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Spectral narrowing of negatively chirped femtosecond pulse by cross-phase modulation in a single-mode optical fiber
Optics Communications 260 (2006) 140–143 www.elsevier.com/locate/optcom
Spectral narrowing of negatively chirped femtosecond pulse by cross-phase modulation in a single-mode optical fiber Lingwei Guo *, Changhe Zhou Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, PR China Graduate School of the Chinese Academy of Sciences, Beijing 100039, PR China Received 11 January 2005; received in revised form 16 August 2005; accepted 4 October 2005
Abstract We demonstrate theoretically that the negatively chirped femtosecond laser pulse can be spectrally narrowed by cross-phase modulation. The new view is well supported by numerical simulation. The negative chirp method in fibers might be useful in all optical wavelength switching applications. 2005 Elsevier B.V. All rights reserved. PACS: 42.81.42.65 Keywords: Cross-phase modulation spectral compression
It is well known that self-phase modulation (SPM) or cross-phase modulation (XPM) will occur when the effective refractive index governing the phase of an optical pulse depends on the intensity of itself or anther co-propagating wave [1]. Generally SPM and XPM result in spectral broadening for pulses that are positive chirped or transform limited [1]. Moreover the effect of XPM can also shift the frequency of the probe pulse [1,2]. However, if the input pulse is negatively chirped, the pulse spectrum will be significantly compressed due to SPM effect [3]. Recently, the method of initially negative chirp have received a great deal of attention in the field of transform-limited pulse producing [4,5], fiber probe [6] and supercontinuum generation [7]. More recently the spectral narrowing induced by degenerate cross-phase modulation (DXPM) has been mentioned and used to pulse temporal measurement [8], but this phenomenon has not been explained clearly in [8]. In this paper, following the works of [2,3], we investigate two same polarization negatively chirped femtosecond optical pulses of different wavelength propagating in a single-mode opti*
Corresponding author. Tel./fax: +86 216 991 8213. E-mail address: [email protected] (L. Guo).
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.10.007
cal fiber. We demonstrate theoretically that a pump pulse with initial negative chirp can narrow the spectrum of a weak signal pulse no matter it is initially negatively chirped or not. We also investigate the frequency shift of the negatively chirped femtosecond laser pulse caused by the XPM. The method we present here is more practical for permitting all optical switch applications in fibers. According to [1], the coupled nonlinear wave equations describing two same polarization nonlinear pulses of different wavelength propagation in single-mode fiber can be written as oU 1 i o2 U 1 þ iN 2 ðjU 1 j2 þ 2jU 2 j2 ÞU 1 ¼ 0; ð1aÞ 2 os2 on oU 2 LD oU 2 i b002 o2 U 2 x2 þ þ iN 2 ðjU 2 j2 þ 2jU 1 j2 ÞU 2 ¼ 0; 2 b001 os2 on Lw os x1 ð1bÞ where Uj is the normalized complex envelope. U1 is pump pulse and U2 is the probe pulses. b00j are group velocity T vg1 vg2 dispersions. LD ¼ T 20 =b00j , s = t/T0, n = z/LD, Lw ¼ vg10 v ; g2 vgj is the group velocities. N is nonlinear factor and N2 = c1P1LD. P1 is initial peak power of the pump. For the
L. Guo, C. Zhou / Optics Communications 260 (2006) 140–143
maximum fiber length we investigate is 60 cm, the third order and higher order dispersion can be neglected. We assume the intensities of the pump much greater than that of the probe: pffiffiffiffi P1 P2, and the ratio are taken to 10. So uð0; tÞ ¼ N sechðTt0 Þ; vð0; tÞ ¼ uð0; tÞ=10. In order to investigate the spectral compression, we need initially negatively chirped pulse, which can be easily done by adding a prism pair [3,4]. So the negative chirped electric fields from prism pairs Uj(x) can be calculated with t U 1 ðxÞ ¼ F N sech expðiðu12 x2 =2 þ u13 x3 =6ÞÞ; T0 N t sech U 2 ðxÞ ¼ F expðiðu22 x2 =2 þ u23 x3 =6ÞÞ; 10 T0 ð2Þ where N sechðTt0 Þ is the initial electric field and F denotes the Fourier transform. Following [3], according to the equations in Table 1 of [9], uj2 and uj3 can be chosen to be u12 = u22 = 4.2 · 104 fs2, u13 = u23 = 33.6 · 