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The dynamics of direction-dependent switching in nonlinear chirped gratings
1 October
1996
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications 130 (1996) 295-301
The dynamics of direction-dependent switching in nonlinear chirped gratings Junmin Liu, Changjun Liao, Songhao Liu, Wencheng Xu Institute of Quantum Electronics, South China Normal University, Guangzhou 510631, China Received 11 October 1995; revised version received 3 April 1996; accepted 3 April 1996
Abstract Nonlinear chirped gratings are numerically demonstrated to have a direction-dependent switching behavior. For the two incident directions, bistability exists with different switching-on powers, and the instable high transmissivity states have different critical powers. The direction-dependent bistability is due to the difference of the energy contained in the gratings for the two incident directions; the direction-dependent critical powers for instability origin from the different modulation instability threshold. Furthermore, it is first reported that for instable output there are spectral side lobes whose frequency shifts relative to carrier frequency have jumps at some input powers.
1. Introduction
Recently, the interest in uniform nonlinear periodic structures has increased for their potential applications, including optical switching [ll, pulse compression [2], pulse train generation [3], and energy storage within the frequency stop band [4]. The underlined mechanism for these various applications is that the nonlinear response will alter the otherwise forbidden gap in the linear case and cause excitation of soliton-like distribution of the electrical field, which will decay exponentially with distance down the structure in linear gratings. In the stable case, Winful et al. demonstrated in 1979 that nonlinear gratings can be seen as bistable or multistable systems [l]. In 1987, Chen and Mills made another breakthrough; they numerically found that the complete transmitting state was linked to the special energy distribution gap soliton [5]. After that, several
authors found out the analytical Bragg soliton solution, of which gap solitons are a special kind [6-81. Meanwhile, a lot of work was done on the study of the dynamics of finite nonlinear gratings [2,3,9-l 11. Among these, De Sterke and coworkers contributed a lot to the understanding of the temporal characteristics. The studies on the temporal characteristics showed that the description without considering temporal variation is incomplete [3,11]: the upper state (high transmissivity state) may be unstable, self-pulsing, even chaos will occur. Nonuniform nonlinear gratings recently have drawn-more attention, because the use of nonuniform gratings will provide extra freedom in the design of grating-based devices. Some authors showed that even in the linear case the nonuniform gratings have to be studied by complicated methods, such as variational technique [12] and effective media method [10,13]. Furthermore, in the nonlinear case one may
0030~4018/96/$12.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved. PII SOO30-4018(96)00239-S
296
J. Liu et al. / Optics Communications
expect some new phenomena. Recently, Scalora et al. found the optical diode effect in a ramped structure with periodic alternating indices layers [ 141. In fact, the optical diode effect can also exist in nonlinear chirped gratings with uniform nonlinearity. It origins from the direction-dependent switching behavior, which has different switching-on power between the two incident directions. The temporal characteristics are also direction-dependent, the power needed for the instability of the upper state is different for the two directions. In this article, we discuss the chirped gratings with uniform nonlinearity. Depending upon the incident direction, the switching behavior is different, both in stable and temporal characteristics. In the stable case, bistability occurs with direction-dependent switching-on powers; this is due to the difference of the energy contained in the energy density envelope for the two incident directions. Considering the temporal characteristics, the high transmissivity state may be unstable, which is caused by the modulation instability. In the following we will discuss them separately, and the underlined mechanism is analyzed.
2. Coupled-mode
theory
We assume that the refractive-index variation is
I30 (1996) 295-301
will be coupled through two mechanisms; one is that the Bragg reflection mixes the two modes, the other is that the third order polarizability P,, = x3 s,EEE * results in an interaction. The resulting coupled-mode equations are [12,13,15] u, + u, =i[6(z)u+~(z)~+y(lu12+21u12)*], u, - ut = -i[S(z)v+~(z)~+y(lv12i-21u12)v],
(1) where the subscripts represent the differential, 6 and K are the effective detuning and coupling coefficient respectively, and y is the nonlinearity coefficient. The unit of time is L/V, V is the group velocity, and z is the normalized length with respect to the device length L. S and K are determined by 6(z)=&)+kn,a(z)-#, K(Z)=hoa(z),
(2)
where k is the free-space wave number of the incident light, 6, = kn, - k, is the detuning parameter, and +’ is the differential of the grating phase with respect to z. In this article, the structure is a chirped grating, i.e. o = 0, 4(z) = Cz2/2, and (+ is a constant along the structure. C is the chirp parameter. Eq. (2) demonstrates that the ramped gratings have the same effect as a properly chirped grating.
