Complex and phase screen methods for studying arbitrary genuine Schell-model partially coherent pulses in nonlinear media

Xiaohan Wang; Xiaohan Wang; Jiahui Tang; Jiahui Tang; Yinghe Wang; Yinghe Wang; Xin Liu; Xin Liu; Chunhao Liang; Chunhao Liang; Chunhao Liang; Lina Zhao; Lina Zhao; Lina Zhao; Bernhard J. Hoenders; Yangjian Cai; Yangjian Cai; Yangjian Cai; Yangjian Cai; Pujuan Ma; Pujuan Ma; Pujuan Ma

doi:10.1364/OE.459928

1. Introduction

Optical coherence, a significant beam characteristic, strongly influences interference effects and further influences beam propagation behavior. With the establishment of optical coherence theory [1–3], we can competently deal with ensemble statistics randomly fluctuating fields in terms of correlation functions. The degree of coherence (DOC), a normalized version of the correlation function, describes the correlation between two points of the random fields in spatial and/or temporal domain(s).

How do we devise genuine partially coherent beams (PCBs) with the prescribed DOC? Gori et al. proposed the necessary and sufficient condition for representing a cross-spectral density function and expressed it in simple integral form [4,5]. This work triggers a series of studies on PCBs, especially in the spatial domain [6–13]. The results demonstrate that spatial PCBs with dominated DOC have multifarious significant applications in optical imaging, particle trapping, optical encryption, image transmission, etc. [14–21]

Inspired by spatial PCBs, the temporal ones [i.e., temporally partially coherent pulses (PCPs)] with different DOC were proposed and explored [22–27]. A few mentions are: optical coherence gratings, cosine-Gaussian correlated Schell-model pulses, and Hermite-Gaussian correlated Schell-model pulses. Compared with the classical PCPs-Gaussian Schell-model pulses [28–34], such PCPs realize pulse self-splitting, self-focusing, self-accelerating, and other fascinating propagation properties in linear second-order dispersive media by controlling the source DOC. Furthermore, we can implement far-zone pulse intensity distributions.

These phenomena are anticipated by the space-time analogy between paraxial propagation of PCBs in the free space and the evolution of PCPs in linear second-order dispersive media. They both have potential applications in pulse shaping, fiber optical communication, ghost imaging, rogue wave generation et al. [35–40]. However, few studies have focused on the coherence-related behaviors of PCPs propagating in nonlinear media, because attaining analytical expressions seems impossible.

To characterize PCPs propagating in nonlinear media, Lajunen et al. proposed in 2010 the pulse-by-pulse (PBP) method, which is based on the coherent-mode representation method and Monte Carlo simulations [41]. For the PCPs whose intensity or DOC has a non-Gaussian profile, it is evident that deriving analytical expressions for coherent modes is a difficult procedure. Until now, there has been no effective and convenient way to study any genuine Schell-model PCPs in nonlinear media.

On the other hand, there are many ways to realize spatial PCBs theoretically and experimentally. Nowadays, two are the most popular. The first technique employs the Van Cittert-Zernike theorem to produce any genuine PCBs of the Schell-model type, using the propagation of an entirely incoherent field distribution generated at some plane with given intensity distribution [9–11]. The other exploits the modes superposition principle to realize such beams [42–51]. This includes the coherent-mode representation, the pseudo-mode representation, and the random mode representation [it mainly refers to the complex screen (CS) and phase screen (PS) methods].

Prior publications have discovered that although random mode representation requires more modes to reach the same precision, it provides a powerful and handy protocol to handle complex PCBs even without the analytical expressions [44]. Studies also revealed that the CS method is simpler and more direct for devising PCBs than the PS method. However, the CS method is useful only in theoretical simulations, and the PS method is mostly applicable to laboratory research. This is because of the commercial availability of phase-only spatial light modulations (SLMs).

Here, we develop the temporal analog of the CS method introduced for the spatial counterpart of a PCB. The simulation results validate that such a method can be accurately employed to explore the beam properties of the Schell-model PCPs, with prescribed DOC, propagating in nonlinear media. The method is then transferred to the PS method via the complex amplitude modulation to make it have laboratory applications.

