of the higher bandgap is approximately twice that of the lower one [4].

In this paper, we employ the idea of double resonance to obtain enhanced SH generation [4]. We present an efficient numerical calculation of SH generation in one-dimensional photonic crystals with quadratic nonlinearity. In order to perform parametric search and optimization of the geometric structures with nonlinear properties, the invariant imbedding method [18, 19] is employed to calculate detailed spectra of transmissivity and electromagnetic mode density in photonic crystals with material dispersion taken into account. We also use a shooting method [20, 21] to find exact solutions of one-dimensional frequency-domain Maxwell equations for fundamental and second-harmonic fields, coupled with each other through the second-order nonlinear susceptibility χ ⁽²⁾. We solve the coupled-mode equations for SH generation in an extensive range of the input field intensity, I _FF, including the pump depletion effect. In this way we find an optimal design of photonic crystals for double resonance. Our approach may be regarded as a bandgap engineering of nonlinear optical devices.

2. Model and methods

Following [4], we consider a one-dimensional photonic crystal with the structure denoted by (ABCD)^N, where N is the number of periods. The elementary cell consists of four sublayers as depicted in Fig. 1. The thicknesses d_A, d_B, d_C and d_D of the four sublayers are given by

\frac{ω_{0}}{c} n_{A} d_{A} = \frac{ω_{0}}{c} n_{D} d_{D} = (1 - α) π, \frac{ω_{0}}{c} n_{B} d_{B} = \frac{ω_{0}}{c} n_{C} d_{C} = α π,

where n_A, n_B, n_C and n_D are the refractive indices of the sublayers and c is the speed of light in a vacuum. We further assume that the structure is made of only two materials, GaAs and AlAs, and n_A=n_C=n ₁ and n_B=n_D=n ₂, where n ₁ and n ₂ are the refractive indices for GaAs and AlAs respectively. The material dispersion is included by assuming that the frequency dependences of n ₁ and n ₂ obey the Sellmeier equation [4, 22].

As the parameter α varies between 0 and 1, the relative thicknesses of the sublayers change and a variety of elementary cell geometries are generated. For example, the choice α=1/2 gives the well-known quarter-wave stack, with its lowest frequency bandgap centered at the frequency 2ω ₀. On the other hand, the choice α=0 corresponds to the half-wave stack, with its lowest frequency bandgap centered at the frequency ω ₀. By choosing suitable values of the reference frequency ω ₀ and the parameter α, we can design the photonic crystal to have desired properties. In particular, we seek a photonic crystal with the band structure such that both the fundamental frequency (FF) with the wavelength λ _FF=3.1 µm and the SH with the wavelength λ _SH=1.55 µm are tuned to be at the edges of photonic bandgaps, where the transmittance and the electromagnetic density of modes take peak values.

Fig. 1. Elementary cell of the photonic crystal under study.

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In order to understand the characteristics of wave propagation through a photonic crystal, we examine the linear transmission and group index spectra. For an s-polarized wave incident on a stratified slab lying in 0≤z≤L and propagating in the xz plane, the complex amplitude of the electric field, E=E(z), satisfies

\frac{d^{2} E}{d z^{2}} - \frac{1}{μ (z)} \frac{d μ}{d z} \frac{d E}{d z} + [{k_{0}}^{2} ε (z) μ (z) - q^{2}] E = 0,

where ε and µ are the dielectric permittivity and the magnetic permeability respectively. k ₀ (=ω/c) is the vacuum wave number and $q (= \sqrt{ε_{1} μ_{1}} k_{0} \sin θ)$ is the x component of the wave vector. θ is the angle of incidence and ε ₁ and µ ₁ are the values of ε and µ in the incident region. We assume that ε and µ vary only along the z axis and the wave is incident on a layer of thickness L from the region where z>L. Then the invariant imbedding method gives the ordinary differential equations for the reflection and transmission coefficients, r and t:

\begin{array}{c} \frac{1}{i p} \frac{d r}{d l} = 2 \frac{μ (l)}{μ_{1}} r (l) + \frac{1}{2} a (l) {[1 + r (l)]}^{2}, \\ \frac{1}{i p} \frac{d t}{d l} = \frac{μ (l)}{μ_{1}} t (l) + \frac{1}{2} a (l) [1 + r (l)] t (l), \end{array}

where

a (l) = \frac{ε (l)}{ε_{1}} - \frac{μ (l)}{μ_{1}} + [\frac{ε (l)}{ε_{1}} - \frac{μ_{1}}{μ (l)}] \tan^{2} θ

and $p = \sqrt{ε_{1} μ_{1}} k_{0} \cos θ$ [18]. When the incident region and the transmitted region consist of the same kind of medium, r and t satisfy very simple initial conditions for any incident angle, r(0)=1 and t(0)=1. We integrate Eq. (3) from l=0 to l=L using these initial conditions and obtain r(L) and t(L).