n( z) = no{1 + cX(z) + 2o( z)cos[%k,z + 24( Z)]}’ where (Y, (+ and # describe the background refractive index change, the amplitude of the grating, and the slowly varying phase of the grating, respectively. In this structure, the electrical field is written in the form
3. Stable characteristics In the stable case, the electrical field is independent of time, the temporal differential terms can be ignored. It is convenient to introduce the following parameters: A,,=~u~~-~v)~,
E( x, y. z) = {u( z)exp[ik,
A2=uv*
z + i+( z)]
+v(z)exp[-ik,z-i~(z)]}F(x,
+u*v,
A1=I~12+Iv12,
As=i(u*v--uv*).
(3)
From Eq. (1) these parameters satisfy the equations Y),
where u(z) and v(z) are the slowly varying amplitude of the forward- and backward-propagating modes respectively and F(x, y) is the common transverse mode field distribution. The two modes
A;=O, A’, = ~KA,, A;=
-26(z)A,-3yA,A3,
A;=2S(z)A2+2~A,+3yA,A2,
(4)
J. Liu et al./Optics
Positivedirection +
z=o
Communications
130 (1996) 295-301
297
Negativedirection +
Z=l
Fig. 1. The geometry used in the calculations. The structure parametersare~=4,6,=3,y=l,C=l.Thechirpedperiodic index variation is indicated by a dotted line. The period of the grating and chirp have been greatly exaggerated for illustrative purpcses.
where the primes represent the differential with respect to z. A,,is also a constant quantity along the device as in uniform gratings. We integrate Eq. (4) using the geometry shown in Fig. 1. The device length and the nonlinearity coefficient are normalized to 1. The boundary conditions for Eq. (4) are A,(O)= 1u(O>l*, A,(O)=A,(O)= 0. When the light incides from the other end of the device in Fig. 1, it is equal to an overturned chirped grating, without other changes. Eq. (4) can be used to numerically calculate the reflectance spectrum with y = 0, which is important for the choice of an appropriate stop gap edge, to which the incident wave should be detuned depending on the sign of the nonlinearity. The analytical calculation of the reflectance spectrum of a nonuniform grating is complicated even in the linear case [12,13]. The reflectance spectrum of the chirped gratings with C = 1 moves for longer wavelengths; the edges of the stop band become about - 1.15 K and 0.85~.One can compare this with the stop gap 1 6 I < K in the chirp-free counterpart gratings. We choose positive nonlinearity, i.e. self-focusing material, and the upper edge of the stop gap. We found that the transmissivity as a function of the incidentlight intensity is different for different incident directions: the bistable behavior will occur regardless of the incident directions, but the switching-on powers are different. This is illustrated in Fig. 2. The solid and dotted lines are for positive and negative direction, respectively. When the light intensity is in the range between 0.58 + 0.01 and 0.70 + 0.01, the optical diode effect is obvious; just above or below the scope only a poor optical diode effect exists. The different switching-on powers for the two directions
0
0.2
0.4
0.6
0.8
1
INPUT LIGHT INTENSITY Fig. 2. Transmissivity
versus the input light intensity.
are a general characteristic. Fig. 3 shows the switching-on power as function of detuning at two values of the coupling coefficient K. Insight into the origin of the directional switching behavior should first refer to the linear case. Different amounts of energy will be stored in the structure for the two directions. For convenience’ sake, the direction of the incident light that induces more energy storage is noted as ‘positive direction,’ the other called ‘negative direction.’ The reason for the different energy storage of positive and negative direction incidence is as follows. The chirped grating will make the incident light find itself in a different 1.6 ~
1.4
-z k
1.2
._ E 1 c 9 p 0.8 I .+z: 0 0.6 R 0.4
0.2 ~,.,,,....I.....‘,..I.........i,,......,I..,..~ 0 1 3 4 2
5
Detuning Fig. 3. The switching-on powers for the two directions versus detuning 6, at (a) K = 4 and (b) K = 5 respectively. The other parameters are the same as in Fig. 1.
