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Chapter 2 New Wavelength Generation Based on PCF with Two Zero-Dispersion Wavelengths (TZDWs) 2.1 Multipole Method Since the PCF has flexible cladding structures, the simulation method for conventional

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Chapter 2 New Wavelength Generation Based on PCF with Two Zero-Dispersion Wavelengths (TZDWs) 2.1 Multipole Method Since the PCF has flexible cladding structures, the simulation method for conventional optical fibers cannot be used for the evaluation of the PCF characteristics accurately. In recent years, several methods have been developed to improve the numerical precision, such as the plane wave expansion method, beam propagation method, finite element method, and multipole method. In this thesis, we mainly use the multipole method to estimate the fiber performances. Compared with other methods, the limited cladding structure is considered in the multipole algorithm, which is especially suitable for the microstructured fibers with circle air holes distributed in the cladding. The calculation about the mode-propagation constant for the PCFs with complex structure can be finished more accurately and faster than other methods. In the multipole method, every air hole in the cladding is considered as a scattering cell. The electromagnetic field components can be expressed by the Bessel functions in the cylindrical coordinate system. The solution of the Helmholtz equation can be obtained by using the boundary conditions. The longitudinal component of the electric field in the nth air hole can be expressed as follows [1, 2]: E z ¼ X1 m¼ 1 am ðnþ J m k? i r n expðimun Þ expðibzþ ð2:1þ The longitudinal component of the electric field in the substrate material around the nth air hole can be expressed as follows: E z ¼ X1 m¼ 1 h bm ðnþ J m k? e r n þ c ðnþ m Hn m ke? r i n expðim/ n Þ expðibzþ ð2:2þ Springer-Verlag Berlin Heidelberg 2016 L. Zhang, Ultra-Broadly Tunable Light Sources Based on the Nonlinear Effects in Photonic Crystal Fibers, Springer Theses, DOI / _2 19 20 2 New Wavelength Generation Based where k i? ¼ðk2 0 n2 i b 2 Þ 1=2 k e? ¼ðk2 0 n2 e b2 Þ 1=2 ð2:3þ where n i = 1 denotes the refractive index of the air, n e denotes the refractive index of the fused silica, k 0 =2π/λ is used to express the wave numbers in the free space, and β denotes the mode-propagation constant. The magnetic field components have the similar expression with the electric field components. The coefficients of ɑ m, b m, and c m can be obtained by using the boundary condition of the electromagnetic field. The effective refractive index n eff and the effective mode area A eff can be obtained by using the expression between the mode-propagation constant and the wave numbers in the free space β = n eff k 0. Then, the nonlinear coefficient, group velocity dispersion, loss, and birefringence can be obtained. 2.2 Parametric Amplification Based on PCF with TZDW The Design and Fabrication of the PCF with TZDW The ZDW is an important parameter for optical fibers. When the pump is located near the ZDW, the nonlinear effects can be increased remarkably, and the optical parametric components can be generated easily. Fiber-based optical parametric generation is the cornerstone of the FOPA, FOPO, and all-optical wavelength converter. The ZDW of the traditional nonlinear fibers (highly-nonlinear fibers, dispersion-shifted fibers, and the highly-nonlinear dispersion-shifted fibers) is located around 1550 nm, and it can be tuned only in a very small wavelength range, which is determined by the fiber structure and the fabricated process. Since the PCF has flexible cladding and core structures, the ZDW can be tuned to nearly any wavelength in the transparent window. It means that the optical parametric components can be generated from visible to the infrared band flexibly. Fiber-based optical parametric generation, also named as modulation instability, refers to the phenomenon that signals from the spontaneous emission are amplified by the pump wave. The shape and the distribution of the optical parametric spectral components are closely related to the nonlinear effect of FWM. Marhic et al. [3] reported a widely tunable optical parametric generation with the anti-stokes spectral components around 1350 nm in a highly-nonlinear dispersion-shifted fiber pumped in the normal dispersion wavelength regime by a tunable diode laser. Wong et al. demonstrated the evolution of the optical parametric spectral components versus the pump wavelength by using a quasi-continuous wave laser source. When the PCF is pumped near the ZDW, a wide-band parametric spectrum can be formed. When the PCF is pumped in the normal dispersion wavelength regime, two narrow parametric bands can be generated far from the pump [4]. Harvey et al. demonstrated that the parametric spectral components can be tuned largely by adjusting the polarization state of the pump in a birefringent PCF [5]. 