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Simulation of optical NOT gate switching by sum-frequency generation in LiNbO3 waveguides
Optics Communications 267 (2006) 187–192 www.elsevier.com/locate/optcom
Simulation of optical NOT gate switching by sum-frequency generation in LiNbO3 waveguides Junqiang Sun *, Jian Wang Wuhan National Laboratory for Optoelectronics, Institute of Optoelectronic Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, Hubei, PR China Received 12 October 2005; received in revised form 23 May 2006; accepted 26 May 2006
Abstract Optically gated switching is simulated numerically by exploiting sum-frequency generation (SFG) in periodically poled LiNbO3 waveguides. The impacts of launched control signal parameters and waveguide length on the switching extinction and speed are analyzed and discussed. The results indicate that high switching extinction and speed can be potentially achieved by choosing appropriate control signal power and its pulse width. The short LiNbO3 waveguide is suitable for enhancing the device performance. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Optical switch; Nonlinear optics; Optical waveguide
1. Introduction Signal processing in optical domain is required in future high bit-rate telecommunication networks to avoid electronics bottleneck. Optically gated switches will be important devices in high speed all-optical communication systems owing to their potential ultrafast response. Several approaches have been proposed to fulfill all-optical switching, such as using the four-wave mixing (FWM) effect in semiconductor optical amplifiers (SOAs) [1] and fibers with high nonlinearity [2], exploiting cascaded sum-frequency generation (SFG) and difference-frequency generation (DFG) [3–5], and cascaded harmonic-frequency generation (SHG) and DFG in periodically poled LiNbO3 (PPLN) waveguides [6,7]. The fiber-based FWM scheme suffers from large switching power and long interaction length due to the weak nonlinearity. The SOA-based FWM approach causes signal degradation due to the additive
*
Corresponding author. Tel.: +86 2787792242x804; fax: 86 2787792225. E-mail address: [email protected] (J. Sun).
0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.05.052
noise arising from the amplified spontaneous emission. In the cascaded SFG and DFG configurations, the switching power is quite high (on the order of several Watts), primarily because they operate far from phase matching in order to simultaneously obtain a large phase shift required for switching and flat spectral response. All-optical switching based on the two-photon absorption in micro-resonators has also manifested some advantages with weak amplified spontaneous emission noise and low switching power [8]. Negative-logic gate switching has been experimentally demonstrated by utilizing signal depletion in second-order nonlinear SFG process and shows advantages of ultrafast response, lower switching power and minimal noise accumulation [9]. But the switching properties including switch speed and signal extinction have not been analyzed and discussed in great detail. In this paper, optically gated switching based on difference-frequency generation process is simulated numerically. The influences of waveguide length, pulse width and launched peak power of the control signal on the properties of the gated switching in terms of operation speed and signal extinction are presented in detail.
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2. Theoretical model Fig. 1 shows the schematic diagram for optically gated switching based on single SFG process. A continuous wave (CW) signal (ks) and a gate control pulse (kc), whose wavelengths are both located in the 1.55 lm band, are simultaneously launched into the PPLN waveguide. Propagating along the waveguide, the control pulse will interact with the signal wave through SFG effect under the perfectly phase-matched condition and the sumfrequency (SF) pulse (wavelength ksf) is generated. The signal wave is depleted during the generation of the sumfrequency, resulting in the implementation of the logic NOT gate switching for the signal wave. This behavior can be described by the well-known coupled mode equations for three-wave mixing. The slowly varying amplitudes As, Ac, Asf of the signal, control, SF waves in the PPLN waveguide are governed by [10] oAs oAs b00 o2 As þ b0s i s ¼ ixs jsf Ac Asf expðiDbsf zÞ ð1Þ oz ot 2 ot2 oAc oAc b00 o2 Ac þ b0c i c ¼ ixc jsf As Asf expðiDbsf zÞ ð2Þ oz ot 2 ot2 oAsf oAsf b00 o2 Asf þ b0sf i sf ¼ ixsf jsf Ac As expðiDbsf zÞ ð3Þ oz ot 2 ot2 ob ðxÞ m ¼ s; c; sf ð4Þ b0m ¼ m ox x¼xm o2 bm ðxÞ b00m ¼ m ¼ s; c; sf ð5Þ ox2 x¼xm 2p ð6Þ Dbsf ¼ bsf bs bc K where bm (m = s, c, sf) are the propagation constants corresponding to the signal, control, and SF fields, respectively. b0m and b00m stand for group velocities and group-velocity dispersions for the signal frequencies xm (m = s, c, sf). Dbsf denotes the phase mismatch for the SFG process and K is the period of the periodically poled structure in PPLN waveguide. By designing an appropriate period K, perfectly phase-matched condition (Dbsf = 0) for SFG process can be satisfied. Thus, the wavelength of the SF wave is determined by the period K through the expression (6). jsf is the coupling coefficient of the SFG process and can be calculated using the following expression sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l0 jsf ¼ d eff ð7Þ cnsf nc ns S sf
where Ssf is the effective areas for SFG interaction, and deff is the effective nonlinear coefficient. nm (m = s, c, sf) are the refractive indexes for the different wavelengths km (m = s, c, sf). l0 is the permeability and c is the light velocity in vacuum. Taking into account the simple case of an exactly phasematched SFG interaction without loss and neglecting group-velocity mismatching and pulse shape distortion, Eqs. (1)–(3) can be simplified as oAs ¼ ixs jsf Ac Asf oz oAc ¼ ixc jsf As Asf oz oAsf ¼ ixsf jsf Ac As oz
ð8Þ ð9Þ ð10Þ
