1. Introduction

All-optical polarization switches that make use of nonlinear optical effects in semiconductor multiple quantum wells have exhibited potentials for the application in many fields such as the high speed communications, signal processing and computing. These switches operate by using a circularly polarized control pulse to induce circular dichroism and birefringence (i.e. difference in absorption coefficient and index of refraction for right and left circularly polarized light) in MQWs. This circular anisotropy changes the polarization state of the linearly polarized signal pulse. The change in the signal polarization is subsequently converted to a modulation of the signal amplitude by using additional polarization sensitive elements.

Bragg-spaced quantum wells (BSQWs), in particular, have exhibited unique linear and nonlinear optical properties, of interest to optical switching [1, 2], which is sensitive to light and has a low switching energy. J. P. Prineas, et al., [3] have investigated the ultrafast suppression and recovery of the active photonic band-gap structure constructed from InGaAs/GaAs BSQWs by making use of control-signal measures. Recently W. J. Johnston, et al. [1] have taken BSQW structure to obtain the coherent polarization switch in a reflection geometry, which has a high contrast ratio, femtosecond switching time and reduced switching energy. However the switches operating at 1.55µm are requisite for the optical communication, which means the BSQWs adopted in the switch have an absorption resonance wavelength around 1.55µm. Here we set up the mathematic model of the switch adopting InGaAs/InP BSQW structure and study the switching process and behavior.

In this paper we outline the transfer matrix approach and calculate the optical response of the BSQWs. The absorption coefficient and indices are obtained from the two-dimensional (2D) susceptibility. At last we have respectively calculated the contrast ratio changing with control intensity, delay time (control-signal delay) and the insert loss changing with delay time.

2. Theoretical model

The BSQW sample consists of 200 periods of In _0.53 Ga _0.47 As quantum wells separated by InP barriers such that the period, d, is equal to one-half the excitonic wavelength in the material shown as Fig. 1. The barrier refractive index [4]n_b=3.03, the quantum well refractive index n_w=3.2003, the interwell width d_w=7nm, the barrier width d_b=233.3 nm, the excitonic resonance energy ħω ₀=0.85eV at 10K [5]. The BSQW sample was placed between two crossed polarizers in reflection geometry [1]. The control pulse which is chosen to be a Gaussian pulse is spectrally centered 3meV below the hh exciton resonance and has 1 ps duration, with a intensity of 1 2 MW/cm ², similar to control pulse the detuning and the duration of the signal pulse is 1meV and 160fs and its intensity is 1% of the control pulse.

The control pulse is right circularly polarized and the linearly polarized signal pulse (say along the x axis) contains equal amounts of right(σ ⁺) and left(σ ^-) circularly polarized light. Both control and signal beams are incident almost normally to the quantum wells’ plane (along the z axis), being tilted slightly from each other. If the signal pulse pass the sample without the control pulse the polarization state of the signal pulse would be unchanged (i.e., the switch is tuned off). But if the control pulse exist, the reflectivity stop band of σ ⁺ polarization component will be partially suppressed and be accompanied by a blueshift meanwhile which of σ ^- polarization will be almost unchanged [1]. It has been shown that this phenomena should be attributed to the optical Stark effect, in the χ ⁽³⁾ regime which involves phase space filling (PSF), Hartree-Fock (HF) mean field effects and higher-order Coulomb exciton–exciton correlations. With the existence of the optical stark effect exciton resonance frequency corresponding to σ ⁺ polarization component change and then the refraction index also vary, which means that the quantum wells are no longer exactly Bragg spaced, coupling to other eigenmodes begins to occur. Consequently, the σ ⁺ and σ ^- component of the signal pulse have a different amplitude and phase. As a result, the signal polarization is rotated and becomes elliptically polarized, and allowing a portion of the signal pulse to pass the analyzer (i.e., the switch is tuned on) [1, 6].

 

Fig. 1. Experimental setup of all-optical polarization switch

Download Full Size | PDF

To calculate the response of the BSQWs to light, a transfer-matrix approach [7] has been used. The transfer matrix through one period of the structure in the basis of incoming and outgoing plane waves can be written in the form

where

is the transfer matirix through the halves of the barriers surrounding the quantum well. Here ϕ_b=ωn_bd_b/c.

The scattering of the electromagnetic wave at the interface between the quantum well and the barrier caused by the mismatch of the indices of refraction of their materials is described by

where ρ is the Fresnel reflection coefficient. Finally,

is the transfer matrix through the quantum well. Here ϕ_w=ωn_wd_w/c. The excitonic contribution to the scattering of the light is described by the function

where Γ is the nonradiative exciton damping rate of a single QW, Γ=0.6meV at 10K [5, 8], Γ₀ is the radiative damping rate of a single QW. The radiative damping rate [9] can be calculated from

where g=|ϕ ^2D _1s (r=0)|²=8/πa ² ₀ is the 2D, 1s exciton probability density of finding the electron and hole in the same location, a ₀ is the Bohr radius of the 1s exciton, d_cv is the dipole moment, E _1s is the 1s-exciton energy, ε ₀ is the permittivity of free space, n_b is the barrier refractive index.

