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Optical fiber-grating compressors utilizing long fibers
1 November 1994
OPTICS COMMUNICATIONS Optics Communications 112 (1994) 48-54
Optical fiber-grating compressors utilizing long fibers Magnus Karlsson institute for Electromagnetic Field Theory, Chalmers University of Technology, S-412 96 Giiteborg, Sweden
Received 6 June 1994; revised manuscript
received
19 August
1994
Abstract
The properties of optical fiber-grating compressors have been investigated numerically, considering fibers ten times longer than in previous studies. It is found that such long fibers yield substantially better compression than the previously investigated short ones. Also considerde are the sensitivity of pulse shape on the compression by using Gaussian-, parabolic- and sechshaped pulses as initial conditions.
When compressing optical pulses into femtosecond durations, the most important method is the two-stage (or fiber-grating) compression scheme. The underlying idea is simple in principle: an initial, nonlinear, stage which broadens the pulse in the spectral domain is followed by a linear delay line which compresses the pulse to its transform-limited width [ 11. In 1981, Nakatsuka et al. [2] showed that single-mode optical fibers, operating in the normal-dispersion regime, could be used as the initial nonlinear stage. A number of record-breaking experiments followed [3,4] and culminated with the experiment by Fork et al. [5] in which the shortest optical pulse to date (6 fs) was produced. The theoretical papers dealing with fiber-grating compressors are almost entirely based on numerical computations [ 6-81. Grischkowsky and Balant [ 61 noted that the combined action of self-phase modulation (SPM) and group velocity dispersion (GVD) in the fiber produces pulses of better quality than if SPM alone was to affect the pulses. In an extensive analysis by Tomlinson et al. [ 71, an optimum fiber length was found, which, for a given input pulse, maximized the compression factor. An extension to consider initially
0030-4018/94/$07.00 @ 1994 Elsevier Science B.V. All rights reserved SSDIOO30-4018(94)00470-6
chirped pulses were found to give only minor quantitative modifications on the compressor properties [ 81. The optimum fiber length found in Ref. [ 71 can be explained qualitatively by the theory of wave breaking in normally dispersive fibers [9-l 11. The wave breaking distance is defined as the propagation distance at which the pulse loses its single pulse shape and achieves sidelobes. This distance is an approximate measure of the transition from an inital, SPM-dominated (nonlinear) propagation to a GVD-dominated (linear) one. It has been pointed out that the optimal fiber length found in Ref. [7] is approximately twice the wave breaking distance [ 6,101, indicating that the pulse can take some amount of wave breaking before the compression properties start to deteriorate. Note, however, that this result has been shown only for sech-shaped pulses. Other pulse shapes have not been studied, although one would expect no qualitative differences for e.g. Gaussians. It was suggested in Ref. [ lo] that pulses of parabolic shape would have good compression properties, since they are approximately wave-breaking-free, i.e. they preserve their shape during the propagation through the fiber. Since all previous results concern sech-shaped pulses, it might be valuable to examine
M. Karlsson/Optics Communications112 (1994) 48-54
49
(1)
R-
b -
Fig. 1. Intensity lu(z, t) I* arising from the input pulse u( 0, I) = 20sech( t) in IQ. ( I). At short distances there are heavy oscillations of the wings (wave-breaking), but this vanishes at longer distances. Asymptotically as z + co, the distribution aproaches a trapezium. The instantaneous frequency ~9arg[ u] /at of the pulse of (a) at z = 0.56 is approximately linear over the whole body of the pulse. The intensity distribution is also plotted for convenience.
also Gaussian and parabolic pulses. Although it can be experimentally difficult to specify a certain pulse shape, it is of theoretical interest to see how the compresssion is affected by different initial pulse shapes. More important, however, is that we have extended the analysis to consider longer fiber lengths than studied in Refs. [ 7,8], with some rather unexpected results. The slowly varying envelope u( z, t) of an optical pulse in a fiber is described by the nonlinear Schriidinger equation, taking fiber nonlinearity and normal GVD into account ’ : 1 For relation to physical units we refer to Refs. [ 11 and [7]. The normalizations of this work only differs with Ref. [ 71 in the definition of the fiber length 5 : 5 = (v/2) (z/& where ( Z/ZQ) is used in Ref. [ 71.