2. Method

This section introduces the CS method. The statistical properties of PCPs with Schell-model type are characterized by the mutual coherent function (MCF) in the space-time domain,

(1)$$\Gamma \left( {{t_1},{t_2}} \right) =\left \langle {T}\left( {{t_1}} \right)T^*\left( {{t_2}} \right) \right \rangle= {E}\left( {{t_1}} \right)E^*\left( {{t_2}} \right)\mu \left( {{t_1} - {t_2}} \right),$$

where $T$ denotes the electric field of the PCPs. The angular bracket stands for the ensemble average. $E$ is a complex amplitude function of the time $t$, $\mu$ is the temporal DOC function, the asterisk denotes complex conjugation. As followed by Van Cittert-Zernike theorem [11,44], for a physically realizable MCF, the DOC can be rewritten as

(2)$$\mu \left( {{t_1} - {t_2}} \right) = \int {\Gamma \left( {{f_1},{f_2}} \right)} \exp \left[ { - i2\pi \left( {{t_1}{f_1} - {t_2}{f_2}} \right)} \right]d{f_1}d{f_2},$$

with

(3)$$\Gamma \left( {{f_1},{f_2}} \right) = \sqrt {p\left( {{f_1}} \right)} \sqrt {p\left( {{f_2}} \right)} \delta \left( {{f_1} - {f_2}} \right),$$

where $f_1$ and $f_2$ are the arbitrary points. $p\left ( {{f}} \right )$ as the power spectral density is non-negative and $\delta$ denotes the Dirac function. Eq. (3) illustrates that $\Gamma \left ( {{f_1},{f_2}} \right )$ could be treated as the MCF of the fully incoherent pulse. Substituting Eqs. (2) and (3) into Eq. (1), the MCF can be rearranged as [44]

(4)$$\Gamma \left( {{t_1},{t_2}} \right) = \left\langle {T\left( {{t_1}} \right){T^*}\left( {{t_2}} \right)} \right\rangle \approx \frac{1}{N}\sum_{n = 1}^N {{T_n}\left( {{t_1}} \right){T_n}^*\left( {{t_2}} \right)} ,$$

with

(5)$${T_n}\left( t \right) = E\left( t \right) \times {\psi _n}\left( t \right),$$

(6)$${\psi _n}\left( t \right) = \int {\sqrt {p\left( f \right)} } {C_n}\left( f \right)\exp \left( { - i2\pi tf} \right)df.$$

Here ${C_n}\left ( f \right )$ is a 1D random complex function, whose real and imaginary parts are independent, unit variance, Gaussian random processes. It satisfies $\left \langle {C_n}\left ( {{f_1}} \right ){C_n}^*\left ( {{f_2}} \right ) \right \rangle =\delta \left ( {{f_1} - {f_2}} \right )$. The angular bracket denotes the ensemble average. To validate Eq. (4), the value of $N$ has to be sufficiently large. Eq. (5) implies that each instantaneous electric field can be achieved by illuminating the screen with a coherent pulse with electric field $E$. The transmittance function of such a screen is characterized by Eq. (6), which is complex. Hence, the method is named as the CS method. We renew the complex circular Gaussian process $C_n$ to refresh the complex screen ($\psi _n$), and to obtain a new instantaneous electric field ($T_n$). On acquiring many instantaneous electric fields, we can correctly describe the MCF by Eq. (4).

3. Validity and application

First, we verify the validity of the CS method in comparison with the results derived by the PBP method. As described in [41], the MCF of a PCP is Hermitian and non-negative definite and is, therefore, represented as

(7)$$\Gamma \left( {{t_1},{t_2}} \right) = \sum_n {{\lambda _n}\varphi _n^*} \left( {{t_1}} \right){\varphi _n}\left( {{t_2}} \right),$$

where $\lambda _n$ and $\varphi _n\left ( {{t}} \right )$ are the eigenvalues and eigenfunctions, respectively, of the Fredholm integral equation,

(8)$$\int {\Gamma \left( {{t_1},{t_2}} \right){\varphi _n}\left( {{t_1}} \right)} d{t_1} = {\lambda _n}{\varphi _n}\left( {{t_2}} \right).$$

The eigenvalues $\lambda _n$ are all positive numbers.

