Abstract

We show the Cross Phase Modulation (XPM) effect between CW probe that operates in bistability region and strong Gaussian pump in a Fiber Bragg Grating (FBG) by Implicit 4th Order Runge-Kutta Method. The XPM effect results in three unique nonlinear switching behaviors of the probe transmission depending on the pump peak intensity and its Full Width Half Maximum (FWHM) value. From this observation, we offer the FBG three potential nonlinear switching applications in all-optical signal processing domain as: a step-up all-optical switching, an all-optical inverter, and an all-optical limiter. The bistability threshold that determines the nonlinear switching behaviors of probe transmission after Gaussian pump injection is defined numerically and shown to be equivalent to the unstable state inside hysteresis loop.

© 2005 Optical Society of America

1. Introduction

Fiber Bragg Grating (FBG) is a device, which exhibits a band gap around the Bragg wavelength where no light is allowed to propagate inside it [1]. For high operating intensity, FBG shows a great variety of nonlinear phenomena such as: optical Bragg Soliton and Gap Soliton, which allow the wave to propagate inside the band gap without changing its shape at a speed lesser than the speed of light by balancing grating dispersion and Kerr nonlinear effect [2, 3].

In 1979, Winful et al. demonstrated optical bistability phenomena in the nonlinear distributed feedback structures [4]. Subsequently, in 2002, Hojoon Lee et al. compared the nonlinear switching performances between uniform and phase-shifted FBG. He showed that optical bistability hysteresis loop exists for nonlinear switching curve of FBG in a certain range of input intensity and detuning. However, unstable state inside the bistability hysteresis loop (middle part of S-curve) which exists analytically was not observed by numerically due to limitation of the simulation too [5]. It is a challenge for us to investigate the unstable state inside the bistability hysteresis loop numerically and to find its physical meaning.

In 1992, C.M. De Sterke introduced Optical Push Broom (OPB). It is the nonlinear phenomena where probe energy inside grating can be swept out by a strong Gaussian pump whose wavelength is far detuned from the Bragg wavelength and therefore propagates unperturbed by the grating [6]. Providing pump pulse width is shorter than grating length and probe wave is operating in negative detuning region for uniform grating structure, it was shown that probe velocity inside the grating can be increased momentarily at every location passed by the pump. The probe energy will finally pile up in front of the pump at the output of FBG. OPB leads to optical pulse compression phenomena. The hallmark of OPB is a high and narrow peak in the probe transmission followed by a much longer dip to maintain the energy conservation law [7]. This hallmark will be useful in giving explanation for our simulation results in later section.

Combining the concept about optical bistability together with probe compression phenomena by Gaussian pump injection from the previous research works, we show that while CW probe intensity is operating in bistability region, through Cross Phase Modulation (XPM) with strong Gaussian pump inside the FBG, three unique nonlinear switching behaviors in the probe transmission can be distinguished depending on the pump peak intensity and its Full Width Half Maximum (FWHM). In other words, the pump FWHM value and its peak intensity are the crucial parameters to control the output probe transmission state to be in low or high state when it is operating in the bistability region. These results offer the FBG three potential applications in optical signal processing domain as: a step-up optical switching, an optical inverter, and an optical limiter. Our results also reveal that the bistability threshold between low and high state is equivalent to the unstable state inside hysteresis loop.

2. Background theory

The schematic of FBG with input pump and probe wave is given in Fig. 1. Unless otherwise stated, all notations in this paper follow the conventions used in [5].

 

Fig. 1. Schematic of the FBG.

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P (z, t)=P (z-Vgt) is the pump that propagates unperturbed inside the grating because its wavelength is far detuned from the Bragg wavelength.

The equation governing refractive index variation along the grating is given by:

n(z)=n¯+n1(z)×cos[2πΛz+θ(z)]+n2×E(z)2

The probe with wavelength is near detuned to the Bragg wavelength can be written as:

E(z,t)=[A+(z,t).exp(iKBz)+A(z,t).exp(iKBz)].exp(iω0t)

With pump included, we can write A+ (z, t) and A- (z, t) in (2) as [6]:

A+(z,t)=P(zVgt)exp(iδpt)+ε+(z,t)exp(iδ0t)
A(z,t)=ε(z,t)exp(iδ0t)

By substituting Eqs. (1) and (2) into Maxwell’s Wave Equations and following the assumptions used in [6] with an additional assumption that the CW probe is operating in nonlinear regime, we arrive at the following sets of Nonlinear Coupled Mode Equations (NLCMEs):

+iε+z+i1Vgε+t+δε++κε+Γ[ε+2+2ε2+2P(zVgt)2]ε+=0
+iεz+i1Vgεt+δε+κ*ε++Γ[ε2+2ε+2+2P(zVgt)2]ε=0

Just like in [5], the NLCMEs are solved by using Implicit 4th Order Runge-Kutta Method [8]. By solving the set of NLCMEs numerically, we can investigate the XPM effect between Gaussian pump and CW probe for operating intensity in bistability region as our region of interest.