Here t denotes the time in a comoving system with t = 0 at the pulse center, z is the axial distance along the fiber, and 4’ is the fiber length. Pulses governed by this equation will broaden monotonically during propagation. Gaussian- and sech-pulses will also change shape, and approximately at the wave breaking distance they become rather rectangular, accompanied with a quadratic phase variation (chirp) and ringings on the pulse wings, see Fig. la. It has been pointed out that the ringing period is of the order of the compressed pulse width [ 111. As the pulse propagates further into the GVD-dominated regime, the ringings gradually disappear, and a pulse having a chirped, trapezium-shaped monotonic intensity distribution emerges. Thus, the oscillations induced by the wave-breaking vanishes at longer distances. The first pulse of Fig. 1 (at z = 0.14) is approximately at the optimum fiber distance for compression predicted in Ref. [ 71 (z = 0.12). We see that it has steep, ringing slopes. The last pulse (at z = 1.05) has a smoother envelope, however, and might therefore be better suited for compression. This is also supported by Fig. lb, showing that the instantaneous frequency varies approximately linearly across the entire pulse body. Another interesting, and to my knowledge unexplained, fact is that this long-distance temporal distribution of pulse intensity and phase becomes similar to the asymptotic spectral distribution. This is true for the parabolic pulse as well. It was shown in Ref. [ lo] that an intense parabolic pulse preserves the parabolic shape during propagation, and that its spectrum becomes parabolic in the limit z + 00. Since the propagation through the delay line is linear it is easily carried out in the Fourier plane: C(w, 5, a) = fi(w, 5) exp[@(w,a)
I,
(2)
where ii(w) denotes the Fourier transform J u(t) exp(iwt) dt, ii(w, 5, a) is the envelope after the linear compressor, and c$( o, a) is the delay function of the compressor. In this work we assume that the delay line consists of a grating pair with the (approximate) delay function +(~,a) = uo2, and therefore called a quadratic compressor. The parameter u is referred to as the compressor constant [ 1,7] and determined by the grating separation. The compressor constant a
50
M. Karlsson /Optics Communications 112 (1994) 48-54
must be optimized to a value aopt in order to create the shortest possible pulse, u( t, 5, aopt). As we will see below, this optimum choice is by no means trivial. Sometimes a trade-off between duration and quality of the compressed pulse has to be made. Following Ref. [ 71, we consider two measures of performance of the compressor; the compression factor F as being the ratio of the widths of the initial and compressed pulses, and the quality factor Q as being [u(a,,tvO)/ui”(0)]2/F, where Uin(t) is the input pulse in Hq. ( 1) . The quality factor is a measure of the energy contained in the compressed pulse relative to its initial energy. All “widths” are measured in the FWHM-sense. The definition of aopt as corresponding to the maximum peak intensity is numerically convenient and was also used in Ref. [7]. As stated above, we will consider three different input pulse shapes, Gaussian, parabolic and hyperbolic secant (sech) . In order to compare these pulses properly we require them to have the same initial energy and FWHM-width. This gives us the initial conditions Uin(t) = Asech(t),
(3a)
Uin( t) = 1.0324A exp( -t2/2.2413),
(3b)
&n(t)
= ApJ1-oZ.