Likewise, for PBP method, the MCF can be represented as Eq. (4), where the instantaneous electric field $T_n\left ( t \right )$ is given by

(9)$${T_n}\left( t \right) = \sum_n {\sqrt {{\lambda _n}} } {C_n}\left( t \right){\varphi _n}\left( t \right),$$

where, $C_n$ is a complex circular Gaussian process. Given $\lambda _n$ and $\varphi _n\left (t\right )$ , we can achieve many instantaneous electric fields $T_n$ by refreshing $C_n$ .

A Gaussian Schell-model pulse (GSMP) is adopted as the input pulse for convenience. Its MCF is shown as

(10)$$\Gamma \left( {{t_1},{t_2}} \right) = \exp \left( { - \frac{{t_1^2 + t_2^2}}{{4t_p^2}}} \right)\exp \left[ { - \frac{{{{\left( {{t_2} - {t_1}} \right)}^2}}}{{2t_c^2}}} \right].$$

Here $t_c$ and $t_p$ denote the coherence time and pulse width, respectively. We employ the CS and PBP methods to achieve the instantaneous electric fields. For the CS method, the pivotal functions $E\left (t\right )$ and $p\left (f\right )$ are given by

(11)$$E\left( t \right) = \exp \left( { - {{{t^2}} \mathord{\left/ {\vphantom {{{t^2}} {4t_p^2}}} \right.} {4t_p^2}}} \right),$$

(12)$$p\left( f \right) = \sqrt {2\pi } {t_c}\exp \left( { - 2{\pi ^2}t_c^2{f^2}} \right),$$

respectively. For the PBP method, the key functions $\varphi _n\left (t\right )$ and values $\lambda _n$ are

(13)$${\varphi _n}\left( t \right) = {\left( {\frac{{2c}}{\pi }} \right)^{1/4}}\frac{1}{{\sqrt {{2^n}n!} }}{H_n}\left( {\sqrt {2c} t} \right)\exp \left( { - c{t^2}} \right),$$

(14)$${\lambda _n} = \sqrt {\frac{\pi }{{a + b + c}}} {\left( {\frac{b}{{a + b + c}}} \right)^n},$$

respectively, with $a=1/4t_p^2$, $b = 1/2t_c^2$ , and $c = \sqrt {{a^2} + 2ab}$. We plot the intensity distributions (${I_n} = {\left | {{T_n}} \right |^2}$ ) of 10 random realizations (or pulses) in the source plane by the CS and PBP methods, shown in Fig. 1(a) and (b), respectively. Since both these methods use complex circular Gaussian variables (or functions), each realization intensity profile exhibits expected fluctuations. We show the average intensity distributions ($\overline I \left ( t \right ) = \sum\limits_{n = 1}^N {{{{I_n}} \mathord {\left /{\vphantom {{{I_n}} N}} \right. } N}}$ ) for a set of $10^4$ random realizations by both methods in Fig. 1(c). As expected, the two results match perfectly.

Fig. 1. Intensity distributions of 10 random realizations (or pulses) were achieved by the CS method in (a) and PBP method in (b). The average intensity distributions are shown for a set of $10^4$ random realizations by both of methods in (c).

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Now we consider the PCPs propagating in the nonlinear Kerr media. The beam behavior is governed by the well-known nonlinear Schrödinger equation (NLSE) [52], given by

(15)$$i\frac{{\partial U}}{{\partial z}} - \frac{{{\beta _2}}}{2}\frac{{{\partial ^2}U}}{{\partial {t^2}}} ={-} \gamma {\left| U \right|^2}U,$$

where $U=U(t,z)$ is the slowly-varying pulse envelope, $\beta _2$ is a group-velocity dispersion, and $\gamma$ is the Kerr nonlinearity coefficient. Further, the time coordinate $t$ is measured in the reference frame moving at the group velocity of the pulse. To explore generic features of the beam behavior, independent of the source and medium particulars, it is convenient to use dimensionless variables, defined as

