Characterizing the polarization and cross-polarization of electromagnetic vortex pulses in the space-time and space-frequency domain

Meilan Luo; Daomu Zhao

doi:10.1364/OE.23.004153

1. Introduction

The coherence theory for stationary random fields has been extensively researched for a long time [1,2] and was well understood, while few studies were carried out with respect to pulsed beams in space-time domain until the theory of coherence for nonstationary fields was established [3–5]. In recent years, a certain class of electromagnetic (EM) pulses, for example, the Gaussian Schell-model (GSM) pulses, has been studied to some extent, including the changes of the temporal and spectral degree of polarization on propagation through a dispersive media [6,7]. However, the work regarding the pulsed vortex beams is much less.

Vortex beams that well merged with ultra-short pulse are of intense interests, as those beams will allow for further study in many fields of optics and photonics, such as temporal imaging, ghost and correlation imaging, as well as ultrafast physical, chemical processes and specific applications in topological spectroscopy [8–11]. Particularly, many of these practical applications originate from the additional controllable degree of freedom of orbital angular momentum (OAM). Therefore, determining the azimuthal index associated with a given vortex field is of importance and may lead to some new studies. To this end, it is very necessary and essential to characterize the correlations of pulsed vortex beams, where the stationary formalism may no longer hold. In addition, most light sources in practice are far from being totally coherent, this underlines the importance of investigating partially coherent pulse. And usually, there are two available approaches for processing the partially coherent nonstationary pulses to study its statistic properties, one is the direct spatiotemporal integral method, which contains four spatial and two temporal coordinates, alternatively, is the spectral way which is obtained by the Fourier transformation of the temporal form and generalizes the coherence theory from the stationary fields into the nonstationary fields in space-frequency domain. Generally speaking, the former way is more convenient to deal with such pulses than the later one.

In this work we will mainly focus on the evolution of the degree of polarization of EM pulsed vortex beams, which is the fundamental characteristic of light and reflects the vectorial nature of statistical EM fields, in dispersive media both in the space-time and space-frequency domains. It aims to show the dependences of the second-order dispersion coefficient, temporal coherence length and topological charge on the temporal degree of polarization and the spectral degree of polarization. Furthermore, the distribution of temporal degree of cross-polarization, which is a function of the location of two points and reduces to the usual degree of polarization when the two points coincide, of vortex pulses in such media is also researched.

2. Propagation formula of mutual coherence function of EM pulsed vortex beams in the space-time domain

Be analogy to the mutual coherence function of the spatially and temporally partially coherent GSM pulsed beams, expressed in a form consisting of separable space- and time-dependent parts [5], the counterpart of the EM pulsed vortex beams without spatiotemporal coupling at the initial plane $z = 0$ is expressed as

Γ_{i j}^{(0)} (x_{10}, y_{10}, t_{10}; x_{20}, y_{20}, t_{20}) = R_{i j} (x_{10}, y_{10}, x_{20}, y_{20}) T_{i j} (t_{10}, t_{20}),

where

R_{i j} (x_{10}, y_{10}, x_{20}, y_{20})

is the spatial dependence characterized by a function corresponding to the stationary EM vortex beams

\begin{array}{l} R_{i j} (x_{10}, y_{10}, x_{20}, y_{20}) = A_{i} A_{j} B_{i j} \frac{{(x_{10} - i y_{10})}^{m} {(x_{20} + i y_{20})}^{m}}{σ^{2 m}} \exp (- \frac{x_{10}^{2} + y_{10}^{2} + x_{20}^{2} + y_{20}^{2}}{σ^{2}}) \\ \times exp [- \frac{{(x_{10} - x_{20})}^{2} + {(y_{10} - y_{20})}^{2}}{δ_{i j}^{2}}], (i = x, y; j = x, y), \end{array}

where

A_{i}

and

σ

refer, respectively, to the characteristic amplitude and beam size of the source,

m

denotes the topological charge of the vortex beam,

δ_{i j}

is a positive constant characterizing the correlation length and the factor

B_{i j}

has the following properties:

B_{i i} = B_{j j} = 1, | B_{i j} | = | B_{j i} | .

