Critical conditions of control light for a silicon-based photonic crystal bistable switching

Chao Li; Hong Wang; Jun-Fang Wu; Wen-Cheng Xu

doi:10.1364/OE.18.017313

1. Introduction

In the last decade, photonic crystal (PC) bistable switching has attracted many attentions for its great potential in all-optical signal processing [1–3]. Commonly, to observe the bistable switching action, one should perform up- and downsweeping power cycles by coupling a continuous wave (CW) signal light into the PC waveguide. Alternatively, this bistable action can also realized under pulsed injection of the control light when the power of the CW signal light is fixed at a certain value within the bistable region [3–5]. Obviously, from a practical point of view, the latter method is more feasible. However, what is the critical condition of the control light (i.e., the minimum values of the pump power and pump time of the control light to trigger a bistable switching to the “on” state)? So far, this question is rarely studied. Of course, one might obtain these critical values by experiments or simulations after many times of attempts. However, it is time consuming and the achieved results cannot be generalized. Therefore, a theoretical investigation on the critical pump conditions becomes very important in PC switching design work.

To date, almost all of the proposed control-light sources are high-peak-power Gaussian pulses [3–6]. Unfortunately, it is very difficult to obtain an analytic solution by this way. In this paper, we present a new trigger method: using a continuous wave (CW) source with suitable launching time to act as control light. We will show this new method is more efficient for the theoretical study on the critical pump conditions, and the achieved results can be readily generalized to the case of pulsed control light. To verify the theoretical predictions, nonlinear finite-difference time-domain (FDTD) method is employed to make simulations. The paper is organized as follows. In section 2, we establish a time-dependent evolution equation to describe the PC bistable switching under the pump of a CW signal light superposed with a CW control light. Based on this model, the critical conditions of the pump power and pump time of the control light are derived in sections 3 and 4, respectively, and FDTD simulations are performed to make verifications. Subsequently, in section 5, these critical conditions for single-beam-CW control light are generalized to the cases of multiple-beam-CW control light and pulsed control light. Finally, a brief summary is given in section 6.

2. Time-dependent evolution equation for a PC bistable switching: using a single-beam-CW control light

As proposed in Refs [7]. and [8], two-photon absorption (TPA) and Kerr effect are two dominant nonlinear factors in a Si system. In our previous work [9], we have established a simple model to study the influence of TPA on the characteristics of bistable switching, and the threshold power (up- and down-jump) of the signal light can be precisely calculated based on this model. Now, we will fix the power of the signal light, and further investigate the critical conditions of the control light.

To describe conveniently, we depict a typical structure of PC bistable switching in Fig. 1 , which is consist of one nonlinear microcavity with two waveguides (WG) connected to the two ends. Suppose the decay rates of the input WG (WG1) and output WG (WG2) are γ ₁ and γ ₂, respectively, and the intrinsic decay rate of the nonlinear cavity is γ _C. Phenomenologically, γ _C can be written as γ _C = γ ₀ + γ _TPA, where γ ₀ is the linear intrinsic decay rate, and γ _TPA is the decay rate correlated with TPA. Therefore, the total decay rate is γ = γ ₁ + γ ₂ + γ ₀ + γ _TPA.

Fig. 1 Sketch of a typical bistable switching composed of a resonator cavity coupled to both input and output WGs.

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For simplicity, we suppose the CW control light has the same frequency ω and initial phase (set to be zero in our case) with the CW signal light. When the signal light superposed with the control light is launched into the nonlinear cavity along WG1, according to the coupled mode theory (CMT) [10], the mode amplitude A can be written as