2.2 Parametric Amplification Based on PCF with TZDW 21 It is clear that the FWM optical parametric spectral components can be easily generated by pumping the PCF near the ZDW. Andersen et al. obtained a widely tuned optical parametric generation in a PCF with two zero-dispersion wavelengths (TZDWs) by using a Ti: sapphire laser as the pump source [6]. Tuan et al. [7] theoretically predicted that largely tuned parametric components can be generated in a PCF with four ZDWs. The largely tuned optical parametric generation has a lot of applications, such as new wavelength laser source, large span wavelength converter, and entangled pair-photon source. In this section, we would like to design and fabricate a PCF with TZDW for the realization of large span optical parametric generation. The PCF structure is designed by using the multipole method. In order to ensure the precision during the fabrication, hexagonal cladding structures are adopted. The fused silica is used as the substrate material. The designed cross-sectional structure of the PCF is shown in Fig The pitch of the air holes is 1.15 μm, the diameter of the air holes is 0.75 μm, and the diameter of the fiber core is 1.55 μm. The effective refractive index and the effective mode area at different wavelength can be calculated by using the multipole method. Based on the relation between the refractive index and the dispersion, the group velocity dispersion of the fundamental mode can be obtained, which is shown in Fig The two ZDWs are located at 701 and 1115 nm, respectively. When a single-longitudinal-mode laser is used as the pump source, the frequencies and the propagation constants of the two pump photons are the same. x 1 ¼ x 2 ¼ x p b 1 ¼ b 2 ¼ b p ð2:4þ ð2:5þ Fig. 2.1 The cross section of the designed PCF 22 2 New Wavelength Generation Based Fig. 2.2 The group velocity dispersion for the designed PCF with TZDW The phase-matching condition for the single pump case can be expressed as follows: 2x p ¼ x s þ x i Dk ¼ Db ¼ b s þ b i 2b p ¼ 0 ð2:6þ ð2:7þ The propagation constant can be calculated by the equation of β = nω/c. The phase-matching condition can be predicted by a combination of the Eqs. (2.6) and (2.7), which is shown in Fig. 2.3 plotted by pump wavelength on the horizontal axis and the wavelengths of the signal and idler on the vertical axis. In the figure, the blue line represents the signal and the red line represents the idler. When the pump is located at 780 nm, the phase-matched signal and idler arise at the telecommunication band of 1550 nm and the visible band of 521 nm, respectively, which is denoted by the purple line in the figure. Fig. 2.3 The phase-matching profile for the designed PCF with TZDW 2.2 Parametric Amplification Based on PCF with TZDW 23 The designed PCF is featured with a small core, which can provide a high nonlinearity. The microstructure of the designed PCF introduces a lot of challenges to the technical fabrication. The group of Prof. Jinyan Li in Huazhong University of Science and Technology fabricated this fiber for us successfully by overcoming lots of difficulties. The SEM image of the cross section of the fabricated PCF is shown in Fig According to the fiber parameters extracted from the SEM image, the dispersion property is calculated, and the TZDWs are located at 723 and 1363 nm, respectively, as shown in Fig The nonlinear coefficient is calculated to be 160 km 1 W 1 at 800 nm. Fig. 2.4 The SEM image of the cross section of the fabricated PCF with TZDW Fig. 2.5 The calculated dispersion for the fabricated PCF 24 2 New Wavelength Generation Based The Relationship Between the Optical Parametric Spectrum and the Pump Wavelength In order to accurately simulate the optical parametric generation, the contribution of the nonlinear effect to the phase matching should be considered, and the phase-matching condition equation should be modified accordingly. For the single pump case, the revised phase-matching condition can be expressed as follows: j ¼ Db þ 2cP ¼ b s þ b i 2b p þ 2cP ¼ 0 ð2:8þ where P denotes the pump power. With the propagation constants β s and β i Taylor expanded around the pump frequency, Eq. (2.9) can be written as follows: j ¼ X 2 b 2 þ 1 12 X4 b 4 þ X6 b 6 þþ2cp ¼ 0 ð2:9þ where Ω denotes the frequency detuning between the pump and the signal. It can be seen that only the even-order dispersion and the nonlinearity have influence on the phase-matching condition. When the linear phase mismatch is located in the region of 4γP Δβ 0, a nonzero parametric gain can be obtained. It requires that the group velocity dispersion parameter β (2) at the pump wavelength is approximately in the range of: 4cP X 2 bð4þ X 2 12 \bð2þ \ bð4þ X 2 12 ð2:10þ where β (4) is the fourth derivative of the propagation parameter β with respect to the pump wavelength. When β (4) 0, the parametric gain can be obtained by pumping the fiber in the normal dispersion region of: b ð2þ \ b ð4þ X 2 =12 ð2:11þ which is near the ZDW, or by pumping in the anomalous dispersion region of: b ð2þ [ 4cP=X 2 b ð4þ X 2 =12 ð2:12þ when β (4) 0, the parametric gain can be obtained by pumping the fiber in the anomalous dispersion region of: 4cP=X 2 b ð4þ X 2 =12\b ð2þ \ b ð4þ X 2 =12 ð2:13þ If the pump wave locates in the normal dispersion region of β (2) β (4) Ω 2 /12, which is far from the ZDW or in the huge negative dispersion region of β (2) 4γP/ Ω 2 β (4) Ω 2 /12, the parametric spectral components cannot be observed [8]. 