ItI <
$9
(3c)
where the normalized amplitude A is the only free parameter. The parabolic pulse deserves a special treatment, however. Using the above criterion of equal energy and width, we find A, = 1.097A and tp = 1.2464. However, an artifact of the parabolic pulse is that it compresses (in the FWHM-sense) even when propagating linearly. More quantitatively it can be shown that the pulse (3~) as input in Eq. (2) have compressed a factor 1.864 at a = 0.087ti. Thus, in order to compare the three pulses in a fair way we choose tp = 1.246x 1.864 = 2.323. The energy in the parabola must be the same as for the other pulses, which yields A, = l.O97A/a = 0.803A. With these considerations we have numerically solved Rqs. ( 1) and (2) with the initial conditions (3) in the two cases A = 10 and A = 20. The numerics involved is rather demanding; in the initial, fiber stage the pulse broadens 20-50 times, whereas in the second dispersive delay line it compresses to lo-30 times its initial width. This evidently puts strong demands on the number of meshpoints. A time window
of 100 units (-50 < t < 50) were used, containing 8192 meshpoints. The number of longitudinal steps in the nonlinear stage was 11200. The convergence was checked by doubling the number of meshpoints and halving the longitudinal step-size. Figs. 2 and 3 show F( 5) and Q( 5) in the two cases A = 10 and A = 20. In each case we have calculated 32 discrete values of 4’ up to the maximum fiber lengths 1.92 and 1.12. For short fiber lengths 4’, there are three separate curves, corresponding to each of the initial conditions (3). Sufficiently far into the GVD-dominated regime, however, the Gaussian and sech-curves split up in two branches each. The two branches correspond to two different values of a,rt, because there are two close maxima of the peak intensity with respect to the grating separation a. As a rule, we find the lower value of qpt to give rise to the higher compression factor, and these curves have been labelled Sechl and Gauss1 . The existence of two possible values of aopt is a rather unexpected feature which has not been reported previously. One reason for this is that previous analyses [ 7,8] have not investigated fiber lengths long enough for this phenomenon to emerge. Furthermore, the two different maxima are found rather close to each other; their separation is typically Au M 0.02. The reason for the presence of two maxima is that the spectral center and the spectral wings have different phase modulations, i.e., different values of aopt, see Fig. 4 below. We see from Figs. 2, 3 that the best compressed pulses are the Gaussians (i.e. Gauss1 ) , and that they need long fiber lengths in order to have good quality factors. This is connected with the fact that the Gaussians do not achieve so heavy ringings as the sechpulses when affected by wave breaking. None the less, the sech-pulses do produce as short pulses, although those seem to require even longer fibers in order to achieve good quality. Note also that the trend of the quality of the sech- and Gauss-pulses is ascending; longer fibers might give even better results. It was the branches labelled Sech2 that was reported in Ref. [ 71, and those have, as we see, relatively low compression factors in comparison with the new branches found here. Using fibers approximately ten times the optimum fiber length of Ref. [7] will therefore produce good quality pulses shorter by a factor of M 2. This finding is of utmost importance for the experiments in this field, since the optimum fiber length
hf. Karlsson/Optics CommunicationsI12 (1994) 48-54 F
51
15.0 Q--+J A--+
Sechl SechP Parab. Gauss1
l-----l
A=10
A=10
1
0.80
0.60
0.50
0.40
0.0
‘I”
0.5
.
I
1.0
”
‘I”
1.5
J
2.0 5
Fig. 2. Compression factor F and quality factor Q for the different pulses of amplitude A = 10. The two branches for the sech and Gaussian pulses represent two different grating separations, aopt.
from Ref. [7] has been used in several experiments, see e.g. Ref. [ 41. Considering the parabolic pulses, their quality is rather constant over the shown interval, the reason be-
ing that these pulses do not change shape during the propagation through the fiber [ lo]. The quality factor starts at 0.73, as a result of its linear compression. The compression factors are not as good as the
52
M. Karlsson / Optics Communications 112 (1994) 48-54
F
30.0
25.0
0.6
0.8
A=20
0.2
0.4
0.6
0.8
1 .o
1.2
c Fig. 3. As in Fig. 2, but with A = 20.
Gaussl/Sechl-curves, but we wish to emphasize that if we neglected the linear compression and the initial width of the parabolic pulses where chosen to be the FWHM of the other pulses, then it would yield values of F higher by approximately a factor of 1.9, which
would be better than the smooth predicted in Ref. [lo]. We will also make a brief of the achievable compression totic spectral broadening of the
pulses. This was also analytical estimation factor. The asympinput pulse Uin( t) =
M. Karlsson/Optics CommunicationsI12 (1994) 48-54
53
Fig. 4. The second derivative of the spectral phase, arg[ E(w) ] ow, of a sech-pulse having A = 20 and t = 0.70. This function determines the delay function of the ideal compressor. The two dashed lines shows how the two optimum quadratic compressors approximate the ideal compressor. The spectral intensity distribution [_!I(o) 1’ is included for convenience.
