# Numerical Study of Mid-IR Ultrashort Pulse Reconstruction Based on Processing of Spectra Converted in Chalcogenide Fibers with High Kerr Nonlinearity

^{*}

*Fibers*)

## Abstract

**:**

_{2}S

_{3}-based glass fibers as nonlinear elements.

## 1. Introduction

_{2}S

_{3}-based glass fibers as nonlinear elements. Such fibers are transparent up to 6.5 µm and have large nonlinear Kerr coefficients, therefore, they are very suitable for the considered problem. The method is based on spectral measurements. Only the initial pulse spectrum and two converted spectra in elements with Kerr nonlinearity are required. These two converted spectra must have different B-integrals characterizing nonlinear phase accumulation. Moreover, there is no need to know the exact values of the B-integrals, only their ratio should be known [13,14]. The hardware implementation of the method is simple and involves the use of elements with third-order rather than second-order nonlinearity. This method requires only a one-dimensional data set (unlike the FROG family of methods), has no moving parts in the optical scheme (unlike most correlators and FROG implementations), and allows working with high-energy as well as low-energy pulses, which makes it promising in terms of potential implementation in devices of integrated photonics and fiber optics [14]. The earlier proposed reconstruction algorithms are based on the Gerchberg–Saxton algorithm [13,14]. However, these Gerchberg–Saxton-like algorithms work correctly if the pulse phase is not large (does not exceed π) [13]. In the case of a large chirp, the global minimum may not be reached; instead, stagnation may occur at a local minimum of an error function characterizing the difference between the original and reconstructed spectra (iterative algorithms often loop around a local minimum). Previously, we applied elements of genetic algorithms to solve this problem [13], which significantly increased the calculation time but did not guarantee to find the global minimum. Here, we propose a very simple and efficient algorithm to overcome this challenge. The algorithm is based on exploiting a polynomial spectral phase and finding polynomial coefficients using an exhaustive search on a reasonable grid of parameters, which allows finding global and all local minima on the preset grid. Moreover, the found polynomial phase corresponding to the global minimum can be used to seed the iterative algorithm. This allows refining the “right” solution and attaining an almost perfect agreement between the original and reconstructed data, which is demonstrated through numerous simulations. Next, we calculate parameters for commercial chalcogenide As

_{2}S

_{3}-based glass fibers and demonstrate their application as nonlinear elements in the analyzed method with the developed algorithm for measuring mid-IR pulses through representative numerical examples.

## 2. Methods

_{D}is much larger than the fiber length (L

_{D}= T

_{0}

^{2}/|β

_{2}| >> L, where T

_{0}is the characteristic pulse duration, β

_{2}is the quadratic dispersion coefficient [15]), the dispersionless approximation can be used to describe pulse propagation, and the evolution of the complex electric field amplitude E is described by a very simple equation [15]:

_{1}and B

_{2}are

_{0}(t) is the field at the input and E

_{1}(t) and E

_{2}(t) are the fields at the fiber output.

_{0}and $\widehat{F}$ is the Fourier transform operator. The corresponding spectral phases are φ

_{0}, φ

_{1}, and φ

_{2}.

_{0}(f) and spectral phase (φ

_{real}) that should be reconstructed. Let us denote the converted spectral intensities at the output of the nonlinear fiber as I

_{1}(f) and I

_{2}(f) measured for two different values of the B-integral (B

_{1}and B

_{2}, respectively). We guess with some accuracy the spectral phase φ

_{0}is close to φ

_{real}(Figure 1a). Using the inverse Fourier transform operator ${\widehat{F}}^{-1}$, we can calculate the corresponding field E

_{0}(t) in the time domain:

_{1}(t) and E

_{2}(t) using expressions (3) and (4). Note that ${\left|{\tilde{E}}_{1}\left(f\right)\right|}^{2}$ and ${\left|{\tilde{E}}_{2}\left(f\right)\right|}^{2}$ will be close to I

_{1}(f) and I

_{2}(f), respectively. Moreover, the smaller the difference between φ

_{0}and φ

_{real}, the better match ${\left|{\tilde{E}}_{j}\left(f\right)\right|}^{2}$ with I

_{j}(f) (j = 1,2). To quantitatively characterize the coincidence of the real and reconstructed spectra, we introduce the error function:

_{q}is the frequency at the q-th point.