298
J. Liu et al./Optics
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130 (1996) 29.5-301
4. Temporal
characteristics
0.8
The dynamics of the direction-dependent switching is more complicated than that in uniform gratings. The incident direction and the structure parameters will combine to affect the temporal characteristics of the switching course. We focus on the instability of the upper state; other features such as the rising time will not be discussed here because they are not so important for the switching applications as the instability. The coupled-mode equations (1) can be numerically integrated by the finite difference 0.2 0
0.5
1
NORMALIZED LENGTH Fig. 4. The energy density distributions the maximum transmissivity.
in the chirped
grating
1
at
position of the local band along the device. For positive direction the initially larger optical energy encounters relatively lower reflectance and as the light continues into the device the eventually small optical energy encounters higher reflectance; for the negative direction the case is the opposite. Although it corresponds to a different energy storage, in the linear case the output does not depend on the incident direction. The involvement of Kerr nonlinearity will make the different energy storage visible in the output. As in uniform gratings, the upper states are always accompanied by excitation of an energy density envelope. When the transmissivity is at its maximum, the input light couples to an object that is somewhat alike the spatial soliton shape along the structure as shown in Fig. 4. However, not like the gap soliton that is centered in the middle of the uniform nonlinear gratings; the object’s maximum is not in the middle, which results in a transmissivity of less than lOO%, otherwise unity in the uniform grating. The energy storage needed for maximum transmissivity is almost the same regardless of the incident direction (see Fig. 4), and for the same input light intensity the energy storages for the positive and negative direction are dissimilar, so to excite the objects larger input light intensity is necessary for the negative direction than for the positive direction. The retard in the light intensity to excite the objects causes the optical diode effect.
t
0-
0
50
loo
I&
Time (L/V) 0.7 .s
0.6
2 a, 0.5 TS .5
0.4
5 fi
0.3
z
0.2
0
50
Time (L/V) Fig. with time time
5. The unstable and stable output for the positive direction, an input light intensity of (a) 0.61 and (b) 0.70. The zero coordinate is chosen as the time just after incidence and 2OCG units after incidence for (a) and (b) respectively.
J. Liu er al. / Optics Communications
scheme outlined in Ref. [16], and we have tested our computer procedure by comparing the transmissivity and switching-on power with the stable case. In order to avoid the shock under the discontinuous initial condition, the input light is applied slowly, other than immediately from zero up to the value at which the system is switched on. When the system reaches its upper state, the output light may be damping or oscillating, which corresponds to stability or instability. This is illustrated in Fig. 5 for the positive direction, where the self-pulsing instability is made sure to occur at 2000 time units stimulation. Similar results can also be obtained for the negative direction. By synthesizing the numerical results, we can conclude that whether the upper states are stable or unstable is determined by the input power. Two critical input powers exist, above which the upper state will be unstable for the two incident directions respectively. For the zero chirp parameter, the two critical values are the same, i.e. the instability is not direction dependent. The critical input light intensities are noted as Pp and P,, for positive and negative direction respectively. Pp and P, are functions of K, 8 and C. For 6 below a special value with fixed K, the critical power may be smaller than the switchingon power. This was discussed in Refs. [3,10], where some of the special S values were numerically calculated With a Certain K. The main temporal characteristics of the switching can be simply described by Pp and P,, the direction dependence is marked by Pp # P,. With the parameters used in Fig. 1, Pp and P,, are 0.657 f 0.002 and 0.721 f 0.002 respectively. Due to the stimulation we believe that P,, > Pp. For either incident direction, the way the electric envelope moves is the direct reason for instability or stability. Fig. 6 shows the typical movement of the maximum of the electric envelope when the upper state is stable and unstable respectively for the positive direction, which is alike for the negative direction. For the stable upper state, the electric envelope will oscillate dampingly and be finally still in the gratings, otherwise the upper state will be unstable with the oscillating electric envelope. The oscillation of the output light can be interpreted as modulation instability, which is a result of an interplay between the nonlinear and dispersive
I30 (1996) 295-301
299
-0.5
0
1
0.5
Frequency (L/V)
0 -1.5
-1
-0.5
0
0.5
1
1.5
Frequency (v/L)
Fig. 6.