A sech(t)
in Eq. ( 1) is
F = J1+z;;ii-
+CU
Aw2(z. + 00) Au2(z
=0)
=
w2]ii(z + oo, w) 12do
+CO X
-I
>
(s
w21fi(z=O,w)l’dw
-co
=1+2A2
(4)
which is easily obtained from the third invariant of Eq. (l),
Z3 = ~,&,y&l’
+ ]uj4dt
-co +CO
=
s
w21ii(w))2dw +
-cc
+CU
J
Ju14dt
(5)
-cc
and the fact that the integral over Iu I4 vanishes as z + co, [ 121. Since the minimum obtainable pulse width (the transform limited width) is the inverse spectral width, it follows that the maximum obtainable compression factor is
= 1.41A,
(6)
for A2 z$ 1, and that this value is reached asymptotically for large 5 (i.e. long fibers). It therefore seems possible to achieve much higher compression ratios than the value F S 0.63A calculated for short fibers and quadratic compressors in Ref. [ 71. The high compression factor 6 is obtained from the so called ideal compressor, in which the linear delay line completely cancels the spectral phase modulation of the pulse. Thus, the delay function of the ideal compressor is defined by &d(W,y) + arg[C(w,[)] = CIW + CO, where the constants co and ci are trivial phase- and frequency-shifts of the compressed pulse. Although we do not consider ideal compressors here, our numerical finding, F ‘2 1.1 A, indicates that the qudratic compressor is closer to the ideal for sufficiently long fibers. This is also clearly seen in Fig. 4, which shows the second derivative of the spectral phase function. This important quantity defines the delay function of the ideal compressor. The dashed lines correspond to the two optimum quadratic compressors. The fiberbroadened pulse spectrum has a slightly higher aopt in its central part than in the spectral wings. The lower value of aopt cancels the phase modulation of the wings (the Sechl-branch) and the higher value is required
54
M. Karlsson/ OpticsCommunications112 (1994) 48-54
to cancel the central phase modulation (the SechZ branch). The shortest of the two pulses is the one containing the highest frequency components, i.e. the Seth 1-branch. In conclusion, we have shown numerically that fiber-grating compressors may be much more efficient than previous calculations have shown. This requires, however, longer fibers than previously suggested. The fiber lengths are not very critical, provided that they are long enough, typically 10 times the previously suggested optimum length; in our notation the normalized fiber length should be 4’ > 25/A. The results for Gaussian and sech-pulses are qualitatively the same, although the Gaussians in general give rise to compressed pulses of better quality, due to their tendency of less wavebreaking. Compression factors closer to the ideal value is thus obtained, and explained by the fact that the optimum grating separation is different for different spectral parts. This finding will have important implications for further idealizations of the dispersive delay in future compressors.
[ll [21 [31
[41
[51 161 171 [81 I91 1101 1111 [121
Gl? Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989) Ch. 6.3, and references therein. H. Nakatsuka, D. Grischkowsky and AC. Balant, Phys. Rev. Lett. 47 (1981) 910. C.V. Shank, R.L. Fork, R. Yen, R.H. Stolen and W.J. Tomlinson, Appl. Phys. I&t. 40 (1982) 761; J.G. Fujimoto, A.M. Weiner and E.I? Ippen, Appl. Phys. Lett. 44 (1984) 832; J.M. Halbout and D. Grischkowsky, Appl. Phys. Lett. 45 (1984) 1281. W.H. Knox, R.L. Fork, M.C. Downer, R.H. Stolen, C.V. Shank and J.A. Valdmanis, Appl. Phys. Lett. 46 (1985) 1120. R.L. Fork, C.H. Brito CNZ, I?C. Becker and C.V. Shank, Optics Lett. 12 (1987) 483. D. Grischkowsky and A.C. Balant, Appl. Phys. Lett. 41 (1982) 1. W.J. Tomlinson, R.H. Stolen and C.V. Shank, J. Opt. Sot. Am. B 1 (1984) 139. D. Mestdagh, Appl. Optics 26 (1987) 5234. D. Anderson, M. Desaix, M. Lisak and M.L. QuirogaTeixeiro, J. Opt. Sot. Am. B 9 (1992) 1358. D. Anderson, M. Desaix, M.Karlsson, M. Lisak and M.L. Quiroga-Teixeiro, J. Opt. Sot. Am. B 10 (1993) 1187. W.J. Tomlinson, R.H. Stolen and A.M. Johnson, Optics I.&. 10 (1985) 457. D. Anderson, M. Lisak and T. Reichel, Optics Lett. 13 (1988) 285.
