Source coherence-induced control of spatiotemporal coherency vortices

Chaoliang Ding; Dmitri Horoshko; Dmitri Horoshko; Olga Korotkova; Olga Korotkova; Chenrui Jing; Xiexing Qi; Liuzhan Pan

doi:10.1364/OE.458666

1. Introduction

In the past three decades, vortex beams with phase singularity and the orbital angular momentum (OAM) have gained a great deal of attention due to their extensive applications in microparticle manipulation [1–3], optical detection [4], optical communications in free space [5] and optical fibers [6], quantum information processing [7] and quantum imaging [8], super-resolution imaging [9], human face recognition [10], etc. Significant theoretical, experimental and applied research results related to vortex beams are obtained [11–19]. During this period, the research on vortices has been extended rapidly from scalar light field to vector one [20], from an integer topological charge to a fractional one [21,22], from an individual vortex to vortex lattices [23], from a classical vortex to a perfect vortex [24]. In particular, the conventional optical vortex in the spatial domain is extended into the space-time dimension, and termed a spatiotemporal optical vortex (STOV), see the seminal work of Sukhorukov et al. in 2005 [25]. It is well known that the common vortex beams with the OAM possess a spatial spiral phase front, where the OAM is longitudinal to the light propagation direction. Whereas the STOV’s OAM is perpendicular to the light propagation direction, being a transverse OAM. Recently, Bliokh and Nori have predicted that the STOV with a transverse OAM can be formed by introducing the temporal variations of phase [26] and Huang et al. demonstrated theoretically the generation of a transverse OAM step by step basing their approach on the diffraction theory [27]. Afterwards, the STOV with a transverse OAM was verified experimentally by the Zhan’s group [28–33] and the Milchberg’s group [34–37], respectively. More recently, Bliokh has performed the accurate theoretical analysis of the STOV’s angular-momentum properties [38]. However, in the aforementioned investigations, partial spatial and temporal coherence of vortex beams have not been considered appropriately.

On the other hand, with the rapid development of the optical coherence theory in the recent two decades [39,40], many researchers pay more and more attention to the design and implementation of the spatial coherence of the light sources. Consequently, the investigation on the optical field vortices was extended to coherency vortices, being the phase singularities of the second-order correlation functions [41–49]. It has been verified that the spatially partially coherent vortex beams possess many unique advantages in optical trapping [50], optical communications [51], information encryption [52], etc [53,54], as compared with their fully coherent counterparts. In addition to the spatial coherence effect, the non-stationary stochastic optical pulses with partial spectral or temporal coherence, representing a more general class of the partially coherent fields, have attracted researchers’ widespread attention due to their potential applications in laser micromachining, ghost imaging, pulse shaping and medical diagnosis [55–68]. Importantly, such optical pulses are radiated by some realistic laser sources, such as random lasers, excimer lasers, free-electron lasers and supercontinuum sources in micro structured fibers [69]. Up to now, the partial spatial and temporal coherence states of the STOVs have not been investigated thoroughly. There are only two papers considering the coherence of the STOVs. The first one is concerned with generation and propagation of specific (twisted) partially coherent sources in the space-frequency and space-time domains [70]. The other one only treats partially temporally coherent STOVs, both theoretically and experimentally, suggesting a convenient and cost-effective setup instead of the traditional use of mode-locked laser sources [71]. Thus, a systematic, simultaneous analysis of temporal and spatial coherence affecting propagation properties of the STOVs is required.

Inspired by the remarkable properties of the STOV [26–36,70,71] we have explored the influence of the spatial and temporal coherence states on the spatiotemporal coherency vortices of the STPCPV beams. In section 2, we have developed the basic theory showing how the source spatial and temporal coherence result in the changes of spatiotemporal coherency vortices of the STPCPV beams propagating through fused silica. In section 3, detailed numerical results regarding the STPCPV beam evolution are presented. In Section 4, the main results of the paper are summarized.

2. Theory

The mutual coherence function (MCF) of a spatially and temporally partially coherent pulsed vortex (STPCPV) beam with a spatiotemporal optical vortex across a source plane has form [34,39]

(1)$$\begin{aligned} \mathrm{\Gamma }_m^{(0)}\textrm{(}{{\boldsymbol r}_1}\textrm{,}{{\boldsymbol r}_2}\textrm{,}{\tau _1}\textrm{,}{\tau _2}\textrm{)} &= \textrm{exp}\left( { - \frac{{{\boldsymbol r}_1^2 + {\boldsymbol r}_2^2}}{{4w_0^2}}} \right)\textrm{exp}\left[ { - \frac{{{{({{{\boldsymbol r}_2} - {{\boldsymbol r}_1}} )}^2}}}{{2\delta_{}^2}}} \right]\textrm{exp}\left( { - \frac{{\tau_1^2 + \tau_2^2}}{{T_0^2}}} \right)\textrm{exp}\left[ { - \frac{{{{({\tau_2^{} - \tau_1^{}} )}^2}}}{{2T_c^2}}} \right]\\ \textrm{ } &\times {\left[ {\frac{{{\tau_1}}}{{{\tau_s}}} - i\textrm{sgn}(m) \cdot \frac{{x_1^{}}}{{x_s^{}}}} \right]^{|m |}} \cdot {\left[ {\frac{{{\tau_2}}}{{{\tau_s}}} + i\textrm{sgn}(m) \cdot \frac{{x_2^{}}}{{x_s^{}}}} \right]^{|m |}}\textrm{exp}[{ - i{\omega_0}(\tau_2^{} - \tau_1^{})} ]\textrm{,} \end{aligned}$$

where r₁= (x₁, y₁), r₂= (x₂, y₂) are the two-dimensional, transverse position vectors, τ₁ and τ₂ are a pair of time instants, sgn (•) is the signum function, m is the topological charge, τ_s and x_s are the temporal and the spatial characteristic widths of the STOV. Further, ω₀ is the pulse’s carrier frequency, while w₀ and δ denote the beam’s waist width and its spatial coherence width, respectively. Also, T₀ and T_c represent the average pulse duration and its coherence time.