The total transfer matrix T_N of a sequence of identical blocks can be calculated by T_N=T ^1/2 _b T^N T ^1/2 _b. Provided $T_N=\left(\begin{array}{cc}{T}_{11}& T_{12}\\ T_{21}& T_{22}\end{array}\right)$ , the total reflection of the Bragg-spaced quantum wells structure including which at the interface of air and cladding can then be calculated:

where r=-T ₂₁/T ₂₂ is the reflection coefficient, r ₀₁=-(n_b-1)/(n_b+1) is the Fresnel reflection coefficient between the air and the cladding. Using Eqs. (7) we can calculate the reflectivity stop band of the 200 period BSQW without the control pulse, as indicated in Fig. 2.

 

Fig. 2. Calculated reflectivity stop band (solid line) of the 200 period BSQWs without the control pulse. A right circularly polarized, 12 MW/cm ², 1 ps control pulse is indicated by the dashed line. A broadband signal pulse, 160 fs is indicated by the dashed-dotted line.

Download Full Size | PDF

In the presence of the control(pump) beam, the reflectivity stop band of σ ⁺ polarization component is suppressed and is accompanied by a blueshift but that of σ ^- polarization is almost unchanged on the time of the control pulse by the ac Stark effect. From Eq. (23) in Ref. [10] the shift of the exciton resonance frequency can be calculated. The excitonic contribution function (Eq. (5)) has been changed and using these calculated results in the transfer-matrix approach above the nonlinear response of the BSQWs to light is obtained.

Equations (8) show the y-polarized portion of the signal light which could be detected after the polarizer with and without control beam respectively.

where r ₊, r _- are the reflection coefficients for right(σ ⁺) and left(σ ^-) circularly polarized light respectively, n ₊, n _- are the indices for right(σ ⁺) and left(σ ^-) circularly polarized light respectively calculated from the two-dimensional (2D) susceptibility [see Eq. (7) in Ref. [9], R ₀(ω) is the reflection without the control pulse, I ₀(ω) is the intensity of the incident signal pulse, k is the wave vector. Using Eqs. (8) we can calculate the contrast ratio and the insert loss, as indicated in Figs. 3–5.

The contrast ratio of the switch is defined as the ratio of transmitted intensity after the analyzer with and without control beam. We have the definition CR=I_y/I _y0, where I _y0 denotes the y-polarized component without control pulse. I _y0 is derived from polarizer leakage which we assume to be 0.006% of the x-polarized signal pulse without control pulse. The contrast ratio is directly related with the intensity of the control pulse. Figure 3 shows the contrast ratio as a function of control intensity.

 

Fig. 3. Simulated contrast ratio as a function of control intensity (MW/cm ²) with τ=0.

Download Full Size | PDF

The contrast ratio increases initially with the control intensity, but was found to saturate at higher control intensities (>100 MW/cm ², corresponding to the maximum contrast ratio ~2000:1). It certifies that the suppression degree of the reflectivity stop band is finite, that is to say, the reflectivity stop band is not further suppressed at higher control intensities.

All numerical calculations above in which we assume the delay times between the control and signal pulse are zero, i.e. the control and signal pulse arrive in the sample simultaneously.

 

Fig. 4. Simulated contrast ratio as a function of control-signal delay (τ) for a 12 MW/cm ², 1 ps control pulse, and a 160 fs signal pulse

Download Full Size | PDF

Figure 4 shows the contrast ratio as a function of control-signal delay (τ) for a 12 MW/cm ², 1 ps control pulse, and a 160 fs signal pulse spectrally positioned as shown in Fig. 2. It is seen that the contrast ratio at its maximum is approximately ~1300:1 (31.1 dB) at zero delay.

Figure 5 shows the insert loss as a function of control-signal delay (τ). A minimum insert loss of ~13 dB is observed at zero delay.

From Fig. 4 and Fig. 5 we can see that when the delay time between the control and signal pulse is zero, the linearly polarized signal pulse has a maximum elliptical change and thus optimal switching effect can be obtained.

 

Fig. 5. Simulated insert loss as a function of control-signal delay (τ) for a 12 MW /cm ², 1 ps control pulse, and a 160 fs signal pulse

Download Full Size | PDF

3. Conclusion

To seek for faster response speed and satisfy the requirement of future high-speed optical communication, we have demonstrated the theoretical model of all-optical polarization switches based on an In _0.53 Ga _0.47 As/InP BSQW structure. After a series of numerical calculations we have obtained the graphs of the contrast ratio changing with control intensity and delay time, the insert loss changing with delay time. The switch adopting InGaAs/InP BSQW structure has a high contrast ratio (~31.1dB),a small insert loss(~13dB), and a small switching energy (~30 MW/cm ²). The contrast ratio(~40dB) and the switching energy(~14uJ/cm²) of the switching in Ref.1 have been simulated by our theory and be agreed with the theory while the values of the corresponding parameters such as 1s-exciton energy and Bohr radius in InGaAs/GaAs BSQWs are adopted. Now our theoretical research points out the potential application of In _0.53 Ga _0.47 As/InP MQWs in this all-optical polarization switching so as to be compatible with the optical communication system.