_{2}/B

_{1}is known. We run the developed algorithms for various B

_{1}and select the value minimizing the error function. It will be demonstrated in Section 3.4 that the found B-integrals coincide with the original B-integrals. The same approach was successfully applied for the processing of experimental data using a Gerchberg–Saxton-like algorithm with the genetic algorithm elements in [13,14].

_{0}

^{opt}. We consider the case of spectral phase approximation taking into account the quadratic and cubic contributions, which is the most important case from the practical point of view and is usually sufficient in the common case of pulse propagation in a dispersive medium [15]. In addition, this case allows a simple and intuitive graphical interpretation, providing visual control of seeking a solution. The scheme of the proposed algorithm is shown in Figure 1b. A polynomial spectral phase is constructed on a reasonable grid of the quadratic and cubic coefficients; spectral complex amplitude ${\tilde{E}}_{0}\left(f\right)$ and corresponding field amplitude E

_{0}(t) is calculated. Next, field amplitudes E

_{1}(t) and E

_{2}(t) are simulated using Equations (3) and (4) and the error function is estimated using Equation (7). An exhaustive search allows for finding quadratic and cubic coefficients giving global and all local minima on the preset grid. If necessary, it will not be difficult to implement this algorithm taking into account the search for a polynomial phase of a higher order. In this work, we purposely study representative cases in which, in addition to the global minimum corresponding to the solution, there are local minima.

## 3. Results

#### 3.1. Reconstruction of “Ideal” Signal

_{2}

^{(0)}= 6.5 ps

^{2}and C

_{3}

^{(0)}= −3.6 ps

^{3}. The spectral intensity profile was sech

^{2}-shaped; the spectral width was FWHM = 0.6 THz (full width at half maximum). We applied the proposed algorithm shown in Figure 1b. We performed an exhaustive search over the range of parameters −10 ps

^{2}≤ C

_{2}≤ 10 ps

^{2}, −10 ps

^{3}≤ C

_{3}≤ 10 ps

^{3}(with steps dC

_{2}= 0.1 ps

^{2}and dC

_{3}= 0.1 ps

^{3}) in which, in addition to the global minimum corresponding to the true solution, a local minimum was also found. The required range of C

_{2}and C

_{3}depends on the pulse spectral width and the maximum expected pulse chirp (which depends on the net dispersion). Figure 2a–d display the numerically found solution (the computational time is a few seconds). As expected for such a case, the numerically found spectral phase ideally coincides with the initially specified one; therefore, the temporal intensity profile and temporal phase as well as the output spectra after conversion in the Kerr fiber with the given B

_{1}and B

_{2}for the reconstructed pulse also perfectly coincide with the original ones. The calculated error function Δ is shown in Figure 2e, where the global minimum is labeled “I” and the local minimum is labeled “II”. We also specially constructed a pulse with a spectral phase corresponding to the local minimum II. In this case, the error is much higher and there is a significant difference in the converted spectra plotted for the initial pulse and for the “wrong” reconstructed one.

#### 3.2. Reconstruction of Signals with Spectral Noise

_{2}

^{(0)}= 6.5 ps

^{2}and C

_{3}

^{(0)}= −3.6 ps

^{3}). Then, we constructed the original spectrum and the nonlinearly converted ones but random noise was added to all three spectra at a level of 5% of the maximum of I

_{0}(f) and we applied the numerical algorithm for finding the polynomial phase (Figure 3a–d). In this case, the results of the reconstruction of the polynomial coefficient C

_{2}coincided with C

_{2}

^{(0)}and the found value C

_{3}= −3.8 ps

^{3}slightly differed from the precisely specified value C

_{3}

^{(0)}(Figure 3a–d). The error was Δ = 0.0072. For comparison, we constructed the pulse for the polynomial phase corresponding to the local minimum II (Figure 3f–i) with a higher error Δ = 0.0201.

_{2}

^{(0)}and C

_{3}

^{(0)}. The spectral intensity profile was also sech

^{2}-shaped and the spectral width was FWHM = 0.6 THz. For each pulse, random spectral noise at a level of 5% of the maximum of I

_{0}(f) was added to the original spectrum and to the converted spectra (for B

_{1}= 1 and B

_{2}= 2), and then the polynomial phase search algorithm was applied (with steps dC

_{2}= 0.1 ps

^{2}, dC

_{3}= 0.1 ps

^{3}). The values of the randomly generated coefficients corresponding to each specific test signal (100 pulses in total) are shown by dots in Figure 4a. Figure 4b shows the error function Δ calculated for each specific test signal, and Figure 4c demonstrates the difference between the reconstructed and original polynomial coefficients. Note that the error function is practically independent of the value of the coefficients C

_{2}and C

_{3}. We also visually checked that in all cases the solution corresponding to the original signal was found.