104 fs3 without losing generality. Here, we also assume the pump and probe pulses are injected into fiber at same time for simplicity. We now solve the Eq. (1), using the split-step Fourier method. We take the parameters given in the experiment of [2]: k1 = 795 nm, k2 = 814 nm, T0 = 130 fs, LD = 0.4, LD/Lw = 6, b001 =b002 1, and we choose the initial peak power P1 to make the nonlinear factor N = 2.5. In Fig. 1, we show that the spectral shape of probe is compressed significantly at the fiber length 32 cm. In order to compare three curves clearly, the peak values of three curves are set at the wavelength of 814 nm. According to the dashed line in Fig. 1(a), when only the probe propagation in the single-mode fiber is present, the spectral of the probe (dashed line) is almost the same as the input probe (dotted line). It is understandable for the intensity of the probe is too weak to induce any significant spectral compression. When the probe and the pump co-propagate in a fiber, the pump will induce strong XPM effect for the probe component. So the significant spectral compression a
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must be caused by XPM. This view is consistent with the result of [3], for SPM and XPM are all caused by the Kerr effect. In Fig. 1(b), we consider another situation where kPump is 1.06 lm, and kProbe is 1.55 lm [10]. b001 ¼ 200 fs2 =cm, LD/Lw = 0, b002 ¼ 200 fs2 =cm, and the other parameters are the same with Fig. 1(a). Fig. 1(b) shows the spectral shape of probe is compressed significantly at the fiber length 42.25 cm. There is no frequency shift and spectrum is narrowed symmetrically for the group velocities of the pump and the probe are equal. So our result can also hold under the condition that the pump is in normal dispersion region, while the probe is in anomalous dispersion region. Furthermore, we have considered the other two situations with the same initial negative chirp and pump power, and other parameters are token from [10,11]. We find our result still hold under the situation where the pump is in anomalous dispersion region while the probe is in normal dispersion region, and the situation where the probe and the pump are all in anomalous dispersion region. So our result is a general result. The theoretical spectral shapes for the different pump peak power with the fixed probe peak power are plotted in Fig. 2. The spectral peak of the probe is shifted to Stokes side with the increasing of the peak power of the pump. The theoretical spectral curves for the different fiber length are plotted in Fig. 3. It can be seen that as fiber length increasing, the spectral peak of the probe is shifted to Stokes side. Both Figs. 2 and 3 show that there is also an optimum fiber length for a given negative chirp and pump peak power as reported in [3]. Those shifts of spectral peaks can easily be explained by the XPM effect. According to [2], we know the walk-off effect between the pump and probe is a key term which will affect the output spectral shape caused by XPM. In our simulation, the pump is slower than the probe. Therefore, the probe sees only the leading edge of the pump, the spectrum of the probe is shifted to the Stokes side only. We simulated the reversed situation where the probe is slower than the pump. The obtained anti-Stokes shifts agree well with our explanations. b
Fig. 1. (a) Spectral shape representations for the probe with pump (solid line) N2 = 6.25 and without pump (dashed line) at the fiber length 32 cm. And spectral shape of the input probe (dotted line). (b) Spectral shape representations for the probe with pump (solid line) N2 = 6.25 and without pump (dashed line) at the fiber length 42.25 cm. And spectral shape of the input probe (dotted line). There is no frequency shift and spectrum is narrowed symmetrically for the group velocities of the pump and the probe are equal.
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L. Guo, C. Zhou / Optics Communications 260 (2006) 140–143
Fig. 2. (a) Theoretical spectral shapes for different pump power at the fixed fiber length 32 cm with u12 = u22 = 4.2 · 104 fs2, u13 = u23 = 33.6 · 104 fs3. (b) Theoretical spectral shapes at different fiber lengths with a fixed pump power (N2 = 6.25), and same initial negative chirp value as (a).