We will now consider propagation of such a pulse from the source plane z = 0 to a plane z > 0. The general relation between the MCFs of the pulse in these planes is well known [39,72,73]:

(2)$$\begin{aligned} \Gamma ({{\boldsymbol \rho }_1}{,}{{\boldsymbol \rho }_2},{t_1},{t_2},z) &= {\left[ {\frac{{{k_0}}}{{2\pi {B_S}}}} \right]^2}\frac{{{\omega _0}}}{{2\pi {B_T}}}\int {\int {\int {\int_{ - \infty }^\infty {\Gamma _m^{(0)}({{\boldsymbol r}_1},{{\boldsymbol r}_2},{\tau _1},{\tau _2})} } } } \\ \textrm{ } &\times \textrm{exp}\left\{ { - \frac{{i{k_0}}}{{2{B_S}}}[{{A_S}({\boldsymbol r}_1^2 - {\boldsymbol r}_2^2) - 2({{\boldsymbol r}_1} \cdot {{\boldsymbol \rho }_1} - {{\boldsymbol r}_2} \cdot {{\boldsymbol \rho }_2}) + {D_S}({\boldsymbol \rho }_1^2 - {\boldsymbol \rho }_2^2)} ]} \right\}d{{\boldsymbol r}_1}d{{\boldsymbol r}_2}\\ \textrm{ } &\times \textrm{exp}\left\{ { - \frac{{i{\omega_0}}}{{2{B_T}}}[{{A_T}(\tau_1^2 - \tau_2^2) - 2({\tau_1}{t_1} - {\tau_2}{t_2}) + {D_T}(t_1^2 - t_2^2)} ]} \right\}d{\tau _1}d{\tau _2}. \end{aligned}$$

Here ρ₁= (u₁, v₁), ρ₂= (u₂, v₂), are the two-dimensional transverse position vectors in plane z > 0, k₀= n(ω₀)ω₀/c denotes the wave number, with c being the speed of light in vacuum, n(ω₀) is the refractive index of the medium at carrier frequency ω₀, t₁ and t₂ are two time instants. Further, A_S, B_S and D_S are the ABCD transfer matrix elements of the optical system in the spatial domain, while A_T, B_T and D_T are those in the temporal domain. These transfer matrices are

(3)$$\left( {\begin{array}{{cc}} {{A_S}}&{{B_S}}\\ {{C_S}}&{{D_S}} \end{array}} \right) = \left( {\begin{array}{{cc}} 1&z\\ 0&1 \end{array}} \right),$$

and

(4)$$\left( {\begin{array}{{cc}} {{A_T}}&{{B_T}}\\ {{C_T}}&{{D_T}} \end{array}} \right) = \left( {\begin{array}{{cc}} 1&{{\omega_0}{\beta_2}z}\\ 0&1 \end{array}} \right),$$

where β₂ denotes the group velocity dispersion coefficient. Without loss of generality, we restrict our attention to one-dimensional case, the two-dimensional extension being straightforward. We set the topological charge to m = ±1. In this case, the MCF in Eq. (1) reduces to expression

(5)$$\begin{aligned} \Gamma _{ {\pm} 1}^{(0)}({x_1},{x_2},{\tau _1},{\tau _2}) &= \textrm{exp}\left( { - \frac{{x_1^2 + x_2^2}}{{4w_0^2}}} \right)\textrm{exp}\left[ { - \frac{{{{({x_2} - {x_1})}^2}}}{{2\delta_{}^2}}} \right]\textrm{exp}\left( { - \frac{{\tau_1^2 + \tau_2^2}}{{T_0^2}}} \right)\exp \left[ { - \frac{{{{(\tau_2^{} - \tau_1^{})}^2}}}{{2T_c^2}}} \right]\\ \textrm{ } &\times \left[ {\frac{{{x_1}{x_2}}}{{x_s^2}} + \frac{{{\tau_1}{\tau_2}}}{{\tau_s^2}} \mp \frac{i}{{x_s^{}\tau_s^{}}}({{x_1}\tau_2^{} - {x_2}\tau_1^{}} )} \right]\exp [{ - i{\omega_0}(\tau_2^{} - \tau_1^{})} ], \end{aligned}$$

where the symbol ‘∓’corresponds to charges m = ±1. Also, the propagating MCF in Eq. (2) can be simplified as

(6)$$\begin{aligned} \Gamma ({u_1},{u_2},{t_1},{t_2},z) &= \frac{{{k_0}}}{{2\pi {B_S}}}\frac{{{\omega _0}}}{{2\pi {B_T}}}\int {\int {\int {\int_{ - \infty }^\infty {\Gamma _{ {\pm} 1}^{(0)}({x_1},{x_2},{\tau _1},{\tau _2})} } } } \\ \textrm{ } &\times \textrm{exp}\left\{ { - \frac{{i{k_0}}}{{2{B_S}}}[{{A_S}(x_1^2 - x_2^2) - 2({x_1} \cdot {u_1} - {x_2} \cdot {u_2}) + {D_S}(u_1^2 - u_2^2)} ]} \right\}d{x_1}d{x_2}\\ &\textrm{ } \times \textrm{exp}\left\{ { - \frac{{i{\omega_0}}}{{2{B_T}}}[{{A_T}(\tau_1^2 - \tau_2^2) - 2({\tau_1}{t_1} - {\tau_2}{t_2}) + {D_T}(t_1^2 - t_2^2)} ]} \right\}d{\tau _1}d{\tau _2}. \end{aligned}$$

On substituting from Eq. (5) into Eq. (6), we obtain for the MCF of a STPCPV beam at plane z the formula

