Ultrashort Pulse Retrieval from Experimental Spectra Transformed in Chalcogenide and Silica Fibers

Abstract

The characterization of ultrashort optical pulses is a highly requested task. The most popular commercially available hardware/software systems are based on interferometric measurements and second-harmonic generation, leading to some ambiguities and limitations. Here we experimentally test the non-interferometric method of pulse retrieval from three spectra: the fundamental spectrum and two spectra that transformed in an element with Kerr nonlinearity and accumulated different nonlinear phases (different Β-integrals). This method has no ambiguities related to time direction, and allows simple hardware/software implementation. We test a novel simple algorithm for experimental data processing based on the search for a polynomial-approximated spectral phase. Two experimental cases are considered. In the first one, we retrieved 160 fs pulses using a chalcogenide arsenic sulfide glass fiber as a nonlinear Kerr element. In the second case, we retrieved 670 fs pulses with a complex spectrum using a piece of silica-based fiber. The results are confirmed by independent measurements using a standard SHG-FROG technique (Second-Harmonic Generation Frequency-Resolved Optical Gating).

1. Introduction

The characterization of ultrashort optical pulses is in demand for solving many problems. The most popular commercially available hardware/software systems, such as autocorrelators and Second-Harmonic Generation Frequency-Resolved Optical Gating (SHG-FROG) are based on interferometric measurements and second-harmonic generation, leading to some ambiguities and limitations [1,2]. The ambiguities concern the direction of the temporal axis and the sign of the phase; the main limitations concern the spectral width, since nonlinear crystals have limited phase-matching bandwidths. To date, there are a lot of different methods, and their modifications for ultrashort pulse characterization have various degrees of complexity, advantages, and shortcomings [2,3,4,5,6,7,8]. Nevertheless, the development of ultrafast metrology with simple hardware/software implementations and without ambiguities is in demand.

A simple non-interferometric method using 3rd-order Kerr nonlinearity rather than 2nd-order nonlinearity was proposed in [9]. It is based on measuring the fundamental (non-transformed) pulse spectrum and two self-phase modulated (SPM) spectra with different values of Β-integrals [9]. The iterative Gerchberg–Saxton-like numerical algorithm was developed for data processing [9]. This method was successfully implemented to characterize high-power single-shot pulses using thin plastic films as nonlinear Kerr elements [9], and for pulses with less power in the telecommunication range using optical fibers as nonlinear elements [10,11,12,13]. It should be emphasized that in previous papers [9,10,11,12,13], various iterative Gerchberg–Saxton-like numerical algorithms were used for spectra processing. However, such numerical algorithms work well for pulses with a relatively small phase (not more than π). For strongly chirped pulses, it may be challenging to find the global, rather than local, minimum of an error function. Very recently, to overcome this challenge, we proposed a novel simple and fast algorithm for processing the fundamental spectrum together with two SPM spectra with different values of Β-integrals [14]. The algorithm is based on searching for the polynomial-approximated spectral phase minimizing the error function, which is the difference between the measured and retrieved SPM spectra [14]. An exhaustive search is run to find optimal polynomial coefficients. Additionally, we numerically showed that this approach can be applied to characterize mid-IR ultrashort pulses using commercial chalcogenide glass fibers [14]. Here we demonstrate experimental implementations of the method using the novel algorithm of searching for a polynomial-approximated spectral phase. As far as we know, this is the first experimental verification of the algorithm on measured (not numerically simulated) data. In particular, we report the pulse retrieval using a nonlinear silica-based fiber and a chalcogenide (arsenic-sulfide-based) fiber. The use of a chalcogenide fiber as a nonlinear element in this method is also demonstrated for the first time. This successful experiment with a chalcogenide fiber in the telecommunication range motivates further studies on pulse retrieval in the wavelength range well beyond 2 µm, where the use of chalcogenide fibers is more justified since silica fibers are not transparent. The obtained results are verified by independent standard SHG-FROG measurements.