Fig. 3. (a) Spectral shapes representations for the input probe (dash line), and the probe (solid line) at the fiber length 24 cm, where there are no initial negative chirp given to the pump and the probe. (b) Spectral shapes representations for the input probe (dash line), and the probe (solid line) at the fiber length 60 cm where only the pump are given a initial negative chirp with u12 = 4.2 · 104 fs2, u13 = 33.6 · 104 fs3. The peak point of the XPM shifted probe is set at the same wavelength with the peak point of the input probe for comparison.
We consider another situation where only the pump is given an initially negative chirp. We choose a higher value for N: N = 7.5, the length of fiber Lfiber = 60 cm, and keep others parameters of Eqs. (2) with u12 = 4.2 · 104 fs2, u22 = 0, u13 = 33.6 · 104 fs3, u23 = 0. In order to present a clear discussion, we compare the result with the result where no initial negative chirp is given to the pump and the probe. In Fig. 3(a), it is a well known result that the XPM effect broaden the spectrum of the probe when the pump and the probe have not been initially negatively chirped [2,9], while in Fig. 3(b) it can be found that even though the probe is not given an initial negative chirp, the spectrum of the probe can still been narrowed by the XPM effect once the pump is given an initial negative chirp(in order to compare three curves clearly, the peak values of curves are set at the wavelength of 814 nm. The wavelength shift induced by the XPM effect is plotted in Fig. 4). It can be easily explained by Eq. (1) that a negative chirp is transported to the probe from the pump by the
Fig. 4. Spectral shapes representations for the input probe (dash line), and the probe (solid line) at the fiber length 60 cm where only the pump are given a initial negative chirp with N = 7.5, u12 = 4.2 · 104 fs2, u13 = 33.6 · 104 fs3. The quality of wavelength switching is 11 nm.
L. Guo, C. Zhou / Optics Communications 260 (2006) 140–143
XPM effect firstly, then the spectrum of probe is narrowed by the XPM effect. According to [2,11], it is known the XPM effect can also shift the spectrum of the probe. In Fig. 4, it shows that a red shift as large as 11 nm can be achieved by the XPM effect. It should be noted that unlike the result of [2,10], the XPM effect compress the spectral of the probe as well as shift the spectral peak of it to the Stokes side. There is almost no spectral overlap between the XPM shift probe and the input probe in Fig. 4. So it is easy to distinguish them. Moreover compared with [2], negative chirp method can make the energy of the probe concentrate to the wavelength region that the probe is shifted to, which means the efficiency of frequency shift is much higher than [2]. Therefore negative chirp method is a more practical method than [2] for all optical switching applications in fibers. In conclusion, spectral compressing through XPM for initially negatively chirped femtosecond laser pulses has been demonstrated in a single-mode optical fiber. Stokes and anti-Stokes frequency shifts of initially negatively chirped laser pulses is attributed to the combined effect of XPM and walk-off effect of the pump and probe pulses in femtosecond pulse propagation. Large frequency shift can be obtained by choosing a proper peak power for the pump and proper initially negative chirps for the pump and the probe, which suggests a more practical method for all optical switching applications in fibers.
143
Acknowledgments The authors acknowledge the support of National Outstanding Youth Foundation of China (60125512) and Shanghai Science and Technology Committee (036105013, 03XD14005). References [1] G.P. Agrawal, Nonlinear Fiber Optics, third ed., Academic, New York, 2001 (Chapters 6 and 7). [2] G.R. Boyer, M.A. Franco, M. Lachgar, B. Gre`zes-Besset, A. Alexandrou, J. Opt. Soc. Am. B 8 (1994) 1451. [3] M. Oberthaler, R.A. Ho¨pfel, Appl. Phys. Lett. 63 (1993) 1017. [4] B.R. Washburn, J.A. Buck, S.E. Ralph, Opt. Lett. 25 (2000) 445. [5] J. Limpert, T. Gabler, A. Liem, H. Zellmer, A. Tu¨nnermann, Appl. Phys. B 74 (2002) 191. [6] M.T. Myaing, J. Urayama, A. Braun, T.B. Norris, Opt. Express 7 (2002) 210. [7] J.W. Nicholson, A.D. Yablon, P.S. Westbrook, K.S. Feder, M.F. Yan, Opt. Express 12 (2004) 3025. [8] L.K. Mouradian, Z. Zohrabyan. Available from:. [9] R.L. Fork, C.H. Brito Cruz, P.C. Becker, C.V. Shank, Opt. Lett. 7 (1987) 483. [10] G.P. Agrawal, P.L. Baldeck, R.R. Alfano, Phys. Rev. A 40 (1989) 5063. [11] C. Yeh, L. Bergman, Phys. Rev. E 57 (1998) 2398.