When the gate control field is assumed undepleted and initially much larger than the signal field, jEc0j2 jEs0j2, and the sum-frequency field is initially zero, jEsfj2 = 0, the evolutions of power in the signal and SF waves are described by [3,11] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P s ðzÞ ¼ P s0 cos2 xs xsf P c0 jsf z ð11Þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi xsf xs xsf P c0 jsf z sin2 ð12Þ P sf ðzÞ ¼ P s0 xs where Ps0 and Pc0 are the launched CW signal and control signal power, respectively. Complete depletion of the signal occurs when the argument of the trigonometric functions equals p/2. Hence, gating of the signal can be accomplished by turning the control power on and off. Physically, a SF photon is generated by combining one signal photon with one control photon, which is called complete power conversion. Therefore, it is reasonable to assume the same photon numbers for the launched signal and control powers. Thus, at the wavelengths of the precisely phase-matched condition, the initial power of the launched control pulse is selected as the following expression P c0 ¼
P s0 ks kc
ð13Þ
Under the condition of (13), Eqs. (8)–(10) can be solved analytically and the power expressions are derived with simple forms [11,12] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P s ðzÞ ¼ P s0 sec h2 xs xsf P c0 jsf z ð14Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xs xsf P c0 jsf z ð15Þ P c ðzÞ ¼ P c0 sec h2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P c0 xsf P sf ðzÞ ¼ xs xsf P c0 jsf z tanh2 ð16Þ xc
s
c
Fig. 1. Schematic illustration of all-optical gated switch based on sumfrequency generation in PPLN waveguide.
It can be found that both signal and the control powers decrease monotonically with the increase of the propagation length and complete depletion of the signal takes place when z ! 1. Therefore, the SFG process is irreversible under equal photon numbers. Furthermore, the signal extinction in the small signal approximation is larger than that under the assumption of complete power conversion.
J. Sun, J. Wang / Optics Communications 267 (2006) 187–192
In fact, the establishment of the optical gate requires an initial trigger of the control pulse. As the control pulse is injected into the waveguide, the CW signal will evolve into a pulsed signal by taking into consideration the SFG process, which gives rise to a pulsed SF wave. Therefore, the influences of the group-velocity mismatching cannot be ignored owing to the control pulse and generated SF pulse belonging to different wavelength bands. Although precisely phase-matched (QPM) condition is satisfied at the central frequencies of the picosecond optical pulses propagating in the PPLN waveguide, the potential phase mismatch will probably occur within entire frequency ranges corresponding to the picosecond optical pulses. Moreover, interactions among the pulses under the case of group-velocity mismatch will lead to walk-off effect and signal pulse distortion. The walk-off effect is usually characterized by the walk-off length through the following expression sc0 Lwalk-off ¼ 0 ð17Þ bsf b0c walk-off length Lwalk-off represents the propagation length for which the temporal walk-off between the control and SF pulses caused by the group-velocity mismatching amounts to sc0. Lwalk-off is approximately 18.3 mm for sc0 = 5 ps, kc = 1.545 lm and ksf = 0.772 lm. The phase mismatch, the group-velocity mismatch and the groupvelocity dispersion in the sum-frequency generation process are taken into consideration in the following theoretical calculations. To reveal the realistic circumstance, numerical calculations are carried out for the coupling equations (1)– (3) by using method of split-step Fourier transform [13]. Once the field amplitudes Am (m = s, c, sf) are obtained, the optical switching properties including switching speed and signal extinction can be calculated and analyzed.
189
The depletion of the signal power is an important parameter for the optically gated switching, which determines the magnitudes of the signal extinction and channel isolation in the optical networks. Here the depletion of the signal power is defined as the ratio of the signal power loss to the initial input signal power. Fig. 2 shows the changes of the signal depletion as a function of waveguide length under different launched peak powers of gate control pulse. The CW signal power is 200 mW and the control pulse width is 10 ps. The signal and control wavelengths are set at 1530 and 1558.3 nm, respectively, which satisfies the precisely phase-matched condition for the SFG process. The depletion of the signal power is increased with the waveguide length and launched control pulse power. Approximately complete (>99%) signal extinction is obtained with input control power of 300 mW (1.5Ps0ks/kc) and waveguide length of 30 mm. It is also found that further increasing the waveguide length will result in the depletion saturation. Fig. 3 illustrates the variations of the signal depletion against the waveguide length under different control pulse widths for the given input CW signal and control powers. The depletion of the signal power is increased with the launched control pulse width. The influences of control pulse width on the signal depletion are rather weak compared to Fig. 2. However, it will become obvious in the following discussions that the control pulse width will mainly impact the time response of the gated switch. The time response of the gated switch dominates the potential applications for high speed network, and here it is defined as the required time for the CW signal power falling from 95% to its minimum value. Fig. 4 plots the relationship of the response time with the waveguide length for different launched control powers with Ps0 = 200 mW and sc0 = 10 ps. The response time increases with the increase of the input control power, meaning that for a high speed switch the input power should be small. Thus, the high
3. Theoretical results and discussions
where sc0 is the full-width at half-maximum (FWHM) and Pc0 the peak power.
