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Continuous wave, dual-wavelength-pumped supercontinuum generation in an all-fiber device Yanbin Wang,1,2,* Chunle Xiong,2,3 Jing Hou,1,4 Jianqiu Cao,1 Ying Li,1 Rui Song,1 and Qisheng Lu1 1

College of Optoelectronic Science and Engineering, National University of Defense Technology, Changsha Hunan 410073, China 2

Centre for Ultrahigh-bandwidth Devices for Optical Systems (CUDOS), Institute for Photonics and Optical Science (IPOS), School of Physics, University of Sydney, New South Wales 2006, Australia 3

e-mail: [email protected] 4

e-mail: [email protected]

*Corresponding author: [email protected] Received 13 December 2010; revised 8 March 2011; accepted 8 March 2011; posted 11 March 2011 (Doc. ID 139489); published 9 June 2011

We propose a continuous wave dual-wavelength-pumped scheme for visible supercontinuum (SC) generation. The scheme is numerically studied in this paper. In the scheme, the dual-wavelength pump source is produced through a four-wave mixing process in a photonic crystal fiber. SC generation is numerically investigated by solving the generalized nonlinear Schrödinger equation. The results verify that the visible SC can be generated by the scheme, which implies that the scheme is promising for generating visible SC with high spectral power densities. © 2011 Optical Society of America OCIS codes: 060.2280, 190.5530.

1. Introduction

Powerful white-light supercontinua (SC) have recently found a wide range of applications, such as chromatic confocal microscopy [1], detection and spectroscopy in highly diffusive media [2], and ultrahigh-resolution optical coherence tomography [3]. Continuous wave (CW) pumped SC have exhibited two remarkable advantages over pulse-pumped SC: higher spectral power densities (SPD) and smoother spectra [4–12]. It has recently been reported that the resulting SC based on CW pump sources have an SPD over 50 mW=nm [5,10], a spectrum within 2:4 dB, spanning from 1520 to 1770 nm [11], and a spectral bandwidth of 600 nm at 8 dB [8]. In the CW 0003-6935/11/172752-07$15.00/0 © 2011 Optical Society of America 2752

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pumped regime, SC demonstrated to date have been generated by using tens- or hundreds-of-watt [5–10] fiber lasers to pump a highly nonlinear photonic crystal fiber (PCF), whose zero-dispersion wavelength (ZDW) is just below the laser wavelength [5–8] or decreases along the fiber length [9]. However, the SC generated in these single-wavelength-pumped schemes cannot cover the visible spectral region [5–10]. In the pulse-pumped regime, the SC wavelength range can be enormously extended by dualwavelength pumping [13,14]. When the two pump wavelengths are group-velocity-matched (GVM) in the PCF, the visible SC can be generated. The dual-wavelength pump source was obtained first by using the frequency-doubling process [13]. However, the walkoff effect induced by the frequencydoubling crystal and the lens dispersion makes it

very difficult to couple the fundamental and second harmonic waves together into an optical fiber [13]. To overcome this difficulty, Xiong et al. [14] replaced the crystal with a fiber to form an all-fiber configuration, which can avoid the coupling problems arising from the lens dispersion and can improve the stability of the optical system. The fiber-based dual-wavelength pump source was effectively produced through a fourwave mixing (FWM) process, and the generated SC can extend from as short as 360 nm to beyond 1750 nm. In spite of that, the pulse-pumped regime limits the output power and SPD of the SC because of the low average power of the pulse-pump source. In this paper, we apply the fiber-based dualwavelength-pumped scheme to a CW-pumped regime and propose a CW dual-wavelength-pumped scheme for visible SC generation with a higher SPD. By solving the generalized nonlinear Schrödinger equation (GNLSE), it is verified numerically that the visible SC can be generated by a CW dual-wavelengthpumped scheme. This paper is arranged as follows: in Section 2, we introduce the theoretical model of solving the GNLSE with the adaptive split-step Fourier method (SSFM). In Section 3, a dual-wavelength pump source operating at 1064=687 nm is effectively produced through the FWM process in the normaldispersion region of a PCF. In Section 4, we couple the generated pump source with two pieces of PCF with different group-velocity dispersion (GVD) and study the physical mechanism of dual-wavelengthpumped SC generation. Section 5 states our conclusion. 2. Theoretical Model

The scheme of fiber-based dual-wavelength-pumped SC generation is illustrated in Fig. 1. The FWM process in PCF-FWM provides the dual-wavelength pump source as the input to PCF-SC to generate the continuum. In our numerical simulation, the temporal output from PCF-FWM, which contains the spectral information of two pump wavelengths by Fourier transform, directly serves as the input to PCF-SC; therefore, we do not need to solve the coupled NLSEs [13], but rather, to change the fiber parameters at the end of PCF-FWM to those of PCF-SC.