Spectral narrowing of negatively chirped femtosecond pulse by cross-phase modulation in a single-mode optical fiber Lingwei Guo *, Changhe Zhou Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, PR China Graduate School of the Chinese Academy of Sciences, Beijing 100039, PR China Received 11 January 2005; received in revised form 16 August 2005; accepted 4 October 2005
Abstract We demonstrate theoretically that the negatively chirped femtosecond laser pulse can be spectrally narrowed by cross-phase modulation. The new view is well supported by numerical simulation. The negative chirp method in fibers might be useful in all optical wavelength switching applications. 2005 Elsevier B.V. All rights reserved. PACS: 42.81.42.65 Keywords: Cross-phase modulation spectral compression
It is well known that self-phase modulation (SPM) or cross-phase modulation (XPM) will occur when the effective refractive index governing the phase of an optical pulse depends on the intensity of itself or anther co-propagating wave [1]. Generally SPM and XPM result in spectral broadening for pulses that are positive chirped or transform limited [1]. Moreover the effect of XPM can also shift the frequency of the probe pulse [1,2]. However, if the input pulse is negatively chirped, the pulse spectrum will be significantly compressed due to SPM effect [3]. Recently, the method of initially negative chirp have received a great deal of attention in the field of transform-limited pulse producing [4,5], fiber probe [6] and supercontinuum generation [7]. More recently the spectral narrowing induced by degenerate cross-phase modulation (DXPM) has been mentioned and used to pulse temporal measurement [8], but this phenomenon has not been explained clearly in [8]. In this paper, following the works of [2,3], we investigate two same polarization negatively chirped femtosecond optical pulses of different wavelength propagating in a single-mode opti*
Corresponding author. Tel./fax: +86 216 991 8213. E-mail address: [email protected] (L. Guo).
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.10.007
cal fiber. We demonstrate theoretically that a pump pulse with initial negative chirp can narrow the spectrum of a weak signal pulse no matter it is initially negatively chirped or not. We also investigate the frequency shift of the negatively chirped femtosecond laser pulse caused by the XPM. The method we present here is more practical for permitting all optical switch applications in fibers. According to [1], the coupled nonlinear wave equations describing two same polarization nonlinear pulses of different wavelength propagation in single-mode fiber can be written as oU 1 i o2 U 1 þ iN 2 ðjU 1 j2 þ 2jU 2 j2 ÞU 1 ¼ 0; ð1aÞ 2 os2 on oU 2 LD oU 2 i b002 o2 U 2 x2 þ þ iN 2 ðjU 2 j2 þ 2jU 1 j2 ÞU 2 ¼ 0; 2 b001 os2 on Lw os x1 ð1bÞ where Uj is the normalized complex envelope. U1 is pump pulse and U2 is the probe pulses. b00j are group velocity T vg1 vg2 dispersions. LD ¼ T 20 =b00j , s = t/T0, n = z/LD, Lw ¼ vg10 v ; g2 vgj is the group velocities. N is nonlinear factor and N2 = c1P1LD. P1 is initial peak power of the pump. For the
L. Guo, C. Zhou / Optics Communications 260 (2006) 140–143
maximum fiber length we investigate is 60 cm, the third order and higher order dispersion can be neglected. We assume the intensities of the pump much greater than that of the probe: pffiffiffiffi P1 P2, and the ratio are taken to 10. So uð0; tÞ ¼ N sechðTt0 Þ; vð0; tÞ ¼ uð0; tÞ=10. In order to investigate the spectral compression, we need initially negatively chirped pulse, which can be easily done by adding a prism pair [3,4]. So the negative chirped electric fields from prism pairs Uj(x) can be calculated with t U 1 ðxÞ ¼ F N sech expðiðu12 x2 =2 þ u13 x3 =6ÞÞ; T0 N t sech U 2 ðxÞ ¼ F expðiðu22 x2 =2 þ u23 x3 =6ÞÞ; 10 T0 ð2Þ where N sechðTt0 Þ is the initial electric field and F denotes the Fourier transform. Following [3], according to the equations in Table 1 of [9], uj2 and uj3 can be chosen to be u12 = u22 = 4.2 · 104 fs2, u13 = u23 = 33.6 · 104 fs3 without losing generality. Here, we also assume the pump and probe pulses are injected