OPTICS COMMUNICATIONS Optics Communications 112 (1994) 48-54
Optical fiber-grating compressors utilizing long fibers Magnus Karlsson institute for Electromagnetic Field Theory, Chalmers University of Technology, S-412 96 Giiteborg, Sweden
Received 6 June 1994; revised manuscript
received
19 August
1994
Abstract
The properties of optical fiber-grating compressors have been investigated numerically, considering fibers ten times longer than in previous studies. It is found that such long fibers yield substantially better compression than the previously investigated short ones. Also considerde are the sensitivity of pulse shape on the compression by using Gaussian-, parabolic- and sechshaped pulses as initial conditions.
When compressing optical pulses into femtosecond durations, the most important method is the two-stage (or fiber-grating) compression scheme. The underlying idea is simple in principle: an initial, nonlinear, stage which broadens the pulse in the spectral domain is followed by a linear delay line which compresses the pulse to its transform-limited width [ 11. In 1981, Nakatsuka et al. [2] showed that single-mode optical fibers, operating in the normal-dispersion regime, could be used as the initial nonlinear stage. A number of record-breaking experiments followed [3,4] and culminated with the experiment by Fork et al. [5] in which the shortest optical pulse to date (6 fs) was produced. The theoretical papers dealing with fiber-grating compressors are almost entirely based on numerical computations [ 6-81. Grischkowsky and Balant [ 61 noted that the combined action of self-phase modulation (SPM) and group velocity dispersion (GVD) in the fiber produces pulses of better quality than if SPM alone was to affect the pulses. In an extensive analysis by Tomlinson et al. [ 71, an optimum fiber length was found, which, for a given input pulse, maximized the compression factor. An extension to consider initially
0030-4018/94/$07.00 @ 1994 Elsevier Science B.V. All rights reserved SSDIOO30-4018(94)00470-6
chirped pulses were found to give only minor quantitative modifications on the compressor properties [ 81. The optimum fiber length found in Ref. [ 71 can be explained qualitatively by the theory of wave breaking in normally dispersive fibers [9-l 11. The wave breaking distance is defined as the propagation distance at which the pulse loses its single pulse shape and achieves sidelobes. This distance is an approximate measure of the transition from an inital, SPM-dominated (nonlinear) propagation to a GVD-dominated (linear) one. It has been pointed out that the optimal fiber length found in Ref. [7] is approximately twice the wave breaking distance [ 6,101, indicating that the pulse can take some amount of wave breaking before the compression properties start to deteriorate. Note, however, that this result has been shown only for sech-shaped pulses. Other pulse shapes have not been studied, although one would expect no qualitative differences for e.g. Gaussians. It was suggested in Ref. [ lo] that pulses of parabolic shape would have good compression properties, since they are approximately wave-breaking-free, i.e. they preserve their shape during the propagation through the fiber. Since all previous results concern sech-shaped pulses, it might be valuable to examine
M. Karlsson/Optics Communications112 (1994) 48-54
49
(1)
R-
b -
Fig. 1. Intensity lu(z, t) I* arising from the input pulse u( 0, I) = 20sech( t) in IQ. ( I). At short distances there are heavy oscillations of the wings (wave-breaking), but this vanishes at longer distances. Asymptotically as z + co, the distribution aproaches a trapezium. The instantaneous frequency ~9arg[ u] /at of the pulse of (a) at z = 0.56 is approximately linear over the whole body of the pulse. The intensity distribution is also plotted for convenience.
also Gaussian and parabolic pulses. Although it can be experimentally difficult to specify a certain pulse shape, it is of theoretical interest to see how the compresssion is affected by different initial pulse shapes. More important, however, is that we have extended the analysis to consider longer fiber lengths than studied in Refs. [ 7,8], with some rather unexpected results. The slowly varying envelope u( z, t) of an optical pulse in a fiber is described by the nonlinear Schriidinger equation, taking fiber nonlinearity and normal GVD into account ’ : 1 For relation to physical units we refer to Refs. [ 11 and [7]. The normalizations of this work only differs with Ref. [ 71 in the definition of the fiber length 5 : 5 = (v/2) (z/& where ( Z/ZQ) is used in Ref. [ 71.