And the remaining temporal part is determined by

T_{i j} (t_{10}, t_{20}) = \exp [- \frac{t_{10}^{2} + t_{20}^{2}}{2 T_{0}^{2}} - \frac{{(t_{10} - t_{20})}^{2}}{2 T_{c i j}^{2}} + i ω_{0} (t_{10} - t_{20})],

where

T_{0}

means the pulse duration,

T_{c i j}

measures the correlation time width, describing for the correlation between two different space-time points, and

ω_{0}

is the carrier frequency.

In much previous work involving the pulsed beams, most cases were restricted to the physical situation in which second-order dispersion coefficient dominates, that controls temporal evolution along the z axis, so only the coordinate z was considered in [12–14] except in [15]. However, the spatial coherence will inevitably affect the dynamics of the mutual coherence function. Consequently, we will take the spatial dependence into account. According to the results obtained in [15], the propagation of partially coherent nonstationary pulsed beams through dispersive media in spatiotemporal domain within the paraxial approximation can be described by the integral form

\begin{array}{l} Γ_{i j} (ρ_{1}, ρ_{2}, t_{1}, t_{2}, z) = {[\frac{k (ω)}{2 π z}]}^{2} \frac{ω_{0}}{2 π a} \iint \iint \iint Γ_{i j}^{(0)} (x_{10}, y_{10}, t_{10}; x_{20}, y_{20}, t_{20}) \\ \times \exp {- \frac{i k (ω)}{2 z} [{(ρ_{1} - r_{1})}^{2} - {(ρ_{2} - r_{2})}^{2}]} d^{2} r_{1} d^{2} r_{2} \exp {\frac{i ω_{0}}{2 a} [{(t_{1} - t_{10})}^{2} - {(t_{2} - t_{20})}^{2}]} d t_{10} d t_{20}, \end{array}

where

a = ω_{0} β_{2} z

, with

β_{2}

representing the second-order dispersion coefficient. Here the assumption that the time coordinate is measured in the reference frame moving at the group velocity of the pulses has been made.

During the generalization of Eq. (5), the $3 \times 3$ spatiotemporal $A B C D$ matrices of the dispersive optical elements reading [16]

A = D = ε, C = 0, B = (\begin{matrix} z / n (ω) & 0 & 0 \\ 0 & z / n (ω) & 0 \\ 0 & 0 & \frac{λ_{0} ω_{0}^{2} z}{v_{g}^{2}} \frac{\partial v_{g}}{\partial ω} \end{matrix})

have been substituted, where

ε

is a unit matrix, with

n (ω)

being the frequency-dependent refractive index,

λ_{0}

is the central wavelength and

v_{g}

denotes the group velocity of the pulse.

On substituting from Eqs. (1), (2) and (4) into Eq. (5), the general output expression of the mutual coherence function of the EM vortex pulses is immediately obtained,

\begin{array}{l} Γ_{i j} (ρ_{1}, ρ_{2}, t_{1}, t_{2}, z) = {(\frac{k (ω)}{2 π z})}^{2} \iint \iint R_{i j} (r_{1}, r_{2}) \exp {- \frac{i k (ω)}{2 z} [{(ρ_{1} - r_{1})}^{2} - {(ρ_{2} - r_{2})}^{2}]} d^{2} r_{1} d^{2} r_{2} \\ \times \frac{ω_{0}}{2 π a} \iint T_{i j} (t_{10}, t_{20}) \exp {\frac{i ω_{0}}{2 a} [{(t_{1} - t_{10})}^{2} - {(t_{2} - t_{20})}^{2}]} d t_{10} d t_{20} \\ = R_{t i j} (ρ_{1}, ρ_{2}, z) Y_{t_{i}} (t_{1}, t_{2}, z) . \end{array}