into fiber at same time for simplicity. We now solve the Eq. (1), using the split-step Fourier method. We take the parameters given in the experiment of [2]: k1 = 795 nm, k2 = 814 nm, T0 = 130 fs, LD = 0.4, LD/Lw = 6, b001 =b002 1, and we choose the initial peak power P1 to make the nonlinear factor N = 2.5. In Fig. 1, we show that the spectral shape of probe is compressed significantly at the fiber length 32 cm. In order to compare three curves clearly, the peak values of three curves are set at the wavelength of 814 nm. According to the dashed line in Fig. 1(a), when only the probe propagation in the single-mode fiber is present, the spectral of the probe (dashed line) is almost the same as the input probe (dotted line). It is understandable for the intensity of the probe is too weak to induce any significant spectral compression. When the probe and the pump co-propagate in a fiber, the pump will induce strong XPM effect for the probe component. So the significant spectral compression a
141
must be caused by XPM. This view is consistent with the result of [3], for SPM and XPM are all caused by the Kerr effect. In Fig. 1(b), we consider another situation where kPump is 1.06 lm, and kProbe is 1.55 lm [10]. b001 ¼ 200 fs2 =cm, LD/Lw = 0, b002 ¼ 200 fs2 =cm, and the other parameters are the same with Fig. 1(a). Fig. 1(b) shows the spectral shape of probe is compressed significantly at the fiber length 42.25 cm. There is no frequency shift and spectrum is narrowed symmetrically for the group velocities of the pump and the probe are equal. So our result can also hold under the condition that the pump is in normal dispersion region, while the probe is in anomalous dispersion region. Furthermore, we have considered the other two situations with the same initial negative chirp and pump power, and other parameters are token from [10,11]. We find our result still hold under the situation where the pump is in anomalous dispersion region while the probe is in normal dispersion region, and the situation where the probe and the pump are all in anomalous dispersion region. So our result is a general result. The theoretical spectral shapes for the different pump peak power with the fixed probe peak power are plotted in Fig. 2. The spectral peak of the probe is shifted to Stokes side with the increasing of the peak power of the pump. The theoretical spectral curves for the different fiber length are plotted in Fig. 3. It can be seen that as fiber length increasing, the spectral peak of the probe is shifted to Stokes side. Both Figs. 2 and 3 show that there is also an optimum fiber length for a given negative chirp and pump peak power as reported in [3]. Those shifts of spectral peaks can easily be explained by the XPM effect. According to [2], we know the walk-off effect between the pump and probe is a key term which will affect the output spectral shape caused by XPM. In our simulation, the pump is slower than the probe. Therefore, the probe sees only the leading edge of the pump, the spectrum of the probe is shifted to the Stokes side only. We simulated the reversed situation where the probe is slower than the pump. The obtained anti-Stokes shifts agree well with our explanations. b
Fig. 1. (a) Spectral shape representations for the probe with pump (solid line) N2 = 6.25 and without pump (dashed line) at the fiber length 32 cm. And spectral shape of the input probe (dotted line). (b) Spectral shape representations for the probe with pump (solid line) N2 = 6.25 and without pump (dashed line) at the fiber length 42.25 cm. And spectral shape of the input probe (dotted line). There is no frequency shift and spectrum is narrowed symmetrically for the group velocities of the pump and the probe are equal.
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L. Guo, C. Zhou / Optics Communications 260 (2006) 140–143
Fig. 2. (a) Theoretical spectral shapes for different pump power at the fixed fiber length 32 cm with u12 = u22 = 4.2 · 104 fs2, u13 = u23 = 33.6 · 104 fs3. (b) Theoretical spectral shapes at different fiber lengths with a fixed pump power (N2 = 6.25), and same initial negative chirp value as (a).