2. Methods

2.1. Numerical Algorithm

The tested method for ultrashort pulse retrieval is based on measuring and processing three spectra: fundamental spectrum I₀(f) (hereinafter f is frequency) and two SPM spectra transformed in a nonlinear fiber. The SPM spectra I₁(f) and I₂(f) correspond to different Β₁ and Β₂ values of Β-integrals ($B={\int }_0^L\gamma P\left(z\right)dz$, where γ is the nonlinear coefficient, Ρ is pulse peak power, z is the longitudinal coordinate, and L is fiber length). The main idea of the method and the block diagram of the tested algorithm are presented in Figure 1a,b, respectively. The details can be found in [14]. In brief, the algorithm finds an optimal polynomial-approximated spectral phase φ₀^opt minimizing the difference between measured and retrieved SPM spectra [14]. An exhaustive search for polynomial coefficients is run on a reasonable grid [14].

The evolution of the complex field amplitude Ε(t, z) (neglecting dispersion) in a nonlinear fiber is given by [15]

$$\frac{\partial {\rm E}\left(t,z\right)}{\partial z}=i\gamma {\left|{\rm E}\left(t,z\right)\right|}^2{\rm E}\left(t,z\right)$$

(1)

The field envelopes Ε₁(t) and Ε₂(t) of the output pulses after propagation in the fiber are

$${{\rm E}}_1\left(t\right)=E_0\left(t\right)\exp (i{{\rm B}}_1|E_0\left(t\right){|}^2/\max (|E_0\left(t\right){|}^2)),$$

(2)

$$E_2\left(t\right)=E_0\left(t\right)\exp (i{{\rm B}}_2|E_0\left(t\right){|}^2/\max (|E_0\left(t\right){|}^2)),$$

(3)

where Ε₀(t) is the complex field envelope of the input pulse. The dispersionless approximation is reasonable when the dispersion length L_D is much longer than L (L_D = τ²/|β₂| >> L, where τ is the pulse duration and β₂ is the second-order dispersion coefficient [15]).

The spectral complex amplitudes are defined as

$${\tilde{E}}_{0,1,2}\left(f\right)=\widehat{F}\left[E_{0,1,2}\left(t\right)\right],$$

(4)

where $\widehat{F}$ is the Fourier transform operator. Assuming that φ₀ is the guessed spectral phase of the characterized pulse, we can write

$$E_0\left(t\right)={\widehat{F}}^{-1}\left[{\tilde{E}}_0\left(f\right)\right]={\widehat{F}}^{-1}\left[\sqrt{I_0\left(f\right)}\exp \left(i{\phi }_0\right)\right].$$

(5)

Next, we can calculate E₁(t) and E₂(t) using Formulas (2) and (3). ${\left|{\tilde{E}}_1\left(f\right)\right|}^2$ and ${\left|{\tilde{E}}_2\left(f\right)\right|}^2$ are close to I₁(f) and I₂(f), respectively, if φ₀ is close to the real spectral phase φ_real (Figure 1a) [9,14]. To quantitatively describe the similarity of the real and retrieved SPM spectra, we use the following error function

$$\mathsf{\Delta }=\frac{1}{2}\frac{{\left[{\sum }_{k=1}^N{\left({\left|{\tilde{E}}_1\left(f_k\right)\right|}^2-I_1\left(f_k\right)\right)}^2+{\sum }_{k=1}^N{\left({\left|{\tilde{E}}_2\left(f_k\right)\right|}^2-I_2\left(f_k\right)\right)}^2\right]}^{1/2}}{{\sum }_{k=1}^N{I}_0\left(f_k\right)},$$

(6)

where N is the number of the chosen frequency points and f_k is the frequency at the k-th point.