{\begin{cases} \frac{d A}{d t} = [j (ω_{0} - γ \frac{{| s_{T} |}^{2}}{p_{0}}) - γ] A + \sqrt{2 γ_{1}} (s_{s} + s_{x}) \\ γ = γ_{0} + γ_{1} + γ_{2} + γ_{T P A} \\ s_{T} = \sqrt{2 γ_{2}} A \\ s_{s} + s_{x} + s_{R} = \sqrt{2 γ_{1}} A \end{cases},

where s _s, s _x, s _T and s _R are the field amplitudes of the incident signal light, control light, transmitted light and reflective light, respectively, ω ₀ is the resonant frequency of the PC cavity, and p ₀ is the characteristic power [4]. Noticing that γ _TPA is proportional to the intensity of energy stored in the PC cavity, γ _TPA can be expressed as [9]

γ_{T P A} = k γ_{1} {| s_{T} |}^{2},

where k is defined as TPA coefficient which denotes the strength of the TPA process, and can be calculated by theory or simulation [9].

Obviously, it is very difficult to solve Eq. (1) directly. However, if the power of the incident light is not very high, we can utilize a perturbation technology and obtain an approximate solution (suppose there is not any photon energy stored in the PC cavity at the initial moment)

A = \frac{\sqrt{2 γ_{1}} (| s_{s} | + | s_{x} |)}{i (ω - ω_{0} + γ {| s_{T} |}^{2} / p_{0}) + γ} [e^{i ω t} - e^{- γ t} \cdot e^{i ω_{0} t}]

and

S_{T} = \frac{\sqrt{4 γ_{1} γ_{2}} (| s_{s} | + | s_{x} |)}{i (ω - ω_{0} + γ {| s_{T} |}^{2} / p_{0}) + γ} (e^{i ω t} - e^{- γ t} \cdot e^{i ω_{0} t}),

where ω is the frequency of the incident light. Thus, the instant values of the energy stored inside the cavity and the transmission rate can be easily calculated by using Eqs. (3) and (4)

| A |^{2} (t) = \frac{{(\sqrt{p_{s}} + \sqrt{p_{x}})}^{2} / 2 γ_{2}}{{(1 + k η {| s_{T} |}^{2} / 2)}^{2}} \cdot \frac{η}{{(δ - {| s_{T} |}^{2} / p_{0})}^{2} + 1} [e^{- 2 γ t} - 2 e^{- γ t} \cos (ω_{0} - ω) t + 1],

T (t) = \frac{| s_{T} |^{2}}{| s_{s} |^{2}} = \frac{{(\sqrt{p_{s}} + \sqrt{p_{x}})}^{2} / p_{s}}{{(1 + k η {| s_{T} |}^{2} / 2)}^{2}} \cdot \frac{η}{{(δ - {| s_{T} |}^{2} / p_{0})}^{2} + 1} [e^{- 2 γ t} - 2 e^{- γ t} \cos (ω_{0} - ω) t + 1],

where

p_{s} = {| s_{s} |}^{2}

and

p_{x} = {| s_{x} |}^{2}

are the incident power of the signal light and control light, respectively;

η = 4 γ_{1} γ_{2} / {(γ_{1} + γ_{2} + γ_{0})}^{2}

is the linear peak transmission rate of the bistable switching;

δ = (ω_{0} - ω) / (γ_{1} + γ_{2} + γ_{0})

is the frequency detuning of the input CW with respect to the cavity mode.

Equations (5) and (6) are the time-dependent evolution equations for PC bistable switching, under the pump of a CW signal light superposed with a CW control light. From these equations, one can see that both the photon energy inside the cavity and the transmission rate will decay with time, and access a steady-state value ultimately.

If the control light is shut off at t ₀, then the instant photon energy stored in the cavity $| A |^{2} (t_{0})$ can be calculated by Eq. (5). Simultaneously, $| A |^{2} (t_{0})$ can be treated as the initial condition for the following evolution of the energy in cavity, and its value will determine whether the final state of the bistable switching is “on” or “off” state.

