Distortionless large-ratio stretcher for ultra-short pulses using photonic crystal fiber

L.-G. Li; L.-S. Yan; G.-Y. Feng; W. Pan; B. Luo; A. Yi; R.-L. Zhu

doi:10.1364/OE.18.012341

1. Introduction

Large-ratio stretching of the ultra-short pulse seed laser is one of the key techniques for ultrahigh peak-power laser systems [1–7]. So far, most stretchers have been made out of free space diffraction pairs (e.g. Öffner-type, Martinez-type and Barty-type) [8–10]. Some of these stretchers can obtain large stretching factor while constructed with components that cause aberrations (mainly color and spherical aberration) and higher order dispersion. Moreover, it is difficult to manage the dispersion of the broadening and compression system. Therefore, such schemes may suffer from rigorous design and manufacture of optical components, intrinsic aberration and pulse distortion. In addition, pulse stretchers can be made based on optical fibers [11,12]. Such all-fiber solutions can avoid free space optics with advantages such as compact size, superior stability and reliability [13].

On the other hand, photonic crystal fibers (PCFs), also called microstructure fibers or holey fibers, constituted of a single material (silica) with an array of air periodic holes along its length, have attracted increasing attentions in the last few years [14–26]. These fibers have been shown desirable characteristics such as endlessly single mode, wide operating wavelength range (350-1800nm), high nonlinear coefficient and high-mode birefringence, etc. In particular, the dispersion and nonlinearity of PCFs can be controlled by changing structural parameters. Therefore, PCFs can easily find many applications such as soliton generation, chromatic dispersion compensation and so on.

In this paper, we propose a PCF structure that can be used as the ultra-short pulse stretcher for ultra-high power laser systems. The PCF can provide more than 300-nm working wavelength range with flattened second-order dispersion β₂ and extremely low higher-order dispersion β₃, suitable for most ultra-short pulse laser sources around 0.8 μm. Through theoretical analysis and simulation, we obtain key parameters of the PCF. Results indicate that more than 10,000 stretching ratio is achievable for ultra-short pulses with pulse width less than 100 femtoseconds. The pulse distortion is negligible as well as the nonlinear effect. Such approach may be more practical in the near future due to its all-fiber nature.

2. PCF model

Figure 1 shows the geometrical diagram of the PCF model, where A is the lattice pitch, d and r are the diameter and radius of air-holes, respectively. The PCF is homogeneous with the air holes surrounded by silica. Full vector effective index method is used to analyze the dispersion properties of PCF. As mentioned in [22, 26], this method is efficient to calculate properties of PCFs, including the dispersion characteristics that agree well with results from the plane-wave expansion method.

Fig. 1 The cross section of triangular lattice of the PCF

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For the PCF made of silica, the boundary conditions are described as [22]:

E_{z} (ρ = R) = H_{z} (ρ = R) \equiv 0

Here R=A/2 is the radius of the field where boundary conditions due to symmetry properties have to be applied. Moreover, we resort to separation of variables and the E_z and H_z field components can be written as:

\begin{array}{l} H_{z} (ρ, θ, z) = H_{l} (ρ) e^{i l θ} e^{- γ z} \\ E_{z} (ρ, θ, z) = E_{l} (ρ) e^{i l θ} e^{- γ z} \end{array}

From boundary conditions in Eq. (1) and the separation of variables method, we can derive dispersion relations for the fundamental modes. Among all the solutions of the characteristic equations, the effective refractive index of EH11 mode exhibits the highest one [22], corresponding to the cladding space-filling mode, with its characteristic equation written as

\frac{I_{2} (w, r)}{I_{1} (w, r)} = - \frac{1}{w \cdot r} - \frac{w \cdot r}{2} (1 + \frac{n_{si}^{2} (λ)}{n_{a i r}^{2}}) g (u) - w \cdot r {[\frac{1}{4} {(1 - \frac{n_{s i}^{2} (λ)}{n_{a i r}^{2}})}^{2} g^{2} (u) + \frac{f (w, u)}{n_{a i r}^{2}}]}^{1 / 2}

where g is a function of u, f is a function of w and u,

n_{a i r}

is the refractive index of the air hole, and

n_{s i} (λ)

is the wavelength-depended refractive index of silica that satisfies with the Sellmeier relationship.