#### 3.3. Reconstruction of Signal with Spectral Phase Perturbations

_{q}to the polynomial phase defined by expression (8). Therefore, the spectra I

_{1}(f) and I

_{2}(f) are strongly modulated. Spectral noise was not added for this test case. Using an exhaustive search for the spectral polynomial coefficients, we reconstruct the pulse (Figure 5a–d, left column). Further, we used this pulse with the polynomial spectral phase to seed the iterative algorithm. The results of its execution are shown in Figure 5e–h (right column). The phase peculiarities were reconstructed very well. The error Δ was reduced by a factor of 27 (from Δ = 0.0269 to Δ = 0.0010) and the original and found converted spectra coincided very well. For comparison, we run the iterative algorithm under other initial conditions. Sometimes we observed that the iterative algorithm could loop near a local minimum (the error functions were similar to the ones shown in Figure 2e and Figure 3e). In these cases, the original and found converted spectra were not in good agreement; the errors Δ were significantly higher than for the solution near the global minimum. Therefore, the use of the hybrid algorithm guaranteed finding the “right” solution.

#### 3.4. Reconstruction of Mid-IR Signals Using Chalcogenide Fiber

_{2}S

_{3}-based) glass fibers as nonlinear elements. Note that such fibers are produced by different companies and are commercially available, which will make it quite easy to implement this method in research laboratories. As an example, we considered IRflex Corporation fibers [17]. Based on technical specifications [17], we estimated the fiber parameters and then simulated pulse reconstruction using these parameters.

_{2}of arsenic sulfide glass fibers, we used the well-known method of finding the propagation constant β and electric fields of fundamental modes for the linear polarization by solving numerically the characteristic equation at different frequencies [15]. The refractive index was given by the Sellmeier equation based on measurements performed by Amorphous Materials Inc. (“AMTIR-6” [18]) and the numerical aperture was NA = 0.3 (corresponding to the difference between the core and cladding refractive indices of about 0.019). The calculated nonlinear coefficients γ and dispersion coefficients β

_{2}are plotted in Figure 6a,b, respectively, for fiber diameters of 5, 6.5, 7, and 9 µm. All the considered fibers had a large γ of about 100 (W⸱km)

^{−1}(Figure 6a). The dispersion curves were almost flat functions of wavelength in the 2.8–4 µm range (Figure 6b). Their calculations were required for estimating L

_{D}.

_{2}S

_{3}fiber with a core diameter of 6.5 µm serves for spectrum conversions. We purposely set spectral phases with leading polynomial contributions due to quadratic and cubic terms and added different regular perturbations. Representative examples are shown in Figure 7. Each column corresponds to an individual pulse with original data (black), data reconstructed with the proposed algorithm for searching an optimal polynomial spectral phase (magenta), and data reconstructed with the iterative algorithm near the global minimum of Δ (blue). We added spectral noise for I

_{0}(f), I

_{1}(f), and I

_{2}(f) at a level of 2% of the maximum of I

_{0}(f). We assumed that B

_{1}is not well known and run the algorithms for various B

_{1}values in the 0.5–2 range. The minima of error functions corresponded to the original B

_{1}values (Figure 7, bottom row).

_{FTL}= 260 fs (Figure 7, left column). Pulses with similar characteristics can be routinely generated with mode-locked fluoride fiber lasers [16]. We set φ

_{real}= −0.4[ps

^{2}]f

^{2}+ 0.15[ps

^{4}]f

^{4}, B

_{1}= 0.9, and B

_{1}= 1.8. Such B-integrals can be achieved for a sub-cm fiber length. For example, B

_{2}= 1.8, if the launched energy is only 0.85 nJ and L = 0.5 cm (γ = 150 (W km)

^{−1}@ 2.8 µm). For a sech

^{2}-shaped spectrum, T

_{0}≈ T

_{FTL}/1.763 = 150 fs, therefore L

_{D}= 9 cm is much longer than 0.5 cm and the dispersionless approximation is correct. Since the proposed algorithm for searching the polynomial phase does not take into account the fourth-order contribution, the mismatch with the original data is visible but not very high. Using the iterative algorithm after the reconstruction of the quadratic phase allows for refining the pulse characteristics (Figure 7, left column). Error Δ was decreased from 0.0059 to 0.0017 (this took eight iterations).

