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Phase engineered wavelength conversion of ultra-short optical pulses in TI:PPLN waveguides
Optics Communications 361 (2016) 143–147
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Phase engineered wavelength conversion of ultra-short optical pulses in TI:PPLN waveguides Amin Babazadeh a,n, Rahman Nouroozi a,b, Wolfgang Sohler c a
Physics Department, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran Optics & Photonics Research Center, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran c University of Paderborn, Faculty of Science, Applied Physics, D-33095 Paderborn, Germany b
art ic l e i nf o
a b s t r a c t
Article history: Received 9 May 2015 Received in revised form 4 September 2015 Accepted 19 September 2015
A phase engineered all-optical wavelength converter for ultra-short pulses (down to 140 fs) in a Tidiffused, periodically poled lithium niobate (Ti:PPLN) waveguide is proposed. The phase engineering, due to the phase conjugation between signal and idler (converted signal) pulses which takes place in the cascaded second harmonic generation and difference frequency generation (cSHG/DFG) based wavelength conversion, already leads to shorter idler pulses. The proposed device consists of an unpoled (passive) waveguide section beside of the PPLN waveguide section in order to compensate pulse broadening and phase distortion of the idler pulses induced by the wavelength conversion (in the PPLN section). For example numerical analysis shows that a 140 fs input signal pulse is only broadened by 1.6% in a device with a combination of 20 mm and 6 mm long periodically poled and unpoled waveguide sections. Thus, cSHG/DFG based wavelength converters of a bandwidth of several Tbits/s can be designed. & 2015 Published by Elsevier B.V.
Keywords: Ultra-short optical Pulses Wavelength conversion Difference Frequency generation Periodically poled lithium niobate
1. Introduction All optical wavelength conversion is of great interest for dense wavelength division multiplexing (DWDM) networks [1]. If inherently combined with optical phase conjugation (OPC), it also enables by mid-span wavelength conversion in (long) fiber-optic links a compensation of nonlinear phase noise and dispersion induced signal distortions in the second half of the link; high bitrate, ultra long-haul transmission has been demonstrated using this concept [2]. Among various wavelength conversion approaches nonlinear χ(2) based difference frequency generation (DFG) in a lithium niobate (LiNbO3, LN) waveguide proved to be very attractive [3]. It offers high efficiency, low noise (quantum limited) and high speed (fs response time) simultaneously and is inherently transparent for signal rate and format. In addition, quasi–phase matching (QPM) of DFG in a periodically poled lithium niobate waveguide (PPLN) of specific periodicity allows operating with any wavelength in the transparency range of the LN crystal. Mainly for practical reasons, a cascaded second harmonic generation (SHG) / DFG process (cSHG/DFG) is usually used [3]; it guarantees a spatial mode selective excitation of the (short wavelength) continuous wave (cw) pump by internal phase matched SHG of a cw fundamental wave. Fig. 1 n
Corresponding author. E-mail address: [email protected] (A. Babazadeh).
http://dx.doi.org/10.1016/j.optcom.2015.09.066 0030-4018/& 2015 Published by Elsevier B.V.
schematically shows this cSHG/DFG-based wavelength conversion/OPC process. cSHG/DFG (OPC) based wavelength conversion in PPLN waveguides has been intensively studied in recent years [e.g. 4,5]. It was shown that group velocity mismatch (GVM) and group velocity dispersion (GVD) lead to a temporal walk off and thereby to distortion and broadening of the interacting pulses. However, negligible pulse broadening for 1.4 ps pulses was experimentally observed enabling all optical wavelength conversion of 40, 160 and 320 Gb/s signal data in the C-band [6–8]. It seems that similar to mid-span wavelength conversion with OPC in a (long) fiber link, the distributed wavelength conversion in the (short) PPLN waveguide yields a (partial) compensation of dispersion induced pulse distortions. It should be emphasized that the acceptance bandwidth of a Ti: PPLN wavelength converter with homogeneous domain grating is more than sufficient even for ultra-short signal pulses for the cSHG/DFG approach studied here. This is in contrast to cSFG/DFG based wavelength converters with two pump waves [e.g. 9,10] where the signal pulses undergo two nonlinear processes (SFG and DFG) limiting strongly the acceptance bandwidth by the QPM conditions. Therefore, chirped domain gratings are used to broaden the bandwidth [9,10]. On the other hand, the cSFG/DFG approach allows a wavelength tuning of the idler (converted signal) even for a fixed signal wavelength, which is not possible for the cSHG/DFG approach. Here, the dispersion properties lead via the QPM condition to a narrow bandwidth of the SHG process in
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A. Babazadeh et al. / Optics Communications 361 (2016) 143–147
poled section
jβ ´´ ∂ 2Ai ∂Ai ∂A α + βi´ i − i = − jκ df Ash As* e−jΔdf z − i Ai 2 ∂t 2 2 ∂z ∂t
unpoled section
(4)
f, sh, s and i denote the fundamental, second harmonic, signal and idler fields, respectively. Am (as a function of time t and position z) represents the slowly varying complex amplitude of the electric field m (m¼f, sh, s, i); Fm(x,y) describes the field distribution of mode m within the waveguide cross section. βm´ ( ∂βm /∂ω at ω = ωm ) stands
Fig. 1. Schematic diagram of the input and output waves in a proposed poled and unpoled sections of PPLN waveguide (top figure). The figure below shows a schematic diagram of cSHG/DFG-based wavelength conversion in the poled section of the waveguide. The cw fundamental wave (λf) generates its second harmonic (λsh) which serves as pump wave for the DFG-process with the signal. The resulting idler (λi) is the wavelength-shifted replica of short (Gaussian shaped) signal pulses (λs).