A. Generalized Nonlinear Schrödinger Equation

A CW field can be broken up into ultrashort pulses as it propagates in dispersive and highly nonlinear fiber, and the propagation of ultrashort pulses is governed by the GNLSE [15]: X ik ∂ k A ∂A α i ∂ β þ A−i ¼ iγ 1 þ k! k ∂T k ∂z 2 ω0 ∂T k≥2 Z þ∞ 0 0 2 RðT ÞjAðz; T − T Þj dT 0 ; × Aðz; TÞ −∞

ð1Þ

where Aðz; tÞ is the electric field envelope, z is the longitudinal PCF coordinate, and T is the retarded time; α is the propagation loss of the fiber, and we ignore it because the propagation loss of silica fiber in the wavelength range of interest (0:4–2 μm) is negligible; βk is the kth-order dispersion coefficient (here we include terms up to β6 in order to obtain high precision) at the pump central frequency ω0, and γ is the nonlinear coefficient. The form of nonlinear response function RðTÞ can be given as [15] RðTÞ ¼ ð1 − f R ÞδðTÞ þ fR

τ21 þ τ22 expð−T=τ2 Þ sinðT=τ1 Þ: τ1 τ22

ð2Þ

Here, τ1 ¼ 12:2 fs and τ2 ¼ 32 fs for SiO2 ; the first term on the right hand represents the Kerr effect, and the second term represents the Raman effect; the factor f R ¼ 0:18 determines the Raman contribution to the nonlinear effects. B. Initial Conditions

In fact, any real laser emits light with random fluctuations of both amplitude and phase. These fluctuations are also called “random noise,” which plays an important role in breaking the original CW into ultrashort pulses. Thus, the input field can be written as [16] Eðz; tÞ ¼ ½A þ δAðz; tÞ expfi½ϕ þ δϕðz; tÞg;

ð3Þ

where A and ϕ are the stationary values for the amplitude and phase, δAðz; tÞ and δϕðz; tÞ, respectively, represent the random noises of the stationary values A and ϕ, such that δAðz; tÞ ≪ Aðz; tÞ, δϕðz; tÞ ≪ ϕðz; tÞ, together with hδAðz; tÞi ¼ 0, and hδϕðz; tÞi ¼ 0; here

Fig. 1. Schematic scheme of CW dual-wavelength-pumped SC generation in an all-fiber device. 10 June 2011 / Vol. 50, No. 17 / APPLIED OPTICS

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the angle brackets denote the ensemble average. In order to simulate random noise, we add one photon with random phase to each frequency mesh point [17]. C.

Table 1.

Adaptive Split-Step Fourier Method

The temporal width of the CW field is considered to be infinite; therefore, we can only simulate a snapshot of the CW field as it propagates [10]. We cautiously select a time window that not only contains sufficient information about the real CW field but also keeps the computation time within reasonable limits. The current literature [10,17,18] has demonstrated that it is sufficient to choose a time window of several hundred picoseconds with the periodic boundary conditions inherent to the SSFM. In the following simulation, we set grid points N ¼ 218 and temporal resolution dt ¼ 1:0 fs over the time window of 262 ps. The GNLSE Eq. (1) is solved by using the adaptive SSFM. The step size can adaptively change based on the relative local error δG [19], which is set to δG ¼ 10−5 in this paper. The simulated wavelength range can cover from 384 to 3000 nm by the relations of Fourier transform,

Parameters of PCFs Used in This Papera

PCF Name

Λ=μm

d=Λ

λ0 =nm

PCF-FWM PCF-SC1 PCF-SC2

3.20 3.20 2.44

0.28 0.75 0.65

1117 850 910

a

The parameters are: pitch, Λ; air hole diameter, d; and zerodispersion wavelength, λ0 .

where c is the speed of light in a vacuum, and λp denotes the pump’s central wavelength, which is set at 1064 nm in this paper.