100 Pc0=0.5Ps0λs/λc Pc0=Ps0λs/λc Pc0=1.5Ps0λs/λc Ps0=200mW;τc0=10ps
90
Depletion of signal power (%)
In the following calculations, an experimental expression of the variations of refractive index with the signal wavelengths at room temperature for LiNbO3 is utilized [14]. The effective area of the PPLN waveguide is assumed 50 lm2. The nonlinear coefficient d33 of the PPLN waveguide is about 27 pm/V and thus the effective nonlinear coefficient (deff = 2d33/p) is approximately 17.2 pm/V. The uniform QPM grating period K is assumed 18.23 lm to meet the exactly phase-matched SFG for sum-frequency signal at the wavelength of 0.772 lm at room temperature. The case of a CW signal and a gate control pulse launched into the waveguide is considered in the numerical simulation. The launched gate control pulse is assumed with sec h2-shape and its amplitudes Ac(0, t) is expressed as pffiffiffiffiffiffiffi 1:76 Ac ð0; tÞ ¼ P c0 sec h t ð18Þ sc0
80 70 60 50 40 30 20 10 0
0
5
10
15
20
25
30
Waveguide Length (mm) Fig. 2. Variations of the signal power depletion with the waveguide length under different launched powers of the control pulse.
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26
100 τc0=5ps τc0=10ps τc0=15ps
80
24 22
Ps0=200mW;Pc0=Ps0λs/λc
70
Response time (ps)
Depletion of signal power (%)
90
60 50 40 30
18 16 14 12
8
10
6
0
5
10 15 20 Waveguide Length (mm)
25
30
Fig. 3. Changes of signal power depletion as a function of the waveguide length for different widths of the launched control pulse.
20 18 16 14 12
Pc0=0.5Ps0λs/λc Pc0=Ps0λs/λc Pc0=1.5Ps0λs/λc Ps0=200mW;τc0=10ps
10 8 6
5
10
15
20
25
30
Waveguide length (mm) Fig. 4. Variations of switch response time with the waveguide length under different launched powers of the control pulse.
speed operation and large signal extinction for the launched control pulse power are conflicted with each other for this kind of gated switch. This implies that there is a trade-off between the switching speed and extinction in choosing the operation parameters. Fig. 5 shows the changes of the response time against the waveguide length for the different control pulse width with Ps0 = 200 mW and Pc0 = 300 mW. Evidently, the response time of the gated switch is determined by the control pulse width. It is also observed that the temporal response window is usually wider than the control pulse width. From Figs. 4 and 5, it can be seen that the response time increases with the waveguide length. Therefore, the shorter waveguide is beneficial for high speed gated switch. To clearly display the gate evolution within the waveguide, the CW signal, con-
τc0=5ps τc0=10ps τc0=15ps
10
20
0
Response time (ps)
20
4
Ps0=200mW;Pc0=Ps0λs/λc 5
10
15 20 25 Waveguide length (mm)
30
Fig. 5. Changes of the switch response time against the waveguide length for different widths of the launched control pulse.
trol, and sum-frequency waves propagating along the waveguide are simulated and shown in Figs. 6(a)–(c), respectively. The operation parameters are the same as in Fig. 2. As mentioned above, the initial CW signal wave gradually evolves into a pulsed gate and leading edge of the pulse overlapping with the control pulse is decreased to generate the sum-frequency pulse. Owing to walk-off caused by the group-velocity mismatching between the control signal and the sum-frequency waves, the sum-frequency pulse will be converted back into the gate signal and control pulse along with its propagation in the waveguide, which gives rise to pulse splitting in control wave and overshoot in the gated signal. It is found that this behavior is reciprocal and reversible among the signal, control and sum-frequency waves. The reverse conversion phenomenon can be ascribed to the fact that according to the expressions (11) and (12) the sum-frequency generation is a periodic process as long as the input optical powers do not satisfy the complete power conversion relationship. As a pulsed control signal is introduced in the optical switching, the walk-off effect will inevitably occur. As a result, the complete power conversion relationship cannot be reached, leading to the reverse conversion of the optical pulse. In the first section of the waveguide, the sum-frequency optical power is increased and the control signal pulse and incident CW signal are decreased, giving rise to the generation of the NOT gate. During this SFG interaction, the walk-off effect between the pulsed control signal and the generated SF pulse is taken place. With the increase of the propagation distance, the control signal power will be decreased to the lowest level and there is no control signal power available for SFG. However, the CW signal power is still strong beside the gated pulse, and in turn difference-frequency generation (DFG) occurs between the SF signal and the CW signal. Therefore, the optical power is reversely transferred from the SF signal to the control signal pulse. Owing
J. Sun, J. Wang / Optics Communications 267 (2006) 187–192
191
(a)
350
Optical power (mW)
300 250 200 150 100 50 0 50
40
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50 10
Time 0 -10 (ps) -20 -30
40 30 20
-40
10 -50
0
e eguid Wa v
m) h (m l engt
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Optical power (mW)
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Time 0 -10 (ps) -20 -30
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40
0
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e e guid Wa v
h lengt
) ( mm
Fig. 6. Optical pulse evolution along the PPLN waveguide: (a) signal wave; (b) control pulse; and (c) sum-frequency pulse. The operation parameters are Ps0 = 200 mW, Pc0 = Ps0ks/kc, and sc0 = 10 ps.