3. Critical pump-power condition of the control light

3.1 Theory

Now, let us discuss the critical pump-power condition of the control light. To describe clearly, we depict a working principle diagram of a bistable switching in Fig. 2 , where ABCDA is the bistable region and the horizontal axis indicates the power of the signal light. The curve of AB denotes the “off” state, while DC denotes the “on” state. It is well known to all, when there is only signal light being incident, the “on” and “off” states can be achieved by upward and downward sweeps of the signal power level. We define A as the down-jump point (D→A), and the corresponding incident power of the signal light is p ₁; B is the up-jump point (B→C), and the corresponding incident power of the signal light is p ₂. Alternatively, we can also fix the power of the signal light at a certain value within the bistable region, and trigger the switching to the “on” state by a control light. Here we should keep in mind that the jump from the “off” state to “on” state is indirect, e.g., the jump from A to D must pass through B and C points, although this process may be very quick.

Fig. 2 Working principle diagram of a bistable switching

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Noticing that the control light has been assumed to have the same frequency and initial phase with the signal light, we can treat them as a whole, i.e., an “equivalent signal light” with incident power of ${(\sqrt{p_{1}} + \sqrt{p_{x}})}^{2}$ . Therefore, to make the switching jump indirectly from A to D, the power of the “equivalent signal light” should be no less than p ₂. Thus we have

{(\sqrt{p_{1}} + \sqrt{p_{x}})}^{2} \geq p_{2},

i.e.,

p_{x} \geq {(\sqrt{p_{2}} - \sqrt{p_{1}})}^{2} .

Equation (8) is the critical pump-power condition of the control light to trigger a bistable switching to “on” state when the signal power is p ₁. That is to say, if p _x is less than the critical value, one will not be able to trigger the bistable switching successfully even if the pump time of the control light is long enough.

3.2 FDTD simulation: verify the theoretical predictions

To verify the above mentioned theoretical predictions, we choose to study a typical structure of 2D PC bistable switching. The structure and the corresponding parameters are the same as the ones presented in our previous work [11], except that in this case the central defect cavity is nonlinear, and the real part of the nonlinear coefficient is 1 x 10⁻⁵ μm²/W, while the imaginary part is set to be 2.5 x 10⁻⁶ μm²/W, denoting TPA effect. To measure the transmission rate, one monitor is set at the output port of WG2. The grid sizes in the horizontal and vertical directions are chosen to be a/20 (where a = 1μm is the lattice constant), and a perfectly matched layer (PML) of 1μm is employed as boundary. Thus, we can begin numerical simulation and focus on the TM modes. By employing nonlinear FDTD technique [12], it is not difficult to obtain the values of p ₁ and p ₂ (these values can also be obtained by theoretical calculations, as presented in Ref [9].).

As an example, when the frequency detuning of the incident signal light is δ = 4.3478, p ₁ and p ₂ are calculated to be 0.120 kW/μm and 0.215 kW/μm, respectively. Thus, the critical pump power of the control light at the down-jump point (p _s = p ₁) can be obtained immediately by using Eq. (8): p _x = 0.0136 kW/μm. To examine the precision of the critical value predicted by theory, we begin simulation by using p _x = 0.0136 kW/μm and p _x = 0.0134 kW/μm as the power of the CW control light, respectively, for comparison. When a CW signal light (δ = 4.3478, p _s = p ₁ = 0.120 kW/μm) superposed with the control light (which has the same frequency and initial phase with the signal light) is launched into the PC switching along WG1, the time-dependent evolution process of the transmission rate recorded by a monitor is shown in Fig. 3 (since the critical pump time keeps unknown at present,to ensure a long enough pump time in this simulation, the pump time of the control light is selected to be 4000(a/c), about 13.33ps). One can see when p _x = 0.0136 kW/μm, the bistable switching is successfully triggered to the “on” state with T = 0.6; while when p _x is slightly weakened to 0.0134 kW/μm, the switching can only works on the “off” state with T = 0.054.

Fig. 3 Evolvement curves of bistable switching for different pump power of the control light. The upper curve: p _x = 0.0136 kW/μm; The lower curve: p _x = 0.0132 kW/μm.