The uniform grating parameters in our simulation are given as follows: grating length (L)=1 cm, =1.45, κ=500/m, n 2=2.6×10-20 m2/W, Bragg wavelength (λB)=1550 nm. The input waves are CW probe with operating intensity at |A+(0, t) |2=18 GW/cm2, coming at a detuning δ=474/m (λ=1549.875 nm) and Gaussian pump is operating at variable peak intensity with a constant FWHM at 75 ps.

3. Three unique nonlinear switching behaviors of probe transmission via Gaussian pump injection in FBG

Figure 2 shows an optical bistability hysteresis loop in the transmissivity curve of uniform FBG for CW input. The output transmissivity for CW input intensity at 18 GW/cm2 are indicated by the cross marks (X) in the figure.

 

Fig. 2. Bistability hysteresis loop in uniform FBG.

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Figure 3 shows three unique nonlinear switching behaviors in probe transmission via Gaussian pump injection with a constant FWHM at 75 ps but variable peak intensities that are categorized into three regions as low, medium, and high. We can see a compression peak followed by a much longer dip in the probe transmission to satisfy the laws of energy conservation after Gaussian pump injection. Similar result is shown in [7] to verify the validity of our results here. To explain how the probe compression phenomena lead to different nonlinear switching behavior, three lines of the output probe transmission are plotted on the same axis in Fig. 4. The plot is zoomed into part just after the probe compression peak, where the dip is located. Bistability threshold at 4.6 GW/cm2 (25.5% of probe input intensity) is drawn. At time when laws of energy conservation is satisfied (energy accumulation in compression peak is exactly compensated by energy reduction in the dip), the probe transmission in the system has to decide whether to go to high or low state. If the dip after satisfying the energy conservation law can recover above the threshold, the probe transmission goes up through an overdamped transient response before it finally settles at high state, or else it goes down to low state. After understanding how the probe transmission behaves after Gaussian pump injection, we will discuss how the three unique nonlinear switching occurs in each of three pump peak intensity regions.

 

Fig. 3. Three unique nonlinear switching behaviors of probe transmission via Gaussian pump injection in FBG.

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Fig. 4. Bistability threshold inside the hysteresis loop at 4.6 GW/cm2.

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Below low region, there is no switching in the output probe transmission. In low region (45 GW/cm2<pump peak intensity ≤55 GW/cm2), the probe compression peak is weak due to the XPM with relatively weak pump. To maintain the energy conservation law, the dip is therefore must be shallow to balance the weak compression peak. After recovery process, the dip is always above the threshold. Therefore, the output probe transmissions always settle at high state in the bistability region. One potential application of the FBG in low region is as a step-up all-optical switching. In medium region (55 GW/cm2<pump peak intensity ≤97.5 GW/cm2), when input is at low state, the compression peak is not high enough to cause the dip falling below the threshold. Hence the output settles at high state. However, when input is at high state, the compression peak is correspondingly high due to strong XPM with the pump. Therefore, the dip is deep enough to force the output to settle at low state. In medium region, the FBG can function as an all-optical inverter due to the inverting nature between the input and output probe transmission. Lastly, by the same reasoning, in high region (pump peak intensity >97.5 GW/cm2), the probe compression peak is very high due to strong XPM with the pump. Correspondingly, the dips always fall below the threshold and finally force the output to settle at low state no matter whether the input is at low or high state. The FBG performs as an all-optical limiter in high region.

Needless to say, when the FWHM of Gaussian pump is changed, the boundaries of the peak intensity in the low, medium, and high region will change accordingly. What really matter is whether the dip after satisfying the energy conservation law can recover above or below the threshold to determine the state of the output probe transmission. The extinction ratio given in our numerical model is calculated roughly to be at 14.8 dB which is more than the minimum typical extinction ratio for switching application that normally range from 8.5 to 10 dB. The nonlinear switching behavior of probe transmission via Gaussian pump injection in the FBG is summarized in Table 1.