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to the walk-off effect accompanying with SFG, there is a frequency offset between the generated and initial control signal pulses. If the residual control power exists after SFG interaction, the reversible behavior will be enhanced, which results in a larger overshoot in the gated signal. This overshoot can be regarded as an optical crosstalk to the adjacent channel signal in optical time division multiplexed (OTDM) communication systems. To suppress the overshoot, the PPLN waveguide with shorter length should be used for designing the SFG-based gated switch. The assumption of the lossless waveguide in our calculation is reasonable. Although the previous results [15] have indicated that the propagation loss leads to the decrease of the conversion efficiency, the propagation loss may not be taken into account when the wavelength length is less than 50 mm. As the PPLN waveguide with the waveguide length less than 30 mm is employed, we find from our results that the depletion ratio of the CW signal can reach more than 90% (see Fig. 2). Our simulation results also reveal that the shorter waveguide is utilized the higher speed optical switching can be realized. In addition, the optical pulses will be distortion when the propagation length is beyond 30 mm (see Fig. 6), meaning that the property of the optical switching will be degraded. Therefore, to achieve high speed NOT gate the waveguide length should be restricted less than 30 mm. Tolerances in the wavelength variations are of important, which dominate the tunable characteristics of the NOT gate. The variable range of the SF wavelength is only 1 nm, determined by the structure parameters of the PPLN waveguide. But it can be changed by heating the PPLN waveguide or by designing a special grating style to broaden the SF wavelength bandwidth [16]. Although tolerances in SF wavelength variation are limited, the tunable property of the NOT gate is not impaired. For a given CW signal, the control signal wavelength can be adjusted to meet the perfectly phase-matched condition. Thus, the tunable NOT gate can be achieved as long as the control signal wavelength is allowed to be varied within a broad wavelength region. It has been proved that the control signal wavelength can be changed within 70 nm [12], meaning that the tunable range of the NOT gate is about 70 nm. 4. Conclusion An optically gated switch based on SFG in PPLN waveguides is simulated numerically. Theoretical analyses
are carried out for the influences of waveguide length and launched control pulse parameters on the performance of the gated switch in terms of the signal extinction and the time response. The results indicate that the launched control power mainly affects the signal depletion and the time response is determined by the control pulse width. Furthermore, there is a trade-off between the switching speed and extinction in optimizing the specific device parameters. The PPLN waveguide with shorter length and proper control signal power is suitable for high speed and powerful all-optical gate applications. Acknowledgement This work was supported by the the National Natural Science Foundation of China under Grant No. 60177015. References [1] D.F. Geraghty, R.B. Lee, K.J. Vahala, M. Verdiell, M. Ziari, A. Mathur, IEEE Photon. Technol. Lett. 9 (4) (1997) 452. [2] S. Bigo, O. Lerclerc, E. Desurvire, IEEE J. Sel. Top. Quant. Electron. 3 (5) (1997) 208. [3] I. Yokohama, M. Asobe, A. Yokoo, H. Itoh, T. Kaino, J. Opt. Soc. Am. B 14 (12) (1997) 3368. [4] H. Kanbara, H. Itoh, M. Asobe, K. Noguchi, H. Miyazawa, T. Yanagawa, I. Yokohama, IEEE Photon. Technol. Lett. 11 (3) (1999) 328. [5] G.S. Kanter, P. Kumar, K.R. Parameswaran, M.M. Fejer, IEEE Photon. Technol. Lett. 13 (4) (2001) 341. [6] Y. Fukuchi, T. Sakamoto, K. Taira, K. Kikuchi, D. Kunimatsu, A. Suzuki, H. Ito, IEEE Photon. Technol. Lett. 14 (9) (2002) 1267. [7] Y. Fukuchi, K. Kikuchi, IEEE Photon. Technol. Lett. 14 (10) (2002) 1409. [8] V. Van, T.A. Ibrahim, K. Ritter, P.P. Absil, F.G. Johnson, R. Grover, J. Goldhar, P.-T. Ho, IEEE Photon. Technol. Lett. 14 (1) (2002) 74. [9] K.R. Parameswaran, M. Fujimura, M.H. Chou, M.M. Fejer, IEEE Photon. Technol. Lett. 12 (6) (2000) 654. [10] B. Chen, C.Q. Xu, IEEE J. Quant. Electron. 40 (3) (2004) 256. [11] T. Suhara, M. Fujimura, Waveguide Nonlinear-Optic Devices, Springer, Berlin, 2003. [12] S. Yu, W. Gu, IEEE J. Quant. Electron. 40 (11) (2004) 1548. [13] G.P. Agrawal, Nonlinear Fiber Optics, third ed., Academic Press, San Diego, 2001. [14] M.V. Hobden, J. Warner, Phys. Lett. 22 (3) (1966) 243. [15] X. Liu, H. Zhang, Y. Guo, Y. Li, IEEE J. Quant. Electron. 38 (9) (2002) 1225. [16] Wei Liu, Junqiang Sun, J. Kurz, Opt. Commun. 216 (1–3) (2003) 239.