_{FTL}= 2 ps (T

_{0}≈ 1.1 ps) (Figure 7, middle column). For such a pulse L

_{D}= 2.8 m, therefore, the L maximum can be about 15–20 cm. We set φ

_{real}= −150[ps

^{2}]f

^{2}+ 450[ps

^{3}]f

^{3}+ 0.05π⸱sin(f/0.01[THz]). We simulated that for B

_{2}= 2.2, L = 20 cm, and γ = 85 (W km)

^{−1}@ 3.5 µm, the peak power is only 130 W, which corresponds to the launched pulse energy of 0.8 nJ (with allowance for the temporal shape of the pulse). Error Δ was decreased from 0.0151 to 0.0012 after running the iterative algorithm (16 iterations).

_{FTL}= 10 ps (T

_{0}≈ 5.7 ps) (Figure 7, right column). For such a pulse, the dispersion length is very large, L

_{D}= 140 m, therefore the maximum fiber length can be about a few meters. We set φ

_{real}= −900[ps

^{2}]f

^{2}+ 600[ps

^{3}]f

^{3}+ 0.05π⸱cos(f/0.005[THz]) and obtain that for B

_{2}= 2.6, γ = 60 (W km)

^{−1}@ 3.9 µm, and L = 5 m, the peak power is as low as 9 W and energy can be lower than 0.2 nJ. Error Δ is decreased from 0.0256 to 0.0030 after running the iterative algorithm (23 iterations).

## 4. Discussion

_{D}. Thus, the peak power of the measured signals should be P >> 1/(γL

_{D}). Note that for sufficiently long pulses with a duration of several ps or tens of ps with a narrowband spectrum, the limiting factor may be not dispersion but optical losses.

_{2}S

_{3}-based glass fibers. Such fibers are transparent in this range and have a huge Kerr nonlinearity. We demonstrated numerically by various representative examples with realistic parameters that the proposed algorithm for searching the polynomial spectral phase gives a fairly good match, and the subsequent application of the iterative algorithm near the global minimum of the error function allows achieving an almost perfect match between the original and reconstructed signals. The next step of the work can be the experimental implementation of the method for measuring mid-IR pulses using chalcogenide fibers. Additionally, note that As

_{2}S

_{3}-based glass fibers can be used to characterize ultrashort pulses in a wider spectral range of up to 5–6 µm. The use of other types of commercial chalcogenide fibers, such as As

_{2}Se

_{3}-based glass fibers or customized chalcogenide fibers based on glasses with special compositions, can provide even more opportunities for characterization of mid-IR ultrashort optical pulses since they can have higher nonlinear coefficients and a wider transparency range [19,20,21,22,23].

## 5. Conclusions

_{2}S

_{3}-based glass fibers.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Illustration of the method for pulse reconstruction based on processing of spectra converted in fibers with Kerr nonlinearity. (

**b**) Block diagram of proposed algorithm of searching for optimal polynomial spectral phase (for certain B-integrals). (

**c**) Block diagram of iterative algorithm (for certain B-integrals). Here, FFT is fast Fourier transform; IFFT is inverse FFT.

**Figure 2.**Search for polynomial spectral phase (“ideal” case). ((

**a**–

**d**), left column) “Right” solution found for the global minimum I of the error function Δ (C

_{2}= 6.5 ps

^{2}and C

_{3}= −3.6 ps

^{3}) shown in (

**e**). ((

**f**–

**i**), right column) “Wrong” solution constructed for the local minimum II of the error function Δ (C

_{2}= 2.2 ps

^{2}and C

_{3}= 4 ps

^{3}) shown in (

**e**). (

**a**,

**f**) Intensity and phase in the time domain; (

**b**,

**g**) input pulse spectrum and spectral phase; (

**c**,

**h**) converted spectra at the output for B

_{1}= 1; (

**d**,

**i**) converted spectra at the output for B

_{2}= 2. Black curves calculated for the initial pulse and color (magenta and blue) curves correspond to numerically reconstructed pulses for subplots (

**a**–

**d**,

**f**–

**i**).

**Figure 3.**Search for polynomial spectral phase (by data with spectral noise). Search for polynomial spectral phase (ideal case). ((