The movement of the maximum of the electric envelope when the upper state is (a) stable and (b) unstable respectively for the positive direction.
effects. The periodic structure provides a very large anomalous GVD(group velocity dispersion) near the upper edge of the stop gap 121. When choosing materials with positive nonlinearity and the incident power above the threshold, the steady-state propagation of the CW wave is inherently unstable. New frequency waves will be generated, which can be explained in terms of four-wave mixing phasematched by SPM and XPM 1171.In Fig. 7, the output spectra of the positive direction are shown for the stable and unstable upper state respectively. For the negative direction, the results are alike. The condition to determine the critical powers to generate Stokes or anti-Stokes waves is that the modulation-
300
J. Liu ef al. / Optics Communications
instability gain is larger than the effective loss of the Stokes or anti-Stokes waves in nonlinear gratings. For the two directions in chirped gratings, as shown in the stable case, the effective loss is different which can be defined as reflectivity because the reflected Stokes or anti-Stokes do not almost couple to the incident wave due to the phase-matching condition, so the critical power Pp # P, is a natural condition. Detailed discussions about the critical power will be presented elsewhere. There is an interesting phenomenon that the spectral side lobes’ frequency shift from the carrier frequency has a jump at some input power, as shown in Fig. 8. When the incident power is about 1.255 + 0.002, the frequency shift jumps about O.l(V/L) for both directions. Although the explanation of this 0.7 t
30
60
90
120
150
Time (L/V)
2
0.6
$
0.5
P E
0.4
0.8 I_
0.4
I
0.6
0.8
1
1.2
1.4
1.6
1.8
2
22
24
Input light power Fig. 8. The frequency shift of the spectral side lobes relative to the carrier frequency versus input power. The solid dots are the results from the numerical stimulations.
I
-0
.s b
130 (1996) 295-301
phenomenon is still questionable, we believe that it origins from the bistability of the wave with a new frequency incident wave. This may be applied in a communication system as a frequency modulator. For practical applications of the direction-dependent switching, we can consider a polydiacetylene PTS waveguide [ 181. It is realistic to consider that its length L is 1 cm and the cross-sectional area is 10 pm2. At a wavelength A = 1.9 pm, the nonlinear index n, is 8 X lo-I6 m2/W. With the parameters 6, L = 3, KL = 4 and C = l/c, the optical diode can operate at a power between about 0.13 and 0.17 W. When the positive direction is used for the optical switching with a power lower than 0.17 W, the device is a switching-isolator. If the power is larger than 0.17 and 0.21 W for the positive and negative direction respectively, the device can be used as a pulse-train generator.
0
5. Conclusions c5
a2
5 ‘W k! g
a1
0.3 i
0’
0
03
I
’
50
la-l
Time (L/V) Fig. 7. The spectra of the output light with a positive input light intensity of (a) 0.61 and (h) 0.70.
direction
The nonlinear chirped gratings are demonstrated to have a direction-dependent switching behavior. In the stable case, the threshold for the system to switch on is different, which depends on the incident direction. When considering the temporal characteristics, the high transmissivity state may be unstable, which is determined by both the structure parameters and
J. Liu et al./Optics
Communications
the incident direction. The modulation instability is responsible for the instability. Furthermore, it is first reported that the unstable output frequency shift has a jump at a certain input power, a phenomenon which remains unexplained.
References [l] H.G. Wmful, J.H. Marberger and E. Garmire, Appl. Phys. Lett. 35 (1979) 379. [2] H.G. Winful, Appl. Phys. Lett. 46 (1985) 527. [3] C.M. de Sterke and J.E. Sipc, Phys. Rev. A 42 (1990) 2858. [4] D.L. Mills and S.E. Trullmger, Phys. Rev. B 36 (1987) 947. [s] W. Chen and D.L. Mills, Phys. Rev. Lett. 58 (1987) 160. 161 D.N. Christodoulides and R.I. Joseph, Phys. Rev. Lett. 62 (1989) 1746. [7] A.B. Aceves and S. Wabnitz, Phys. Lett. A 141 (1987) 37. [8] J.E. Sipe and H.G. Wmful, Optics Lett. 13 (1988) 132;
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C.M. de Sterke and J.E. Sipe, Phys. Rev. A 39 (1989) 5163; 38 (1988) 5145. 191 H.G. Winful, R. Zamir and S. Feldman, Appl. Phys. Lett. 58 (1991) 1001; J. Peyraud and J. Cost, Phys. Rev. B 40 (1989) 12201. [lo] C.M. de Sterke, Phys. Rev. A 45 (1992) 2012. [l 11 C.M. de Sterke, Phys. Rev. A 45 (1992) 8252. [12] L. Poladian, J. Opt. Sot. Am. A 11 (1994) 1846. [13] J.E. Sipe, L. Poladian and C.M. de Sterke, J. Opt. Sot. Am. A 11 (1994) 1307. [14] M. Scalora, J.P. Dowling, C.M. Bowden and M.J. Bloemer, J. Appl. Phys. 76 (1994) 2023. [ 151 For example, H.G. Winful and G.D. Cooperman, Appl. Phys. Lett. 40 (1982) 298. [16] C.M. de Sterke, K.R. Jackson and B.D. Robert, J. Opt. Sot. Am. B 8 (1991) 403. 1171 G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, Boston, San Diego, New York, 1989) ch. 10, p. 289. [18] A. Mecozzi, S. Trill0 and S. Wabnitz, Optics Lett. 12 (1987) 1008.