In Fig. 2, we plot the transmittance T(=|t|²) versus the normalized frequency Ω(=ω/ω ₀) for the (ABCD)^N structure made of GaAs and AlAs with N=40, λ ₀=2πc/ω ₀=2.9872 µm, α=0.313 and ε ₁=µ ₁=1. We observe that the FF and the SH are precisely at the lower edges of bandgaps. The spectral bandwidth (FWHM) allowed by the band edge resonance at the FF frequency in Fig. 2 is about 60 GHz. When pulses are used, it is required that the spectral bandwidth of the pulse should be shorter than or comparable to the band edge resonance bandwidth [1]. Based on this condition, we estimate the pulse width to be in the range of 10 ps.

Fig. 2. Transmittance T versus the normalized frequency Ω(=ω/ω ₀) for the (ABCD)^N structure made of GaAs and AlAs with N=40, λ ₀=2πc/ω ₀=2.9872 µm and α=0.313. The FF and the SH are at the lower edges of bandgaps.

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Once we obtain the complex transmission coefficient t(ω)=x(ω)+iy(ω) as a function of the frequency ω, the electromagnetic density of modes, ρ(=dk/dω), can be calculated from

ρ = \frac{1}{L} \frac{y^{'} x - x^{'} y}{x^{2} + y^{2}},

where the prime denotes differentiation with respect to ω [23]. In Fig. 3(a), we plot the group index n_g (=cρ), which is the normalized density of modes, versus Ω for the (ABCD)^N structure with N=40, λ ₀=2.9872 µm, α=0.313 and ε ₁=µ ₁=1. In Figs. 3(b) and 3(c), we plot n_g versus vacuum wavelength near the FF and the SH. From these figures, we clearly see that there is a double resonance. The group index spectrum shows very sharp peaks at the band edges corresponding to λ _FF=3.1 µm and λ _SH=1.55 µm.

Two coupled nonlinear wave equations satisfied by the FF and SH fields, E _FF and E _SH, in stratified media have the form [24]

\begin{array}{c} \frac{d^{2} E_{FF}}{d z^{2}} = - \frac{ω_{{FF}^{2}}}{c^{2}} ε_{FF} E_{FF} - 2 χ^{(2)} \frac{ω_{{FF}^{2}}}{c^{2}} E_{FF}^{*} E_{SH}, \\ \frac{d^{2} E_{SH}}{d z^{2}} = - \frac{ω_{{SH}^{2}}}{c^{2}} ε_{SH} E_{SH} - χ^{(2)} \frac{ω_{{SH}^{2}}}{c^{2}} E_{{FF}^{2}}, \end{array}

where χ ⁽²⁾ is the second-order nonlinear optical susceptibility and ε _FF and εSH are the values of the linear dielectric permittivity at the fundamental frequency and the SH frequency respectively: ε _FF=ε(ω=ω _FF)=[n(ω=ω _FF)]² and ε _SH=ε(ω=ω _SH)=[n(ω=ω _SH)]². We solve these equations in a numerically exact manner using the shooting method of Midrio [20, 21] and compute the reflected and transmitted SH powers. We note that this formulation is free of the slowly varying amplitude approximation and can include the pump depletion effect properly [25].

Fig. 3. (a) Group index versus normalized frequency for the (ABCD)^N structure with N=40, λ ₀=2.9872 µm, α=0.313 and ε ₁=µ ₁=1. Group index versus vacuum wavelength (b) near the fundamental frequency and (c) near the SH frequency.

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It is convenient to rewrite Eq. (6) as

\begin{array}{c} \frac{d^{2} u_{FF}}{d ξ^{2}} = - \frac{ω_{{FF}^{2}}}{c^{2}} ε_{FF} u_{FF} - 2 β \frac{ω_{{FF}^{2}}}{c^{2}} u_{FF}^{*} u_{SH}, \\ \frac{d^{2} u_{SH}}{d ξ^{2}} = - \frac{ω_{{SH}^{2}}}{c^{2}} ε_{SH} u_{SH} - β \frac{ω_{{SH}^{2}}}{c^{2}} u_{{FF}^{2}}, \end{array}