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In Table 1 , we make a further comparison between the theoretical predictions and FDTD simulation results, with respect to different frequency detuning, where $p_{x}^{T h e o r y}$ is calculated by Eq. (8), and $p_{x}^{F D T D}$ is obtained by FDTD simulation. One can see clearly that in a great range of frequency detuning, the theoretical predictions agree with the FDTD simulation results very well.

Table 1. Critical pump power of the control light with respect to different frequency detuning.

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4. Critical pump-time condition of the control light

In section 3, we have obtained the critical pump-power condition of the control light. However, this sole condition cannot ensure the bistable switching to be successfully triggered to an “on” state, because the decisive factor is the photon energy stored inside the cavity, which is surely related with the pump time. Now, let us make a discussion on the critical pump-time condition of the control light.

To make the switching jump indirectly from A to D, the photon energy inside the cavity before the control light is shut off should be accumulated to exceed the energy value at D point, say, ${| A |}_{D}^{2} = p_{1} T_{1} / (2 γ_{2})$ , where T ₁ is the steady-state transmission rate at D point after the control light is shut off. Thus, we obtain the critical condition of the energy stored inside the cavity for the incident signal power p ₁

{| A |}^{2} \geq p_{1} T_{1} / (2 γ_{2}) .

Obviously, Eqs. (8) and (9) can be readily generalized to the arbitrary signal power in the bistable region by substituting p ₁ with p _s.

Since the dynamic evolution process of the energy in cavity is determined by Eq. (5), thus, the critical pump-time condition of the control light can be directly derived by using Eqs. (5) and (9):

\frac{{(\sqrt{p_{s}} + \sqrt{p_{x}})}^{2}}{{(1 + k η {| s_{T} |}^{2} / 2)}^{2}} \cdot \frac{η}{{(δ - {| s_{T} |}^{2} / p_{0})}^{2} + 1} [e^{- 2 γ t} - 2 e^{- γ t} \cos (ω_{0} - ω) t + 1] \geq T_{s} p_{s},

where p_s is the incident signal power within the bistable region and T _s is the corresponding steady transmission rate of “on” state.

Specially, let us make a discussion on the critical pump time of the control light with a critical power at the down-jump point (p _s = p ₁, T _s = T ₁). In this case, we have

{(\sqrt{p_{1}} + \sqrt{p_{x}})}^{2} = p_{2} .

Substituting formula (11) into Eq. (10), we obtain

\frac{p_{2} / p_{1}}{{(1 + k η {| s_{T} |}^{2} / 2)}^{2}} \cdot \frac{η}{{(δ - {| s_{T} |}^{2} / p_{0})}^{2} + 1} [e^{- 2 γ t} - 2 e^{- γ t} \cos (ω_{0} - ω) t + 1] \geq T_{1} .

From Eq. (12) the desired critical pump time of the control light can be calculated. Noticing that the left part of Eq. (12) is actually the time-dependent transmission rate under the pump of the signal light and control light, Eq. (12) can be rewritten as a more brief form

T (t) \geq T_{1} .

Obviously, it is rather difficult to obtain an analytic solution of the critical pump time of the control light directly from Eq. (12), since both ${| s_{T} |}^{2}$ and γ are time dependent. Therefore, we have to fall back on a numerical method: plot the evolution curve of T(t) and a horizon line of T = T ₁, and the first intersection point of them reads the critical value, as shown in Fig. 4(a) (δ = 4.3478). One can see clearly that the critical pump time of the control light is t _s = 3522(a/c) [before we know this critical value, the pump time of the control light should be long enough, e.g., 4000(a/c) in our case]. To verify it, in Fig. 4(b) we make a FDTD simulation on the bistable switching when the control light is shut off at the critical value of t _s = 3522(a/c) (about 11.74ps). One can see that the bistable switching is successfully triggered to the “on” state with transmission rate of 0.6.