Simulation of optical NOT gate switching by sum-frequency generation in LiNbO3 waveguides Junqiang Sun *, Jian Wang Wuhan National Laboratory for Optoelectronics, Institute of Optoelectronic Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, Hubei, PR China Received 12 October 2005; received in revised form 23 May 2006; accepted 26 May 2006
Abstract Optically gated switching is simulated numerically by exploiting sum-frequency generation (SFG) in periodically poled LiNbO3 waveguides. The impacts of launched control signal parameters and waveguide length on the switching extinction and speed are analyzed and discussed. The results indicate that high switching extinction and speed can be potentially achieved by choosing appropriate control signal power and its pulse width. The short LiNbO3 waveguide is suitable for enhancing the device performance. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Optical switch; Nonlinear optics; Optical waveguide
1. Introduction Signal processing in optical domain is required in future high bit-rate telecommunication networks to avoid electronics bottleneck. Optically gated switches will be important devices in high speed all-optical communication systems owing to their potential ultrafast response. Several approaches have been proposed to fulfill all-optical switching, such as using the four-wave mixing (FWM) effect in semiconductor optical amplifiers (SOAs) [1] and fibers with high nonlinearity [2], exploiting cascaded sum-frequency generation (SFG) and difference-frequency generation (DFG) [3–5], and cascaded harmonic-frequency generation (SHG) and DFG in periodically poled LiNbO3 (PPLN) waveguides [6,7]. The fiber-based FWM scheme suffers from large switching power and long interaction length due to the weak nonlinearity. The SOA-based FWM approach causes signal degradation due to the additive
*
Corresponding author. Tel.: +86 2787792242x804; fax: 86 2787792225. E-mail address: [email protected] (J. Sun).
0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.05.052
noise arising from the amplified spontaneous emission. In the cascaded SFG and DFG configurations, the switching power is quite high (on the order of several Watts), primarily because they operate far from phase matching in order to simultaneously obtain a large phase shift required for switching and flat spectral response. All-optical switching based on the two-photon absorption in micro-resonators has also manifested some advantages with weak amplified spontaneous emission noise and low switching power [8]. Negative-logic gate switching has been experimentally demonstrated by utilizing signal depletion in second-order nonlinear SFG process and shows advantages of ultrafast response, lower switching power and minimal noise accumulation [9]. But the switching properties including switch speed and signal extinction have not been analyzed and discussed in great detail. In this paper, optically gated switching based on difference-frequency generation process is simulated numerically. The influences of waveguide length, pulse width and launched peak power of the control signal on the properties of the gated switching in terms of operation speed and signal extinction are presented in detail.