**a**–

**d**), left column) “Right” solution found for the global minimum I of the error function Δ (C

_{2}= 6.5 ps

^{2}and C

_{3}= −3.8 ps

^{3}) shown in (

**e**). ((

**f**–

**i**), right column) “Wrong” solution constructed for the local minimum II of the error function Δ (C

_{2}= 2.4 ps

^{2}and C

_{3}= 3.9 ps

^{3}) shown in (

**e**). (

**a**,

**f**) Intensity and phase in the time domain; (

**b**,

**g**) input pulse spectrum and spectral phase; (

**c**,

**h**); converted spectra at the output for B

_{1}= 1; (

**d**,

**i**) converted spectra at the output for B

_{2}= 2. Black curves calculated for the initial pulse and color (magenta and blue) curves correspond to numerically reconstructed pulses for subplots (

**a**–

**d**,

**f**–

**i**).

**Figure 4.**Reconstruction of 100 pulses with noisy sech

^{2}-shaped spectrum with spectral width FWHM = 0.6 THz and randomly generated spectral phases. Each dot in panel (

**a**) indicates polynomial coefficients for a single test pulse. Calculated error Δ for reconstructed pulses (

**b**). Difference between reconstructed and original polynomial coefficients of spectral phases (

**c**).

**Figure 5.**Reconstruction of the pulse with random addition to the spectral phase. ((

**a**–

**d**), left column) Solution for polynomial spectral phase found for the global minimum of error function Δ (C

_{2}= 6.3 ps

^{2}and C

_{3}= −3.5 ps

^{3}). ((

**e**–

**h**), right column) Solution found for the global minimum using the iterative algorithm. (

**a**,

**e**) Intensity and phase in the time domain; (

**b**,

**f**) input pulse spectrum and spectral phase; (

**c**,

**g**); converted spectra at the output for B

_{1}= 1; (

**d**,

**h**) converted spectra at the output for B

_{2}= 2. Black curves calculated for the initial pulse and color (magenta and blue) curves correspond to numerically reconstructed pulses.

**Figure 6.**Calculated nonlinear Kerr coefficients (

**a**) and quadratic dispersion coefficients (

**b**) for step-index arsenic sulfide glass fibers with NA = 0.3 and different core diameters.

**Figure 7.**Reconstruction of mid-IR 3 µm class laser pulses with realistic parameters for commercially available arsenic sulfide glass fibers as nonlinear elements for spectrum conversion. Each column corresponds to an individual pulse. The upper row shows intensity profiles and phases; the second row shows the initial spectra and spectral phases; the third and fourth rows show converted spectra for different B-integrals. Solid black curves correspond to original signals, magenta curves correspond to pulses obtained by the proposed algorithm for polynomial spectral phase searching (dashed—temporal or spectral intensities, dash-dotted—temporal or spectral phases), and blue curves are obtained by the iterative algorithm near the global minimum of error Δ (dashed—temporal or spectral intensities, dash-dotted—temporal or spectral phases). Bottom row shows error functions calculated for different B

_{1}values (the minima correspond to the original B

_{1}values). Magenta graphs (left axes) are for algorithm optimizing polynomial phase and blue graphs (right axes) are for iterative algorithm.

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**MDPI and ACS Style**

Sorokin, A.A.; Andrianov, A.V.; Anashkina, E.A.
Numerical Study of Mid-IR Ultrashort Pulse Reconstruction Based on Processing of Spectra Converted in Chalcogenide Fibers with High Kerr Nonlinearity. *Fibers* **2022**, *10*, 81.
https://doi.org/10.3390/fib10100081

**AMA Style**

Sorokin AA, Andrianov AV, Anashkina EA.
Numerical Study of Mid-IR Ultrashort Pulse Reconstruction Based on Processing of Spectra Converted in Chalcogenide Fibers with High Kerr Nonlinearity. *Fibers*. 2022; 10(10):81.
https://doi.org/10.3390/fib10100081

**Chicago/Turabian Style**

Sorokin, Arseny A., Alexey V. Andrianov, and Elena A. Anashkina.
2022. "Numerical Study of Mid-IR Ultrashort Pulse Reconstruction Based on Processing of Spectra Converted in Chalcogenide Fibers with High Kerr Nonlinearity" *Fibers* 10, no. 10: 81.
https://doi.org/10.3390/fib10100081