Tables Icon

Table 1. The summary of nonlinear switching behaviors of probe transmission via Gaussian pump injection in the FBG

4. Bistability threshold inside hysteresis loop of FBG

What is missing in Section 3 is the detail description about the bistability threshold inside hysteresis loop. What is the bistability threshold value corresponds to? Using the same method as the one used in Section 3, the bistability thresholds for various probe intensities are plotted in Fig. 5. The cross mark indicates the bistability threshold in Fig. 4. Figure 5 shows that the bistability threshold is equivalent to the unstable state that is missing in the previous numerical study [5]. For the first time, the role of the unstable state inside hysteresis loop as bistability threshold between low and high state is reported in this paper.

 

Fig. 5. Bistability threshold inside hysteresis loop of FBG.

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5. Some practical considerations

Some major challenges from the experimental point of view to verify the existence of bistability threshold inside hysteresis loop of nonlinear FBG are given as follows: firstly, the feasibility of the high power fiber laser (HPFL) for CW probe and Gaussian pump. The GW/cm2 range of the CW probe and pump peak operating intensity are needed for FBG to operate in the bistability region and perform the three nonlinear switching operations as described in Section three. Taking the effective cross-sectional area (AEFF) of the fiber in the unit of µm2 range, the CW probe and pump peak power must be in Kilo Watt (KW) range. The availability of Gaussian pulse with peak power in KW range can easily be realized by a commercial actively mode locked Nd:YLF laser emitting at 1053 nm, and the KW CW high power fiber laser (HPFL) can also be easily realized by IPG photonics commercial product YLR-HP series emitting at 1070 nm. Correspondingly the Bragg wavelength in the grating must be redesigned to be closed to the CW emitting wavelength around 1070 nm. The availability of the Bi-NLF fiber with nonlinear coefficient as high as 1360 W-1 km-1 with core AEFF as small as 3.3µm2 can reduce the nonlinear switching intensity up to 30 times smaller from the switching intensity value provided in our numerical simulation for the normal single mode fiber (SMF) assumption [9]. From the availability of the commercial HPFL, high nonlinear fiber, and the vast progress in optical fiber technology, it is feasible to verify the three nonlinear switching applications of FBG and to find the bistability threshold experimentally in near future.

Secondly, other detrimental nonlinear effects such as Stimulated Brillouin Scattering (SBS), Stimulated Raman Scattering (SRS), and Four Wave mixing (FWM) may come into the picture in the experiment for a KW power CW fiber laser operation. The experiments to show the OPB by XPM between the weak CW probe and strong Gaussian pump in FBG have been done by Broderick et al. [7]. Only cross phase modulation (XPM) nonlinear effect was observed and other detrimental nonlinear effects were never reported in the paper. However, the conditions will be much different when both the Gaussian pump and CW probe are strong in FBG. The SRS will be too weak for a few cm length of the FBG. The SBS for nonlinear pulse propagation in a FBG has been investigated by K. Oguzu [10]. It is found that SBS will be significant to deform the pulse shape if its peak intensity is in KW and its FWHM is more than 1 ns for a few cm short propagation in the FBG. Therefore, the SBS will not be significant in our Gaussian pump case as its FWHM is in ps range. However, SBS could be significant for the strong CW probe because its power is in KW range. The grating parameters must be designed carefully such that the spectrum of the Stokes pulse generated through SBS will fall entirely within its stop band for the SBS effect to be suppressed as proposed in [11]. Due to the flexibility in determining the Gaussian pump wavelength in equations (5) and (6) [6], we can choose the wavelength such that the interfering FWM signals spectrum will also fall entirely inside the stop band of nonlinear FBG.

The main point in this paper is to show the existence of the bistability threshold between low and high state inside hysteresis loop of nonlinear FBG. We conclude that the grating parameters and pump wavelength must be chosen judiciously such that other undesired nonlinear effects such as SBS and FWM are suppressed inside the stop band of nonlinear FBG for the practical considerations.

6. Conclusions

Three unique nonlinear switching behaviors of probe transmission with operating intensity in bistability region can be achieved by XPM with strong Gaussian pump in FBG. The parameters that distinguish the probe transmission switching behaviors are the pump peak intensity and its FWHM value. For a constant FWHM, three pump peak intensity regions are classified as: low, medium, and high, where the FBG offers three unique nonlinear switching applications in each of the three pump peak intensity regions respectively: in low region as a step-up all-optical switching, in medium region as an all-optical inverter, and in high region as an all-optical limiter. We show for the first time that the bistability threshold between the low and high state inside hysteresis loop of FBG is equivalent to the unstable state which was missing in the previous numerical study. The main challenges to verify the three nonlinear switching applications and the existence of the bistability threshold experimentally are to design the grating parameters and pump wavelength judiciously such that the undesired nonlinear effects such as SBS and FWM could be suppressed inside the stop band of nonlinear FBG.