Fig. 4 Critical pump time of the control light for δ = 4.3478. (a) Illustration for how to obtain the critical pump time of the control light. (b) Evolvement curve of transmission when the control light is shut off at t = 3522(a/c).

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To further examine the precision of the critical pump time calculated by Eq. (12), we make a comparison between the theoretically-calculated value $t_{s}^{T h e o r y}$ and FDTD simulation result $t_{s}^{F D T D}$ in a very wide range of the frequency detuning, as shown in Table 2 , and a perfect agreement is achieved between them.

Table 2. Critical pump time of the control light with respect to different frequency detuning.

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Form Tables 1 and 2, it is also found that with the increase of the frequency detuning, the critical pump power of the control light will rise while the corresponding critical pump time is shortened.

5. Critical condition of multiple-beam-CW control light and pulsed control light

So far, we have discussed the critical conditions for single-beam-CW control light. Before the further investigation on the case of pulsed control light, we would like to continue our discussion on the multiple-beam-CW-control-light case, since it is the basis for the former, as will be seen below.

When a bistable switching is simultaneously triggered by N beams of CW control light, for convenience, we suppose these beams of control light have the same frequency and initial phase with the signal light. Therefore, the total power of the incident light is

p_{i n} = {(\sqrt{p_{s}} + \sum_{i = 1}^{N} \sqrt{p_{x_{i}}})}^{2},

where

p_{x_{i}}

denotes the power of the ith beam of control light.

The following discussion is exactly the same with the one performed for single-beam-CW control light provided that $\sqrt{p_{x}}$ is substituted by $\sum_{i = 1}^{N} \sqrt{p_{x_{i}}}$ . Thus, Eq. (8) can be directly transformed as

{(\sum_{i = 1}^{N} \sqrt{p_{x_{i}}})}^{2} \geq {(\sqrt{p_{2}} - \sqrt{p_{1}})}^{2} .

Equation (15) is the critical pump-power condition of the multiple-beam-CW control light, and the corresponding critical pump power is determined by

f (t) \exp (i ω_{0} t)

Since

{(\sum_{i = 1}^{N} \sqrt{p_{x_{i}}})}^{2} \geq \sum_{i = 1}^{N} p_{x_{i}}

, from Eq. (16) we obtain

\sum_{i = 1}^{N} p_{x_{i}} \leq {(\sqrt{p_{2}} - \sqrt{p_{1}})}^{2} .

Comparing Eq. (17) with (11), it is not difficult to deduce that under the same conditions, the critical total power of the multiple-beam-CW control light is less than the one of the single-beam-CW control light, especially for a greater N. Enlightened by this find, if the power of the control-light source in lab is unfortunate to be lower than the critical value, we still have opportunity to trigger the bistable switching by dividing the control light into several beams. Obviously, this method is promising in actual applications.

Additionally, the critical pump time of multiple-beam-CW control light can also be calculated by Eq. (12). As for the other incident signal power in bistable region, the corresponding critical conditions can also be readily obtained only by substituting p ₁with p _s in Eqs. (12) and (15).

Finally, let us make a brief discussion on the case of pulsed control light.

We suppose the pulse can be expressed as “ $f (t) \exp (i ω_{0} t)$ ”, where f(t) is the envelopment function in time domain, and ω ₀ is the central frequency. By using Fourier transformation, the pulse can be approximately regarded as a superposition of a series of CWs with frequencies near ω ₀ and appropriate weights:

f (t) \exp (i ω_{0} t) = \sum_{i} g (ω_{i}) \exp (i ω_{i} t) Δ ω,

where ω _i and g(ω _i) are the frequency and field amplitude of the ith CW component, respectively, Δω is the step width in frequency domain. Thus, we can use “|g(ω _i)|²Δω” as the corresponding weight factor, and a smaller Δω will lead to a more precise result.