g (u) = \frac{1}{ω r} \frac{J_{0} (u r) Y_{1} (u R) - Y_{0} (u r) J_{1} (u R)}{J_{1} (u r) Y_{1} (u R) - Y_{1} (u r) J_{1} (u R)} - \frac{1}{u^{2} r^{2}}

f (w, u) = \frac{1}{r^{4}} (\frac{1}{u^{2}} + \frac{1}{w^{2}}) (\frac{n_{s i}^{2}}{u^{2}} + \frac{n_{a i r}^{2}}{w^{2}})

Here J_m and Y_m are the first and second kind of m-order Bessel functions, respectively. I_m is the m-order modified Bessel function of the first kind. Parameters w and u are

w^{2} = ω^{2} (n_{e f f}^{2} - n_{a i r}^{2}) / c^{2}

u^{2} = ω^{2} (n_{s i}^{2} - n_{e f f}^{2}) / c^{2}

where c is velocity of light and n_eff is the effective index of the space filling mode. Equations (3-7) can be solved numerically to find n_eff. Then we can calculate the effective index of the PCF (

n_{e f f}^{'}

) using the step theory of normal optical fiber. The dispersion parameter D and the GVD parameter β₂ are defined as [27]

D = \frac{λ}{c} \frac{d^{2} n_{e f f}^{'}}{d λ^{2}} & β_{2} = - \frac{λ^{2}}{2 π c} D

For ultra-short pulse stretching, our goal is to have a broadband wavelength region with dispersion parameters as flat as possible. In addition, although the output power of most seed laser is not high, we still need take into account the nonlinear coefficient of the PCF in case that the nonlinearity may have significant effects on the pulse. The nonlinear coefficient γ is defined as [27]

γ (λ) = \frac{2 π}{λ} \frac{\iint_{s} n_{2} (x, y) I^{2} (x, y, λ) d x d y}{{(\iint_{s} I (x, y, λ) d x d y)}^{2}} = \frac{2 π}{λ} \frac{n_{2} (λ)}{A_{e f f} (λ)}

where I is the transverse electric field vector, λ is the operating wavelength, S denotes the whole fiber cross section, n₂ is the effective nonlinear refractive index (approximately 2.76×10⁻²⁰m²/W at 0.8 μm) and the effective mode area A_eff can be calculated from [19]

A_{e f f} (λ) = \frac{{(\iint {| I (x, y, λ |}^{2} d x d y)}^{2}}{\iint {| I (x, y, λ |}^{4} d x d y} = π R^{2} = \frac{π a_{e f f}^{2}}{\ln V_{N L}}

where R is the spot size, a_eff is the effective core radius of the PCF. The parameter V_NL is defined as

V_{N L} = \frac{V}{\sqrt{1 - k_{0}^{2} n_{c o}^{2} n_{2} R^{2} {| A |}^{2} / 8 Z_{0}}}

where k₀ is the free-space wave number, Z₀ is the free-space impedance, n_co is the equivalent refractive index of the core, A is the amplitude of the field and V is the normalized frequency.

3. PCF design

As mentioned before, for short-pulse stretcher applications, large wavelength range with flat second-order dispersion β₂ and low higher-order dispersion are required around the operation wavelength. Based on the structure in Fig. 1 and above theory (in fact, our simulation model is firstly verified using parameters in [18, 22] before investigating the pulse stretching application), we simulate characteristics of the PCF as we vary different parameters (i.e. the radius of air holes r and the lattice pitch A). Especially we concentrate on the dispersion around 0.8μm. Figures 2(a) & (b) illustrate the GVD parameter β₂ as a function of operating wavelength for different r (Fig. 2a) and A (Fig. 2b) values. The case of pure silica is also included for comparison. Apparently wide wavelength regions with flat dispersion variations are available.

Fig. 2 Simulation results of effective dispersion for the proposed PCF with (a) different radius of air holes and (b) different lattice pitch. The black curve corresponds to the material dispersion of pure silica.

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From Fig. 2, it is observed that, around 0.8-μm wavelength range, there is a region of interest with flat second-order dispersion β₂ over 300 nm when we carefully choose the lattice pitch and the radius of the air hole. Subsequently relative optimal PCF parameters around 0.8 μm wavelength can be obtained, i.e. the lattice pitch is 1.27μm and the radius of air holes is 0.265μm. Once we fix these two parameters, the dispersion parameters (β₂ and the 3rd-order β₃) versus wavelength are shown in Fig. 3 . For the wavelength around 0.8μm, we have β₂≈35.72ps²/km and β₃≈ 0.001 ps³/km. Therefore, the third-order effect can be neglected and the value of β₂ dominates the pulse stretching.