We assume that the spectral phase can be approximated by a cubic polynomial with sufficient accuracy; ${\phi }_0\left(f\right)=C_2{f}^2+C_3{f}^3$. We minimize the error function using an exhaustive search for C₂ and C₃ (Figure 1b) [14]. The values of C₂⁽⁰⁾ and C₃⁽⁰⁾ correspond to the global minimum of Δ; ${\phi }_0^{opt}\left(f\right)=C_2^{\left(0\right)}{f}^2+C_3^{\left(0\right)}{f}^3$.

As a rule, Β-integrals are not exactly known but their ratio Β₂/Β₁ is known. Therefore, the algorithm is run for various Β₂ and after that the value minimizing Δ is selected [14]. Note that the ratio of B-integrals should not be very close to 1. Otherwise, the SPM spectra are very similar, which may lead to errors in their processing, especially with allowance for noise. We set Β₂/Β₁ as ≥ 1.5.

2.2. Experimental Schemes

For experimental testing of the algorithm optimizing the polynomial-approximated spectral phase, we used two setups based on a femtosecond Er-doped fiber laser. The schemes are shown in Figure 2a,b. In both cases, the systems start from the Er-doped fiber master oscillator with passive mode locking using a nonlinear polarization rotation. The laser generates ultrashort pulses at a central wavelength of 1.56 µm with a repetition rate of 50 MHz.

In the first investigated case, “Er:fiber laser system I” contains a single-fiber amplifier. The maximum output pulse energy is 4 nJ. The phase is not flat due to the nonlinear and dispersive effects in the amplifier. This system is similar to the one reported in [16]. Here, a chalcogenide As₂S₃ glass fiber is used as a nonlinear Kerr element. Laser pulses from the output of “Er:fiber laser system I” are coupled to 1.5 cm As₂S₃ fiber through a lens (Figure 2a). The coupling efficiency is regulated by tuning the lens focus relative to the chalcogenide fiber input. This allows us to control the ratio of Β-integrals by monitoring average output powers (Β₂/B₁ is proportional to the ratio of average powers). The coupling efficiency was of the order of a few percent. The standard SHG-FROG technique was used to independently measure the tested pulses (Figure 2a).

In the second case, we characterized the pulses from the output of “Er: fiber laser system II” delivering maximum energy of about 10 µJ (although this value was attenuated down to ~1 µJ). The system configuration is similar to that reported in [17]. This system operates in a chirped pulse amplification regime. A fiber stretcher is used after the master oscillator before two fiber preamplifiers and the final amplifying cascade. The pulse repetition rate is decreased from 50 MHz to 100 kHz by an acousto-optic modulator inserted between two preamplifiers. This is conducted to reduce the average power. The pulses at the output of the final fiber amplifying cascade pass through a grating compressor and after that are propagated in a short piece of a standard telecommunication fiber (of order 1 cm) to broaden the spectrum and to accumulate an additional phase due to SPM. This is conducted to test the applicability of the method for retrieving the additional phase. Furthermore, as in the previous scheme, the pulses are coupled through the lens into a segment of a nonlinear fiber playing a key role in the tested method (Figure 2b). In this experiment, we used a silica-based fiber with a smaller diameter compared to SMF28e and with a larger nonlinear Kerr coefficient due to a smaller mode field area. The SHG-FROG technique was also used to verify the results obtained with the tested method.