Here

\begin{array}{l} R_{t i j} (ρ_{1}, ρ_{2}, z) \\ = \frac{A_{i} A_{j} B_{i j} z^{2 m}}{Δ_{i j}^{m + 1} k {(ω)}^{2 m} σ^{2 m}} \exp [- \frac{β_{i j} ρ_{1}^{2} + α_{i j} ρ_{2}^{2} - 2 (x_{1} x_{2} + y_{1} y_{2}) / δ_{i j}^{2}}{Δ_{i j}} - \frac{i k (ω)}{2 z} (ρ_{1}^{2} - ρ_{2}^{2})] \\ \times \sum_{n = 0}^{m} {(C_{m}^{n})}^{2} \frac{n! 4^{n}}{δ_{i j}^{2 n}} {[\frac{- 4 (β_{i j} ρ_{1}^{2} + α_{i j} ρ_{2}^{2})}{Δ_{i j} δ_{i j}^{2}} + i \frac{k {(ω)}^{2}}{z^{2}} (x_{1} y_{2} - x_{2} y_{1}) + (\frac{k {(ω)}^{2}}{z^{2}} + \frac{8}{Δ_{i j} δ_{i j}^{4}}) (x_{1} x_{2} + y_{1} y_{2})]}^{m - n}, \end{array}

where we have introduced the following notation:

α_{i j} = \frac{1}{σ^{2}} + \frac{1}{δ_{i j}^{2}} + \frac{i k (ω)}{2 z}

,

β_{i j} = \frac{1}{σ^{2}} + \frac{1}{δ_{i j}^{2}} - \frac{i k (ω)}{2 z}

,

Δ_{i j} = 1 + \frac{4 z^{2}}{σ^{2} k {(ω)}^{2}} (\frac{1}{σ^{2}} + \frac{2}{δ_{i j}^{2}})

, the Eq. (8) just characterizes the propagation of the stationary EM vortex beams in dispersive media. And the temporal part is

Y_{t_{i}} (t_{1}, t_{2}, z) = \frac{1}{\sqrt{Λ_{i}}} \exp [- \frac{{(t_{1} + t_{2})}^{2}}{4 T_{0}^{2} Λ_{i}}] \exp [\frac{i ω_{0}}{2 a} (1 - \frac{1}{Λ_{i}}) (t_{1}^{2} - t_{2}^{2})] \exp [- \frac{α_{t i}}{Λ_{i}} {(t_{1} - t_{2})}^{2}],

with

α_{t i} = \frac{1}{4 T_{0}^{2}} + \frac{1}{2 T_{c i}^{2}}

and

Λ_{i} = 1 + \frac{4 α_{t i} a^{2}}{T_{0}^{2} ω_{0}^{2}}

, which is also defined as the temporal broadening factor. It should be noted that the term of

\exp [i ω_{0} (t_{10} - t_{20})]

has been omitted during the derivation of Eq. (9), due to the slowly varying envelope of the field.

The degree of polarization in space-time domain can be characterized by the expression of mutual coherence function as [17]

P_{t} (ρ, t, z) = \sqrt{1 - \frac{4 Det [\overset{\leftrightarrow}{Γ} (ρ, ρ, t, t, z)]}{{[Tr \overset{\leftrightarrow}{Γ} (ρ, ρ, t, t, z)]}^{2}}}

where Det and Tr stand for the determinant and the trace, respectively. If the off-diagonal elements of the mutual coherence function in the source plane are set as zero, the temporal degree of polarization is simplified to be

P_{t} (ρ, t, z) = | \frac{Γ_{x x} (ρ, ρ, t, z) - Γ_{y y} (ρ, ρ, t, z)}{Γ_{x x} (ρ, ρ, t, z) + Γ_{y y} (ρ, ρ, t, z)} |,

with

\begin{array}{l} Γ_{i i} (ρ, ρ, t, z) = \frac{A_{i}^{2} z^{2 m}}{Δ_{i i}^{m + 1} k {(ω)}^{2 m} σ^{2 m}} \frac{1}{\sqrt{Λ_{i}}} \exp (- \frac{β_{i i} + α_{i i} - 2 / δ_{i i}^{2}}{Δ_{i i}} ρ^{2}) \\ \times \sum_{n = 0}^{m} {(C_{m}^{n})}^{2} \frac{n! 4^{n}}{δ_{i i}^{2 n}} {[[\frac{- 4 (β_{i i} + α_{i i})}{Δ_{i i} δ_{i i}^{2}} + \frac{k {(ω)}^{2}}{z^{2}} + \frac{8}{Δ_{i i} δ_{i i}^{4}}] ρ^{2}]}^{m - n} \exp [- \frac{t^{2}}{T_{0}^{2} Λ_{i}}] . \end{array}