Fig. 3 Dispersion parameter (a)β₂ and (b)β₃ versus wavelength for the optimal PCF

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In addition, we calculate the nonlinear coefficient γ according to Eqs. (9-11). For the optimal PCF at the wavelength of 0.8μm, the nonlinear coefficient is ~36.4 (W^-1•km⁻¹).

4. Large-ratio pulse stretching

To numerically model the PCF stretcher for ultra-short pulses in the presence of possible nonlinear effects, we use the Nonlinear Schrödinger equation (NLSE) [11, 27]:

\frac{\partial A}{\partial z} = - \frac{α}{2} - \frac{i}{2} β_{2} \frac{\partial^{2} A}{\partial T^{2}} + \frac{1}{6} β_{3} \frac{\partial^{3} A}{\partial T^{3}} + i γ | A |^{2} A

where A is the amplitude of the pulse envelope, and β₂, β₃, γ and α are the second-order dispersion parameter, third-order dispersion parameter, nonlinear and attenuation coefficients of the PCF, respectively. Higher-order (i.e. beyond third-order) dispersion is neglected.

The NLSE is solved using the standard split-step method to evaluate the ultashort pulse stretching ratio/factor. We choose 1-km PCF in our simulation, and assume its nonlinear coefficient and attenuation coefficient as 36.4 (W•km)⁻¹, 0.5dB/km [17], respectively. The dispersion parameter β₂ and β₃ are 35.72 ps²/km and 0.001 ps³/km, respectively, which are calculated from the optimal PCF parameters (the lattice pitch and air holes radius are 1.27μm and 0.265μm, respectively, see Fig. 2). For the input ultra-short pulse, we assume it has a Gaussian envelop with the peak power of 10mW and a certain pulse width (we define the FWHM of the pulse as ΔT). The center wavelength is set to 0.8μm mostly used in ultra-high peak-power laser systems.

First, we investigate the pulse stretching factor (it is defined as the ratio of the FWHM of the input and output pulses.) directly. Figure 4 illustrates the stretching ratio as a function of the pulse width ΔT, where we can see that the stretching ratio increases as the pulse width decreases. More than 10,000 stretching ratio can be readily achieved. As examples, Inserted Figs. 4(a) and (b) show stretched pulses for the original pulses with ΔT of 40fs and 90fs, respectively. The corresponding stretching ratios are 52,523and 10,380, respectively As we look into the zoomed version of these stretched pulses, the distortion due to higher-order dispersion is negligible, which demonstrates the feasibility of such PCF-based approach.

Fig. 4 Stretching ratio as a function of the pulse width ΔT after 1-km PCF, inserted (a) ΔT=40fs and (b) ΔT=90fs

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Generally fiber nonlinear effects have to be considered for ultra-short pulses. Therefore, we simulate the pulse evolution under different nonlinear coefficient. As mentioned, the coefficient γ is calculated as 36.4 (W⁻¹•km⁻¹) for the center wavelength (i.e. 0.8μm), we vary the coefficient from 10 to 60 W⁻¹•km⁻¹ for reference. Figure 5 shows different broadened waveforms as the nonlinear coefficient varies. We can see that the nonlinear coefficient γ has limited effect on the waveform broadening comparing with the dispersion as the peak power of the input pulse is limited.

Fig. 5 Waveforms for different nonlinear coefficient γ values (ΔT=60fs)

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It should be noted that, as mentioned in [28–30], the randomness of the input ultra-short pulse, such as temporal jitter, amplitude ripple and noise, may affect the stretching performance significantly. The impact of randomness in the input pulse on the output pulse has been observed in our simulation indeed as the peak power increases dramatically, which is under our further investigation.

5. Conclusion

We proposed a PCF-based large-ratio stretcher for ultra-short pulse applications. Through properly choosing structure parameters of the PCF, ultra-short pulse with 3-dB width less than 100-femtosecond can be broadened more than 10,000 times with negligible distortion due to either dispersion or nonlinearity. Such approach will be of great potential values for ultra-high peak-power laser systems due to its all-fiber nature and wide wavelength operation range.

Acknowledgement

The research is supported by the National Natural Science Foundation of China (No. 10876022, 60972003) and the Program for New Century Excellent Talents in University (NCET-08-0821), Ministry of Education, China.

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Distortionless large-ratio stretcher for ultra-short pulses using photonic crystal fiber

Abstract

1. Introduction

2. PCF model

3. PCF design

4. Large-ratio pulse stretching

5. Conclusion

Acknowledgement

References and links

Cited By

Figures (5)

Equations (12)

Optics Express