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2. Theoretical model Fig. 1 shows the schematic diagram for optically gated switching based on single SFG process. A continuous wave (CW) signal (ks) and a gate control pulse (kc), whose wavelengths are both located in the 1.55 lm band, are simultaneously launched into the PPLN waveguide. Propagating along the waveguide, the control pulse will interact with the signal wave through SFG effect under the perfectly phase-matched condition and the sumfrequency (SF) pulse (wavelength ksf) is generated. The signal wave is depleted during the generation of the sumfrequency, resulting in the implementation of the logic NOT gate switching for the signal wave. This behavior can be described by the well-known coupled mode equations for three-wave mixing. The slowly varying amplitudes As, Ac, Asf of the signal, control, SF waves in the PPLN waveguide are governed by [10] oAs oAs b00 o2 As þ b0s i s ¼ ixs jsf Ac Asf expðiDbsf zÞ ð1Þ oz ot 2 ot2 oAc oAc b00 o2 Ac þ b0c i c ¼ ixc jsf As Asf expðiDbsf zÞ ð2Þ oz ot 2 ot2 oAsf oAsf b00 o2 Asf þ b0sf i sf ¼ ixsf jsf Ac As expðiDbsf zÞ ð3Þ oz ot 2 ot2 ob ðxÞ m ¼ s; c; sf ð4Þ b0m ¼ m ox x¼xm o2 bm ðxÞ b00m ¼ m ¼ s; c; sf ð5Þ ox2 x¼xm 2p ð6Þ Dbsf ¼ bsf bs bc K where bm (m = s, c, sf) are the propagation constants corresponding to the signal, control, and SF fields, respectively. b0m and b00m stand for group velocities and group-velocity dispersions for the signal frequencies xm (m = s, c, sf). Dbsf denotes the phase mismatch for the SFG process and K is the period of the periodically poled structure in PPLN waveguide. By designing an appropriate period K, perfectly phase-matched condition (Dbsf = 0) for SFG process can be satisfied. Thus, the wavelength of the SF wave is determined by the period K through the expression (6). jsf is the coupling coefficient of the SFG process and can be calculated using the following expression sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l0 jsf ¼ d eff ð7Þ cnsf nc ns S sf
where Ssf is the effective areas for SFG interaction, and deff is the effective nonlinear coefficient. nm (m = s, c, sf) are the refractive indexes for the different wavelengths km (m = s, c, sf). l0 is the permeability and c is the light velocity in vacuum. Taking into account the simple case of an exactly phasematched SFG interaction without loss and neglecting group-velocity mismatching and pulse shape distortion, Eqs. (1)–(3) can be simplified as oAs ¼ ixs jsf Ac Asf oz oAc ¼ ixc jsf As Asf oz oAsf ¼ ixsf jsf Ac As oz
ð8Þ ð9Þ ð10Þ
When the gate control field is assumed undepleted and initially much larger than the signal field, jEc0j2 jEs0j2, and the sum-frequency field is initially zero, jEsfj2 = 0, the evolutions of power in the signal and SF waves are described by [3,11] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P s ðzÞ ¼ P s0 cos2 xs xsf P c0 jsf z ð11Þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi xsf xs xsf P c0 jsf z sin2 ð12Þ P sf ðzÞ ¼ P s0 xs where Ps0 and Pc0 are the launched CW signal and control signal power, respectively. Complete depletion of the signal occurs when the argument of the trigonometric functions equals p/2. Hence, gating of the signal can be accomplished by turning the control power on and off. Physically, a SF photon is generated by combining one signal photon with one control photon, which is called complete power conversion. Therefore, it is reasonable to assume the same photon numbers for the launched signal and control powers. Thus, at the wavelengths of the precisely phase-matched condition, the initial power of the launched control pulse is selected as the following expression P c0 ¼
P s0 ks kc
ð13Þ
Under the condition of (13), Eqs. (8)–(10) can be solved analytically and the power expressions are derived with simple forms [11,12] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P s ðzÞ ¼ P s0 sec h2 xs xsf P c0 jsf z ð14Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xs xsf P c0 jsf z ð15Þ P c ðzÞ ¼ P c0 sec h2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P c0 xsf P sf ðzÞ ¼ xs xsf P c0 jsf z tanh2 ð16Þ xc
s
c
Fig. 1. Schematic illustration of all-optical gated switch based on sumfrequency generation in PPLN waveguide.
It can be found that both signal and the control powers decrease monotonically with the increase of the propagation length and complete depletion of the signal takes place when z ! 1. Therefore, the SFG process is irreversible under equal photon numbers. Furthermore, the signal extinction in the small signal approximation is larger than that under the assumption of complete power conversion.
J. Sun, J. Wang / Optics Communications 267 (2006) 187–192
In fact, the establishment of the optical gate requires an initial trigger of the control pulse. As the control pulse is injected into the waveguide, the CW signal will evolve into a pulsed signal by taking into consideration the SFG process, which gives rise to a pulsed SF wave. Therefore, the influences of the group-velocity mismatching cannot be ignored owing to the control pulse and generated SF pulse belonging to different wavelength bands. Although precisely phase-matched (QPM) condition is satisfied at the central frequencies of the picosecond optical pulses propagating in the PPLN waveguide, the potential phase mismatch will probably occur within entire frequency ranges corresponding to the picosecond optical pulses. Moreover, interactions among the pulses under the case of group-velocity mismatch will lead to walk-off effect and signal pulse distortion. The walk-off effect is usually characterized by the walk-off length through the following expression sc0 Lwalk-off ¼ 0 ð17Þ bsf b0c walk-off length Lwalk-off represents the propagation length for which the temporal walk-off between the control and SF pulses caused by the group-velocity mismatching amounts to sc0. Lwalk-off is approximately 18.3 mm for sc0 = 5 ps, kc = 1.545 lm and ksf = 0.772 lm. The phase mismatch, the group-velocity mismatch and the groupvelocity dispersion in the sum-frequency generation process are taken into consideration in the following theoretical calculations. To reveal the realistic circumstance, numerical calculations are carried out for the coupling equations (1)– (3) by using method of split-step Fourier transform [13]. Once the field amplitudes Am (m = s, c, sf) are obtained, the optical switching properties including switching speed and signal extinction can be calculated and analyzed.