(16)$$U' = {U \mathord{\left/ {\vphantom {U {\sqrt {\left\langle {{P_0}} \right\rangle } }}} \right. } {\sqrt {\left\langle {{P_0}} \right\rangle } }},{\rm{ }}t' = {t \mathord{\left/ {\vphantom {t {{t_p}}}} \right. } {{t_p}}},{\rm{ }}z' = {z \mathord{\left/ {\vphantom {z L}} \right. } L},$$

where $\left \langle {{P_0}} \right \rangle$ is the average peak power, $t_p$ is the pulse width, and $L = \sqrt {{L_{NL}}{L_D}}$. Here ${L_{NL}} = 1/{\gamma {P_0}}$ and ${L_D} = t_p^2/\left | {{\beta _2}} \right |$ denote the usual nonlinear and dispersion lengths, respectively. In consequence, the dimensionless NLSE is rearranged as

(17)$$i\sigma \frac{{\partial U'}}{{\partial z'}} - {\mathop{\rm sgn}} \left( {{\beta _2}} \right)\frac{{{\sigma ^2}}}{2}\frac{{{\partial ^2}U'}}{{\partial t{'^2}}} ={-} {\left| {U'} \right|^2}U',$$

where sgn is a sign function. $\sigma = \sqrt {{{{L_D}} \mathord {\left / {\vphantom {{{L_D}} {{L_{NL}}}}} \right. } {{L_{NL}}}}}$ the soliton parameter, determines the dynamic behavior of the pulse propagating in the nonlinear media. $\sigma \gg 1$ and $\sigma \ll 1$ respectively indicate that nonlinearity and dispersion are the dominant factors.

As shown in Fig. 1, each random pulse (or realization) is adopted as the input pulse in Eq. (17). We apply the split-step Fourier method to Eq. (17) to study the pulse propagation behavior [50]. The random pulses propagate through the nonlinear medium and are recorded as illustrated in Fig. 2. We use $T_{nz}$ to represent the electric field of the $n^{th}$ random pulse in the receiving plane. Then, we employ ${\overline I _z}\left ( t \right ) = \sum \limits_{n = 1}^N {{{{{\left | {{T_{nz}}} \right |}^2}} \mathord {\left / {\vphantom {{{{\left | {{T_{nz}}} \right |}^2}} N}} \right. } N}}$ to study the intensity evolution of the PCPs during propagation.

Fig. 2. Sketch of the random pulses (or realizations) propagating through nonlinear media (here we take optical fiber as the example).

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We have demonstrated that the average intensity distributions achieved by the two methods in the source plane are completely consistent in Fig. 1. To further illustrate the validity of the CS method, we study the evolution of the intensity distributions of a GSMP by the two methods in nonlinear media with anomalous dispersion. The relevant parameters are chosen as ${t_c} = 1{\rm {ps}}$ , ${t_p} = 10{\rm {ps}}$, ${\beta _2} = - 20{{{\rm {p}}{{\rm {s}}^2}} \mathord {\left / {\vphantom {{{\rm {p}}{{\rm {s}}^2}} {{\rm {km}}}}} \right. } {{\rm {km}}}}$, $\gamma = 0.1{{\rm {W}}^{ - 1}}{\rm {k}}{{\rm {m}}^{ - 1}}$ and $\left \langle {{P_0}} \right \rangle = 50{\rm {W}}$. The number of realizations is set as $3{\times }10^4$. The GSMP suffers from beam spreading during propagation. The scenarios shown in Fig. 3(a) and (b) are achieved by the CS and PBP methods, respectively. We plot the crosslines of the intensity distributions generated by the two methods at three different propagation distances $z_1$=0.2km, $z_2$=1km, and $z_3$=1.8km, in Fig. 3(c), (d), (e), respectively. The locations are labeled by the dashed white lines in Fig. 3(a) and (b). For all propagation distances, they are in agreement with each other. Figs. (1) and (3) prove that the CS method works well for PCPs propagating in nonlinear media. Both CS and PBP methods are based on random variables. Therefore, the time consumption of the two methods is almost the same.