Here t denotes the time in a comoving system with t = 0 at the pulse center, z is the axial distance along the fiber, and 4’ is the fiber length. Pulses governed by this equation will broaden monotonically during propagation. Gaussian- and sech-pulses will also change shape, and approximately at the wave breaking distance they become rather rectangular, accompanied with a quadratic phase variation (chirp) and ringings on the pulse wings, see Fig. la. It has been pointed out that the ringing period is of the order of the compressed pulse width [ 111. As the pulse propagates further into the GVD-dominated regime, the ringings gradually disappear, and a pulse having a chirped, trapezium-shaped monotonic intensity distribution emerges. Thus, the oscillations induced by the wave-breaking vanishes at longer distances. The first pulse of Fig. 1 (at z = 0.14) is approximately at the optimum fiber distance for compression predicted in Ref. [ 71 (z = 0.12). We see that it has steep, ringing slopes. The last pulse (at z = 1.05) has a smoother envelope, however, and might therefore be better suited for compression. This is also supported by Fig. lb, showing that the instantaneous frequency varies approximately linearly across the entire pulse body. Another interesting, and to my knowledge unexplained, fact is that this long-distance temporal distribution of pulse intensity and phase becomes similar to the asymptotic spectral distribution. This is true for the parabolic pulse as well. It was shown in Ref. [ lo] that an intense parabolic pulse preserves the parabolic shape during propagation, and that its spectrum becomes parabolic in the limit z + 00. Since the propagation through the delay line is linear it is easily carried out in the Fourier plane: C(w, 5, a) = fi(w, 5) exp[@(w,a)
I,
(2)
where ii(w) denotes the Fourier transform J u(t) exp(iwt) dt, ii(w, 5, a) is the envelope after the linear compressor, and c$( o, a) is the delay function of the compressor. In this work we assume that the delay line consists of a grating pair with the (approximate) delay function +(~,a) = uo2, and therefore called a quadratic compressor. The parameter u is referred to as the compressor constant [ 1,7] and determined by the grating separation. The compressor constant a
50
M. Karlsson /Optics Communications 112 (1994) 48-54
must be optimized to a value aopt in order to create the shortest possible pulse, u( t, 5, aopt). As we will see below, this optimum choice is by no means trivial. Sometimes a trade-off between duration and quality of the compressed pulse has to be made. Following Ref. [ 71, we consider two measures of performance of the compressor; the compression factor F as being the ratio of the widths of the initial and compressed pulses, and the quality factor Q as being [u(a,,tvO)/ui”(0)]2/F, where Uin(t) is the input pulse in Hq. ( 1) . The quality factor is a measure of the energy contained in the compressed pulse relative to its initial energy. All “widths” are measured in the FWHM-sense. The definition of aopt as corresponding to the maximum peak intensity is numerically convenient and was also used in Ref. [7]. As stated above, we will consider three different input pulse shapes, Gaussian, parabolic and hyperbolic secant (sech) . In order to compare these pulses properly we require them to have the same initial energy and FWHM-width. This gives us the initial conditions Uin(t) = Asech(t),
(3a)
Uin( t) = 1.0324A exp( -t2/2.2413),
(3b)
&n(t)
= ApJ1-oZ.