the homogeneous domain grating assumed. However, this is not a problem as no short pulses but a narrow line width cw wave is frequency doubled; therefore, chirped gratings to broaden the bandwidth of the SHG process [e.g. 11,12] are not necessary. Is a design of an inherently dispersion compensated wavelength converter of nominally unlimited bandwidth possible? This question is theoretically investigated in this letter for cSHG/DFG (OPC) based wavelength conversion of 1.4 ps, 200 fs and 140 fs signal pulses in a standard Ti-indiffused PPLN waveguide. After introducing the mathematical methods in section II, the results are presented and discussed in Section 3. The evolution of signal and idler pulses along the waveguide is analyzed in detail by studying pulse widths and phase distributions. Based on the results, an engineered device is proposed to compensate idler pulse broadening and phase distortion as far as possible. This phase engineering by an unpoled (passive) waveguide section causes wavelength conversion of minimum distortion.
2. Device Modeling The following coupled partial differential equations (Eqs. 1–4) describe the interacting waves and their evolution during cSHG/DFG inside a PPLN waveguide [4]. These equations have been obtained using the plane wave and the slowly varying envelope approximations. The latter can be applied if the bandwidth Δω of the pulses to be investigated is significantly smaller than the carrier frequency ω0, i. e. Δω/ω0 « 1. For our simulations we get Δω/ω0 ¼0.0036 (0.036) using pulses of 1.4 ps (140 fs) width, respectively.
E (x, y , z, t ) =
1 2
∑ Am (z, t ) Fm (x, y) exp ( jωt − jβm z) m
+ c. c
for reciprocal group velocities and βm´´ ( ∂ 2βm /∂ω2 at ω = ωm ) for group velocity dispersions. βm are the propagation constants. κsh and κdf are the coupling coefficients of SHG and DFG processes, and we assume κsh E κdf for the case of ωf E ωs. Λ is the period of the domain inverted grating for QPM. Finally, Δsh and Δdf represent the deviations from SHG and DFG quasi phase mismatching conditions, respectively. All the calculations in this paper are done with Δsh ¼ Δdf ¼ 0 ( Δsh = βsh − 2βf − 2Λπ & Δdf = βsh − βs − βi − 2Λπ ) assuming perfect quasi phase matching. The launched fundamental and, consequently, the internally generated pump waves are assumed to be continuous waves (cw); therefore, βf´, βsh´, βf´´ and βsh´´ are negligible. The signal wave is assumed to be a Gaussian pulse at the beginning of the waveguide (z¼ 0):
As (0, t ) =
⎛ −(1 − jC ) t 2 ⎞ ⎟ Ps0 exp ⎜ 2 Ts20 ⎝ ⎠
(5)
where Ps0 is the peak power and Ts0 is related to the Full Width at Half Maximum (FWHM) of the initial signal pulse by the relation τs0 = 2 ln (2) Ts0 . The parameter C is the frequency chirping factor which is zero in our simulation except where otherwise noted. The (slowly varying) phase evolution with position and time along the waveguide can be written as:
⎛ Im ( Am (z, t ) ) ⎞ C (z ) 2 ⎟⎟ = t φm (z, t ) = tan−1 ⎜⎜ 2 Ts0 ⎝ Re (Am (z, t ) ⎠
The coupled Eqs. (1)–(4) cannot be solved analytically. The Split Step Fourier method [13] with parameters listed in Table 1 for the specific waveguide investigated [8] is used to solve them numerically. The refractive index, the group velocity and group velocity dispersion are taken from [14]. It is important to mention that for cSHG/DFG-based wavelength conversion within the C-band, the QPM determined bandwidth for the fundamental wave of the SHG process is relatively narrow (about a nm) [15]. On the other hand, due to the dispersion properties of LN, the bandwidth for the signal wave is broad enough to cover the whole C-band [8]. Here the wavelength of 1551 nm is chosen for the signal wave to enable a comparison with published experimental data [8]. Table 1 Simulation parameters.