Fig. 3. Density map of GSM pulses propagating for 2km in a nonlinear medium with anomalous dispersion by the CS method and PBP method in (a) and (b), respectively. (c)-(e) show the crosslines of the intensity profiles at three different propagation distances $z_1$=0.2km, $z_2$=1km, and $z_3$=1.8km for both methods.

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Next, to demonstrate the CS method’s power, we adopt one complex pulse - a Cosine-Gaussian correlated Schell-model pulse (CGSMP) as the input. Its MCF in dimensionless form is given by [22]

(18)$$\Gamma \left( {t{'_1},t{'_2}} \right) = \exp \left( { - \frac{{{t'_1}^2 + {t'_2}^2}}{2}} \right)\cos \left[ {\frac{{n\sqrt {2\pi } \left( {t{'_2} - t{'_1}} \right)}}{{{\varepsilon _c}}}} \right]\exp \left[ { - \frac{{{{\left( {t{'_2} - t{'_1}} \right)}^2}}}{{2\varepsilon _c^2}}} \right],$$

where ${\varepsilon _c} = {{{t_c}} \mathord {\left / {\vphantom {{{t_c}} {{t_p}}}} \right. } {{t_p}}}$ is a source coherence parameter and $n$ is the beam order. It is challenging to obtain accurate coherent modes ($\varphi _n$ ) for the PBP method. As for the CS method, we rearrange Eqs. (1)–(3) and after a simple calculation [22], we obtain the corresponding power spectrum density $p$ as follows

(19)$$p\left( {f'} \right) = \sqrt {2\pi } {\varepsilon _c}\exp \left( { - \pi {n^2}} \right)\exp \left( { - 2\varepsilon _c^2{\pi ^2}f{'^2}} \right)\cosh \left( {\sqrt {2\pi } 2\pi n{\varepsilon _c}f'} \right),$$

with $f' = f \times {t_p}$. The function $E$ in Eq. (5) is expressed as $E\left ( {t'} \right ) = \exp \left ( { - {{t{'^2}} \mathord {\left / {\vphantom {{t{'^2}} 2}} \right. } 2}} \right )$ .

Substituting Eq. (19) into Eqs. (5) and (6), we obtain the input set of random pulses for the NLSE, viz. Eq. (17). The intensity evolution of CGSMPs in the nonlinear media is exhibited in Fig. 4. The relevant parameters are adopted as ${t_c} = 10{\rm {ps}}$ , ${t_p} = 10{\rm {ps}}$ , ${\beta _2} = - 20{{{\rm {p}}{{\rm {s}}^2}} \mathord {\left / {\vphantom {{{\rm {p}}{{\rm {s}}^2}} {{\rm {km}}}}} \right. } {{\rm {km}}}}$ , $\gamma = 0.1{{\rm {W}}^{ - 1}}{\rm {k}}{{\rm {m}}^{ - 1}}$ and the propagation distance is 8km. We choose three different input average peak powers $\left \langle {{P_0}} \right \rangle = 2{\rm {W}}$, $\left \langle {{P_0}} \right \rangle = 8{\rm {W}}$ , $\left \langle {{P_0}} \right \rangle = 50{\rm {W}}$ , corresponding to the soliton parameter $\sigma = 1$, $\sigma = 3$ , $\sigma = 5$ , in Fig. 4(a), (b), (c), respectively. The number of realizations is set as $3{\times }10^4$. It is discovered that the pulses suffer from shape distortion during propagation because of the source cosine-Gaussian correlated function. They split into two pulses in the far-zone. Such behavior is similar to the linear case [22]. In Fig. 4(d), we plot the crosslines of the pulse intensity profiles at $z$=8km in Fig. 4(a), (b), and (c). As the nonlinear effect increases, the two split pulses converge, as expected. The intensity profile fluctuations for $\sigma = 5$, shown by a black dotted curve in Fig. 4(d), are mainly caused by nonlinearity.