(7)$$\Gamma ({u_1},{u_2},{t_1},{t_2},z) = {\Gamma _1}({u_1},{u_2},{t_1},{t_2},z) + {\Gamma _2}({u_1},{u_2},{t_1},{t_2},z) + {\Gamma _3}({u_1},{u_2},{t_1},{t_2},z)\textrm{,}$$

where

(8)$$\begin{aligned} {\Gamma _1}({u_1},{u_2},{t_1},{t_2},z) &= \frac{{{k_0}}}{{2\pi {B_S}}}\frac{{{\omega _0}}}{{2\pi {B_T}}}\int {\int {\int {\int_{ - \infty }^\infty {\frac{{{x_1}{x_2}}}{{x_s^2}}\textrm{exp}\left( { - \frac{{x_1^2 + x_2^2}}{{4w_0^2}}} \right)\textrm{exp}\left[ { - \frac{{{{({x_2} - {x_1})}^2}}}{{2\delta_{}^2}}} \right]} } } } \\ \textrm{ } &\times \textrm{exp}\left\{ { - \frac{{i{k_0}}}{{2{B_S}}}[{{A_S}(x_1^2 - x_2^2) - 2({x_1} \cdot {u_1} - {x_2} \cdot {u_2}) + {D_S}(u_1^2 - u_2^2)} ]} \right\}d{x_1}d{x_2}\\ \textrm{ } &\times \textrm{exp}\left( { - \frac{{\tau_1^2 + \tau_2^2}}{{T_0^2}}} \right)\exp \left[ { - \frac{{{{(\tau_2^{} - \tau_1^{})}^2}}}{{2T_c^2}}} \right]\textrm{exp}[{ - i{\omega_0}(\tau_2^{} - \tau_1^{})} ]\\ \textrm{ } &\times \textrm{exp}\left\{ { - \frac{{i{\omega_0}}}{{2{B_T}}}[{{A_T}(\tau_1^2 - \tau_2^2) - 2({\tau_1}{t_1} - {\tau_2}{t_2}) + {D_T}(t_1^2 - t_2^2)} ]} \right\}d{\tau _1}d{\tau _2}, \end{aligned}$$

(9)$$\begin{aligned}{\Gamma _2}({u_1},{u_2},{t_1},{t_2},z) &= \frac{{{k_0}}}{{2\pi {B_S}}}\frac{{{\omega _0}}}{{2\pi {B_T}}}\int {\int {\int {\int_{ - \infty }^\infty {\textrm{exp}\left( { - \frac{{x_1^2 + x_2^2}}{{4w_0^2}}} \right)\textrm{exp}\left[ { - \frac{{{{({x_2} - {x_1})}^2}}}{{2\delta_{}^2}}} \right]} } } } \\ \textrm{ } &\times \textrm{exp}\left\{ { - \frac{{i{k_0}}}{{2{B_S}}}[{{A_S}(x_1^2 - x_2^2) - 2({x_1} \cdot {u_1} - {x_2} \cdot {u_2}) + {D_S}(u_1^2 - u_2^2)} ]} \right\}d{x_1}d{x_2}\\ \textrm{ } &\times \frac{{{\tau _1}{\tau _2}}}{{\tau _s^2}}\textrm{exp}\left( { - \frac{{\tau_1^2 + \tau_2^2}}{{T_0^2}}} \right)\exp \left[ { - \frac{{{{(\tau_2^{} - \tau_1^{})}^2}}}{{2T_c^2}}} \right]\textrm{exp}[{ - i{\omega_0}(\tau_2^{} - \tau_1^{})} ]\\ \textrm{ } &\times \textrm{exp}\left\{ { - \frac{{i{\omega_0}}}{{2{B_T}}}[{{A_T}(\tau_1^2 - \tau_2^2) - 2({\tau_1}{t_1} - {\tau_2}{t_2}) + {D_T}(t_1^2 - t_2^2)} ]} \right\}d{\tau _1}d{\tau _2}, \end{aligned}$$

(10)$${0.9}{$\begin{aligned} {\Gamma _3}({u_1},{u_2},{t_1},{t_2},z) &= \frac{{{k_0}}}{{2\pi {B_S}}}\frac{{{\omega _0}}}{{2\pi {B_T}}}\int {\int {\int {\int_{ - \infty }^\infty {\left[ { \mp \frac{i}{{x_s^{}\tau_s^{}}}({{x_1}\tau_2^{} - {x_2}\tau_1^{}} )} \right]\textrm{exp}\left( { - \frac{{x_1^2 + x_2^2}}{{4w_0^2}}} \right)\textrm{exp}\left[ { - \frac{{{{({x_2} - {x_1})}^2}}}{{2\delta_{}^2}}} \right]} } } } \\ \textrm{ } &\times \textrm{exp}\left\{ { - \frac{{i{k_0}}}{{2{B_S}}}[{{A_S}(x_1^2 - x_2^2) - 2({x_1} \cdot {u_1} - {x_2} \cdot {u_2}) + {D_S}(u_1^2 - u_2^2)} ]} \right\}d{x_1}d{x_2}\\ \textrm{ } &\times \textrm{exp}\left( { - \frac{{\tau_1^2 + \tau_2^2}}{{T_0^2}}} \right)\exp \left[ { - \frac{{{{(\tau_2^{} - \tau_1^{})}^2}}}{{2T_c^2}}} \right]\textrm{exp}[{ - i{\omega_0}(\tau_2^{} - \tau_1^{})} ]\\ \textrm{ } &\times \textrm{exp}\left\{ { - \frac{{i{\omega_0}}}{{2{B_T}}}[{{A_T}(\tau_1^2 - \tau_2^2) - 2({\tau_1}{t_1} - {\tau_2}{t_2}) + {D_T}(t_1^2 - t_2^2)} ]} \right\}d{\tau _1}d{\tau _2}. \end{aligned}$}$$

We assume that the medium is linearly dispersive [74], whose refractive index is given by expression n(ω) = n_aω + n_b, where n_a = β₂c/2 and n_b = c/v_g – β₂ω₀c. Here v_g is the group velocity of the pulse and β₂ is the group velocity dispersion coefficient at the central frequency of the pulse (it was already defined in Eq. (4)). Further, we set the time coordinate as being measured in the reference frame moving at the group velocity of the pulse, and for computational convenience we introduce the following average and difference coordinates:

(11)$$\bar{t} = ({t_1} + {t_2})/2\textrm{, }\Delta t = {t_2} - {t_1},\bar{\tau } = ({\tau _1} + {\tau _2})/2\textrm{, }\Delta \tau = {\tau _2} - {\tau _1},$$

(12)$$\bar{u} = ({u_1} + {u_2})/2\textrm{, }\Delta u = {u_2} - {u_1},\bar{x} = ({x_1} + {x_2})/2\textrm{, }\Delta x = {x_2} - {x_1}.$$