jβf´´ ∂ 2Af ∂Af ∂Af α * A*f Ash e−jΔsh z − f Af = − jκsh + βf´ − 2 ∂t 2 2 ∂z ∂t
(1)
jβ´´ ∂ 2Ash ∂Ash ∂Ash + βsh − sh ´ 2 ∂t 2 ∂z ∂t ω α j Δ = − jκsh A*f Af e sh z − j sh κ df* As Ai e jΔdf z − sh Ash 2 ωi
(2)
jβ´´ ∂ 2As ∂As ∂A ω α + βs´ s − s = − j s κ df Ash Ai* e−jΔdf z − s As 2 ∂t 2 2 ∂z ∂t ωi
(3)
(6)
Parameter
Value
Ps0 Pi0 λf λs τs0 αf ¼αs ¼αi αsh κsh Eκdf βf Eβs E βi βsh βs´ Eβi´ βs´´ βi´´ Λ
100 mW 0 mW 1.546 μm 1.551 μm 1.4 ps, 200 fs, 140 fs 0.15 dB/cm 0.3 dB/cm 0.55 1/cmW1/2 8.723 1/μm 17.825 1/μm 7.34 ns/m 110 ps2/km 113 ps2/km 16.6 μm
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3. Results and discussion 3.1. Evolution of signal and idler pulses The evolution of the normalized pulse width of signal and idler within the PPLN waveguide is shown in Fig. 2 as function of the interaction length. The normalization is done by dividing the propagating signal or idler pulse width τ by the initial signal pulse width τs0. Three different input pulses with τs0 ¼ 1.4 ps, 200 fs and 140 fs are investigated; the (cw) fundamental power is 55 mW. In the insets three signal and idler pulses are plotted after propagating 2, 4 and 6 cm, respectively, for τs0 ¼ 140 fs. The FWHM of the signal pulses with τs0 ¼ 1.4 ps remains nearly constant as function of the interaction length (Fig. 2, left). However, it grows significantly for the shorter pulses with τs0 ¼ 200 fs and 140 fs. This is demonstrated in more detail by the temporal shape of the pulses presented in the inset as examples. Also an attenuation of the signal pulses can be observed, which is mainly due to waveguide propagation losses. The attenuation due to nonlinear wavelength conversion is ≤4% at the 55 mW fundamental power level considered here. As a consequence, the evolution of the signal pulses can essentially be understood by linear waveguide dispersion and attenuation. Group Velocity Dispersion (GVD) causes a broadening and a corresponding phase distribution within the pulses which is the larger the shorter the input pulses are. It can be described by the chirping parameter C of a Gaussian pulse (see also Eq. (5)) which propagates inside a waveguide without nonlinear interactions [16]:
C=
z βs′′ 2 Ts0
(7)
As examples, C ¼ 0.0033 (0.33) for Ts0 ¼ 1.4 ps (140 fs) in a 60 mm long waveguide. The evolution of the idler pulses within the waveguide is presented in Fig. 2 on the right. In contrast to the signal pulses, they are generated by cSHG/DFG within the waveguide; they start from zero and strongly grow with propagation length. As expected, the FWHM of the idler pulses remains nearly constant as long as the pulses are in the picosecond regime (black curve). This is in agreement with experimental work [8]. But it experiences a
Fig. 3. Calculated phase distributions for 140 fs signal (left) and idler (right) pulses at zi ¼0, 5, 20, 40 and 60 mm during propagation inside a PPLN waveguide.
significant broadening in the femtosecond regime (red & green curves), which is, however, considerably smaller than the broadening of fs-signal pulses. As Eq. (4) shows, the conjugate of the electric field amplitude of the signal (As*) contributes to generate the idler wave. Therefore, the sign of the chirp distribution is reversed in comparison with the signal pulse. This leads to a partial compensation of the GVD effects and a weaker pulse broadening as result. To better understand the pulse evolutions as described above, also the phase distributions of the signal (As) and idler (Ai) are calculated for successive propagation lengths in the waveguide. The results are presented in Fig. 3 for signal input pulses of a FWHM of 140 fs and zero phase distribution (cyan curve). As mentioned, the signal pulses are mainly distorted by GVD in the normal dispersive material PPLN leading to a phase dependence of positive curvature [17]. On the other hand, the idler pulse gets a phase dependence of negative curvature which is due to the contribution of the conjugated signal to the generation process of the idler (see Eq. (4)). The signal pulse (As) gets a positive frequency chirp (frequency grows with time) whereas the idler pulse (Ai) gets a negative one. The chirping parameter of the signal pulse at zi ¼0, 5, 20, 40 and 60 mm, induced by GVD, is described by C ¼0, 0.03, 0.12, 0.22 and 0.32, respectively. However, the idler pulse experiences a chirping of reverse sign in comparison with the signal pulse: C ¼ 0, 0.01, 0.04, 0.07 and 0.09, respectively. These data have been obtained by fitting quadratic polynomials (see Eq. 6) to the graphs presented in Fig. 3. 3.2. Proposed device
Fig. 2. Normalized signal (left) and idler (right) pulse width (FWHM) versus propagation length in a 60 mm long PPLN waveguide for coupled signal pulses with τs0 ¼ 1.4 ps, 200 fs and 140 fs. Insets show the signal and idler pulses after 20, 40 & 60 mm propagation for an initial 140 fs signal pulse and 55 mW fundamental power. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
In this section, based on the results presented above, a kind of so called phase engineered wavelength converter is proposed and numerically investigated. The proposed device consists of a periodically poled waveguide section followed by an unpoled (passive) one. In fact, the nonlinear induced phase distribution of the idler pulse generated in the PPLN section can be compensated up to a certain degree by the GVD induced phase distortion in the passive waveguide section. As a result, a wavelength converted idler pulse at the output with nearly undistorted phase distribution and pulse width is expected. Therefore, the phase evolution of the idler pulse is investigated in a waveguide with a 10, 20 and 30 mm long periodically poled section followed by an unpoled section of appropriate length (see Fig.1 top). In the passive section pulse propagation is determined by GVD alone without any nonlinear interaction. The rare presented in Fig. 4.