And the degree of cross-polarization for two contrary position vectors is given by [18]

P_{t} (ρ, - ρ, t_{1}, t_{2}, z) = | \frac{Γ_{x x} (ρ, - ρ, t_{1}, t_{2}, z) - Γ_{y y} (ρ, - ρ, t_{1}, t_{2}, z)}{Γ_{x x} (ρ, - ρ, t_{1}, t_{2}, z) + Γ_{y y} (ρ, - ρ, t_{1}, t_{2}, z)} | .

Based on Eqs. (11)-(13), the properties of the degree of polarization and the degree of cross-polarization of spatially and temporally partially coherent EM vortex pulses can be studied in the following. We assume a linearly dispersive media whose refractive index is $n (ω) = n_{a} ω + n_{b}$ , where $n_{a}$ and $n_{b}$ are constants [19]. The second-order dispersion coefficient $β_{2}$ depends on $n_{a}$ by $β_{2} = n_{a} / c$ . In the numerical calculations, unless specified otherwise, the parameters are chosen as: $A_{x} = 1$ , $A_{y} = \sqrt{0.5}$ , $σ = 1 cm$ , $δ_{x x} = 5 mm$ , $δ_{y y} = 2 mm$ , $ω_{0} = 2.36 rad / fs$ , $T_{0} = 1 ps$ .

In order to study the contribution of second-order dispersion coefficient and temporal coherence length, the degree of polarization of the spatially and temporally partially coherent EM vortex pulses with topological charge $m = 1$ as a function of the propagation distance z is displayed in Fig. 1. As seen in Fig. 1, both of the variables result in the more rapid changes of the degree of temporal polarization. It can be explained from the temporal broadening factor, which is related to $β_{2}$ and $T_{c x}$ with the expression $Λ_{i} = 1 + (\frac{1}{T_{0}^{4}} + \frac{2}{T_{c i}^{2} T_{0}^{2}}) β_{2}^{2} z^{2}$ . Consequently, the temporal widths of x and y components broaden at different rates, and the temporal width in x component has a slower expansion with the increasing of $β_{2}$ or $T_{c x}$ . Comparing parts (a) and (b) of Fig. 1, the difference is that the degree of temporal polarization approaches to its asymptotic value at a large distance, which is the same regardless of the choice of $β_{2}$ while is depending on the temporal coherence. This agrees fairly well with our expectation as the fact that second-order dispersion coefficient plays a more vital role when the pulse spreading is relatively small. After propagating some distance, the pulse broadening is so pronounced that the coherence length gradually dominates and the second-order dispersion becomes less prominent. However, it is not the case for temporal coherence length, rather, the better the temporal coherence is, the higher the degree of polarization. This is well understood since the degree of polarization is defined as the energy in the polarized part over the total beam energy for a fixed point at the same time, while the coherence length represents the strength of the correlations of the field.

Fig. 1 Changes in the temporal degree of polarization on-axis as a function of z for the choice of (a) second-order dispersion coefficient $β_{2}$ while $T_{c x} = 0.4 ps$ , $T_{c y} = 0.1 ps$ and (b) temporal coherence length $T_{c x}$ with $β_{2} = 20 {ps}^{2} / km$ .