With the help of Eqs. (11) and (12), Eq. (6) can be expressed as

(13)$$\begin{aligned} {\Gamma _1}({u_1},{u_2},{t_1},{t_2},z) &= \frac{{{k_0}}}{{2\pi {B_S}x_s^2}}\textrm{exp}\left( {\frac{{i{k_0}{D_S}}}{{B_S^{}}}\Delta u \cdot \bar{u}} \right)\\ \textrm{ } &\times \int {\int_{ - \infty }^\infty {\left( {{{\bar{x}}^2} - \frac{1}{4}\Delta {x^2}} \right)\textrm{exp}\left( { - \frac{1}{{2w_0^2}}{{\bar{x}}^2}} \right)\textrm{exp}\left[ {\frac{{i{k_0}}}{{{B_S}}}({{A_S}\Delta x - \Delta u} )\bar{x}} \right]} } d\bar{x}\\ \textrm{ } &\times \textrm{exp}\left[ { - \left( {\frac{1}{{8w_0^2}} + \frac{1}{{2\delta_{}^2}}} \right)\Delta {x^2}} \right]\textrm{exp}\left( { - \frac{{ik}}{{B_S^{}}}\bar{y}\Delta x} \right)d\Delta x\\ \textrm{ } &\times \frac{{{\omega _0}}}{{2\pi {B_T}}}\textrm{exp}\left( {\frac{{i{\omega_0}{D_T}}}{{B_T^{}}}\Delta t \cdot \bar{t}} \right)\int {\int_{ - \infty }^\infty {\textrm{exp}\left( { - \frac{2}{{T_0^2}}{{\bar{\tau }}^2}} \right)\exp \left[ {\frac{{i{\omega_0}}}{{{B_T}}}({{A_T}\Delta \tau - \Delta t} )\bar{\tau }} \right]\textrm{d}\bar{\tau }} } \\ \textrm{ } &\times \textrm{exp}\left[ { - \left( {\frac{1}{{2T_0^2}} + \frac{1}{{2T_c^2}}} \right)\Delta {\tau^2}} \right]\textrm{exp}\left( { - \frac{{i{\omega_0}}}{{B_T^{}}}\bar{t}\Delta \tau } \right)d\Delta \tau . \end{aligned}$$

Performing integrations in Eq. (13) yields

(14)$$\begin{aligned} {\Gamma _1}({u_1},{u_2},{t_1},{t_2},z) &= \frac{{{k_0}w_0^3}}{{{B_S}x_s^2}}\textrm{exp}\left( {\frac{{i{k_0}{D_S}}}{{B_S^{}}}\Delta u \cdot \bar{u}} \right)\textrm{exp}\left( { - \frac{{k_0^2w_0^2}}{{2B_S^2}}\Delta {u^2}} \right)\\ \textrm{ } &\times \frac{1}{{\sqrt a }}\exp \left( { - \frac{{{b^2}}}{{2a}}} \right)\left[ {\frac{1}{a}\left( {\frac{1}{{\delta_{}^2}} + \frac{1}{{4w_0^2}}\frac{{{b^2}}}{a}} \right) + \frac{{k_0^2w_0^2}}{{B_S^2}}{{\left( {\frac{b}{a}A_S^{} - i\Delta u} \right)}^2}} \right]\\ \textrm{ } &\times \frac{{{\omega _0}{T_0}}}{{2{B_T}}}\textrm{exp}\left( {\frac{{i{\omega_0}{D_T}}}{{B_T^{}}}\Delta t \cdot \bar{t}} \right)\textrm{exp}\left( { - \frac{{\omega_0^2T_0^2}}{{8B_T^2}}\Delta {t^2}} \right)\frac{1}{{\sqrt {c^{\prime}} }}\exp \left( { - \frac{{{d^2}}}{{2c^{\prime}}}} \right) \end{aligned},$$

where

(15)$$a = \frac{1}{{4w_0^2}} + \frac{1}{{\delta _{}^2}} + \frac{{k_0^2w_0^2A_S^2}}{{B_S^2}},$$

(16)$$b = \frac{{{k_0}}}{{{B_S}}}\bar{u} + i\frac{{k_0^2w_0^2A_S^{}}}{{B_S^2}}\Delta u,$$

(17)$$c^{\prime} = \frac{1}{{T_0^2}} + \frac{1}{{T_c^2}} + \frac{{\omega _0^2T_0^2A_T^2}}{{4B_T^2}},$$

(18)$$d = \frac{{{\omega _0}}}{{{B_T}}}\bar{t} + i\frac{{\omega _0^2T_0^2A_T^{}}}{{4B_T^2}}\Delta t.$$

Using similar calculations while performing tedious evaluations of the multiple integrals, we obtain for Eqs. (9) and (10) the expressions

(19)$$\begin{aligned} {\Gamma _2}({u_1},{u_2},{t_1},{t_2},z) &= \frac{{{k_0}w_0^{}}}{{{B_S}}}\textrm{exp}\left( {\frac{{i{k_0}{D_S}}}{{B_S^{}}}\Delta u \cdot \bar{u}} \right)\textrm{exp}\left( { - \frac{{k_0^2w_0^2}}{{2B_S^2}}\Delta {u^2}} \right)\frac{1}{{\sqrt a }}\exp \left[ { - \frac{{{b^2}}}{{2a}}} \right]\\ \textrm{ } &\times \frac{{{\omega _0}T_0^3}}{{8{B_T}\tau _s^2}}\textrm{exp}\left( {\frac{{i{\omega_0}{D_T}}}{{B_T^{}}}\Delta t \cdot \bar{t}} \right)\textrm{exp}\left( { - \frac{{\omega_0^2T_0^2}}{{8B_T^2}}\Delta {t^2}} \right)\\ \textrm{ } &\times \frac{1}{{\sqrt {c^{\prime}} }}\exp \left( { - \frac{{{d^2}}}{{2c^{\prime}}}} \right)\left[ {\frac{1}{{c^{\prime}}}\left( {\frac{1}{{T_c^2}} + \frac{1}{{T_0^2}}\frac{{{d^2}}}{{c^{\prime}}}} \right) + \frac{{\omega_0^2T_0^2}}{{4B_T^2}}{{\left( {\frac{d}{{c^{\prime}}}A_T^{} - i\Delta t} \right)}^2}} \right], \end{aligned}$$