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Fig. 4. Calculated phase distributions (a, b and c) of the idler at different positions ( zi ) in a passive waveguide section after a periodically poled 10, 20 and 30 mm long PPLN waveguide. The idler was generated by cSHG/DFG with a 140 fs signal input pulse and 55 mW fundamental power. The second row (d, e and f) shows the evolution of the idler pulse width in the combined waveguide structure with poled and unpoled sections. After 13, 26 and 40 mm propagation length the idler pulse width achieves its minimum FWHM corresponding to a phase distribution similar to that of the initially launched signal pulse (phase E0, blue curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
It is remarkable that indeed the phase distribution of the generated idler can be compensated up to a certain extent by a passive section of specific length, which depends on the length of the nonlinear one (Fig. 4(a–c)). By longer passive sections the influence of GVD over-compensates the chirp induced by the nonlinear generation process. In addition, Fig. 4 (d–f) also shows the evolution of the corresponding FWHM of the idler pulses with minima at 3, 6 and 10 mm length in the passive section after the 10, 20 and 30 mm poled PPLN waveguide, respectively. These results demonstrate that wavelength conversion (signal-idler) with negligible pulse distortion is possible. Since the nonlinear interaction yields a negligible contribution to the evolution of the signal pulse, its propagation is essentially determined by GVD only (see Fig. 2, left). However, there is a tradeoff between pulse broadening and conversion efficiency: the longer the periodically poled waveguide section is, the higher the conversion efficiency becomes at the expense of a pulse broadening which cannot completely be compensated in the passive waveguide section. This behavior is shown in Fig. 4 (d–f) which allow to determine the optimum length of the passive section for three different lengths of the active waveguide with a periodical domain grating. This dependence is plotted in Fig. 5 for 140 fs signal pulses as example.
Fig. 5. The length of the unpoled section versus the length of the poled one to achieve minimum pulse distortion of a 140 fs signal pulses launched into the device.
4. Conclusions In conclusion, cSHG/DFG based wavelength conversion of ultrashort optical pulses (1.4 ps, 200 fs, 140 fs) in Ti:PPLN waveguides has been theoretically analyzed. It could be shown that the
A. Babazadeh et al. / Optics Communications 361 (2016) 143–147
wavelength converted idler pulses have a significantly smaller halfwidth than the transmitted residual signal pulses which are mainly broadened by GVD. To get a deeper insight into the wavelength conversion process, the evolutions of signal and idler phase distributions along the nonlinear waveguide have been studied. It became evident that optical phase conjugation during wavelength conversion by cSHG/DFG is essential and-in combination with a compensating effect by GVD – responsible for the small halfwidth of the generated idler pulses. This finding stimulated considering phase engineering by an unpoled (passive) section after the (active) periodically poled waveguide leading to wavelength converted (idler) pulses of minimum distortion and minimum broadening (o 10%) with respect to signal input pulses of 140 fs FWHM. Thus, cSHG/DFG based wavelength converters of a bandwidth of several Tbit/s can be designed.
References [1] S. Yoo, Wavelength conversion technologies for WDM network applications, J. Light. Technol. 14 (1996) 955–966. [2] S. Jansen, D. van den Borne, C.C. Monsalve, S. Spälter, P.M. Krummrich, G. D. Khoe, H. de Waardt, Reduction of Gordon-Mollenauer phase noise by midlink spectral inversion, IEEE Photonics Technol. Lett. 17 (2005) 923–925. [3] M.H. Chou, I. Brener, M.M. Fejer, E.E. Chaban, S.B. Christman, 1.5-μm-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides, IEEE Photonics Technol. Lett. 11 (1999) 653–655. [4] H. Ishizuki, T. Suhara, H. Nishihara, Numerical analysis of ultra-short pulse wavelength conversion characteristics of LiNbO3 waveguide nonlinear-optic devices, Electron. Commun. Jpn. Part II Electron. 86 (2003) 11–20. [5] Y. Wang, J. Fonseca-Campos, C.-Q. Xu, S. Yang, E. a Ponomarev, X. Bao, Picosecond-pulse wavelength conversion based on cascaded second-harmonic generation-difference frequency generation in a periodically poled lithium niobate waveguide, Appl. Opt. 45 (2006) 5391–5403.