1996
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications 130 (1996) 295-301
The dynamics of direction-dependent switching in nonlinear chirped gratings Junmin Liu, Changjun Liao, Songhao Liu, Wencheng Xu Institute of Quantum Electronics, South China Normal University, Guangzhou 510631, China Received 11 October 1995; revised version received 3 April 1996; accepted 3 April 1996
Abstract Nonlinear chirped gratings are numerically demonstrated to have a direction-dependent switching behavior. For the two incident directions, bistability exists with different switching-on powers, and the instable high transmissivity states have different critical powers. The direction-dependent bistability is due to the difference of the energy contained in the gratings for the two incident directions; the direction-dependent critical powers for instability origin from the different modulation instability threshold. Furthermore, it is first reported that for instable output there are spectral side lobes whose frequency shifts relative to carrier frequency have jumps at some input powers.
1. Introduction
Recently, the interest in uniform nonlinear periodic structures has increased for their potential applications, including optical switching [ll, pulse compression [2], pulse train generation [3], and energy storage within the frequency stop band [4]. The underlined mechanism for these various applications is that the nonlinear response will alter the otherwise forbidden gap in the linear case and cause excitation of soliton-like distribution of the electrical field, which will decay exponentially with distance down the structure in linear gratings. In the stable case, Winful et al. demonstrated in 1979 that nonlinear gratings can be seen as bistable or multistable systems [l]. In 1987, Chen and Mills made another breakthrough; they numerically found that the complete transmitting state was linked to the special energy distribution gap soliton [5]. After that, several
authors found out the analytical Bragg soliton solution, of which gap solitons are a special kind [6-81. Meanwhile, a lot of work was done on the study of the dynamics of finite nonlinear gratings [2,3,9-l 11. Among these, De Sterke and coworkers contributed a lot to the understanding of the temporal characteristics. The studies on the temporal characteristics showed that the description without considering temporal variation is incomplete [3,11]: the upper state (high transmissivity state) may be unstable, self-pulsing, even chaos will occur. Nonuniform nonlinear gratings recently have drawn-more attention, because the use of nonuniform gratings will provide extra freedom in the design of grating-based devices. Some authors showed that even in the linear case the nonuniform gratings have to be studied by complicated methods, such as variational technique [12] and effective media method [10,13]. Furthermore, in the nonlinear case one may
0030~4018/96/$12.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved. PII SOO30-4018(96)00239-S
296
J. Liu et al. / Optics Communications
expect some new phenomena. Recently, Scalora et al. found the optical diode effect in a ramped structure with periodic alternating indices layers [ 141. In fact, the optical diode effect can also exist in nonlinear chirped gratings with uniform nonlinearity. It origins from the direction-dependent switching behavior, which has different switching-on power between the two incident directions. The temporal characteristics are also direction-dependent, the power needed for the instability of the upper state is different for the two directions. In this article, we discuss the chirped gratings with uniform nonlinearity. Depending upon the incident direction, the switching behavior is different, both in stable and temporal characteristics. In the stable case, bistability occurs with direction-dependent switching-on powers; this is due to the difference of the energy contained in the energy density envelope for the two incident directions. Considering the temporal characteristics, the high transmissivity state may be unstable, which is caused by the modulation instability. In the following we will discuss them separately, and the underlined mechanism is analyzed.