and

(20)$$\begin{aligned} {\Gamma _3}({u_1},{u_2},{t_1},{t_2},z) &={\pm} \frac{{{k_0}w_0^{}}}{{{B_S}{x_s}}}\textrm{exp}\left( {\frac{{i{k_0}{D_S}}}{{B_S^{}}}\Delta u \cdot \bar{u}} \right)\textrm{exp}\left( { - \frac{{k_0^2w_0^2}}{{2B_S^2}}\Delta {u^2}} \right)\frac{b}{{a\sqrt a }}\exp \left[ { - \frac{{{b^2}}}{{2a}}} \right]\\ \textrm{ } &\times \frac{{\omega _0^2T_0^3}}{{8B_T^2\tau _s^{}}}\textrm{exp}\left( {\frac{{i{\omega_0}{D_T}}}{{B_T^{}}}\Delta t \cdot \bar{t}} \right)\textrm{exp}\left( { - \frac{{\omega_0^2T_0^2}}{{8B_T^2}}\Delta {t^2}} \right)\frac{1}{{\sqrt {c^{\prime}} }}\exp \left( { - \frac{{{d^2}}}{{2c^{\prime}}}} \right)\left( {\frac{d}{{c^{\prime}}}A_T^{} - i\Delta t} \right)\\ &\textrm{ } \mp \frac{{k_0^2w_0^3}}{{B_S^2{x_s}}}\textrm{exp}\left( {\frac{{i{k_0}{D_S}}}{{B_S^{}}}\Delta u \cdot \bar{u}} \right)\textrm{exp}\left( { - \frac{{k_0^2w_0^2}}{{2B_S^2}}\Delta {u^2}} \right)\frac{1}{{\sqrt a }}\exp \left[ { - \frac{{{b^2}}}{{2a}}} \right]\left( {\frac{b}{a}A_S^{} - i\Delta u} \right)\\ \textrm{ } &\times \frac{{\omega _0^{}T_0^{}}}{{2B_T^{}\tau _s^{}}}\textrm{exp}\left( {\frac{{i{\omega_0}{D_T}}}{{B_T^{}}}\Delta t \cdot \bar{t}} \right)\textrm{exp}\left( { - \frac{{\omega_0^2T_0^2}}{{8B_T^2}}\Delta {t^2}} \right)\frac{d}{{c^{\prime}\sqrt {c^{\prime}} }}\exp \left( { - \frac{{{d^2}}}{{2c^{\prime}}}} \right), \end{aligned}$$

respectively. In the above calculation, the following tabulated integrals are used:

(21)$$\int_{ - \infty }^\infty {\exp ({ - p{x^2} + 2qx} )} dx = \sqrt {\frac{\pi }{p}} \exp \left( {\frac{{{q^2}}}{p}} \right),$$

(22)$$\int_{ - \infty }^\infty {xexp ({ - p{x^2} + 2qx} )} dx = \frac{q}{p}\sqrt {\frac{\pi }{p}} \exp \left( {\frac{{{q^2}}}{p}} \right),$$

(23)$$\int_{ - \infty }^\infty {{x^2}\exp ({ - p{x^2} + 2qx} )} dx = \frac{1}{p}\left( {\frac{1}{2} + \frac{{{q^2}}}{p}} \right)\sqrt {\frac{\pi }{p}} \exp \left( {\frac{{{q^2}}}{p}} \right).$$

On using the definition of the complex degree of coherence of pulsed beams [39] for the MCF in Eqs. (7), (14), (19) and (20), we obtain expression

(24)$$\mu ({u_1},{u_2},{t_1},{t_2}{,}z) = \frac{{\Gamma ({u_1},{u_2},{t_1},{t_2}{,}z)}}{{\sqrt {\Gamma ({u_1},{u_1},{t_1},{t_1}{,}z)} \sqrt {\Gamma ({u_2},{u_2},{t_2},{t_2}{,}z)} }}$$

which enables us to further study the evolution of the spatiotemporal coherency vortices of the STPCPV beams. Conventionally, the position of a spatiotemporal coherency vortex is determined by relations

(25)$${\textrm{Re}} [{\mu ({u_1}\textrm{,}{u_2},{t_1},{t_2}{,}z)} ]= 0,$$

(26)$${\mathop{\rm Im}\nolimits} [{\mu ({u_1}\textrm{,}{u_2},{t_1},{t_2}{,}z)} ]= 0,$$

where Re and Im denote the real and imaginary parts of a complex number respectively. For a spatiotemporal coherency vortex, the topological charge and its sign are determined by the vorticity of phase contours around the singularity [75]. Equations (25) and (26) can be conveniently employed for determining the location of the coherency vortex. Usually, for a fully coherent light field, the optical vortex arises at the position where the field’s amplitude is equal to zero [42]. However, for a partially coherent light field, the coherency vortex takes place at the position at which the complex degree of coherence of the field is equal to zero [76].

According to Eqs. (24)-(26), one can investigate the evolution of the spatiotemporal coherency vortices in the STPCPV beams in dispersive media. In particular, for u₁= u₂= u, t₁= t₂= t, the spatiotemporal intensity of the STPCPV beams can be determined as