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[6] R. Nouroozi, R. Ludwig, B. Hü ttl, C. Schmidt-Langhorst, H. Suche, W. Sohler, C. Schubert, Simultaneous Polarization-Insensitive Wavelength Conversion of 80-Gb/s RZ-DQPSK Signal and 40-Gb/s RZ-OOK Signal in a Ti:PPLN Waveguide, J. Lightw. Technol. 29 (2011) 1092–1097. [7] H. Hu, R. Nouroozi, R. Ludwig, C. Schmidt-Langhorst, H. Suche, W. Sohler, C. Schubert, 110 km transmission of 160 Gbit/s RZ-DQPSK signals by midspan polarization-insensitive optical phase conjugation in a Ti:PPLN waveguide, Opt. Lett. 35 (2010) 2867–2869. [8] H. Hu, R. Nouroozi, R. Ludwig, B. Huettl, C. Schmidt-Langhorst, H. Suche, W. Sohler, C. Schubert, Polarization-insensitive all-optical wavelength conversion of 320 Gb/s RZ-DQPSK signals using a Ti:PPLN waveguide, Appl. Phys. B 101 (2010) 875–882. [9] A. Tehranchi, R. Morandotti, R. Kashyap, Efficient flattop ultra-wideband wavelength converters based on double-pass cascaded sum and difference frequency generation using engineered chirped gratings, Opt. Express 19 (2011) 22528. [10] G.-W. Lu, S. Shinada, H. Furukawa, N. Wada, T. Miyazaki, H. Ito, 160-Gb/s alloptical phase-transparent wavelength conversion through cascaded SFG-DFG in a broadband linear-chirped PPLN waveguide, Opt. Express 18 (2010) 6064–6070. [11] A. Tehranchi, R. Kashyap, Design of novel unapodized and apodized stepchirped quasi-phase matched gratings for broadband frequency converters based on second-harmonic generation, J. Lightw. Technol 26 (2008) 343–349. [12] Y.L. Lee, Y.-C. Noh, C. Jung, T. Yu, D.-K. Ko, J. Lee, Broadening of the secondharmonic phase-matching bandwidth in a temperature-gradient-controlled periodically poled Ti:LiNbO3 channel waveguide, Opt. Express 11 (2003) 2813–2819. [13] G.Agrawal, Nonlinear Fiber Optics, Third Edit (Academic Press, 2007). [14] G.J. Edwards, M. Lawrence, A temperature-dependent dispersion equation for congruently grown lithium niobate, Opt. Quantum Electron. 16 (1984) 373–375. [15] R. Nouroozi, All Optical Wavelength Conversion and Parametric Amplification in Ti : PPLN Channel Waveguides for Telecommunication Applications (Ph. D. dissertation), University of Paderborn, Germany, 2010. [16] G. Agrawal, Lightwave Technology: Telecommunication Systems, Wiley, 2005. [17] R. Nouroozi, H. Suche, Hao Hu, T. Richter, A. Babazadeh, C. Schubert, W. Sohler, Efficient ultra-fast all-optical wavelength converters with Ti: PPLN waveguides, Transparent Optical Networks (ICTON), 16th International Conference on (2014).
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Phase engineered wavelength conversion of ultra-short optical pulses in TI:PPLN waveguides Amin Babazadeh a,n, Rahman Nouroozi a,b, Wolfgang Sohler c a
Physics Department, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran Optics & Photonics Research Center, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran c University of Paderborn, Faculty of Science, Applied Physics, D-33095 Paderborn, Germany b
art ic l e i nf o
a b s t r a c t
Article history: Received 9 May 2015 Received in revised form 4 September 2015 Accepted 19 September 2015
A phase engineered all-optical wavelength converter for ultra-short pulses (down to 140 fs) in a Tidiffused, periodically poled lithium niobate (Ti:PPLN) waveguide is proposed. The phase engineering, due to the phase conjugation between signal and idler (converted signal) pulses which takes place in the cascaded second harmonic generation and difference frequency generation (cSHG/DFG) based wavelength conversion, already leads to shorter idler pulses. The proposed device consists of an unpoled (passive) waveguide section beside of the PPLN waveguide section in order to compensate pulse broadening and phase distortion of the idler pulses induced by the wavelength conversion (in the PPLN section). For example numerical analysis shows that a 140 fs input signal pulse is only broadened by 1.6% in a device with a combination of 20 mm and 6 mm long periodically poled and unpoled waveguide sections. Thus, cSHG/DFG based wavelength converters of a bandwidth of several Tbits/s can be designed. & 2015 Published by Elsevier B.V.
Keywords: Ultra-short optical Pulses Wavelength conversion Difference Frequency generation Periodically poled lithium niobate
1. Introduction All optical wavelength conversion is of great interest for dense wavelength division multiplexing (DWDM) networks [1]. If inherently combined with optical phase conjugation (OPC), it also enables by mid-span wavelength conversion in (long) fiber-optic links a compensation of nonlinear phase noise and dispersion induced signal distortions in the second half of the link; high bitrate, ultra long-haul transmission has been demonstrated using this concept [2]. Among various wavelength conversion approaches nonlinear χ(2) based difference frequency generation (DFG) in a lithium niobate (LiNbO3, LN) waveguide proved to be very attractive [3]. It offers high efficiency, low noise (quantum limited) and high speed (fs response time) simultaneously and is inherently transparent for signal rate and format. In addition, quasi–phase matching (QPM) of DFG in a periodically poled lithium niobate waveguide (PPLN) of specific periodicity allows operating with any wavelength in the transparency range of the LN crystal. Mainly for practical reasons, a cascaded second harmonic generation (SHG) / DFG process (cSHG/DFG) is usually used [3]; it guarantees a spatial mode selective excitation of the (short wavelength) continuous wave (cw) pump by internal phase matched SHG of a cw fundamental wave. Fig. 1 n
Corresponding author. E-mail address: [email protected] (A. Babazadeh).
http://dx.doi.org/10.1016/j.optcom.2015.09.066 0030-4018/& 2015 Published by Elsevier B.V.