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The depletion of the signal power is an important parameter for the optically gated switching, which determines the magnitudes of the signal extinction and channel isolation in the optical networks. Here the depletion of the signal power is defined as the ratio of the signal power loss to the initial input signal power. Fig. 2 shows the changes of the signal depletion as a function of waveguide length under different launched peak powers of gate control pulse. The CW signal power is 200 mW and the control pulse width is 10 ps. The signal and control wavelengths are set at 1530 and 1558.3 nm, respectively, which satisfies the precisely phase-matched condition for the SFG process. The depletion of the signal power is increased with the waveguide length and launched control pulse power. Approximately complete (>99%) signal extinction is obtained with input control power of 300 mW (1.5Ps0ks/kc) and waveguide length of 30 mm. It is also found that further increasing the waveguide length will result in the depletion saturation. Fig. 3 illustrates the variations of the signal depletion against the waveguide length under different control pulse widths for the given input CW signal and control powers. The depletion of the signal power is increased with the launched control pulse width. The influences of control pulse width on the signal depletion are rather weak compared to Fig. 2. However, it will become obvious in the following discussions that the control pulse width will mainly impact the time response of the gated switch. The time response of the gated switch dominates the potential applications for high speed network, and here it is defined as the required time for the CW signal power falling from 95% to its minimum value. Fig. 4 plots the relationship of the response time with the waveguide length for different launched control powers with Ps0 = 200 mW and sc0 = 10 ps. The response time increases with the increase of the input control power, meaning that for a high speed switch the input power should be small. Thus, the high
3. Theoretical results and discussions
where sc0 is the full-width at half-maximum (FWHM) and Pc0 the peak power.
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In the following calculations, an experimental expression of the variations of refractive index with the signal wavelengths at room temperature for LiNbO3 is utilized [14]. The effective area of the PPLN waveguide is assumed 50 lm2. The nonlinear coefficient d33 of the PPLN waveguide is about 27 pm/V and thus the effective nonlinear coefficient (deff = 2d33/p) is approximately 17.2 pm/V. The uniform QPM grating period K is assumed 18.23 lm to meet the exactly phase-matched SFG for sum-frequency signal at the wavelength of 0.772 lm at room temperature. The case of a CW signal and a gate control pulse launched into the waveguide is considered in the numerical simulation. The launched gate control pulse is assumed with sec h2-shape and its amplitudes Ac(0, t) is expressed as pffiffiffiffiffiffiffi 1:76 Ac ð0; tÞ ¼ P c0 sec h t ð18Þ sc0
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speed operation and large signal extinction for the launched control pulse power are conflicted with each other for this kind of gated switch. This implies that there is a trade-off between the switching speed and extinction in choosing the operation parameters. Fig. 5 shows the changes of the response time against the waveguide length for the different control pulse width with Ps0 = 200 mW and Pc0 = 300 mW. Evidently, the response time of the gated switch is determined by the control pulse width. It is also observed that the temporal response window is usually wider than the control pulse width. From Figs. 4 and 5, it can be seen that the response time increases with the waveguide length. Therefore, the shorter waveguide is beneficial for high speed gated switch. To clearly display the gate evolution within the waveguide, the CW signal, con-
τc0=5ps τc0=10ps τc0=15ps
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Fig. 5. Changes of the switch response time against the waveguide length for different widths of the launched control pulse.
trol, and sum-frequency waves propagating along the waveguide are simulated and shown in Figs. 6(a)–(c), respectively. The operation parameters are the same as in Fig. 2. As mentioned above, the initial CW signal wave gradually evolves into a pulsed gate and leading edge of the pulse overlapping with the control pulse is decreased to generate the sum-frequency pulse. Owing to walk-off caused by the group-velocity mismatching between the control signal and the sum-frequency waves, the sum-frequency pulse will be converted back into the gate signal and control pulse along with its propagation in the waveguide, which gives rise to pulse splitting in control wave and overshoot in the gated signal. It is found that this behavior is reciprocal and reversible among the signal, control and sum-frequency waves. The reverse conversion phenomenon can be ascribed to the fact that according to the expressions (11) and (12) the sum-frequency generation is a periodic process as long as the input optical powers do not satisfy the complete power conversion relationship. As a pulsed control signal is introduced in the optical switching, the walk-off effect will inevitably occur. As a result, the complete power conversion relationship cannot be reached, leading to the reverse conversion of the optical pulse. In the first section of the waveguide, the sum-frequency optical power is increased and the control signal pulse and incident CW signal are decreased, giving rise to the generation of the NOT gate. During this SFG interaction, the walk-off effect between the pulsed control signal and the generated SF pulse is taken place. With the increase of the propagation distance, the control signal power will be decreased to the lowest level and there is no control signal power available for SFG. However, the CW signal power is still strong beside the gated pulse, and in turn difference-frequency generation (DFG) occurs between the SF signal and the CW signal. Therefore, the optical power is reversely transferred from the SF signal to the control signal pulse. Owing
J. Sun, J. Wang / Optics Communications 267 (2006) 187–192
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Fig. 6. Optical pulse evolution along the PPLN waveguide: (a) signal wave; (b) control pulse; and (c) sum-frequency pulse. The operation parameters are Ps0 = 200 mW, Pc0 = Ps0ks/kc, and sc0 = 10 ps.