2.2 Parametric Amplification Based on PCF with TZDW 25 Fig. 2.6 The phase-matching contour for the nonlinear mismatch term of γp =0. Reprinted from Ref. [8], copyright 2013, with permission from Elsevier i (μm) Sideband wavelength λ s & λ Pump wavelength λ p (μm) signal idler The phase-matching contour for the fundamental mode of the PCF with nonlinear mismatch term γp = 0 is shown in Fig The range of the pump wavelength, in which phase-matched signal and idler pair exist, is from 710 to 1346 nm, beginning at the normal dispersion regime near the first ZDW of 723 nm, and ending at the anomalous dispersion regime close to the second ZDW of 1363 nm. For each pump wave in the regions from 710 to 832 nm and from 1120 to 1346 nm, two groups of phase-matched signal and idler pairs exist. For example, for the pump wavelength of 1200 nm, the outer pair of the phase-matched points is marked with 1 and 2, and the inner pair of the phase-matched points is marked with 3 and 4, as shown in the vertical solid line on Fig For each pump wave in the region from 832 to 1120 nm, only one group of phase-matched signal and idler pair exists, and the signal and the idler bands have a large interval. Figure 2.7 shows the evolution of the phase-matched sidebands versus the pump wavelength with different pump power. The black slash shows the location of the pump wavelength in the vertical Fig. 2.7 The phase-matching contours for different pump power. Reprinted from Ref. [8], copyright 2013, with permission from Elsevier i (μm) λ s & λ Sideband wavelength W 5000W 10000W 20000W 50000W Pump wavelengthλ (μm) p 26 2 New Wavelength Generation Based coordinate. The curves above the slash indicate the wavelengths of the signal sidebands, and the curves below the slash indicate the wavelengths of the idler sidebands. When the peak pump power is 500 W, for each pump wavelength in the region from 711 to 1330 nm, two groups of phase-matched signal and idler pairs exist. The inner and outer phase-matched pairs constitute a ring shape on each side of the slash of the pump wavelength. With the peak pump power increased, the nonlinear phase mismatch will seriously affect the phase-matching condition. The pump wavelength region in which the phase-matched waves pair can appear becomes smaller. The frequency detuning of the inner pair from the pump wave increases, and the frequency detuning of the outer pair from the pump wave decreases. The gain bandwidth is a critical parameter to the parametric amplification. The frequencies distributed in the region of 4γP Δβ(Ω) 0 will experience a nonzero gain. Figure 2.8 shows the phase-matching contours for the fundamental mode of the PCF when the linear phase mismatch Δβ equals to 4γP, 2γP and zero, respectively, and the peak pump power P is 20,000 W. For a given pump wavelength, the gain band covers the wavelengths that satisfy the condition of 4γP Δβ(Ω) 0. The phase-matching curve with Δβ = 2γP indicates the signals and idlers with the maximal gain. Based on the profiles of the gain spectrum, the pump wavelengths are divided into six regions. In the region 1, for a given pump wave, there are four separated gain bands, corresponding to the signal and idler gain bands of the inner and outer pairs, respectively. For example, with the pump wavelength of 800 nm, the four gain bands are a, b, c, and d, respectively, which are marked in the Fig In the region 2, for a given pump, the signal and idler bands of the inner pair are connected to each other, and a broad band is formed near the pump wavelength except for two separated signal and idler bands of the outer i (μm) λ s & λ Sideband wavelength a b c Pump wavelength d λ p (μm) Δβ = 0 Δβ = 2γ P Δβ = 4γ P Fig. 2.8 The Phase-matching contours for the high group-index mode of the PCF with the linear phase mismatch Δβ of 4γP, 2γP and zero, respectively, when the pump power is 20,000 W. Reprinted from Ref. [8], copyright 2013, with permission from Elsevier 2.2 Parametric Amplification Based on PCF with TZDW 27 pair. In the region 3, for a given pump, the four gain bands are all connected to each other and form a superbroad gain band similar to a supercontinuum spectrum. In the region 4, for a given pump, the idler or signal of the outer and inner gain bands are combined together. Either the signal or the idler band includes two gain peaks since two fulfilled phase-matching wavelengths are existed. In the region 5, the fulfilled phase-matched wavelengths do not exist, and the parametric gain is very weak. The region 6 belongs to zero gain region, and no parametric gain exists. It can be predicted