(27)$$\begin{aligned} &I(u,t,z)= \Gamma ({u_1} = {u_2} = u,{t_1} = {t_2} = t,z)\\ &= \frac{{{k_0}w_0^3}}{{{B_S}x_s^2}}\frac{1}{{\sqrt a }}\textrm{exp}\left( { - \frac{1}{{2a}}\frac{{k_0^2}}{{B_S^2}}{u^2}} \right)\left[ {\frac{1}{a}\left( {\frac{1}{{{\delta^2}}} + \frac{1}{{4w_0^2}}\frac{{k_0^2}}{{aB_S^2}}{u^2}} \right) + \frac{{k_0^4w_0^2}}{{B_S^4}}\frac{{A_S^2}}{{{a^2}}}{u^2}} \right]\frac{{{\omega _0}{T_0}}}{{2{B_T}}}\frac{1}{{\sqrt {c^\prime} }}\textrm{exp}\left( { - \frac{1}{{2{c^ \prime}}}\frac{{\omega_0^2}}{{B_T^2}}{t^2}} \right)\\ &+ \frac{{{k_0}{w_0}}}{{{B_S}}}\frac{1}{{\sqrt a }}\textrm{exp}\left( { - \frac{1}{{2a}}\frac{{k_0^2}}{{B_S^2}}{u^2}} \right)\frac{{{\omega _0}T_0^3}}{{8{B_T}\tau _s^2}}\frac{1}{{\sqrt {c^\prime} }}\textrm{exp}\left( { - \frac{1}{{2{c^\prime}}}\frac{{\omega_0^2}}{{B_T^2}}{t^2}} \right)\left[ {\frac{1}{{c^\prime}}\left( {\frac{1}{{T_c^2}} + \frac{1}{{T_0^2}}\frac{{\omega_0^2}}{{{c^ \prime}B_T^2}}{t^2}} \right) + \frac{{\omega_0^4T_0^2}}{{4B_T^4}}\frac{{A_T^2}}{{{{c^\prime}^2}}}{t^2}} \right]\\ &\pm \frac{{{k_0}{w_0}}}{{{B_S}{x_s}}}\frac{1}{{a\sqrt a }}\textrm{exp}\left( { - \frac{1}{{2a}}\frac{{k_0^2}}{{B_S^2}}{u^2}} \right)\left( {\frac{{{k_0}}}{{{B_S}}}u} \right)\frac{{{\omega _0}{T_0}}}{{2{B_T}{\tau _S}}}\frac{1}{{{c^ \prime}\sqrt {c^ \prime} }}\textrm{exp}\left( { - \frac{1}{{2{c^ \prime}}}\frac{{\omega_0^2}}{{B_T^2}}{t^2}} \right)\left( {\frac{{{\omega_0}}}{{{B_T}}}t} \right)\left( {\frac{{{\omega_0}T_0^2{A_T}}}{{4{B_T}}} - \frac{{{k_0}w_0^2{A_S}}}{{{B_S}}}} \right). \end{aligned}$$

3. Coherence control of spatiotemporal coherency vortices

In this section, we will illustrate the possibility of achieving the control of the evolving spatiotemporal coherency vortices of the STPCPV beams via adjusting the coherence state of the source. For the considered numerical examples, the following values of the beam’s parameters are chosen to be λ₀= 532 nm, w₀= 2 mm, δ = 4 mm, T₀= 3 ps, T_c= 5 ps, x_s= 1 mm, τ_s= 1 ps, u₁= 10 mm, t₁= 20 ps, unless other values are specified. For the dispersive medium, we choose fused silica, a typical material of the optical fiber. The group velocity dispersion parameter and the group velocity refractive index of fused silica at 20°C are β₂= 65.781 ps²/km and n_g= c/v_g= 1.4853, respectively [77].

The evolution of the spatiotemporal coherency vortices of a STPCPV beam is specified by Eqs. (24)-(26). In Figs. 1 (a)–1(c) we show the spatiotemporal intensity distribution of the STPCPV beams with topological charge m = +1 for different propagation distances z = 0, z = 0.2 km and z = 0.8 km. As shown in Fig. 1(a), at the source plane z = 0, zero spatiotemporal intensity value appears at the spatiotemporal center (u₂= 0, t₂= 0), while the phase is singular. With the increasing propagation distance, for example, at z = 0.2 km in Fig. 1(b), a spatiotemporal intensity profile acquires a center with a non-zero minimum. For larger distances from the source, e.g., Fig. 1(c), the central dip disappears and the intensity evolves into a Gaussian profile. In addition, the intensity profile rotates anticlockwise along the z axis because of the different spreading in temporal dispersion as compared to transverse beam diffraction. Figures 1 (d)-(f) show the phase distribution of µ of a STPCPV beam with topological charge m = +1, for different propagation distances: z = 0, z = 0.2 km and z = 0.8 km. In addition, the level curves Re µ = 0 and Im µ = 0 are given in Figs. 1 (g)–1(i). It is shown that, at the spatiotemporal center (u₂= 0, t₂= 0) in the u-t plane, there is a spatiotemporal coherency vortex whose phase increases anticlockwise along a closed loop from -π to π and returns to the origin. With the increasing propagation distance, the phase structure and the position of coherent vortex change. However, there always exists a spatiotemporal optical vortex near the spatiotemporal center of the beam and the topological charge is conserved along the propagation path. This phenomenon is quite different from the case of partially coherent beams propagation in random media, whose topological charge is not generally conserved [78].

Fig. 1. (a)-(c) Spatiotemporal intensity, (d)-(f) Phase distribution of µ and (g)-(i) Curves of Re µ = 0 (solid line) and Im µ = 0 (dashed line) of a STPCPV beam with topological charge m = +1 for different propagation distances.

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Figure 2 is organized the same way as Fig. 1 and uses the same parameters but illustrates the case m = -1. Comparison of insets (b) and (c) of Figs. 1 and 2 shows that the spatiotemporal intensity profile changes the sense of rotation: for the latter case the phase of the spatiotemporal coherency vortex increases clockwise. In both figures the phase structure and the position of coherent vortex change with increasing the propagation distance. In addition, from Figs. 1(g) and 2(g), one can see two straight lines with negative and positive slopes. The crossover point of two straight lines satisfies the equations Re µ = 0 and Im µ = 0 simultaneously, i.e., it is the core’s position of a spatiotemporal coherency vortex. Usually, one can judge the position of a spatiotemporal coherency vortex by the crossover point. We note that the two straight lines are not chosen along the u₂ and t₂ axes. This is due to the fact that in the plane z = 0, with fixed m = ±1, the complex degree of coherence of the pulsed beam can be expressed as

(28)$$\mu ({u_1},{u_2},{t_1},{t_2}{,}0) = \mu ^{\prime}\left[ {\frac{{{u_1}{u_2}}}{{x_s^2}} + \frac{{{t_1}{t_2}}}{{\tau_s^2}} \mp \frac{i}{{x_s^{}\tau_s^{}}}({{u_1}t_2^{} - {u_2}t_1^{}} )} \right]$$

where

(29)$$\mu ^{\prime} = {{\textrm{exp}\left[ { - \frac{{{{({u_2} - {u_1})}^2}}}{{2\delta_{}^2}}} \right]\exp \left[ { - \frac{{{{(t_2^{} - t_1^{})}^2}}}{{2T_c^2}}} \right]} / {\sqrt {\left( {\frac{{t_1^2}}{{\tau_s^2}} + \frac{{u_1^2}}{{x_s^2}}} \right)\left( {\frac{{t_2^2}}{{\tau_s^2}} + \frac{{u_2^2}}{{x_s^2}}} \right)} }}$$

Fig. 2. (a)-(c) Spatiotemporal intensity, (d)-(f) Phase distribution of the complex degree of coherence µ and (g)-(i) Curves of Re µ = 0 (solid line) and Im µ = 0 (dashed line) of a STPCPV beam with topological charge m = -1 for different propagation distances.