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to the walk-off effect accompanying with SFG, there is a frequency offset between the generated and initial control signal pulses. If the residual control power exists after SFG interaction, the reversible behavior will be enhanced, which results in a larger overshoot in the gated signal. This overshoot can be regarded as an optical crosstalk to the adjacent channel signal in optical time division multiplexed (OTDM) communication systems. To suppress the overshoot, the PPLN waveguide with shorter length should be used for designing the SFG-based gated switch. The assumption of the lossless waveguide in our calculation is reasonable. Although the previous results [15] have indicated that the propagation loss leads to the decrease of the conversion efficiency, the propagation loss may not be taken into account when the wavelength length is less than 50 mm. As the PPLN waveguide with the waveguide length less than 30 mm is employed, we find from our results that the depletion ratio of the CW signal can reach more than 90% (see Fig. 2). Our simulation results also reveal that the shorter waveguide is utilized the higher speed optical switching can be realized. In addition, the optical pulses will be distortion when the propagation length is beyond 30 mm (see Fig. 6), meaning that the property of the optical switching will be degraded. Therefore, to achieve high speed NOT gate the waveguide length should be restricted less than 30 mm. Tolerances in the wavelength variations are of important, which dominate the tunable characteristics of the NOT gate. The variable range of the SF wavelength is only 1 nm, determined by the structure parameters of the PPLN waveguide. But it can be changed by heating the PPLN waveguide or by designing a special grating style to broaden the SF wavelength bandwidth [16]. Although tolerances in SF wavelength variation are limited, the tunable property of the NOT gate is not impaired. For a given CW signal, the control signal wavelength can be adjusted to meet the perfectly phase-matched condition. Thus, the tunable NOT gate can be achieved as long as the control signal wavelength is allowed to be varied within a broad wavelength region. It has been proved that the control signal wavelength can be changed within 70 nm [12], meaning that the tunable range of the NOT gate is about 70 nm. 4. Conclusion An optically gated switch based on SFG in PPLN waveguides is simulated numerically. Theoretical analyses
are carried out for the influences of waveguide length and launched control pulse parameters on the performance of the gated switch in terms of the signal extinction and the time response. The results indicate that the launched control power mainly affects the signal depletion and the time response is determined by the control pulse width. Furthermore, there is a trade-off between the switching speed and extinction in optimizing the specific device parameters. The PPLN waveguide with shorter length and proper control signal power is suitable for high speed and powerful all-optical gate applications. Acknowledgement This work was supported by the the National Natural Science Foundation of China under Grant No. 60177015. References [1] D.F. Geraghty, R.B. Lee, K.J. Vahala, M. Verdiell, M. Ziari, A. Mathur, IEEE Photon. Technol. Lett. 9 (4) (1997) 452. [2] S. Bigo, O. Lerclerc, E. Desurvire, IEEE J. Sel. Top. Quant. Electron. 3 (5) (1997) 208. [3] I. Yokohama, M. Asobe, A. Yokoo, H. Itoh, T. Kaino, J. Opt. Soc. Am. B 14 (12) (1997) 3368. [4] H. Kanbara, H. Itoh, M. Asobe, K. Noguchi, H. Miyazawa, T. Yanagawa, I. Yokohama, IEEE Photon. Technol. Lett. 11 (3) (1999) 328. [5] G.S. Kanter, P. Kumar, K.R. Parameswaran, M.M. Fejer, IEEE Photon. Technol. Lett. 13 (4) (2001) 341. [6] Y. Fukuchi, T. Sakamoto, K. Taira, K. Kikuchi, D. Kunimatsu, A. Suzuki, H. Ito, IEEE Photon. Technol. Lett. 14 (9) (2002) 1267. [7] Y. Fukuchi, K. Kikuchi, IEEE Photon. Technol. Lett. 14 (10) (2002) 1409. [8] V. Van, T.A. Ibrahim, K. Ritter, P.P. Absil, F.G. Johnson, R. Grover, J. Goldhar, P.-T. Ho, IEEE Photon. Technol. Lett. 14 (1) (2002) 74. [9] K.R. Parameswaran, M. Fujimura, M.H. Chou, M.M. Fejer, IEEE Photon. Technol. Lett. 12 (6) (2000) 654. [10] B. Chen, C.Q. Xu, IEEE J. Quant. Electron. 40 (3) (2004) 256. [11] T. Suhara, M. Fujimura, Waveguide Nonlinear-Optic Devices, Springer, Berlin, 2003. [12] S. Yu, W. Gu, IEEE J. Quant. Electron. 40 (11) (2004) 1548. [13] G.P. Agrawal, Nonlinear Fiber Optics, third ed., Academic Press, San Diego, 2001. [14] M.V. Hobden, J. Warner, Phys. Lett. 22 (3) (1966) 243. [15] X. Liu, H. Zhang, Y. Guo, Y. Li, IEEE J. Quant. Electron. 38 (9) (2002) 1225. [16] Wei Liu, Junqiang Sun, J. Kurz, Opt. Commun. 216 (1–3) (2003) 239.