that, according to band gain contours, various parametric gain shapes can be obtained by adjusting the pump wavelength The Optical Parametric Generation in a PCF with TZDW The optical parametric spectra are measured for the PCF with TZDW. The experimental setup is shown in Fig Experimentally, a Ti: sapphire pulse laser can emit a pulse train with the full width at half maximum (FWHM) of 130 fs, at the repetition rate of 76 MHz. The pump pulse train is coupled into 1.0-m PCF mentioned above through a 40 microscope objective lens with the numerical aperture of The central wavelength of the pump wave is set to be 800 nm. The pump power can be adjusted by a neutral-density filter wheel. The light emitting from the fiber is collimated by a 40 microscope objective lens and then sent to two optical fiber spectrometers (Avaspec and Avaspec-NIR ) with the measurement scopes from 200 to 1100 nm and from 900 to 2500 nm. Since the signal band distributing in the wavelength region of nm, a suitable signal source is not accessible. We used amplified spontaneous emission (ASE) from the pump laser as the seed source and inferred the FOPA gain spectrum from the measurement of the output ASE spectrum. The experimentally observed spectra for the peak pump power of 20,000 W and the pump wavelengths of 760, 800 and 815 nm are shown in Fig The idler bands are shown in Fig. 2.10a, c, and e, in which two gain bands in visible region can be clearly seen. When the pump is operated at 800 nm, the region of nm corresponds to the idler wave of the inner pair (IWIP) of the sideband, and the region of nm corresponds to the idler wave of the outer pair (IWOP) of the sideband. When the pump wavelength increases, both the IWIP and IWOP move to the longer Fig. 2.9 The experimental setup 28 2 New Wavelength Generation Based Fig The observed output spectra of the idler waves, with the pump wavelengths of a 0.76 μm, c 0.8 μm, and e μm, respectively. The observed output spectra of the signal waves, with the pump wavelengths of b 0.76 μm, d 0.8 μm, and f μm, respectively. The peak pump power is set at P = 20,000 W. Reprinted from Ref. [8], copyright 2013, with permission from Elsevier wavelength, which is shown in Fig In particular, when the pump wavelength is 760 nm, the idler wave extends down to the ultraviolet region of 300 nm. The signal waves are shown in Fig. 2.10b, d, and f. Two gain bands corresponding to the signal waves of the inner and outer pairs (SWIP and SWOP) can be observed clearly. For example, when the pump wavelength is 800 nm, the two gain bands distribute in bands of nm and nm, respectively. Each band has two peaks resulted from the birefringence of the PCF, since only one half-wave plate is used and the polarization state of pump is not aligned properly with the principle axis of the PCF. Theoretically, a half-wave plate and two quarter-wave 2.2 Parametric Amplification Based on PCF with TZDW 29 Fig Sideband wavelength of the two pairs of FWM versus the pump wavelength, the peak pump power is set at 20,000 W. Reprinted from Ref. [8], copyright 2013, with permission from Elsevier i (μm) λ s & λ Sideband wavelength IWOP IWIP SWOP SWIP Pump wavelength λp (μm) plates should be used simultaneously so that all the polarization states can be reached. The evolution of the signal band versus the pump wavelength is also shown in Fig When the pump wavelength is 815 nm, the signal band extends to the mid-infrared region of 2190 nm. The two pairs of gain bands are approximately in agreement with the simulation results in Fig In order to further study the evolution of the two pairs FWM gain bands versus the propagation distance, we simulate the sech 2 pulses with peak power of 20,000 W and 130 fs FWHM propagating in 1-m PCF using the split-step Fourier method to solve the generalized nonlinear Schrödinger equation [9]. The spectral evolution is shown in Fig It is clear that two pairs of sideband are generated at the propagation distance about 3 mm. As shown in Fig. 2.12, the outer pair of FWM gain bands is composed of A and B, and the inner pair is composed of C and D. The wavelengths of them remain relatively fixed with the increasing of the propagation distance to 1 m. Fig Simulated spectral evolution of a pulse launched at 800 nm with the peak pump power of 20,000 W. Reprinted from Ref. [8], copyright 2013, with permission from Elsevier 30 2 New Wavelength Generation Based Reeves et al. [10] theoretically predicted that two pairs of gain bands can be existed in particular dispersion-engineered PCFs as early as This phenomenon is clearly observed in this experiment for the first time. The relationship between the optical parametric spectrum and the pump wavelength provides more effective interpretations to the SC generation. 2.3 Dispersive Wave Generation The Principle of Dispersive Wave Generation When optical pulses are transmitted in the anomalous dispersion regime, multiple solitons can be generated by the soliton fissio

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