meters of PCF-FWM and the other PCFs used in this paper are given in Table 1. Figure 2 shows the calculated GVD curve of PCF-FWM. Its ZDW is located at ∼1117 nm, and the calculated dispersion coefficients for the pump wavelength at 1064 nm are β2 ¼ 5:1275 × 10−27 s2 =m, β3 ¼ 3:6276 × 10−41 s3 =m, β4 ¼ −5:5305 × 10−56 s4 =m, β5 ¼ −1:8764 × 10−71 s5 =m, and β6 ¼ −3:1227 × 10−85 s6 =m, respectively. The average power of the CW field is set at 100 W; Fig. 3(a) shows the evolution of output spectra with the length of PCF-FWM, and Fig. 3(b) shows the output spectra from a 50 m PCF-FWM. As we can see in Fig. 3(b), the signal/idler wavelength shows up at 687=2365 nm and the first-order Stokes line of the 687 nm wave and the 1064 nm wave are located at 708 nm and 1120 nm, respectively, owing to stimulated Raman scattering (SRS). The idler wavelength at 2365 nm was actually absent [14] because of confinement loss and material absorption; therefore, the residual pump wave at 1064 nm and the generated signal wave at 687 nm can build up a dual-wavelength pump source at 687=1064 nm.

3. Dual-Wavelength Pump Source

4. Supercontinuum Generation

The propagation of long pulses in the normaldispersion region of PCF can generate a FWM process [14,20], where long pulses can be considered quasiCW and some nonlinear processes, such as self-phase modulation and walkoff effect can be negligible. This situation also holds for the CW field, so we can obtain the CW dual-wavelength pump source through the FWM process. To choose a suitable PCF-FWM to generate the secondary pump wave, we need to balance the wavelength of the signal wave, the generation efficiency, and the fiber loss; and finally we select the PCF-FWM that can generate the signal wave at 687 nm. For generation of a signal wave shorter than 687 nm, the zero-dispersion wavelength of the PCFFWM needs to be further away from the 1064 nm pump wave. The PCF-FWM needs to either have smaller holes or a bigger core, or both. All of these designs will decrease the confinement, which will reduce the nonlinearity and increase the confinement loss and bending loss. Also, when the signal wavelength becomes shorter, the idler wavelength will be very long, and the idler becomes lossy in silica fibers, which will decrease the FWM efficiency. Furthermore, the shortest signal wavelength by FWM cannot be shorter than 532 nm, because this is second-order harmonic generation, which is unlikely to occur in silica fibers. Therefore, one cannot get much benefit from a shorter wavelength signal wave. The para-

After the formation of the dual-wavelength pump source in PCF-FWM, the next stage is to couple into the suitable PCF-SC for visible SC generation. For comparison, we provide two pieces of PCF-SC (PCF-SC1 and PCF-SC2 in Table 1) with different GVD properties. Their group-velocity index curves, computed by the empirical relation [21], are shown in Fig. 4, and the indices at 687 nm and 1064 nm are also marked by the blue crosses. The group-velocity

λmin ¼

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λp 1þ

λp 2cdt

;

λmax ¼

λp

λ

p 1 − 2cdt

;


ð4Þ

Fig. 2. (Color online) Calculated GVD curve of PCF-FWM.

Fig. 3. (Color online) FWM process in PCF-FWM: (a) evolution of output spectra with the length of PCF-FWM; (b) output spectra from 50 m PCF-FWM.

indices of PCF-SC2 at 687 nm and 1064 nm are 1.4778 and 1.4752, respectively, with a little mismatch. Compared with PCF-SC2, the group-velocity index curve of PCF-SC1 has two features: on one hand, the group-velocity indices at 687 nm and 1064 nm are 1.4802 and 1.4801, which are well matched; on the other hand, the group-velocity index of a short-wavelength range of 400–600 nm matches that of a long-wavelength range of 1200–2100 nm. Figure 5 shows the evolution of the dualwavelength pump source with the length of (a) PCFSC1 and (b) PCF-SC2, and the output spectra at the short wavelengths from the 50 m PCF-SC1 (superimposed green curve) and PCF-SC2 (background red curve) are compared in Fig. 5(c). As the 1064 nm wave propagates in the anomalous-dispersion region of PCF-SC, random noises are amplified and modulation instability (MI) [15] can lead to spontaneous breakup of the original CW field into multiple ultrashort pulses in the time domain. These ultrashort

Fig. 4. (Color online) Group-velocity index curves of PCF-SC1 (upper red curve) and PCF-SC2 (lower green curve). The indices at 687 nm and 1064 nm are marked by the blue crosses.