2. Coupled-mode
theory
We assume that the refractive-index variation is
I30 (1996) 295-301
will be coupled through two mechanisms; one is that the Bragg reflection mixes the two modes, the other is that the third order polarizability P,, = x3 s,EEE * results in an interaction. The resulting coupled-mode equations are [12,13,15] u, + u, =i[6(z)u+~(z)~+y(lu12+21u12)*], u, - ut = -i[S(z)v+~(z)~+y(lv12i-21u12)v],
(1) where the subscripts represent the differential, 6 and K are the effective detuning and coupling coefficient respectively, and y is the nonlinearity coefficient. The unit of time is L/V, V is the group velocity, and z is the normalized length with respect to the device length L. S and K are determined by 6(z)=&)+kn,a(z)-#, K(Z)=hoa(z),
(2)
where k is the free-space wave number of the incident light, 6, = kn, - k, is the detuning parameter, and +’ is the differential of the grating phase with respect to z. In this article, the structure is a chirped grating, i.e. o = 0, 4(z) = Cz2/2, and (+ is a constant along the structure. C is the chirp parameter. Eq. (2) demonstrates that the ramped gratings have the same effect as a properly chirped grating.
n( z) = no{1 + cX(z) + 2o( z)cos[%k,z + 24( Z)]}’ where (Y, (+ and # describe the background refractive index change, the amplitude of the grating, and the slowly varying phase of the grating, respectively. In this structure, the electrical field is written in the form
3. Stable characteristics In the stable case, the electrical field is independent of time, the temporal differential terms can be ignored. It is convenient to introduce the following parameters: A,,=~u~~-~v)~,
E( x, y. z) = {u( z)exp[ik,
A2=uv*
z + i+( z)]
+v(z)exp[-ik,z-i~(z)]}F(x,
+u*v,
A1=I~12+Iv12,
As=i(u*v--uv*).
(3)
From Eq. (1) these parameters satisfy the equations Y),
where u(z) and v(z) are the slowly varying amplitude of the forward- and backward-propagating modes respectively and F(x, y) is the common transverse mode field distribution. The two modes
A;=O, A’, = ~KA,, A;=
-26(z)A,-3yA,A3,
A;=2S(z)A2+2~A,+3yA,A2,
(4)
J. Liu et al./Optics
Positivedirection +
z=o
Communications
130 (1996) 295-301
297
Negativedirection +
Z=l
Fig. 1. The geometry used in the calculations. The structure parametersare~=4,6,=3,y=l,C=l.Thechirpedperiodic index variation is indicated by a dotted line. The period of the grating and chirp have been greatly exaggerated for illustrative purpcses.
where the primes represent the differential with respect to z. A,,is also a constant quantity along the device as in uniform gratings. We integrate Eq. (4) using the geometry shown in Fig. 1. The device length and the nonlinearity coefficient are normalized to 1. The boundary conditions for Eq. (4) are A,(O)= 1u(O>l*, A,(O)=A,(O)= 0. When the light incides from the other end of the device in Fig. 1, it is equal to an overturned chirped grating, without other changes. Eq. (4) can be used to numerically calculate the reflectance spectrum with y = 0, which is important for the choice of an appropriate stop gap edge, to which the incident wave should be detuned depending on the sign of the nonlinearity. The analytical calculation of the reflectance spectrum of a nonuniform grating is complicated even in the linear case [12,13]. The reflectance spectrum of the chirped gratings with C = 1 moves for longer wavelengths; the edges of the stop band become about - 1.15 K and 0.85~.One can compare this with the stop gap 1 6 I < K in the chirp-free counterpart gratings. We choose positive nonlinearity, i.e. self-focusing material, and the upper edge of the stop gap. We found that the transmissivity as a function of the incidentlight intensity is different for different incident directions: the bistable behavior will occur regardless of the incident directions, but the switching-on powers are different. This is illustrated in Fig. 2. The solid and dotted lines are for positive and negative direction, respectively. When the light intensity is in the range between 0.58 + 0.01 and 0.70 + 0.01, the optical diode effect is obvious; just above or below the scope only a poor optical diode effect exists. The different switching-on powers for the two directions
0
0.2
0.4
0.6
0.8
1
INPUT LIGHT INTENSITY Fig. 2. Transmissivity
versus the input light intensity.
are a general characteristic. Fig. 3 shows the switching-on power as function of detuning at two values of the coupling coefficient K. Insight into the origin of the directional switching behavior should first refer to the linear case. Different amounts of energy will be stored in the structure for the two directions. For convenience’ sake, the direction of the incident light that induces more energy storage is noted as ‘positive direction,’ the other called ‘negative direction.’ The reason for the different energy storage of positive and negative direction incidence is as follows. The chirped grating will make the incident light find itself in a different 1.6 ~
1.4
-z k
1.2
._ E 1 c 9 p 0.8 I .+z: 0 0.6 R 0.4
0.2 ~,.,,,....I.....‘,..I.........i,,......,I..,..~ 0 1 3 4 2
5
Detuning Fig. 3. The switching-on powers for the two directions versus detuning 6, at (a) K = 4 and (b) K = 5 respectively. The other parameters are the same as in Fig. 1.