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The extent of variations of the temporal degree of polarization on-axis versus time coordinate t is calculated for a fixed propagation distance $z = 20 m$ , the results are exhibited in Fig. 2. It can be found from Fig. 2(a) that the degree of polarization of pulses with higher topological charge changes more gently than the lower topological charge across a pulse. This may due to the fact that the pulsed beams with higher topological charge expand more quickly, hence it’s easier to become a nearly coherent pulsed beam during propagation. And it is well known that the degree of polarization of a coherent electromagnetic beam remains invariant on propagation in free space [20]. Therefore, the light tends to become completely polarized across the pulse for a larger topological charge. We now proceed with the analysis concerning the temporal coherence length. As shown in Fig. 2(b), the temporal degree of polarization changes with the evolution of time except the particular case of $T_{c x} = T_{c y}$ , this is consistent with the indication of Eqs. (11) and (12). Since in this circumstance, $α_{t x} = α_{t y}$ , so that $Λ_{x} = Λ_{y}$ , and the temporal degree of polarization reduced to the form of $P_{t} (ρ, t, z) = | \frac{R_{t x x} (ρ, z) - R_{t y y} (ρ, z)}{R_{t x x} (ρ, z) + R_{t y y} (ρ, z)} |$ , which is independent of $T_{c i}$ or $t$ . In addition, pulses with lower coherence need to propagate longer to make the degree of polarization stable.

Fig. 2 Behaviors in the temporal degree of polarization on-axis at the propagation distance $z = 20 m$ versus t when $β_{2} = 20 {ps}^{2} / km$ for different parameters of (a) topological charge m while $T_{c x} = 0.4 ps$ , $T_{c y} = 0.1 ps$ and (b) temporal coherence length $T_{c x}$ with $m = 1$ .

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At last the distribution of the temporal degree of cross-polarization for two contrary position vectors is researched. Similar with the method we proposed in a previous work for measuring the topological charge of stationary electromagnetic vortex beams through the distribution of the degree of cross-polarization [21], we will examine whether it is applicable for cases of nonstationary vortex pulses by this way. Our results are presented in Fig. 3. Interestingly, there are also some bright ring dislocations appearing in the distribution of the temporal degree of cross-polarization. Besides, the number of the bright ring dislocations is just equal to the topological charge of a given EM pulsed vortex beams. Therefore, on that basis it can be concluded that measuring the azimuthal index $m$ according to the distribution of the degree of cross-polarization in the space-time domain is feasible

Fig. 3 The temporal degree of cross-polarization of spatially and temporally partially coherent EM vortex pulses for different topological charges (a) $m = 1$ , (b) $m = 2$ , (c) $m = 3$ at $z = 20 m$ with $t_{1} = 0.1 ps$ , $t_{2} = 0.2 ps$ , and $T_{c x} = 0.4 ps$ , $T_{c y} = 0.1 ps$ .

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3. Propagation of cross-spectral density function of EM pulsed vortex beams in dispersive media in the space-frequency domain

The discussion above has been concerned with the pulsed vortex beams in space-time domain. On the other hand, since the refractive index of such a media intimately depends on the frequency, so the correlation properties are most naturally examined in the frequency domain. To achieve a comparative and systematic treatment for vortex pulses in such a media, we continue the question of propagation of such a beam in the space-frequency domain. The temporal and spectral coherence matrices are connected via the integral transforms [17]

\begin{array}{l} W_{i j}^{(0)} (x_{10}, y_{10}, ω_{1}; x_{20}, y_{20}, ω_{2}) = \frac{1}{{(2 π)}^{2}} \\ \times \iint Γ_{i j}^{(0)} (x_{10}, y_{10}, t_{10}; x_{20}, y_{20}, t_{20}) \exp [- i (ω_{1} t_{10} - ω_{2} t_{20})] d t_{10} d t_{20}, \end{array}

the double integrals with respect to time show the difference with the stationary case, where the corresponding relations between the cross-spectral density matrix and the temporal coherence matrix are formulated using a single integration with respect to time delay or frequency.