Fig. 4. The intensity evolution of CGSM pulse in nonlinear dispersive medium with the soliton parameter (a) $\sigma = 1$ , (b) $\sigma = 3$ , and (c) $\sigma = 5$. (d) The crosslines of the pulse intensity profiles at $z$=8km in (a), (b), and (c).

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Lastly, we transfer the complex screen/function to the phase screen/function (namely, the CS method to the PS method) through complex amplitude modulation to make it applicable for the commercial availability of phase-only SLMs. Rosales-Guzmãn and Forbes provided the arguments showing the feasibility of such an application [53]. According to this protocol, the desired phase grating, loading on the phase-only SLM, is specified by

(20)$${\phi _{SLM}} = {A_f}\sin \left[ {{\rm{Arg}}\left( {{\psi_n}} \right) + 2\pi {f_0}t} \right],$$

where $A_f$ is obtained by the numerical inversion of ${J_1}\left ( {{A_f}} \right ) = {\rm {Abs}}\left ( {{\psi _n}} \right )$ [53]. $J_1$ is the first-order Bessel function of the first kind. We use "Abs" and "Arg" operations on the function $\psi _n$ to achieve its amplitude and phase, where $\psi _n$ is a characteristic function of the complex screen described by Eq. (6). $f_0$ denotes the grating frequency. In the PS method, a coherent pulse with an electric field $E$ [given in Eq. (5)] illuminates the grating (or SLM) characterized by Eq. (20). The reflected or transmitted pulse produces diffraction patterns of different orders, owing to the grating with blazed type [54]. We obtain the desired pulse with the instantaneous electric field $T_n$ described by Eq. (5) in the positive or negative $1^{st}$ order diffraction. In the simulation, we select out the desired pulse by multiplying the electric field of all different orders of diffraction pulses by a circular function circ(r/a) with the suitable radius $a$. The retained pulse is adopted as the input pulse for Eq. (17). We employ the PS method to study the intensity evolution of the CGSMP in nonlinear media. All parameters are similar to those used in Fig. 4(a). To validate the PS method, we compare it with the CS method. The result is given in Fig. 5. The intensity distributions obtained by the two methods are completely consistent at different propagation distances. It shows that the PS method also works for any Schell-model PCPs. Hence, we are able to further study the behavior of many coherence-related phenomena in nonlinear media with the help of CS or PS method.

Fig. 5. The intensity profiles of a CGSMP during propagation in nonlinear media at different propagation distances, obtained by the PS method and the CS method.

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4. Conclusions

We developed the temporal analogue of the complex screen method introduced for the spatial counterpart of partially coherent beams. To validate the method, we followed a procedure to study the propagation of Gaussian Schell-mode pulses in nonlinear media and discovered that the results were completely consistent with those obtained by the well-known pulse-by-pulse method (based on the coherent-mode representation method and Monte Carlo simulations). Next, we employed the complex screen method to study a complex, partially coherent pulse. Such pulses are hard to achieve as accurate values for the coherent modes, needed for the pulse-by-pulse method, are not easily obtainable. The achieved results demonstrated the power and convenience of the complex screen method. Further, to make the complex screen method applicable in the lab, this method is transferred to the phase screen method using complex amplitude modulation. The results derived by the phase screen method confirmed its effectiveness. It is worth noting that to ensure the accuracy of the proposed methods, we needed a sufficiently large ensemble of the pulses. In the numerical simulation, we increased the number of realizations to examine the statistical properties (e.g., the intensity, the degree of coherence) of the output pulse. If they do not change anymore, it means that the realization number is sufficient. These methods are powerful tools and are expected to explore the behavior of many coherence-related phenomena in nonlinear media both theoretically and experimentally.

Funding

National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11974218, 12004220, 12192254, 91750201); Local Science and Technology Development Project of the Central Government (YDZX20203700001766); Innovation Group of Jinan (2018GXRC010); China Postdoctoral Science Foundation (2019M662424).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.

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Complex and phase screen methods for studying arbitrary genuine Schell-model partially coherent pulses in nonlinear media

Abstract

1. Introduction

2. Method

3. Validity and application

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (20)

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