298
J. Liu et al./Optics
Communications
130 (1996) 29.5-301
4. Temporal
characteristics
0.8
The dynamics of the direction-dependent switching is more complicated than that in uniform gratings. The incident direction and the structure parameters will combine to affect the temporal characteristics of the switching course. We focus on the instability of the upper state; other features such as the rising time will not be discussed here because they are not so important for the switching applications as the instability. The coupled-mode equations (1) can be numerically integrated by the finite difference 0.2 0
0.5
1
NORMALIZED LENGTH Fig. 4. The energy density distributions the maximum transmissivity.
in the chirped
grating
1
at
position of the local band along the device. For positive direction the initially larger optical energy encounters relatively lower reflectance and as the light continues into the device the eventually small optical energy encounters higher reflectance; for the negative direction the case is the opposite. Although it corresponds to a different energy storage, in the linear case the output does not depend on the incident direction. The involvement of Kerr nonlinearity will make the different energy storage visible in the output. As in uniform gratings, the upper states are always accompanied by excitation of an energy density envelope. When the transmissivity is at its maximum, the input light couples to an object that is somewhat alike the spatial soliton shape along the structure as shown in Fig. 4. However, not like the gap soliton that is centered in the middle of the uniform nonlinear gratings; the object’s maximum is not in the middle, which results in a transmissivity of less than lOO%, otherwise unity in the uniform grating. The energy storage needed for maximum transmissivity is almost the same regardless of the incident direction (see Fig. 4), and for the same input light intensity the energy storages for the positive and negative direction are dissimilar, so to excite the objects larger input light intensity is necessary for the negative direction than for the positive direction. The retard in the light intensity to excite the objects causes the optical diode effect.
t
0-
0
50
loo
I&
Time (L/V) 0.7 .s
0.6
2 a, 0.5 TS .5
0.4
5 fi
0.3
z
0.2
0
50
Time (L/V) Fig. with time time
5. The unstable and stable output for the positive direction, an input light intensity of (a) 0.61 and (b) 0.70. The zero coordinate is chosen as the time just after incidence and 2OCG units after incidence for (a) and (b) respectively.
J. Liu er al. / Optics Communications
scheme outlined in Ref. [16], and we have tested our computer procedure by comparing the transmissivity and switching-on power with the stable case. In order to avoid the shock under the discontinuous initial condition, the input light is applied slowly, other than immediately from zero up to the value at which the system is switched on. When the system reaches its upper state, the output light may be damping or oscillating, which corresponds to stability or instability. This is illustrated in Fig. 5 for the positive direction, where the self-pulsing instability is made sure to occur at 2000 time units stimulation. Similar results can also be obtained for the negative direction. By synthesizing the numerical results, we can conclude that whether the upper states are stable or unstable is determined by the input power. Two critical input powers exist, above which the upper state will be unstable for the two incident directions respectively. For the zero chirp parameter, the two critical values are the same, i.e. the instability is not direction dependent. The critical input light intensities are noted as Pp and P,, for positive and negative direction respectively. Pp and P, are functions of K, 8 and C. For 6 below a special value with fixed K, the critical power may be smaller than the switchingon power. This was discussed in Refs. [3,10], where some of the special S values were numerically calculated With a Certain K. The main temporal characteristics of the switching can be simply described by Pp and P,, the direction dependence is marked by Pp # P,. With the parameters used in Fig. 1, Pp and P,, are 0.657 f 0.002 and 0.721 f 0.002 respectively. Due to the stimulation we believe that P,, > Pp. For either incident direction, the way the electric envelope moves is the direct reason for instability or stability. Fig. 6 shows the typical movement of the maximum of the electric envelope when the upper state is stable and unstable respectively for the positive direction, which is alike for the negative direction. For the stable upper state, the electric envelope will oscillate dampingly and be finally still in the gratings, otherwise the upper state will be unstable with the oscillating electric envelope. The oscillation of the output light can be interpreted as modulation instability, which is a result of an interplay between the nonlinear and dispersive
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The movement of the maximum of the electric envelope when the upper state is (a) stable and (b) unstable respectively for the positive direction.