ItI <
$9
(3c)
where the normalized amplitude A is the only free parameter. The parabolic pulse deserves a special treatment, however. Using the above criterion of equal energy and width, we find A, = 1.097A and tp = 1.2464. However, an artifact of the parabolic pulse is that it compresses (in the FWHM-sense) even when propagating linearly. More quantitatively it can be shown that the pulse (3~) as input in Eq. (2) have compressed a factor 1.864 at a = 0.087ti. Thus, in order to compare the three pulses in a fair way we choose tp = 1.246x 1.864 = 2.323. The energy in the parabola must be the same as for the other pulses, which yields A, = l.O97A/a = 0.803A. With these considerations we have numerically solved Rqs. ( 1) and (2) with the initial conditions (3) in the two cases A = 10 and A = 20. The numerics involved is rather demanding; in the initial, fiber stage the pulse broadens 20-50 times, whereas in the second dispersive delay line it compresses to lo-30 times its initial width. This evidently puts strong demands on the number of meshpoints. A time window
of 100 units (-50 < t < 50) were used, containing 8192 meshpoints. The number of longitudinal steps in the nonlinear stage was 11200. The convergence was checked by doubling the number of meshpoints and halving the longitudinal step-size. Figs. 2 and 3 show F( 5) and Q( 5) in the two cases A = 10 and A = 20. In each case we have calculated 32 discrete values of 4’ up to the maximum fiber lengths 1.92 and 1.12. For short fiber lengths 4’, there are three separate curves, corresponding to each of the initial conditions (3). Sufficiently far into the GVD-dominated regime, however, the Gaussian and sech-curves split up in two branches each. The two branches correspond to two different values of a,rt, because there are two close maxima of the peak intensity with respect to the grating separation a. As a rule, we find the lower value of qpt to give rise to the higher compression factor, and these curves have been labelled Sechl and Gauss1 . The existence of two possible values of aopt is a rather unexpected feature which has not been reported previously. One reason for this is that previous analyses [ 7,8] have not investigated fiber lengths long enough for this phenomenon to emerge. Furthermore, the two different maxima are found rather close to each other; their separation is typically Au M 0.02. The reason for the presence of two maxima is that the spectral center and the spectral wings have different phase modulations, i.e., different values of aopt, see Fig. 4 below. We see from Figs. 2, 3 that the best compressed pulses are the Gaussians (i.e. Gauss1 ) , and that they need long fiber lengths in order to have good quality factors. This is connected with the fact that the Gaussians do not achieve so heavy ringings as the sechpulses when affected by wave breaking. None the less, the sech-pulses do produce as short pulses, although those seem to require even longer fibers in order to achieve good quality. Note also that the trend of the quality of the sech- and Gauss-pulses is ascending; longer fibers might give even better results. It was the branches labelled Sech2 that was reported in Ref. [ 71, and those have, as we see, relatively low compression factors in comparison with the new branches found here. Using fibers approximately ten times the optimum fiber length of Ref. [7] will therefore produce good quality pulses shorter by a factor of M 2. This finding is of utmost importance for the experiments in this field, since the optimum fiber length
hf. Karlsson/Optics CommunicationsI12 (1994) 48-54 F
51
15.0 Q--+J A--+
Sechl SechP Parab. Gauss1
l-----l
A=10
A=10
1
0.80
0.60
0.50
0.40
0.0
‘I”
0.5
.
I
1.0
”
‘I”
1.5
J
2.0 5
Fig. 2. Compression factor F and quality factor Q for the different pulses of amplitude A = 10. The two branches for the sech and Gaussian pulses represent two different grating separations, aopt.
from Ref. [7] has been used in several experiments, see e.g. Ref. [ 41. Considering the parabolic pulses, their quality is rather constant over the shown interval, the reason be-
ing that these pulses do not change shape during the propagation through the fiber [ lo]. The quality factor starts at 0.73, as a result of its linear compression. The compression factors are not as good as the
52
M. Karlsson / Optics Communications 112 (1994) 48-54
F
30.0
25.0
0.6
0.8
A=20
0.2
0.4
0.6
0.8
1 .o
1.2
c Fig. 3. As in Fig. 2, but with A = 20.
Gaussl/Sechl-curves, but we wish to emphasize that if we neglected the linear compression and the initial width of the parabolic pulses where chosen to be the FWHM of the other pulses, then it would yield values of F higher by approximately a factor of 1.9, which
would be better than the smooth predicted in Ref. [lo]. We will also make a brief of the achievable compression totic spectral broadening of the
pulses. This was also analytical estimation factor. The asympinput pulse Uin( t) =
M. Karlsson/Optics CommunicationsI12 (1994) 48-54
53
Fig. 4. The second derivative of the spectral phase, arg[ E(w) ] ow, of a sech-pulse having A = 20 and t = 0.70. This function determines the delay function of the ideal compressor. The two dashed lines shows how the two optimum quadratic compressors approximate the ideal compressor. The spectral intensity distribution [_!I(o) 1’ is included for convenience.