Inserting Eqs. (1), (2) and (4) into Eq. (14) and performing the integrals, the elements of cross-spectral density of the spatially and spectrally partially coherent EM vortex pulses at the initial plane yield

\begin{array}{l} W_{i j}^{(0)} (x_{10}, y_{10}, ω_{1}; x_{20}, y_{20}, ω_{2}) = \frac{A_{i} A_{j} B_{i j} T_{0}}{2 π Ω_{0 i j} σ^{2 m}} {(x_{10} - i y_{10})}^{m} {(x_{20} + i y_{20})}^{m} \\ \times \exp (- \frac{x_{10}^{2} + y_{10}^{2} + x_{20}^{2} + y_{20}^{2}}{σ^{2}}) exp [- \frac{{(x_{10} - x_{20})}^{2} + {(y_{10} - y_{20})}^{2}}{δ_{i j}^{2}}] \\ \times \exp [- \frac{{(ω_{1} - ω_{0})}^{2} + {(ω_{2} - ω_{0})}^{2}}{2 Ω_{0 i j}^{2}} - \frac{{(ω_{1} - ω_{2})}^{2}}{2 Ω_{c i j}^{2}}] . \end{array}

The parameters

Ω_{0 i j}

and

Ω_{c i j}

characterize the spectral width and the degree of spectral coherence between the i and j components of the electric vector, respectively, depending on the corresponding time domain parameters by equations

Ω_{0 i j} = \sqrt{\frac{1}{T_{0}^{2}} + \frac{2}{T_{c i j}^{2}}}

, and

Ω_{c i j} = \frac{T_{c i j}}{T_{0}} Ω_{0 i j}

.

The propagation of the cross-spectral density function for spatially and spectrally partially coherent EM vortex pulses at the plane $z > 0$ in the dispersive media can be described by the formula [1]

\begin{array}{l} W_{i j} (ρ_{1}, ρ_{2}, ω_{1}, ω_{2}, z) = \frac{k_{1} (ω_{1}) k_{2} (ω_{2})}{4 π^{2} z^{2}} \exp {i [k_{2} (ω_{2}) - k_{1} (ω_{1})] z} \\ \times \iint \iint W_{i j}^{(0)} (r_{1}, r_{2}, ω_{1}, ω_{2}) \exp {- \frac{i}{2 z} [k_{1} (ω_{1}) {(ρ_{1} - r_{1})}^{2} - k_{2} (ω_{2}) {(ρ_{2} - r_{2})}^{2}]} d^{2} r_{1} d^{2} r_{2} . \end{array}

Here

k_{i} (ω_{i})

is the wave number related to the frequency by

k_{i} (ω_{i}) = n (ω_{i}) ω_{i} / c

, and

n (ω_{i})

measures the frequency-dependent refractive index, with c being the velocity of light in vacuum.

The propagated cross-spectral density function of EM spatially and spectrally partially coherent vortex pulses is obtained, according to Eqs. (15) and (16)

W_{i j} (ρ_{1}, ρ_{2}, ω_{1}, ω_{2}, z) = R_{w i j} (ρ_{1}, ρ_{2}, z) Y_{w i j} (ω_{1}, ω_{2}, z) .

where

\begin{array}{l} R_{w i j} (ρ_{1}, ρ_{2}, z) = \frac{A_{i} A_{j} B_{i j} z^{2 m}}{Δ_{w i j}^{m + 1} k_{1}^{m} (ω_{1}) k_{2}^{m} (ω_{2}) σ^{2 m}} \sum_{n = 0}^{m} {(C_{m}^{n})}^{2} \frac{n! 4^{n}}{δ_{i j}^{2 n}} \\ \times \exp [- \frac{β_{w i j} ρ_{1}^{2} + α_{w i j} ρ_{2}^{2} - 2 (x_{1} x_{2} + y_{1} y_{2}) / δ_{i j}^{2}}{Δ_{w i j}} - \frac{i}{2 z} [k_{1} (ω_{1}) ρ_{1}^{2} - k_{2} (ω_{2}) ρ_{2}^{2}]] \\ \times {[\frac{- 4 (β_{w i j} ρ_{1}^{2} + α_{w i j} ρ_{2}^{2})}{Δ_{w i j} δ_{i j}^{2}} + i \frac{k_{1} (ω_{1}) k_{2} (ω_{2})}{z^{2}} (x_{1} y_{2} - x_{2} y_{1}) + [\frac{k_{1} (ω_{1}) k_{2} (ω_{2})}{z^{2}} + \frac{8}{Δ_{w i j} δ_{i j}^{4}}] (x_{1} x_{2} + y_{1} y_{2})]}^{m - n} . \end{array}