effects. The periodic structure provides a very large anomalous GVD(group velocity dispersion) near the upper edge of the stop gap 121. When choosing materials with positive nonlinearity and the incident power above the threshold, the steady-state propagation of the CW wave is inherently unstable. New frequency waves will be generated, which can be explained in terms of four-wave mixing phasematched by SPM and XPM 1171.In Fig. 7, the output spectra of the positive direction are shown for the stable and unstable upper state respectively. For the negative direction, the results are alike. The condition to determine the critical powers to generate Stokes or anti-Stokes waves is that the modulation-
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instability gain is larger than the effective loss of the Stokes or anti-Stokes waves in nonlinear gratings. For the two directions in chirped gratings, as shown in the stable case, the effective loss is different which can be defined as reflectivity because the reflected Stokes or anti-Stokes do not almost couple to the incident wave due to the phase-matching condition, so the critical power Pp # P, is a natural condition. Detailed discussions about the critical power will be presented elsewhere. There is an interesting phenomenon that the spectral side lobes’ frequency shift from the carrier frequency has a jump at some input power, as shown in Fig. 8. When the incident power is about 1.255 + 0.002, the frequency shift jumps about O.l(V/L) for both directions. Although the explanation of this 0.7 t
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Input light power Fig. 8. The frequency shift of the spectral side lobes relative to the carrier frequency versus input power. The solid dots are the results from the numerical stimulations.
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phenomenon is still questionable, we believe that it origins from the bistability of the wave with a new frequency incident wave. This may be applied in a communication system as a frequency modulator. For practical applications of the direction-dependent switching, we can consider a polydiacetylene PTS waveguide [ 181. It is realistic to consider that its length L is 1 cm and the cross-sectional area is 10 pm2. At a wavelength A = 1.9 pm, the nonlinear index n, is 8 X lo-I6 m2/W. With the parameters 6, L = 3, KL = 4 and C = l/c, the optical diode can operate at a power between about 0.13 and 0.17 W. When the positive direction is used for the optical switching with a power lower than 0.17 W, the device is a switching-isolator. If the power is larger than 0.17 and 0.21 W for the positive and negative direction respectively, the device can be used as a pulse-train generator.
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Time (L/V) Fig. 7. The spectra of the output light with a positive input light intensity of (a) 0.61 and (h) 0.70.
direction
The nonlinear chirped gratings are demonstrated to have a direction-dependent switching behavior. In the stable case, the threshold for the system to switch on is different, which depends on the incident direction. When considering the temporal characteristics, the high transmissivity state may be unstable, which is determined by both the structure parameters and
J. Liu et al./Optics
Communications
the incident direction. The modulation instability is responsible for the instability. Furthermore, it is first reported that the unstable output frequency shift has a jump at a certain input power, a phenomenon which remains unexplained.
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C.M. de Sterke and J.E. Sipe, Phys. Rev. A 39 (1989) 5163; 38 (1988) 5145. 191 H.G. Winful, R. Zamir and S. Feldman, Appl. Phys. Lett. 58 (1991) 1001; J. Peyraud and J. Cost, Phys. Rev. B 40 (1989) 12201. [lo] C.M. de Sterke, Phys. Rev. A 45 (1992) 2012. [l 11 C.M. de Sterke, Phys. Rev. A 45 (1992) 8252. [12] L. Poladian, J. Opt. Sot. Am. A 11 (1994) 1846. [13] J.E. Sipe, L. Poladian and C.M. de Sterke, J. Opt. Sot. Am. A 11 (1994) 1307. [14] M. Scalora, J.P. Dowling, C.M. Bowden and M.J. Bloemer, J. Appl. Phys. 76 (1994) 2023. [ 151 For example, H.G. Winful and G.D. Cooperman, Appl. Phys. Lett. 40 (1982) 298. [16] C.M. de Sterke, K.R. Jackson and B.D. Robert, J. Opt. Sot. Am. B 8 (1991) 403. 1171 G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, Boston, San Diego, New York, 1989) ch. 10, p. 289. [18] A. Mecozzi, S. Trill0 and S. Wabnitz, Optics Lett. 12 (1987) 1008.