Specially, if the control pulse is relatively wide (e.g., tens of ps), the frequency of each CW component is approximately equal to ω ₀. In this case, the problem can be converted to the one in the multiple-beam-CW-control-light case, which has been above discussed. While for an ultrashort control pulse (e.g, several fs), the method presented above is not a good approximation anymore (due to the further broadening of the frequency spectrum), and a more precise model including multiple-beam-CW control light with deferent frequencies is required. However, we can still predict that the critical pump power of CWs with deferent frequencies must be greater than the one with same frequency, as has been testified by our FDTD simulations.

6. Conclusion

In summary, by establishing a time-dependent evolution equation for PC bistable switching, we have derived the critical conditions of the pump power and pump time of the control light, respectively, and the theoretical predictions are precisely verified by FDTD simulations. It is found that with the increase of the frequency detuning of the incident light, the critical power of the control light will rise, while the corresponding critical time will decrease. It is also found that under the same conditions, the critical total power of the multiple-beam-CW control light is less than the one of the single-beam-CW control light. The theory and technology presented in this paper will be helpful for actual designs of PC switching.

Acknowledgments

This paper is supported by Key Technologies R&D Program of Guangdong Province, Major Research Plan (No. 2009A080301013), and Key Technologies R&D Program of Guangzhou City, Major Plan Program (No. 2009A1-D081), and National Natural Science Foundation of China (No. 60835001). Chao Li and Hong Wang contributed equally to this work.

References and links

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δ	p ₁ (kW/μm)	p ₂ (kW/μm)	$p_{x}^{T h e o r y}$ (kW/μm)	$p_{x}^{F D T D}$ (kW/μm)	$p_{x}^{T h e o r y}$ / $p_{x}^{F D T D}$
4.3478	0.120	0.215	0.0136	0.0135	1.007
8.695	0.353	1.517	0.402	0.386	1.041
13.04	0.741	5.63	2.28	2.13	1.07
17.39	1.341	14.78	7.217	7.21	1.001

δ	p ₁ (kW/μm)	p _x (kW/μm)	$t_{s}^{T h e o r y}$ (ps)	$t_{s}^{F D T D}$ (ps)	$t_{s}^{T h e o r y} / t_{s}^{F D T D}$
4.3478	0.120	0.014	11.74	11.07	1.061
8.695	0.353	0.402	5.47	5.47	1.000
13.04	0.741	2.280	2.51	2.45	1.024
17.39	1.341	7.217	1.56	1.52	1.026

δ	p ₁ (kW/μm)	p ₂ (kW/μm)	$p_{x}^{T h e o r y}$ (kW/μm)	$p_{x}^{F D T D}$ (kW/μm)	$p_{x}^{T h e o r y}$ / $p_{x}^{F D T D}$
4.3478	0.120	0.215	0.0136	0.0135	1.007
8.695	0.353	1.517	0.402	0.386	1.041
13.04	0.741	5.63	2.28	2.13	1.07
17.39	1.341	14.78	7.217	7.21	1.001

δ	p ₁ (kW/μm)	p _x (kW/μm)	$t_{s}^{T h e o r y}$ (ps)	$t_{s}^{F D T D}$ (ps)	$t_{s}^{T h e o r y} / t_{s}^{F D T D}$
4.3478	0.120	0.014	11.74	11.07	1.061
8.695	0.353	0.402	5.47	5.47	1.000
13.04	0.741	2.280	2.51	2.45	1.024
17.39	1.341	7.217	1.56	1.52	1.026

Critical conditions of control light for a silicon-based photonic crystal bistable switching

Abstract

1. Introduction

2. Time-dependent evolution equation for a PC bistable switching: using a single-beam-CW control light

3. Critical pump-power condition of the control light

3.1 Theory

3.2 FDTD simulation: verify the theoretical predictions

4. Critical pump-time condition of the control light

5. Critical condition of multiple-beam-CW control light and pulsed control light

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Tables (2)

Equations (18)

Optics Express