where u _FF (=E _FF/|E ₀|) and u _SH (=E _SH/|E ₀|) are the electric fields normalized by the magnitude of the incident FF field amplitude E ₀ and ξ(=z/Λ) is the spatial coordinate in the direction of propagation normalized by the period of the photonic crystal, Λ=d_A+d_B+d_C+d_D. E ₀ is related to the input field intensity I _FF by I _FF=2n|E ₀|²/Z ₀, where n is the refractive index in the incident region and $Z_{0} = \sqrt{μ_{0} / ε_{0}} \approx 377 Ω$ . We have introduced a nonlinearity parameter β=χ ⁽²⁾|E ₀|. We notice that the parameters ε _FF, ε _SH, χ ⁽²⁾(=2d _eff) and β are periodic functions of ξ. For numerical calculations, we use d _eff(GaAs)=170 pm/V [25] and d _eff(AlAs)=0.23 d _eff(GaAs)=39.1 pm/V [26]. The frequency dependence of the nonlinear susceptibility is ignored.

Given a complex amplitude of the incident wave A ₀ as an initial condition for each mode, the wave equations above are solved to find the complex amplitudes of the reflected wave, B ₀, and those of the transmitted wave, A _out. B ₀ at the incident face serves as the shooting parameter, while A _out is to satisfy the shooting condition at the output face [20]. In the code that we developed, we used the well-known Newton’s method to solve the multidimensional zero-finding problem.

3. Results

In Fig. 4, we show the forward and backward SH conversion efficiencies, η_f and η_b, as a function of the normalized frequency Ω for the (ABCD)^N structure made of GaAs and AlAs with N=40, λ ₀=2.9872 µm and α=0.313. We assume that the incident and transmitted regions consist of GaAs. The pump intensity I _FF is equal to 10 MW/cm². We find that both curves are sharply peaked at the frequency corresponding to λ _FF=3.1 µm. The peak values of η _f and η_b are 7.29 % and 6.38 % respectively. In a previous theoretical calculation, a peak efficiency of 1.3 % has been reported in a similar structure for the same pump intensity [4]. In addition, there is another example of the SH generation in a mixed half-quarter-wave structure composed of GaAs and AlAs layers with material dispersion. For a pump field with the intensity I _FF=10 GW/cm² and the wavelength of 3 µm, conversion efficiencies of the order of 10^-2 -10^-3 were theoretically obtained in a 20-period structure [1]. Although with different parameters, it is clear that our study opens a venue for improved SH generation in photonic crystals with material dispersion.

Fig. 4. Forward and backward SH conversion efficiencies, η _f and η_b, versus the normalized frequency Ω for the (ABCD)^N structure with N=40, λ ₀=2.9872 µm and α=0.313. The incident and transmitted regions consist of GaAs. The pump intensity I _FF is equal to 10 MW/cm². Both curves are peaked at the frequency corresponding to λ _FF=3.1 µm.

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In Fig. 5, we plot the spatial distributions of the normalized FF and SH field intensities, |u _FF|² and |u _SH|², inside the 40-period photonic crystal at double resonance. The wave is assumed to be incident from the left. Both fields are nearly standing waves, which explains the fact that the forward and backward SH conversion efficiencies in Fig. 4 are close to each other [4]. The maximum field intensity is amplified by more than an order of magnitude by linear interference effects of backward- and forward-traveling components. In view of the magnitude of the field intensity, it is clear that the SH field overlaps well with the pump field. The fact that a single envelope can be identified for the SH field inside the structure is due to that the SH signal is tuned to the band edge. In Fig. 5(c), we show expanded field distributions near the center of the structure.

Fig. 5. Spatial distributions of (a) the normalized FF field intensity, |u _FF|², and (b) the normalized SH field intensity, |u _SH|², inside the 40-period photonic crystal at double resonance, when the pump intensity I _FF is equal to 10 MW/cm². The wave is incident from the left. (c) Expanded field distributions near the center of the structure.

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In Fig. 6, we plot the forward and backward SH conversion efficiencies versus the pump intensity I _FF. We also plot the reflectance R _FF and the transmittance T _FF of the FF wave. For relatively low levels of I _FF, the pump field is not depleted and the SH conversion efficiencies grow rapidly. The forward SH conversion efficiency grows until about I _FF=430 MW/cm² to 26.4 % and then decreases gradually. Similarly, the backward SH conversion efficiency grows until about I _FF=270 MW/cm² to 20.7 % and then decreases. The forward and backward SH conversion efficiencies are enhanced by roughly the same order, which is due to the strong feedback resulting from the high contrast in the refractive indices between layers [10]. At high pump-wave power, the SH conversion efficiencies do not saturate in both the forward and backward directions but decreases slowly. When the input intensity is 3200 MW/cm², η_f and η_b are equal to 21.4 % and 12.9 % respectively, and thus a total conversion efficiency of 34.3 % is obtained. In the calculated parameter range, T _FF decreases and R _FF increases monotonically. We note that the forward FF channel is strongly coupled to the backward FF channel, leading to the prominent rise of R _FF as I _FF increases [10]. We have verified numerically that the law of energy conservation, ηf+ηb+T _FF+R _FF=1, always holds precisely. Our results clearly suggest that there is an optimal operation point to obtain the best SH generation in this photonic crystal.