Acknowledgments

This work is supported by Agency for Science, Technology, and Research (A*STAR), Singapore (http://www.a-star.edu.sg/astar/index.do).

References

1. Pochi Yeh, Optical Waves in Layered Media, (Wiley, New York, 1988).

2. C. M. de Sterke and J. E. Sipe, “Gap Solitons,” Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1994).

3. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg Grating Solitons,” Phys. Lett 76, 1627–30 (1996). [CrossRef]  

4. H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979). [CrossRef]  

5. Hojoon Lee and Govind P. Agrawal, “Nonlinear Switching of Optical Pulses in Fiber Bragg Gratings,” IEEE J. Quantum Electron.39 (2003).

6. C. M. de Sterke, “Optical Push Broom,” Opt. Lett. 17, 914–916 (1992). [CrossRef]   [PubMed]  

7. Neil G. R. Broderick, Domino Taverner, David J. Richardson, and Morten Ibsen, “Cross Phase Modulation Effects in nonlinear Fiber Bragg Gratings,” J. Opt. Soc. Am. B 17, 345–353 (2000) [CrossRef]  

8. C.M. de Sterke, K.R. Jackson, and B. D. Robert, “Nonlinear Coupled-Mode Equations on a finite interval: A numerical Procedure,” J. Opt. Soc. Am. B 8, 403–412 (1991). [CrossRef]  

9. J. H. Lee, T. Tanemura, T. Nagashima, T. Hasegawa, S. Ohara, N. Sugimoto, and K. Kikuchi, “Use of 1-m Bi2O3 nonlinear fiber for 160-Gbit/s optical-time division demultiplexing based on polarization rotation and wavelength shift induced by cross-phase modulation,” Opt. Lett. 30, 3144–3149 (2005). [CrossRef]  

10. K. Ogusu, “Effect of stimulated Brillouin scattering on nonlinear pulse propagation in fiber Bragg gratings,” J. Opt. Soc. Am. B 17, 769–774 (2000). [CrossRef]  

11. Hojoon Lee and Govind P. Agrawal, “Suppression of stimulated Brillouin scattering in optical fibers using fiber Bragg gratings,” Opt. Express 11, 3467–3472 (2003). [CrossRef]   [PubMed]  

References

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  1. Pochi Yeh, Optical Waves in Layered Media, (Wiley, New York, 1988).
  2. C. M. de Sterke and J. E. Sipe, “Gap Solitons,” Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1994).
  3. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg Grating Solitons,” Phys. Lett 76, 1627–30 (1996).
    [Crossref]
  4. H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
    [Crossref]
  5. Hojoon Lee and Govind P. Agrawal, “Nonlinear Switching of Optical Pulses in Fiber Bragg Gratings,” IEEE J. Quantum Electron.39 (2003).
  6. C. M. de Sterke, “Optical Push Broom,” Opt. Lett. 17, 914–916 (1992).
    [Crossref] [PubMed]
  7. Neil G. R. Broderick, Domino Taverner, David J. Richardson, and Morten Ibsen, “Cross Phase Modulation Effects in nonlinear Fiber Bragg Gratings,” J. Opt. Soc. Am. B 17, 345–353 (2000)
    [Crossref]
  8. C.M. de Sterke, K.R. Jackson, and B. D. Robert, “Nonlinear Coupled-Mode Equations on a finite interval: A numerical Procedure,” J. Opt. Soc. Am. B 8, 403–412 (1991).
    [Crossref]
  9. J. H. Lee, T. Tanemura, T. Nagashima, T. Hasegawa, S. Ohara, N. Sugimoto, and K. Kikuchi, “Use of 1-m Bi2O3 nonlinear fiber for 160-Gbit/s optical-time division demultiplexing based on polarization rotation and wavelength shift induced by cross-phase modulation,” Opt. Lett. 30, 3144–3149 (2005).
    [Crossref]
  10. K. Ogusu, “Effect of stimulated Brillouin scattering on nonlinear pulse propagation in fiber Bragg gratings,” J. Opt. Soc. Am. B 17, 769–774 (2000).
    [Crossref]
  11. Hojoon Lee and Govind P. Agrawal, “Suppression of stimulated Brillouin scattering in optical fibers using fiber Bragg gratings,” Opt. Express 11, 3467–3472 (2003).
    [Crossref] [PubMed]

2005 (1)

2003 (1)

2000 (2)

1996 (1)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg Grating Solitons,” Phys. Lett 76, 1627–30 (1996).
[Crossref]

1992 (1)

1991 (1)

1979 (1)

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[Crossref]

Agrawal, Govind P.