3. Results

3.1. Pulse Retrieval from Measurements Using Chalcogenide Fiber

In the first experiment, we used the scheme presented in Figure 2a. We took a small piece of a chalcogenide fiber, 1.5 cm long, as an element for spectrum transformation due to SPM. Here, the estimated dispersion length of the order of 10 cm is much longer than the fiber length (L >> L_D), thus the dispersionless approximation is reasonable. For this fiber, the calculated nonlinear Kerr coefficient is γ~300 (W km)⁻¹. We measured the fundamental and two transformed SPM spectra, and after that applying the described algorithm for finding the polynomial-approximated spectral phase, we retrieved the temporal intensity profile (Figure 3a) and the temporal phase (Figure 3b). The error function is visualized in Figure 3c. In this case, the optimal polynomial coefficients minimizing the error are C₂⁽⁰⁾ = 0.044 ps² and C₃⁽⁰⁾ = −0.001 ps³. The fundamental spectrum of the signal is shown in Figure 3d. The SPM measured and retrieved spectra are plotted in Figure 3e,f for two different values of the Β-integrals: Β₁ = 0.8 and Β₂ = 1.3. The pulse duration is 160 fs (full width at half maximum, FWHM), which is in excellent agreement with the results of independent SHG-FROG measurements also shown in Figure 3a. The temporal phases retrieved using the tested method and SHG-FROG also coincide almost perfectly (Figure 3b). It should be noted that the SHG-FROG method exhibits ambiguity in the choice of the direction of the time axis and the phase sign. To eliminate this ambiguity, we took into account the results of our method, which produced an unambiguous answer. Note that the execution time of the tested algorithm for processing experimental spectra was only a few seconds (to calculate the whole error map in Figure 3c).

3.2. Pulse Retrieval from Measurements Using a Silica Fiber

In the second experiment, we used the scheme sketched in Figure 2b. A 2 cm silica-based fiber was chosen as a nonlinear Kerr element. Its nonlinear Kerr coefficient is γ~4 (W km)⁻¹. As in the first experiment, here we also measured the fundamental and two transformed SPM spectra, and after that, we applied the tested algorithm for finding the polynomial-approximated spectral phase. In contrast to the previous case, when the Β-integrals were of order 1, here the Β-integrals were large (of order 10) and the transformed spectra were significantly broadened and strongly modulated. We found that the tested algorithm works correctly in this case as well. The retrieved temporal intensity profile and temporal phase are shown in Figure 4a,b, respectively. The error function is demonstrated in Figure 4c. The optimal polynomial coefficients are C₂⁽⁰⁾ = 1.3 ps² and C₃⁽⁰⁾ = −0.015 ps³. The fundamental spectrum and two SPM spectra with Β₁ = 6.5 and Β₂ = 11 are presented in Figure 4d–f, respectively. The obtained pulse with a FWHM duration of 670 fs agrees sufficiently well with the SHG-FROG pulse (Figure 4a). The temporal phases retrieved from the tested method and the SHG-FROG are also close to each other (Figure 4b).

4. Discussion and Conclusions

We demonstrated the experimental implementations of the simple non-interferometric method of pulse retrieval from fundamental and two SPM spectra using the novel algorithm of searching for a polynomial spectral phase. The algorithm was proposed and numerically investigated in [14]; here, we presented its first experimental verification on measured data. Namely, we reported the retrieval of (i) 670 fs pulses using a nonlinear silica-based fiber and (ii) 160 fs pulses using a chalcogenide (arsenic-sulfide-based) fiber. The obtained results are in a good agreement with the independent standard SHG-FROG measurements. Use of a chalcogenide fiber as a nonlinear element for the tested method is also demonstrated for the first time. Although, pulses were retrieved in the range of about 1.5 µm (not in the mid-IR range, as was numerically studied in [14]), the successful application of chalcogenide fibers allows us to propose their use as nonlinear Kerr elements for characterizing ultrashort pulses in a wider wavelength range corresponding to the transparency band of such fibers.

We worked in the dispersionless approximation, but the fiber dispersion should be taken into account for shorter pulses (when the relation L_D >> L is not satisfied). Previously, we realized a Gerchberg–Saxton-like algorithm with allowance for the dispersion and demonstrated its experimental verification [10]. The novel algorithm of searching for a polynomial spectral phase considered here can be also modified in a similar way. E₁(t) and E₂(t) should be calculated from the nonlinear Schrödinger equation with the dispersion term [15] instead of Formulas (2) and (3) originated from Equation (1).