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Thus, by solving the equations Re[µ] = 0 and Im[µ] = 0, one can obtain the linear equations

(30)$${u_2} ={-} \frac{{{t_1}}}{{{u_1}}}\frac{{x_s^2}}{{\tau _s^2}}{t_2},$$

(31)$${u_2} = \frac{{{u_1}}}{{{t_1}}}{t_2}. $$

It is apparent from Eqs. [30] and [31] that there are two straight lines in the u₂ and t₂ coordinate system appearing for the fixed values of u₁, t₁, x_s and τ_s.

Figure 3 shows the phase distribution of the complex degree of coherence µ of a STPCPV beam having different values of temporal coherent length: T_c= 1ps, T_c= 2ps and T_c → ∞, for two values of topological charge: m = ±1, at z = 0.2km. Figure 3 implies that the coherence time T_c plays a crucial part in determining the phase structure of the spatiotemporal coherency vortices of the STPCPV beams. In addition, the position of the spatiotemporal coherency vortex changes appreciably with increasing T_c. When T_c → ∞, i.e., for a spatially partially and temporally fully coherent pulsed vortex (STPCPV) beam, the position of spatiotemporal coherency vortex tends to the spatiotemporal center (u₂= 0, t₂= 0). Thus, both the phase structure and the position of the spatiotemporal coherency vortex can be conveniently controlled by adjusting the source coherence time T_c.

Fig. 3. Phase distributions of µ of a STPCPV beam with different coherence time values: T_c= 1ps, T_c= 2ps and T_c → ∞ for different topological charges m = ±1 at z = 0.2 km.

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Figure 4 shows the evolution of the positions of the spatiotemporal coherency vortices with coherence time set to T_c= 1ps and for the limiting case T_c → ∞, for two values of the topological charge m = ±1. As can be seen from Fig. 4(a), the position’s behavior exhibits a noticeable difference m = ±1. For m = +1, the position coordinate changes non-monotonically with the growing propagation distance z. Whereas for m = -1, it changes monotonically with growing z. In addition, for fully temporally coherent case T_c→∞ shown in Fig. 4(b), the position’s evolution is qualitatively similar for m = ±1, exhibiting, in both cases, the non-monotonic changes. Regardless of the value of T_c, the position coordinates deviate farther from the spatiotemporal center for sufficiently large distances. Comparing overall Figs. 4(a) and 4(b), we see that the coherence time T_c has a remarkable influence on the dynamics of the position of the spatiotemporal coherency vortex.

Figure 5 gives the spatial 5(a) and the temporal 5(b) positions of the spatiotemporal coherency vortices of a STPCPV beam as a function of coherence time T_c for topological charges m = ±1 at z = 0.2km. As can be seen from Fig. 5(a), spatial position u₂ increases monotonically with increasing T_c for m = +1, whereas spatial position u₂ increases first, then decreases with increasing T_c. Moreover, spatial position u₂ approaches an asymptotic value when T_c is large enough. For example, depending on topological charge m, the asymptotic value is equal to 2.481 mm and 2.657 mm for m = +1 and m = -1, respectively. However, temporal position t₂ of the spatiotemporal coherency vortex decreases with increasing T_c and approaches two asymptotic values, t₂= 1.371ps and t₂= -1.984ps for m = +1 and m = -1, respectively, when T_c is large enough.

Fig. 4. Positions evolution of spatiotemporal coherency vortices of STPCPV beams with different coherence time T_c= 1ps (a) and T_c→∞ (b) for different topological charges m=±1.

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Fig. 5. Spatial (a) and temporal (b) positions of spatiotemporal coherency vortices of a STPCPV beam as functions of coherence time T_c for topological charges m = ±1 and z = 0.2 km.

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Figure 6 shows the phase distributions of the complex degree of coherence µ of a STPCPV beam with different spatial coherence widths δ = 1mm, δ = 2mm and δ → ∞, for two topological charges m = ±1 at z = 0.2km. Just like T_c in Fig. 3, the spatial coherence width δ also plays an important role in determining the phase structure of the spatiotemporal coherency vortices of the STPCPV beams. The position of a spatiotemporal coherency vortex substantially changes with increasing δ. When δ → ∞, i.e., for spatially fully and temporally partially coherent pulsed vortex beams, the position of the spatiotemporal coherency vortex approaches the spatiotemporal center (u₂= 0, t₂= 0). Thus, the phase structure and the position of the spatiotemporal coherency vortex of a STPCPV beam can also be controlled by adjusting the spatial coherence width δ.

Fig. 6. Phase distributions of the complex degree of coherence µ of a STPCPV beam with different spatial coherent widths: δ = 1mm, δ = 2mm and δ → ∞ for topological charges m = ±1 at z = 0.2 km.

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Figure 7 illustrates the dynamics of the positions of the spatiotemporal coherency vortices of the STPCPV beams with coherence widths δ = 1mm and δ → ∞ for two topological charge values, m = ±1. Figure 7(a), shows that in the low spatial coherence case δ = 1mm, the position’s evolution is similar for m = ±1, in both cases changing non-monotonically with increasing z. However, for fully spatially coherent case δ → ∞ shown in Fig. 7(b), the position’s evolution presents a significant difference for m = ±1, especially for the propagation distances smaller than 0.3 km. Moreover, regardless of the spatial coherence width, the position coordinate deviates farther from the spatiotemporal center as z → ∞.