Hojoon Lee and Govind P. Agrawal, “Suppression of stimulated Brillouin scattering in optical fibers using fiber Bragg gratings,” Opt. Express 11, 3467–3472 (2003).
[Crossref] [PubMed]

Hojoon Lee and Govind P. Agrawal, “Nonlinear Switching of Optical Pulses in Fiber Bragg Gratings,” IEEE J. Quantum Electron.39 (2003).

Broderick, Neil G. R.

de Sterke, C. M.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg Grating Solitons,” Phys. Lett 76, 1627–30 (1996).
[Crossref]

C. M. de Sterke, “Optical Push Broom,” Opt. Lett. 17, 914–916 (1992).
[Crossref] [PubMed]

C. M. de Sterke and J. E. Sipe, “Gap Solitons,” Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1994).

de Sterke, C.M.

Eggleton, B. J.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg Grating Solitons,” Phys. Lett 76, 1627–30 (1996).
[Crossref]

Garmire, E.

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[Crossref]

Hasegawa, T.

Ibsen, Morten

Jackson, K.R.

Kikuchi, K.

Krug, P. A.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg Grating Solitons,” Phys. Lett 76, 1627–30 (1996).
[Crossref]

Lee, Hojoon

Hojoon Lee and Govind P. Agrawal, “Suppression of stimulated Brillouin scattering in optical fibers using fiber Bragg gratings,” Opt. Express 11, 3467–3472 (2003).
[Crossref] [PubMed]

Hojoon Lee and Govind P. Agrawal, “Nonlinear Switching of Optical Pulses in Fiber Bragg Gratings,” IEEE J. Quantum Electron.39 (2003).

Lee, J. H.

Marburger, J. H.

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[Crossref]

Nagashima, T.

Ogusu, K.

Ohara, S.

Richardson, David J.

Robert, B. D.

Sipe, J. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg Grating Solitons,” Phys. Lett 76, 1627–30 (1996).
[Crossref]

C. M. de Sterke and J. E. Sipe, “Gap Solitons,” Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1994).

Slusher, R. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg Grating Solitons,” Phys. Lett 76, 1627–30 (1996).
[Crossref]

Sugimoto, N.

Tanemura, T.

Taverner, Domino

Winful, H. G.

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[Crossref]

Yeh, Pochi

Pochi Yeh, Optical Waves in Layered Media, (Wiley, New York, 1988).

Appl. Phys. Lett. (1)

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[Crossref]

J. Opt. Soc. Am. B (3)

Opt. Express (1)

Opt. Lett. (2)

Phys. Lett (1)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg Grating Solitons,” Phys. Lett 76, 1627–30 (1996).
[Crossref]

Other (3)

Hojoon Lee and Govind P. Agrawal, “Nonlinear Switching of Optical Pulses in Fiber Bragg Gratings,” IEEE J. Quantum Electron.39 (2003).

Pochi Yeh, Optical Waves in Layered Media, (Wiley, New York, 1988).

C. M. de Sterke and J. E. Sipe, “Gap Solitons,” Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1994).

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of the FBG.

Fig. 2.
Fig. 2.

Bistability hysteresis loop in uniform FBG.

Fig. 3.
Fig. 3.

Three unique nonlinear switching behaviors of probe transmission via Gaussian pump injection in FBG.

Fig. 4.
Fig. 4.

Bistability threshold inside the hysteresis loop at 4.6 GW/cm2.

Fig. 5.
Fig. 5.

Bistability threshold inside hysteresis loop of FBG.

Tables (1)

Tables Icon

Table 1. The summary of nonlinear switching behaviors of probe transmission via Gaussian pump injection in the FBG

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

n ( z ) = n ¯ + n 1 ( z ) × cos [ 2 π Λ z + θ ( z ) ] + n 2 × E ( z ) 2
E ( z , t ) = [ A + ( z , t ) . exp ( iK B z ) + A ( z , t ) . exp ( iK B z ) ] . exp ( i ω 0 t )
A + ( z , t ) = P ( z V g t ) exp ( i δ p t ) + ε + ( z , t ) exp ( i δ 0 t )
A ( z , t ) = ε ( z , t ) exp ( i δ 0 t )
+ i ε + z + i 1 V g ε + t + δ ε + + κ ε + Γ [ ε + 2 + 2 ε 2 + 2 P ( z V g t ) 2 ] ε + = 0
+ i ε z + i 1 V g ε t + δ ε + κ * ε + + Γ [ ε 2 + 2 ε + 2 + 2 P ( z V g t ) 2 ] ε = 0

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