In Fig. 7, we plot the spatial distributions of the normalized FF and SH field intensities inside the 40-period photonic crystal at double resonance, when the pump intensity I _FF is equal to 3200 MW/cm². Compared to Fig. 5, the FF field intensity is greatly reduced, while the SH field intensity is substantially enhanced in the central part of the structure. We observe that the shape of the |u _FF|² curve remains almost the same in the right half of the structure close to the output face, while the curve rises to higher values in the left half close to the incident face. It appears that the high input intensity “pulls up” the pump field at the incident face, while the “soliton” [27] prefers to stay in place in the other part of the structure. On the other hand, the SH field profile shows almost the same standing wave pattern as that obtained in the smaller input intensity. This implies that the interaction between the pump field and the SH field may not be uniform inside the photonic crystal. If this trend persists or gets stronger as the pump intensity increases, the conversion efficiency may not be able to take its maximum possible value for the given intensity. The profiles also suggest that the conversion between the FF and SH fields on the incident side is not as good as that on the output side, since the SH field distribution is relatively symmetric while the FF field distribution is severely skewed. Thus the backward SH conversion efficiency η_b is always lower than the forward SH conversion efficiency η_f.

Fig. 6. (a) Forward and backward SH conversion efficiencies η_f and η_b versus the pump intensity I _FF. (b) Reflectance R _FF and transmittance T _FF of the FF wave versus I _FF.

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4. Conclusion

In summary, we have demonstrated that an optimal reference frequency ω ₀ and the corresponding thicknesses of the sublayers of the elementary cell can be chosen by observing the SH conversion efficiencies while varying λ ₀=2πc/ω ₀ and a geometric parameterα. The enhancement of the SH conversion efficiencies found in previous studies was confirmed, while we searched for an improved design of photonic crystals to enhance SH generation further. The (ABCD)^N structure was employed and searches for better combinations of the critical parameters λ ₀, λ _FF and α were conducted. We have thus found a new configuration for efficient doubly-resonant SH generation, which is different from that found in [4] and obtained an increased total SH conversion efficiency of 13.7 % at a relatively low pump intensity I _FF=10 MW/cm². The FF and the SH frequency are tuned to be precisely at the edges of photonic bandgaps, where the electromagnetic density of modes takes very high peak values.

Simulation of electromagnetic fields inside nonlinear photonic crystals is difficult, and thus one is often forced to rely on simplifying approximations, such as the piecewise continuous modeling and iteration. This paper has presented a set of powerful numerical methods which can handle such difficult problems in a numerically exact manner. And their usefulness has been demonstrated by finding an optimal design of a photonic crystal that enhances SH generation significantly.

Fig. 7. Spatial distributions of (a) the normalized FF field intensity and (b) the normalized SH field intensity inside the 40-period photonic crystal at double resonance, when the pump intensity I _FF is equal to 3200 MW/cm².

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Our 40-period structure has the total length of 38.7 µm and the smallest thickness of a sublayer is 0.14 µm. State-of-the-art microfabrication techniques such as the molecular beam epitaxy (MBE) or the metal-organic chemical vapor deposition (MOCVD) can be employed to fabricate our structure with the accuracy of a few °A or a fraction of nm. In order to estimate the effects of fabrication errors, we have added small random variations in the thickness of layers and run our code. We found no significant change in the SH generation behavior. Thus we believe that it will be possible to find a good correspondence between our numerical results and experimental measurements.

Acknowledgments

S.K. acknowledges helpful discussions with Dr. G. T. Kiehne and Prof. Jae Jin Lee. This work has been supported by the Korea Research Foundation grant (MOEHRD, KRF-2007-412-J04002) and by the National Research Foundation of Korea grant (No. R0A-2007-000-20113-0), funded by the Korean Government.

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Computational design of one-dimensional nonlinear photonic crystals with material dispersion for efficient second-harmonic generation

Abstract

1. Introduction

2. Model and methods

3. Results

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Equations (7)

Optics Express