Figure 8 demonstrates the spatial 8(a) and the temporal 8(b) positions of the spatiotemporal coherency vortices of the STPCPV beams as a function of coherence width δ, for topological charges m = ±1 at z = 0.2 km. As can be seen from Fig. 8(a), spatial position u₂ decreases monotonically with increasing δ for m = ±1. And spatial position u₂ approaches asymptotic values -0.516mm and 0.437mm for m = +1 and m = -1, respectively. However, temporal position t₂ exhibits a non-monotonical change with the increasing coherence width δ, as shown in Fig. 8(b). Moreover, for m = +1, temporal position t₂ increases first, then decreases with increasing δ, and approaches an asymptotic value t₂= 2.383ps. Nevertheless, for m = -1, the situation is the opposite, i.e., temporal position t₂ decreases first, then increases with increasing δ, and approaches the asymptotic value t₂= 2.383ps. As shown in Fig. 8, the coherence width δ can efficiently control the spatial and the temporal positions of the spatiotemporal coherency vortices of the STPCPV beams.

Fig. 7. Position evolution of spatiotemporal coherency vortices of STPCPV beams with coherence widths δ = 1mm and δ → ∞ for topological charges m = ±1.

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Fig. 8. Spatial (a) and temporal (b) positions of spatiotemporal coherency vortices of a STPCPV beam as functions of coherence width δ for topological charges m = ±1 and z = 0.2 km.

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Figure 9 shows the phase distributions of the complex degree of coherence µ and curves of Re µ = 0 and Im µ = 0 of spatially fully and temporally fully coherent pulsed vortex beams with topological charge m = ±1, for propagation distances z = 0, z = 0.2km and z = 0.8km, respectively. As is well known, in stationary beams, the coherency vortex reduces to an optical field vortex when the degree of spatial coherence approaches infinity. Similarly to the case of stationary beams, the spatiotemporal coherency vortex reduces to a spatiotemporal optical field vortex when both coherence width δ and coherence time T_c approach infinity. As shown in Fig. (9), the position of the spatiotemporal optical field vortex of a fully coherent pulsed vortex beam always remains at the spatiotemporal center (u₂= 0, t₂= 0). In addition, the phase structure remains invariant with the increasing propagation distance. Thus, we conclude that the discovered dynamics in the spatiotemporal coherency vortex’ structure and position are entirely the source correlation-induced.

Fig. 9. Phase distributions of the complex degree of coherence µ and curves of Re µ = 0 and Im µ = 0 of spatially fully and temporally fully coherent pulsed vortex beams with topological charge m = ±1 for propagation distances z = 0, z = 0.2km and z = 0.8km, respectively.

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As is widely known, optical vortices and STOVs have symmetries in the properties of the ∓ 1 topological charges. However, for a stationary partially coherent light field, the symmetry is broken. This is because the coherency vortex takes place at the position where the spectral degree of coherence µ (r₁, r₂, ω) is equal to zero. The phase distribution of µ depends tremendously on the choice of the position variable r₁ [79]. Physically, µ (r₁, r₂, ω) is related to the cross-spectral density (CSD) W(r₁, r₂, ω). The CSD of a partially coherent light field is characterized by the ensemble averaging of the individual electric field realizations, which includes intrinsic random phase fluctuations.

Similarly to the case of a stationary light field, for a partially coherent pulsed field of our interest, the symmetry is broken too, because the spatiotemporal coherency vortex takes place at the position where the complex degree of coherence µ (u₁, u₂, t₁, t₂, z) of the field is equal to zero. The phase distribution of µ also depends on the choice of the position variable (u₁, t₁), which is affected by the topological charges. In addition, the position-related spatiotemporal intensity also depends on the topological charges shown in Eq. (27).

4. Conclusion

In this paper, we have proposed a novel method to achieve the coherence control of the spatiotemporal coherency vortices in the STPCPV beams. Specifically, the analytical expressions for the mutual coherence functions of the STPCPV beams with topological charge m = ±1 have been derived and used to study the evolution of the spatiotemporal coherency vortices of these beams propagating through fused silica. It has been found that the coherence width δ and the coherence time T_c play an important role in determining the phase distribution and position of spatiotemporal coherency vortices. For topological charge m = +1, the spatial position u₂ grows monotonically with increasing coherence time T_c. For topological charge m = -1, the spatial position u₂ changes non-monotonically with increasing coherence time T_c. In addition, temporal position t₂ changes non-monotonically with increasing coherence width δ for both topological charges, m = ±1. Both the dependency of the temporal position t₂ on T_c and that of the spatial position u₂ on δ are monotonically decreasing functions for m = ±1. By taking advantage of these special properties, one can control the phase distribution and the position of the spatiotemporal coherency vortices. In other words, the spatial and temporal coherence are found to be very efficient tools for controlling the phase distribution and the position of the spatiotemporal coherency vortices of the STPCPV beams. One possible extension of the present work is related to using nonzero coefficients C_S and C_T in Eqs. (3) and (4), which would correspond to spatial focusing and temporal compression [80] of the pulse, respectively, as it was done recently for pulses with a different spatiotemporal form [81].

It should be pointed out that the results obtained in this paper will degenerate into spatiotemporal optical field vortices of fully coherent pulsed beams when both the coherence width δ and the coherence time T_c approach infinity. The method introduced in our paper will open new opportunities for the theoretical analysis of spatiotemporal optical vortices with partial spatial and temporal coherence. Furthermore, the results obtained in this paper will benefit a number of technologies relating to light-matter interaction, such as optical trapping, quantum entanglement, spatiotemporal spin-orbit angular momentum coupling, and many others [28].

Funding

National Natural Science Foundation of China (12174171); Central Plains Talents Program of Henan (ZYYCYU202012144); China Scholarship Council (202008410578); Excellent Overseas Visiting Scholar Program of Henan (026); Belarusian Republican Foundation for Fundamental Research (F21TURG-003); University of Miami via the Cooper Fellowship.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are available from the corresponding authors upon reasonable request.

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Source coherence-induced control of spatiotemporal coherency vortices

Abstract

1. Introduction

2. Theory

3. Coherence control of spatiotemporal coherency vortices

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (31)

Optics Express