In Eq. (18),

α_{w i j} = \frac{1}{σ^{2}} + \frac{1}{δ_{i j}^{2}} + \frac{i k_{1} (ω_{1})}{2 z}

,

β_{w i j} = \frac{1}{σ^{2}} + \frac{1}{δ_{i j}^{2}} - \frac{i k_{2} (ω_{2})}{2 z}

,

Δ_{w i j} = 1 + \frac{4 z^{2}}{σ^{2} k_{1} (ω_{1}) k_{2} (ω_{2})} (\frac{1}{σ^{2}} + \frac{2}{δ_{i j}^{2}})

.

The spectral part is

Y_{w i j} (ω_{1}, ω_{2}, z) = \frac{T_{0}}{2 π Ω_{0 i j}} \exp {i [k_{2} (ω_{2}) - k_{1} (ω_{1})] z} \exp [- \frac{{(ω_{1} - ω_{0})}^{2} + {(ω_{2} - ω_{0})}^{2}}{2 Ω_{0 i j}^{2}} - \frac{{(ω_{1} - ω_{2})}^{2}}{2 Ω_{c i j}^{2}}] .

If setting

ρ_{1} = ρ_{2} = ρ

and

ω_{1} = ω_{2} = ω

, the expression

\begin{array}{l} W_{i j} (ρ, ρ, ω, z) = \frac{A_{i} A_{j} B_{i j} z^{2 m} T_{0}}{2 π Ω_{0 i j} Δ_{w i j}^{m + 1} k {(ω)}^{2 m} σ^{2 m}} \exp [- \frac{β_{w i j} + α_{w i j} - 2 / δ_{i j}^{2}}{Δ_{w i j}} ρ^{2}] \\ \times \sum_{n = 0}^{m} {(C_{m}^{n})}^{2} \frac{n! 4^{n}}{δ_{i j}^{2 n}} {[\frac{- 4 (β_{w i j} + α_{w i j})}{Δ_{w i j} δ_{i j}^{2}} + \frac{k {(ω)}^{2}}{z^{2}} + \frac{8}{Δ_{w i j} δ_{i j}^{4}}] ρ^{2}}^{m - n} \exp [- \frac{{(ω - ω_{0})}^{2}}{Ω_{0 i j}^{2}}] \end{array}

is obtained for the elements of cross-spectral density matrix at points with the same position and frequency. In view of Eq. (20), it can be seen that there only exists the spatial expansion during the pulses passing though the dispersive media.

And the expressions for the degree of polarization and the degree of cross-polarization in space-frequency domain are related to the cross-spectral density matrix [22,23]

P_{ω} (ρ, ω, z) = | \frac{W_{x x} (ρ, ρ, ω, z) - W_{y y} (ρ, ρ, ω, z)}{W_{x x} (ρ, ρ, ω, z) + W_{y y} (ρ, ρ, ω, z)} |,

P_{ω} (ρ, - ρ, ω_{1}, ω_{2}, z) = | \frac{W_{x x} (ρ, - ρ, ω_{1}, ω_{2}, z) - W_{y y} (ρ, - ρ, ω_{1}, ω_{2}, z)}{W_{x x} (ρ, - ρ, ω_{1}, ω_{2}, z) + W_{y y} (ρ, - ρ, ω_{1}, ω_{2}, z)} | .

Based on the results obtained above, the changes related to the degree of polarization and the degree of cross-polarization of spatially and spectrally partially coherent EM vortex pulses can be investigated.