schematically shows this cSHG/DFG-based wavelength conversion/OPC process. cSHG/DFG (OPC) based wavelength conversion in PPLN waveguides has been intensively studied in recent years [e.g. 4,5]. It was shown that group velocity mismatch (GVM) and group velocity dispersion (GVD) lead to a temporal walk off and thereby to distortion and broadening of the interacting pulses. However, negligible pulse broadening for 1.4 ps pulses was experimentally observed enabling all optical wavelength conversion of 40, 160 and 320 Gb/s signal data in the C-band [6–8]. It seems that similar to mid-span wavelength conversion with OPC in a (long) fiber link, the distributed wavelength conversion in the (short) PPLN waveguide yields a (partial) compensation of dispersion induced pulse distortions. It should be emphasized that the acceptance bandwidth of a Ti: PPLN wavelength converter with homogeneous domain grating is more than sufficient even for ultra-short signal pulses for the cSHG/DFG approach studied here. This is in contrast to cSFG/DFG based wavelength converters with two pump waves [e.g. 9,10] where the signal pulses undergo two nonlinear processes (SFG and DFG) limiting strongly the acceptance bandwidth by the QPM conditions. Therefore, chirped domain gratings are used to broaden the bandwidth [9,10]. On the other hand, the cSFG/DFG approach allows a wavelength tuning of the idler (converted signal) even for a fixed signal wavelength, which is not possible for the cSHG/DFG approach. Here, the dispersion properties lead via the QPM condition to a narrow bandwidth of the SHG process in
144
A. Babazadeh et al. / Optics Communications 361 (2016) 143–147
poled section
jβ ´´ ∂ 2Ai ∂Ai ∂A α + βi´ i − i = − jκ df Ash As* e−jΔdf z − i Ai 2 ∂t 2 2 ∂z ∂t
unpoled section
(4)
f, sh, s and i denote the fundamental, second harmonic, signal and idler fields, respectively. Am (as a function of time t and position z) represents the slowly varying complex amplitude of the electric field m (m¼f, sh, s, i); Fm(x,y) describes the field distribution of mode m within the waveguide cross section. βm´ ( ∂βm /∂ω at ω = ωm ) stands
Fig. 1. Schematic diagram of the input and output waves in a proposed poled and unpoled sections of PPLN waveguide (top figure). The figure below shows a schematic diagram of cSHG/DFG-based wavelength conversion in the poled section of the waveguide. The cw fundamental wave (λf) generates its second harmonic (λsh) which serves as pump wave for the DFG-process with the signal. The resulting idler (λi) is the wavelength-shifted replica of short (Gaussian shaped) signal pulses (λs).
the homogeneous domain grating assumed. However, this is not a problem as no short pulses but a narrow line width cw wave is frequency doubled; therefore, chirped gratings to broaden the bandwidth of the SHG process [e.g. 11,12] are not necessary. Is a design of an inherently dispersion compensated wavelength converter of nominally unlimited bandwidth possible? This question is theoretically investigated in this letter for cSHG/DFG (OPC) based wavelength conversion of 1.4 ps, 200 fs and 140 fs signal pulses in a standard Ti-indiffused PPLN waveguide. After introducing the mathematical methods in section II, the results are presented and discussed in Section 3. The evolution of signal and idler pulses along the waveguide is analyzed in detail by studying pulse widths and phase distributions. Based on the results, an engineered device is proposed to compensate idler pulse broadening and phase distortion as far as possible. This phase engineering by an unpoled (passive) waveguide section causes wavelength conversion of minimum distortion.
2. Device Modeling The following coupled partial differential equations (Eqs. 1–4) describe the interacting waves and their evolution during cSHG/DFG inside a PPLN waveguide [4]. These equations have been obtained using the plane wave and the slowly varying envelope approximations. The latter can be applied if the bandwidth Δω of the pulses to be investigated is significantly smaller than the carrier frequency ω0, i. e. Δω/ω0 « 1. For our simulations we get Δω/ω0 ¼0.0036 (0.036) using pulses of 1.4 ps (140 fs) width, respectively.