A sech(t)
in Eq. ( 1) is
F = J1+z;;ii-
+CU
Aw2(z. + 00) Au2(z
=0)
=
w2]ii(z + oo, w) 12do
+CO X
-I
>
(s
w21fi(z=O,w)l’dw
-co
=1+2A2
(4)
which is easily obtained from the third invariant of Eq. (l),
Z3 = ~,&,y&l’
+ ]uj4dt
-co +CO
=
s
w21ii(w))2dw +
-cc
+CU
J
Ju14dt
(5)
-cc
and the fact that the integral over Iu I4 vanishes as z + co, [ 121. Since the minimum obtainable pulse width (the transform limited width) is the inverse spectral width, it follows that the maximum obtainable compression factor is
= 1.41A,
(6)
for A2 z$ 1, and that this value is reached asymptotically for large 5 (i.e. long fibers). It therefore seems possible to achieve much higher compression ratios than the value F S 0.63A calculated for short fibers and quadratic compressors in Ref. [ 71. The high compression factor 6 is obtained from the so called ideal compressor, in which the linear delay line completely cancels the spectral phase modulation of the pulse. Thus, the delay function of the ideal compressor is defined by &d(W,y) + arg[C(w,[)] = CIW + CO, where the constants co and ci are trivial phase- and frequency-shifts of the compressed pulse. Although we do not consider ideal compressors here, our numerical finding, F ‘2 1.1 A, indicates that the qudratic compressor is closer to the ideal for sufficiently long fibers. This is also clearly seen in Fig. 4, which shows the second derivative of the spectral phase function. This important quantity defines the delay function of the ideal compressor. The dashed lines correspond to the two optimum quadratic compressors. The fiberbroadened pulse spectrum has a slightly higher aopt in its central part than in the spectral wings. The lower value of aopt cancels the phase modulation of the wings (the Sechl-branch) and the higher value is required
54
M. Karlsson/ OpticsCommunications112 (1994) 48-54
to cancel the central phase modulation (the SechZ branch). The shortest of the two pulses is the one containing the highest frequency components, i.e. the Seth 1-branch. In conclusion, we have shown numerically that fiber-grating compressors may be much more efficient than previous calculations have shown. This requires, however, longer fibers than previously suggested. The fiber lengths are not very critical, provided that they are long enough, typically 10 times the previously suggested optimum length; in our notation the normalized fiber length should be 4’ > 25/A. The results for Gaussian and sech-pulses are qualitatively the same, although the Gaussians in general give rise to compressed pulses of better quality, due to their tendency of less wavebreaking. Compression factors closer to the ideal value is thus obtained, and explained by the fact that the optimum grating separation is different for different spectral parts. This finding will have important implications for further idealizations of the dispersive delay in future compressors.
[ll [21 [31
[41
[51 161 171 [81 I91 1101 1111 [121
Gl? Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989) Ch. 6.3, and references therein. H. Nakatsuka, D. Grischkowsky and AC. Balant, Phys. Rev. Lett. 47 (1981) 910. C.V. Shank, R.L. Fork, R. Yen, R.H. Stolen and W.J. Tomlinson, Appl. Phys. I&t. 40 (1982) 761; J.G. Fujimoto, A.M. Weiner and E.I? Ippen, Appl. Phys. Lett. 44 (1984) 832; J.M. Halbout and D. Grischkowsky, Appl. Phys. Lett. 45 (1984) 1281. W.H. Knox, R.L. Fork, M.C. Downer, R.H. Stolen, C.V. Shank and J.A. Valdmanis, Appl. Phys. Lett. 46 (1985) 1120. R.L. Fork, C.H. Brito CNZ, I?C. Becker and C.V. Shank, Optics Lett. 12 (1987) 483. D. Grischkowsky and A.C. Balant, Appl. Phys. Lett. 41 (1982) 1. W.J. Tomlinson, R.H. Stolen and C.V. Shank, J. Opt. Sot. Am. B 1 (1984) 139. D. Mestdagh, Appl. Optics 26 (1987) 5234. D. Anderson, M. Desaix, M. Lisak and M.L. QuirogaTeixeiro, J. Opt. Sot. Am. B 9 (1992) 1358. D. Anderson, M. Desaix, M.Karlsson, M. Lisak and M.L. Quiroga-Teixeiro, J. Opt. Sot. Am. B 10 (1993) 1187. W.J. Tomlinson, R.H. Stolen and A.M. Johnson, Optics I.&. 10 (1985) 457. D. Anderson, M. Lisak and T. Reichel, Optics Lett. 13 (1988) 285.