pulses then evolve into optical solitons in the interplay of minus GVD and self-phase modulation (SPM), which correspond to the dark spots in the long-wavelength region in Figs. 5(a) and 5(b). The spectral width of optical solitons is large enough that the Raman gain can amplify their low-frequency spectral components, with their high-frequency components acting as a pump; therefore, these optical solitons undergo intrapulse Raman scattering leading to the soliton self-frequency shift (SSFS) [5,13], generating a long-wavelength soliton Raman continuum. Figure 6 shows that the Raman-induced frequency shift grows linearly with the length of PCF-SC1, which can be written as [15] ΩðzÞ ¼ −

8T R γPS z: 15T 20

ð5Þ

Here, parameter T R ¼ 3 fs, and PS and T 0 are soliton’s peak power and pulse width, respectively. The outermost soliton peaks and the corresponding fiber length are also marked by red circles. In the meantime, the 687 nm wave propagates in the normal dispersion region of PCF-SC, but the evolution of the 687 nm wave exhibits different properties as the lengths of PCF-SC1 and PCF-SC2 change, corresponding to Figs. 5(a) and 5(b). The power at the wavelength ranges 820–1000 nm and 384–560 nm in Fig. 5(a) is obviously higher than the power in Fig. 5(b), while the power at the wavelength range 687–820 nm in Fig. 5(a) is lower than that in Fig. 5(b). The reason is that the major nonlinear process is SRS for the propagation of the 687 nm wave along PCF-SC2, and the spectral broadening induced by SPM is negligible, as can be seen by observing the spectral evolution of the 687 nm wave. The Raman shift of the 687 nm wave increases the power at the wavelength range of 687–820 nm, as we can see in Fig. 5(b). The circumstances in PCF-SC1 are a little different from those in PCF-SC2. Apart from SRS induced by the 687 nm wave, cross-phase modulation 10 June 2011 / Vol. 50, No. 17 / APPLIED OPTICS

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Fig. 6. (Color online) Soliton self-frequency shift with length of PCF-SC1. The outermost soliton peaks and the corresponding fiber length are marked by red circles.

701 nm in Fig. 7. Their frequency offset from the central frequency of the 687 nm wave can be calculated as ∼9:5 THz by [15] Ωmax ¼

2γP0 jβ2 j

1=2

;

ð6Þ

where β2 is the second-order dispersion coefficient at the central frequency of the 687 nm wave. With further propagation, the nonlinear phase-shift grows and multiple sidebands are generated. XPM-induced MI causes the breakup of the 687 nm wave into multiple ultrashort pulses. As the 687 nm wave is located in the normal-dispersion region of PCF-SC1, red components of these ultrashort pulses around 687 nm travel faster than their blue components. Thus, the interaction between the leading edge of these pulses and optical solitons can produce a lot of redshifted spectral components and increase the power at wavelength range 687–1000 nm in Fig. 5(a); at the same time, the interaction between

Fig. 5. (Color online) Evolution of the dual-wavelength pump source along (a) PCF-SC1 and (b) PCF-SC2; (c) output spectra from 50 m PCF-SC1 (superimposed green curve) and PCF-SC2 (background red curve).

(XPM) plays an important role due to the absolute group-velocity matching between the 687 nm wave and the 1064 nm wave in PCF-SC1. As optical solitons emerge in the anomalousdispersion region, they can impose a nonlinear phase-shift on the 687 nm wave through XPM, owing to their high peak power. The nonlinear phase-shift can lead to XPM-induced MI around the 687 nm wave at 2 m PCF-SC1, which can be proved by the appearance of two definite peaks at 671 nm and 2756


Fig. 7. XPM-induced MI.

5. Conclusion

Fig. 8. (Color online) GVM and phase-matched curves for PCF-SC1. The outermost soliton peaks and the corresponding DW wavelengths are marked by blue circles.

these ultrashort pulses and optical solitons results in the generation of dispersive waves (DW) at the blue side of 687 nm. In order to clarify the relation between optical solitons and the generated DW below 687 nm, we plot group-velocity-matched (GVM) and phase-matched (PM) curves for PCF-SC1 in Fig. 8. Phase match between the soliton and the DW is governed by the well-known resonance condition [22] X βn ðwS Þ n≥2

n!