Figure 4 shows the changes of the spectral degree of polarization on propagation. Due to the temporal coherence effects on the polarization properties in the time domain and the frequency domain are, in general, different. It can reasonably be expected that the change of the spectral degree of polarization influenced by temporal coherence length distinguishes from that in space-time domain. This is indeed the case, as seen in Fig. 4 and, moreover, holds true for the situation of second-order dispersion coefficient. In space-frequency domain, the large $β_{2}$ slows down the process of changes in the degree of polarization on-axis. This can be interpreted by using Eqs. (20) and (21), and letting $P (0, ω, z) = 0$ , from which the following equation

\frac{A_{x}^{2}}{Ω_{0 x x} Δ_{w x x}^{m + 1} δ_{x x}^{2 m}} \exp [- \frac{{(ω - ω_{0})}^{2}}{Ω_{0 x x}^{2}}] = \frac{A_{y}^{2}}{Ω_{0 y y} Δ_{w y y}^{m + 1} δ_{y y}^{2 m}} \exp [- \frac{{(ω - ω_{0})}^{2}}{Ω_{0 y y}^{2}}]

need to be satisfied. To illustrate the dependence of the spectral degree of polarization on the

β_{2}

for simplicity, we consider the general case of

m = 0

, therefore, Eq. (23) reduces to

\frac{Δ_{w x x}}{Δ_{w y y}} = {(\frac{A_{x}}{A_{y}})}^{2} \frac{Ω_{0 y}}{Ω_{0 x}} \exp [\frac{{(ω - ω_{0})}^{2}}{Ω_{0 y}^{2}} - \frac{{(ω - ω_{0})}^{2}}{Ω_{0 x}^{2}}] .

Inserting the corresponding expression we immediately obtain

z_{P = 0} = \sqrt{\frac{M - 1}{G_{x x} - M G_{y y}}}

, where

M = \frac{A_{x}^{2}}{A_{y}^{2}} \frac{Ω_{0 y}}{Ω_{0 y}}

,

G_{i i} = \frac{4 c^{2}}{σ^{2} ω^{2} n {(ω)}^{2}} (\frac{1}{σ^{2}} + \frac{2}{δ_{i i}^{2}})

. This indicates that

z_{P = 0}

is a nonmonotonic function of

n (ω)

(or

β_{2}

) and

T_{c x}

. In the physical view, the spectral widths of x and y components are the same when

T_{c x} = T_{c y}

. Otherwise, it is inversely proportional to the coherence length.

Fig. 4 Variations in the spectral degree of polarization on-axis as a function of z, the other parameters are the same with Fig. 1.

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As the spectral degree of polarization varying with the frequency and the spectral degree of cross-polarization have a qualitatively similar behavior as the temporal ones, here we don’t discuss it in detail to avoid the tediousness.

4. Conclusion

This work concentrates on the dynamics related to the degree of polarization and the degree of cross-polarization of nonstationary EM pulsed vortex beams in dispersive media in both space-time and space-frequency domains. Results show that the characteristic parameters of the media and the source, including the second-order dispersion coefficient, temporal coherence length and topological charge, have fundamentally different effects on the evolution of the temporal degree of polarization and spectral degree of polarization. For instance, the larger value of the second-order dispersion coefficient leads to the more rapid changes in space-time domain, in contrast with that in the frequency domain, and a similar situation happens to the coherence length. Also, the vortex pulses with higher topological charge are more prone to being fully polarized across a pulse. At last, we also find the mapping relation between the bright ring dislocations of the degree of cross-polarization and the number of topological charge for a given EM pulsed vortex beam. On the basis of the correlativity, it may enable us to access the orbital angular momentum of such beams. As the degree of cross-polarization of a beam can be determined from the knowledge of the elements of the cross-spectral density matrix are measurable [1]. Our measurement method should also prove practically applicable in related fields of research and application, such as quantum information processing by ultra-broadband optical vortex pulses.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) (11274273 and 11474253).

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Characterizing the polarization and cross-polarization of electromagnetic vortex pulses in the space-time and space-frequency domain

Abstract

1. Introduction

2. Propagation formula of mutual coherence function of EM pulsed vortex beams in the space-time domain

3. Propagation of cross-spectral density function of EM pulsed vortex beams in dispersive media in the space-frequency domain

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (24)

Optics Express