E (x, y , z, t ) =
1 2
∑ Am (z, t ) Fm (x, y) exp ( jωt − jβm z) m
+ c. c
for reciprocal group velocities and βm´´ ( ∂ 2βm /∂ω2 at ω = ωm ) for group velocity dispersions. βm are the propagation constants. κsh and κdf are the coupling coefficients of SHG and DFG processes, and we assume κsh E κdf for the case of ωf E ωs. Λ is the period of the domain inverted grating for QPM. Finally, Δsh and Δdf represent the deviations from SHG and DFG quasi phase mismatching conditions, respectively. All the calculations in this paper are done with Δsh ¼ Δdf ¼ 0 ( Δsh = βsh − 2βf − 2Λπ & Δdf = βsh − βs − βi − 2Λπ ) assuming perfect quasi phase matching. The launched fundamental and, consequently, the internally generated pump waves are assumed to be continuous waves (cw); therefore, βf´, βsh´, βf´´ and βsh´´ are negligible. The signal wave is assumed to be a Gaussian pulse at the beginning of the waveguide (z¼ 0):
As (0, t ) =
⎛ −(1 − jC ) t 2 ⎞ ⎟ Ps0 exp ⎜ 2 Ts20 ⎝ ⎠
(5)
where Ps0 is the peak power and Ts0 is related to the Full Width at Half Maximum (FWHM) of the initial signal pulse by the relation τs0 = 2 ln (2) Ts0 . The parameter C is the frequency chirping factor which is zero in our simulation except where otherwise noted. The (slowly varying) phase evolution with position and time along the waveguide can be written as:
⎛ Im ( Am (z, t ) ) ⎞ C (z ) 2 ⎟⎟ = t φm (z, t ) = tan−1 ⎜⎜ 2 Ts0 ⎝ Re (Am (z, t ) ⎠
The coupled Eqs. (1)–(4) cannot be solved analytically. The Split Step Fourier method [13] with parameters listed in Table 1 for the specific waveguide investigated [8] is used to solve them numerically. The refractive index, the group velocity and group velocity dispersion are taken from [14]. It is important to mention that for cSHG/DFG-based wavelength conversion within the C-band, the QPM determined bandwidth for the fundamental wave of the SHG process is relatively narrow (about a nm) [15]. On the other hand, due to the dispersion properties of LN, the bandwidth for the signal wave is broad enough to cover the whole C-band [8]. Here the wavelength of 1551 nm is chosen for the signal wave to enable a comparison with published experimental data [8]. Table 1 Simulation parameters.
jβf´´ ∂ 2Af ∂Af ∂Af α * A*f Ash e−jΔsh z − f Af = − jκsh + βf´ − 2 ∂t 2 2 ∂z ∂t
(1)
jβ´´ ∂ 2Ash ∂Ash ∂Ash + βsh − sh ´ 2 ∂t 2 ∂z ∂t ω α j Δ = − jκsh A*f Af e sh z − j sh κ df* As Ai e jΔdf z − sh Ash 2 ωi
(2)
jβ´´ ∂ 2As ∂As ∂A ω α + βs´ s − s = − j s κ df Ash Ai* e−jΔdf z − s As 2 ∂t 2 2 ∂z ∂t ωi
(3)
(6)
Parameter
Value
Ps0 Pi0 λf λs τs0 αf ¼αs ¼αi αsh κsh Eκdf βf Eβs E βi βsh βs´ Eβi´ βs´´ βi´´ Λ
100 mW 0 mW 1.546 μm 1.551 μm 1.4 ps, 200 fs, 140 fs 0.15 dB/cm 0.3 dB/cm 0.55 1/cmW1/2 8.723 1/μm 17.825 1/μm 7.34 ns/m 110 ps2/km 113 ps2/km 16.6 μm
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3. Results and discussion 3.1. Evolution of signal and idler pulses The evolution of the normalized pulse width of signal and idler within the PPLN waveguide is shown in Fig. 2 as function of the interaction length. The normalization is done by dividing the propagating signal or idler pulse width τ by the initial signal pulse width τs0. Three different input pulses with τs0 ¼ 1.4 ps, 200 fs and 140 fs are investigated; the (cw) fundamental power is 55 mW. In the insets three signal and idler pulses are plotted after propagating 2, 4 and 6 cm, respectively, for τs0 ¼ 140 fs. The FWHM of the signal pulses with τs0 ¼ 1.4 ps remains nearly constant as function of the interaction length (Fig. 2, left). However, it grows significantly for the shorter pulses with τs0 ¼ 200 fs and 140 fs. This is demonstrated in more detail by the temporal shape of the pulses presented in the inset as examples. Also an attenuation of the signal pulses can be observed, which is mainly due to waveguide propagation losses. The attenuation due to nonlinear wavelength conversion is ≤4% at the 55 mW fundamental power level considered here. As a consequence, the evolution of the signal pulses can essentially be understood by linear waveguide dispersion and attenuation. Group Velocity Dispersion (GVD) causes a broadening and a corresponding phase distribution within the pulses which is the larger the shorter the input pulses are. It can be described by the chirping parameter C of a Gaussian pulse (see also Eq. (5)) which propagates inside a waveguide without nonlinear interactions [16]:
C=
z βs′′ 2 Ts0
(7)
As examples, C ¼ 0.0033 (0.33) for Ts0 ¼ 1.4 ps (140 fs) in a 60 mm long waveguide. The evolution of the idler pulses within the waveguide is presented in Fig. 2 on the right. In contrast to the signal pulses, they are generated by cSHG/DFG within the waveguide; they start from zero and strongly grow with propagation length. As expected, the FWHM of the idler pulses remains nearly constant as long as the pulses are in the picosecond regime (black curve). This is in agreement with experimental work [8]. But it experiences a
Fig. 3. Calculated phase distributions for 140 fs signal (left) and idler (right) pulses at zi ¼0, 5, 20, 40 and 60 mm during propagation inside a PPLN waveguide.
significant broadening in the femtosecond regime (red & green curves), which is, however, considerably smaller than the broadening of fs-signal pulses. As Eq. (4) shows, the conjugate of the electric field amplitude of the signal (As*) contributes to generate the idler wave. Therefore, the sign of the chirp distribution is reversed in comparison with the signal pulse. This leads to a partial compensation of the GVD effects and a weaker pulse broadening as result. To better understand the pulse evolutions as described above, also the phase distributions of the signal (As) and idler (Ai) are calculated for successive propagation lengths in the waveguide. The results are presented in Fig. 3 for signal input pulses of a FWHM of 140 fs and zero phase distribution (cyan curve). As mentioned, the signal pulses are mainly distorted by GVD in the normal dispersive material PPLN leading to a phase dependence of positive curvature [17]. On the other hand, the idler pulse gets a phase dependence of negative curvature which is due to the contribution of the conjugated signal to the generation process of the idler (see Eq. (4)). The signal pulse (As) gets a positive frequency chirp (frequency grows with time) whereas the idler pulse (Ai) gets a negative one. The chirping parameter of the signal pulse at zi ¼0, 5, 20, 40 and 60 mm, induced by GVD, is described by C ¼0, 0.03, 0.12, 0.22 and 0.32, respectively. However, the idler pulse experiences a chirping of reverse sign in comparison with the signal pulse: C ¼ 0, 0.01, 0.04, 0.07 and 0.09, respectively. These data have been obtained by fitting quadratic polynomials (see Eq. 6) to the graphs presented in Fig. 3. 3.2. Proposed device
Fig. 2. Normalized signal (left) and idler (right) pulse width (FWHM) versus propagation length in a 60 mm long PPLN waveguide for coupled signal pulses with τs0 ¼ 1.4 ps, 200 fs and 140 fs. Insets show the signal and idler pulses after 20, 40 & 60 mm propagation for an initial 140 fs signal pulse and 55 mW fundamental power. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
In this section, based on the results presented above, a kind of so called phase engineered wavelength converter is proposed and numerically investigated. The proposed device consists of a periodically poled waveguide section followed by an unpoled (passive) one. In fact, the nonlinear induced phase distribution of the idler pulse generated in the PPLN section can be compensated up to a certain degree by the GVD induced phase distortion in the passive waveguide section. As a result, a wavelength converted idler pulse at the output with nearly undistorted phase distribution and pulse width is expected. Therefore, the phase evolution of the idler pulse is investigated in a waveguide with a 10, 20 and 30 mm long periodically poled section followed by an unpoled section of appropriate length (see Fig.1 top). In the passive section pulse propagation is determined by GVD alone without any nonlinear interaction. The rare presented in Fig. 4.
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Fig. 4. Calculated phase distributions (a, b and c) of the idler at different positions ( zi ) in a passive waveguide section after a periodically poled 10, 20 and 30 mm long PPLN waveguide. The idler was generated by cSHG/DFG with a 140 fs signal input pulse and 55 mW fundamental power. The second row (d, e and f) shows the evolution of the idler pulse width in the combined waveguide structure with poled and unpoled sections. After 13, 26 and 40 mm propagation length the idler pulse width achieves its minimum FWHM corresponding to a phase distribution similar to that of the initially launched signal pulse (phase E0, blue curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
It is remarkable that indeed the phase distribution of the generated idler can be compensated up to a certain extent by a passive section of specific length, which depends on the length of the nonlinear one (Fig. 4(a–c)). By longer passive sections the influence of GVD over-compensates the chirp induced by the nonlinear generation process. In addition, Fig. 4 (d–f) also shows the evolution of the corresponding FWHM of the idler pulses with minima at 3, 6 and 10 mm length in the passive section after the 10, 20 and 30 mm poled PPLN waveguide, respectively. These results demonstrate that wavelength conversion (signal-idler) with negligible pulse distortion is possible. Since the nonlinear interaction yields a negligible contribution to the evolution of the signal pulse, its propagation is essentially determined by GVD only (see Fig. 2, left). However, there is a tradeoff between pulse broadening and conversion efficiency: the longer the periodically poled waveguide section is, the higher the conversion efficiency becomes at the expense of a pulse broadening which cannot completely be compensated in the passive waveguide section. This behavior is shown in Fig. 4 (d–f) which allow to determine the optimum length of the passive section for three different lengths of the active waveguide with a periodical domain grating. This dependence is plotted in Fig. 5 for 140 fs signal pulses as example.
Fig. 5. The length of the unpoled section versus the length of the poled one to achieve minimum pulse distortion of a 140 fs signal pulses launched into the device.
4. Conclusions In conclusion, cSHG/DFG based wavelength conversion of ultrashort optical pulses (1.4 ps, 200 fs, 140 fs) in Ti:PPLN waveguides has been theoretically analyzed. It could be shown that the
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wavelength converted idler pulses have a significantly smaller halfwidth than the transmitted residual signal pulses which are mainly broadened by GVD. To get a deeper insight into the wavelength conversion process, the evolutions of signal and idler phase distributions along the nonlinear waveguide have been studied. It became evident that optical phase conjugation during wavelength conversion by cSHG/DFG is essential and-in combination with a compensating effect by GVD – responsible for the small halfwidth of the generated idler pulses. This finding stimulated considering phase engineering by an unpoled (passive) section after the (active) periodically poled waveguide leading to wavelength converted (idler) pulses of minimum distortion and minimum broadening (o 10%) with respect to signal input pulses of 140 fs FWHM. Thus, cSHG/DFG based wavelength converters of a bandwidth of several Tbit/s can be designed.
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