ðwDW − wS Þn ¼

γðwS ÞpS ; 2

ð7Þ

where wS and wDW are the soliton and the DW’s central frequency, and PS is the soliton’s peak power. The power-dependent term represents the nonlinear phase-shift and is often small compared to the other terms in Eq. (7). The outermost soliton peaks and the corresponding DW wavelengths are marked by blue circles. The generated DW matches the PM curve at the wavelength range 600–687 nm, while agreeing with the GVM curve below 600 nm in Fig. 8. GVM is very small between these ultrashort pulses at the range 600–687 nm and redshifted solitons, so the walkoff effect can be negligible and FWM plays an important role. As solitons are further redshifted to longer wavelengths, their group velocity gradually decreasea and cannot catch up with the ultrashort pulses. The walkoff effect limits the efficiency of the FWM process. However, these ultrashort pulses can be trapped as optical solitons experience SSFS toward longer wavelengths [5,23] and the wavelengths of the trapped pulse are shifted toward shorter wavelengths to satisfy the condition of GVM. As we can see in Figs. 5(a) and 8, as the solitons’ peak gradually moves to longer wavelengths with the increase of PCF-SC1’s length, the blueshifted DWs further shift down to ∼400 nm. The combined effect of the FWM process, soliton trapping, and SSFS through XPM effectively improves the power and flatness of the continuum at the visible wavelengths.

In this paper, we have proposed and numerically investigated a CW dual-wavelength-pumped scheme for visible SC generation. The dual-wavelength pump source at 1064=687 nm has been quantitatively produced through the FWM process in the normal-dispersion region of PCF-FWM. Our simulation has demonstrated that the visible SC can be generated. It is found that the combined effects of the FWM process, soliton trapping, and SSFS can effectively improve the power and flatness of the continuum at the visible wavelengths. Considering the advantage of a CW pump in generating the highSPD SC, we believe that the CW dual-wavelengthpumped scheme is promising for high-SPD visible SC generation. Related experiments will be reported in the future. This research was supported by National Natural Science Foundation of China (NNSF) under grants 61077076 and 61007037, and by the Program for New Century Excellent Talents in University under grant NCET-08-0142. We acknowledge the support of Prof. Eggleton from CUDOS at the University of Sydney. References 1. K. Shi, P. Li, S. Yin, and Z. Liu, “Chromatic confocal microscopy using supercontinuum light,” Opt. Express 12, 2096–2101 (2004). 2. K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy,” Phys. Rev. Lett. 93, 037401 (2004). 3. I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and S. Windeler, “Ultrahigh resolution optical coherence tomography using continuum generation in an air–silica microstructure optical fiber,” Opt. Lett. 26, 608–610 (2001). 4. J. W. Nicholson, A. K. Abeeluck, C. Headley, M. F. Yan, and C. G. Jorgensen, “Pulsed and continuous wave supercontinuum generation in highly nonlinear, dispersion-shifted fibers,” Appl. Phys. B 77, 211–218 (2003). 5. B. A. Cumberland, J. C. Travers, S. V. Popov, and J. R. Taylor, “29 W High power CW supercontinuum source,” Opt. Express 16, 5954–5962 (2008). 6. A. V. Avdokhin, S. V. Popov, and J. R. Taylor, “Continuous wave, high-power, Raman continuum generation in holey fibers,” Opt. Lett. 28, 1353–1355 (2003). 7. J. C. Travers, R. E. Kennedy, S. V. Popov, and J. R. Taylor, “Extended continuous wave supercontinuum generation in a low-water-loss holey fiber,” Opt. Lett. 30, 1938–1940 (2005). 8. B. A. Cumberland, J. C. Travers, S. V. Popov, and J. R. Taylor, “Toward visible cw-pumped supercontinua,” Opt. Lett. 33, 2122–2124 (2008). 9. A. Kudlinski and A. Mussot, “Visible cw-pumped supercontinuum,” Opt. Lett. 33, 2407–2409 (2008). 10. J. C. Travers, A. B. Rulkov, B. A. Cumberland, S. V. Popov, and J. R. Taylor, “Visible supercontinuum generation in photonic crystal fibers with a 400 W continuous wave fiber laser,” Opt. Express 16, 14435–14447 (2008). 11. A. K. Abeeluck and C. Headley, “High-power supercontinuum generation in highly nonlinear, dispersion-shifted fibers by use of a continuous wave Raman fiber laser,” Opt. Lett. 29, 2163–2165 (2004). 10 June 2011 / Vol. 50, No. 17 / APPLIED OPTICS

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