Fig. 3. (a) Spectral shapes representations for the input probe (dash line), and the probe (solid line) at the fiber length 24 cm, where there are no initial negative chirp given to the pump and the probe. (b) Spectral shapes representations for the input probe (dash line), and the probe (solid line) at the fiber length 60 cm where only the pump are given a initial negative chirp with u12 = 4.2 · 104 fs2, u13 = 33.6 · 104 fs3. The peak point of the XPM shifted probe is set at the same wavelength with the peak point of the input probe for comparison.
We consider another situation where only the pump is given an initially negative chirp. We choose a higher value for N: N = 7.5, the length of fiber Lfiber = 60 cm, and keep others parameters of Eqs. (2) with u12 = 4.2 · 104 fs2, u22 = 0, u13 = 33.6 · 104 fs3, u23 = 0. In order to present a clear discussion, we compare the result with the result where no initial negative chirp is given to the pump and the probe. In Fig. 3(a), it is a well known result that the XPM effect broaden the spectrum of the probe when the pump and the probe have not been initially negatively chirped [2,9], while in Fig. 3(b) it can be found that even though the probe is not given an initial negative chirp, the spectrum of the probe can still been narrowed by the XPM effect once the pump is given an initial negative chirp(in order to compare three curves clearly, the peak values of curves are set at the wavelength of 814 nm. The wavelength shift induced by the XPM effect is plotted in Fig. 4). It can be easily explained by Eq. (1) that a negative chirp is transported to the probe from the pump by the
Fig. 4. Spectral shapes representations for the input probe (dash line), and the probe (solid line) at the fiber length 60 cm where only the pump are given a initial negative chirp with N = 7.5, u12 = 4.2 · 104 fs2, u13 = 33.6 · 104 fs3. The quality of wavelength switching is 11 nm.
L. Guo, C. Zhou / Optics Communications 260 (2006) 140–143
XPM effect firstly, then the spectrum of probe is narrowed by the XPM effect. According to [2,11], it is known the XPM effect can also shift the spectrum of the probe. In Fig. 4, it shows that a red shift as large as 11 nm can be achieved by the XPM effect. It should be noted that unlike the result of [2,10], the XPM effect compress the spectral of the probe as well as shift the spectral peak of it to the Stokes side. There is almost no spectral overlap between the XPM shift probe and the input probe in Fig. 4. So it is easy to distinguish them. Moreover compared with [2], negative chirp method can make the energy of the probe concentrate to the wavelength region that the probe is shifted to, which means the efficiency of frequency shift is much higher than [2]. Therefore negative chirp method is a more practical method than [2] for all optical switching applications in fibers. In conclusion, spectral compressing through XPM for initially negatively chirped femtosecond laser pulses has been demonstrated in a single-mode optical fiber. Stokes and anti-Stokes frequency shifts of initially negatively chirped laser pulses is attributed to the combined effect of XPM and walk-off effect of the pump and probe pulses in femtosecond pulse propagation. Large frequency shift can be obtained by choosing a proper peak power for the pump and proper initially negative chirps for the pump and the probe, which suggests a more practical method for all optical switching applications in fibers.
143
Acknowledgments The authors acknowledge the support of National Outstanding Youth Foundation of China (60125512) and Shanghai Science and Technology Committee (036105013, 03XD14005). References [1] G.P. Agrawal, Nonlinear Fiber Optics, third ed., Academic, New York, 2001 (Chapters 6 and 7). [2] G.R. Boyer, M.A. Franco, M. Lachgar, B. Gre`zes-Besset, A. Alexandrou, J. Opt. Soc. Am. B 8 (1994) 1451. [3] M. Oberthaler, R.A. Ho¨pfel, Appl. Phys. Lett. 63 (1993) 1017. [4] B.R. Washburn, J.A. Buck, S.E. Ralph, Opt. Lett. 25 (2000) 445. [5] J. Limpert, T. Gabler, A. Liem, H. Zellmer, A. Tu¨nnermann, Appl. Phys. B 74 (2002) 191. [6] M.T. Myaing, J. Urayama, A. Braun, T.B. Norris, Opt. Express 7 (2002) 210. [7] J.W. Nicholson, A.D. Yablon, P.S. Westbrook, K.S. Feder, M.F. Yan, Opt. Express 12 (2004) 3025. [8] L.K. Mouradian, Z. Zohrabyan. Available from: