M2 factor of a vector Schell-model beam

Milo W. Hyde; Mark F. Spencer

doi:10.1117/1.OE.58.7.074101

10 July 2019 M² factor of a vector Schell-model beam

Milo W. Hyde, Mark F. Spencer

Author Affiliations +

Optical Engineering, Vol. 58, Issue 7, 074101 (July 2019). https://doi.org/10.1117/1.OE.58.7.074101

Abstract

Extending existing scalar Schell-model source work, we derive the M² factor for a general electromagnetic or vector Schell-model source to assess beam quality. In particular, we compute the M² factors for two vector Schell-model sources found in the literature. We then describe how to synthesize vector Schell-model beams in terms of specified, desired M² and present Monte Carlo simulation results to validate our analysis.

In the early 1990s, Siegman proposed the $M^{2}$ factor as a metric to assess laser beam quality.¹ Siegman defined the $M^{2}$ factor as the ratio of a test beam’s space-beamwidth product to that of an ideal Gaussian ${TEM}_{00}$ beam. In so doing, he showed that the test beam’s spot size $W_{x} (z)$ (in the $x$ direction as a function of range $z$ ) obeyed the quadratic propagation formula:

Eq. (1)

W_{x}^{2} (z) = W_{x}^{2} (z_{0}) + M_{x}^{4} \frac{λ^{2}}{π^{2} W_{x}^{2} (z_{0})} {(z - z_{0})}^{2},

where

λ

is the wavelength,

z_{0}

is the

z

location of the waist plane, and

W_{x} (z_{0})

is the test beam’s spot size in the waist plane, i.e.,

W_{x} (z_{0})

is the test beam’s waist. Expressed in this form, the physical interpretation of

M^{2}

becomes clear:

M^{2} \geq 1

is the test beam’s far-zone (FZ) beam spread relative to an ideal Gaussian beam with a waist equal to that of the test beam.

Siegman’s $M^{2}$ quickly gained acceptance and became the laser industry’s standard for assessing beam quality. It was soon extended to include hard-aperture,²^,³ vortex,⁴ and stochastic, or partially coherent beams.⁵^–⁸ The latter, which are germane to this work, focus exclusively on scalar partially coherent beams. There have been several papers that derived the $M^{2}$ factors for specific electromagnetic (EM) or vector partially coherent beams, e.g., EM Gaussian Schell-model (EGSM) beams.⁹^–¹² However, none to date have derived an expression for the $M^{2}$ factor of a general EM Schell-model beam.

This analysis is useful considering the wide-spread use of $M^{2}$ to assess beam quality and a large number of vector Schell-model sources that have been developed for applications, such as free-space/underwater optical communications, directed energy, optical trapping/tweezers, etc.¹³^–²⁶ For directed energy, in particular, the use of $M^{2}$ to describe beam quality is pervasive, and there has been recent work in modeling dynamic, stochastic, noncommon-path phase errors in high-energy-laser systems in terms of $M^{2}$ .²⁷ Most importantly, having a relation for the $M^{2}$ factor of a general EM Schell-model source allows one to consider beam quality in the design of new vector Schell-model beams.

For the reasons stated above, here, we extend the scalar partially coherent source $M^{2}$ analysis presented in Refs. 5 and 6 to vector Schell-model beams. Starting with Siegman’s $M^{2}$ definition, we derive an expression for the $M^{2}$ factor of a vector partially coherent source in terms of the vector component beam quality factors. We then apply the results in Refs. 5 and 6 to derive a simple and physical relation for the $M^{2}$ factor of a general vector Schell-model beam. We present examples, where we calculate $M^{2}$ for two EM Schell-model sources using our $M^{2}$ relation and describe how to synthesize vector Schell-model beams in terms of specified, desired $M^{2}$ .

In Sec. 2, we present Monte Carlo simulations, where we generate two examples of EM Schell-model sources in terms of $M^{2}$ to validate our analysis. We then study the convergence of the stochastic vector field realizations to the specified, desired $M^{2}$ . Last, we conclude this paper with a summary of the work presented herein.

1. Theory

In this section, we first review the scalar $M^{2}$ theory found in Refs. 5 and 6. Next, we calculate the $M^{2}$ factors for two example vector Schell-model sources. We then describe how to generate vector Schell-model beams in terms of specified, desired $M^{2}$ .

1.1.

Siegman’s $M^{2}$

As defined by Siegman,¹ the $M^{2}$ factor in the $x$ direction $M_{x}^{2}$ is as follows:

Eq. (2)

M_{x}^{2} = 4 π σ_{x} σ_{f_{x}},

with a similar definition for

M_{y}^{2}

. The normalized beam widths

σ_{x}

and

σ_{f_{x}}

are as follows:

Eq. (3)

σ_{x}^{2} = \frac{\iint_{- \infty}^{\infty} {(x - ⟨ x ⟩)}^{2} Tr [\underline{W} (ρ, ρ)] d^{2} ρ}{\iint_{- \infty}^{\infty} Tr [\underline{W} (ρ, ρ)] d^{2} ρ} = \frac{\iint_{- \infty}^{\infty} {(x - ⟨ x ⟩)}^{2} S_{x} (ρ) d^{2} ρ}{\iint_{- \infty}^{\infty} S_{x} (ρ) d^{2} ρ + \iint_{- \infty}^{\infty} S_{y} (ρ) d^{2} ρ} + \frac{\iint_{- \infty}^{\infty} {(x - ⟨ x ⟩)}^{2} S_{y} (ρ) d^{2} ρ}{\iint_{- \infty}^{\infty} S_{x} (ρ) d^{2} ρ + \iint_{- \infty}^{\infty} S_{y} (ρ) d^{2} ρ} σ_{f_{x}}^{2} = \frac{\iint_{- \infty}^{\infty} {(f_{x} - ⟨ f_{x} ⟩)}^{2} Tr [\underline{\tilde{W}} (f, f)] d^{2} f}{\iint_{- \infty}^{\infty} Tr [\underline{\tilde{W}} (f, f)] d^{2} f} = \frac{\iint_{- \infty}^{\infty} {(f_{x} - ⟨ f_{x} ⟩)}^{2} {\tilde{S}}_{x} (f) d^{2} f}{\iint_{- \infty}^{\infty} {\tilde{S}}_{x} (f) d^{2} f + \iint_{- \infty}^{\infty} {\tilde{S}}_{y} (f) d^{2} f} + \frac{\iint_{- \infty}^{\infty} {(f_{x} - ⟨ f_{x} ⟩)}^{2} {\tilde{S}}_{y} (f) d^{2} f}{\iint_{- \infty}^{\infty} {\tilde{S}}_{x} (f) d^{2} f + \iint_{- \infty}^{\infty} {\tilde{S}}_{y} (f) d^{2} f},

where

\underline{W}

is the cross-spectral density (CSD) matrix¹⁷^,²⁸^,²⁹ of the beam at its waist location,

Tr

is the trace, and

S_{α} (ρ) = W_{α α} (ρ, ρ)

are the spectral densities (SDs)¹⁷^,²⁸^,²⁹ of the beam’s

α = x, y

polarization components.

Continuing to define the symbols in Eq. (3), $\underline{\tilde{W}}$ is the Fourier transform of $\underline{W}$ , i.e.:

Eq. (4)

\underline{\tilde{W}} (f_{1}, f_{2}) = ⨌_{- \infty}^{\infty} \underline{W} (ρ_{1}, ρ_{2}) \exp [- j 2 π (ρ_{1} \cdot f_{1} - ρ_{2} \cdot f_{2})] d^{2} ρ_{1} d^{2} ρ_{2},

where

{\tilde{S}}_{α}

is as follows:

Eq. (5)

{\tilde{S}}_{α} (f) = ⨌_{- \infty}^{\infty} W_{α α} (ρ_{1}, ρ_{2}) \exp [- j 2 π (ρ_{1} - ρ_{2}) \cdot f] d^{2} ρ_{1} d^{2} ρ_{2},

where

ρ = \hat{x} x + \hat{y} y

and

f = \hat{x} f_{x} + \hat{y} f_{y}

. Note that the denominators of

σ_{x}^{2}

and

σ_{f_{x}}^{2}

are equal due to Parseval’s theorem. As such, we define and use the symbol

P = P_{x} + P_{y}

hereafter to represent this normalization factor.

Last, $⟨ x ⟩$ and $⟨ f_{x} ⟩$ are the normalized beam “first moments” at the beam’s waist location and in the spatial frequency domain, respectively, such that

Eq. (6)

⟨ x ⟩ = \frac{\iint_{- \infty}^{\infty} x S_{x} (ρ) d^{2} ρ}{P_{x} + P_{y}} + \frac{\iint_{- \infty}^{\infty} x S_{y} (ρ) d^{2} ρ}{P_{x} + P_{y}} ⟨ f_{x} ⟩ = \frac{\iint_{- \infty}^{\infty} f_{x} {\tilde{S}}_{x} (f) d^{2} f}{P_{x} + P_{y}} + \frac{\iint_{- \infty}^{\infty} f_{x} {\tilde{S}}_{y} (f) d^{2} f}{P_{x} + P_{y}} .

Following the scalar partially coherent source analysis presented in Refs. 5 and 6, we choose the $z$ axis—the mean propagation direction of the random beam—such that $⟨ x ⟩ = ⟨ f_{x} ⟩ = 0$ . This permits us to write Eq. (3) as follows:

Eq. (7)

σ_{x}^{2} = \frac{P_{x}}{P_{x} + P_{y}} σ_{x, x}^{2} + \frac{P_{y}}{P_{x} + P_{y}} σ_{x, y}^{2} σ_{f_{x}}^{2} = \frac{P_{x}}{P_{x} + P_{y}} σ_{f_{x}, x}^{2} + \frac{P_{y}}{P_{x} + P_{y}} σ_{f_{x}, y}^{2},

where

σ_{x, α}^{2}

and

σ_{f_{x}, α}^{2}

are as follows:

Eq. (8)

σ_{x, α}^{2} = \frac{1}{P_{α}} \iint_{- \infty}^{\infty} x^{2} S_{α} (ρ) d^{2} ρ σ_{f_{x}, α}^{2} = \frac{1}{P_{α}} \iint_{- \infty}^{\infty} f_{x}^{2} {\tilde{S}}_{α} (f) d^{2} f .

Substituting Eq. (7) into Eq. (2) and simplifying produce as follows:

Eq. (9)

M_{x}^{4} = \frac{P_{x}}{{(P_{x} + P_{y})}^{2}} (P_{x} + \frac{σ_{x, y}^{2}}{σ_{x, x}^{2}} P_{y}) {[4 π σ_{x, x} σ_{f_{x}, x}]}^{2} + \frac{P_{y}}{{(P_{x} + P_{y})}^{2}} (P_{y} + \frac{σ_{x, x}^{2}}{σ_{x, y}^{2}} P_{x}) {[4 π σ_{x, y} σ_{f_{x}, y}]}^{2} .

The bracketed quantities in Eq. (9) are the vector component beam quality factors, and therefore,

Eq. (10)

M_{x}^{4} = \frac{1}{P_{x} + P_{y}} (\frac{P_{x}}{P_{x} + P_{y}} σ_{x, x}^{2} + \frac{P_{y}}{P_{x} + P_{y}} σ_{x, y}^{2}) (\frac{P_{x}}{σ_{x, x}^{2}} M_{x, x}^{4} + \frac{P_{y}}{σ_{x, y}^{2}} M_{x, y}^{4}),

where

M_{x, α}^{2} = 4 π σ_{x, α} σ_{f_{x}, α}

are the beam quality factors for the

α = x, y

vector components.

1.2.

Prior $M_{x}^{2}$ Scalar Analysis

Through extensive analysis and clever mathematics, Refs. 5 and 6 showed that $M_{x}^{2}$ for a scalar beam of any state of coherence is as follows:

Eq. (11)

M_{x}^{4} = 16 π^{2} σ_{x}^{2} σ_{f_{x}}^{2} - J^{2},

where

σ_{x}^{2}

is given in Eq. (8) but computed in the source plane,

σ_{f_{x}}^{2}

is as follows:

Eq. (12)

σ_{f_{x}}^{2} = {\frac{1}{4 π^{2} P} \iint_{- \infty}^{\infty} \frac{\partial^{2} W (ρ_{1}, ρ_{2})}{\partial x_{1} \partial x_{2}} |}_{ρ, ρ} d^{2} ρ

and

J

is

Eq. (13)

J = \frac{1}{P} Im [\iint_{- \infty}^{\infty} {f_{x} \frac{\partial \tilde{W} (f_{1}, f_{2})}{\partial f_{1 x}} |}_{f, f} d^{2} f] .

The

|_{ρ, ρ}

and

|_{f, f}

denote that

ρ_{1} = ρ_{2} = ρ

and

f_{1} = f_{2} = f

after computing the partial derivatives, respectively.

For Schell-model sources,¹⁷^,²⁸ Ref. 6 showed that Eq. (11) further reduces to

Eq. (14)

M_{x}^{4} = {(M_{x}^{c})}^{4} - {4 σ_{x}^{2} \frac{\partial^{2} a (ρ_{d})}{\partial x_{d}^{2}} |}_{ρ_{d} = 0},

where

{(M_{x}^{c})}^{2}

is the beam quality factor of the corresponding coherent source,

ρ_{d} = \hat{x} x_{d} + \hat{y} y_{d} = \hat{x} (x_{1} - x_{2}) + \hat{y} (y_{1} - y_{2})

, and

a (ρ_{d})

is related to the source’s spectral autocorrelation function by

Eq. (15)

μ (ρ_{d}) = a (ρ_{d}) \exp [j ψ (ρ_{d})] .

Both

a

and

ψ

are real functions. Since

μ (0) = 1

and

μ (- ρ_{d}) = μ^{*} (ρ_{d})

,

a (0) = 1

,

ψ (0) = 0

,

a (- ρ_{d}) = a (ρ_{d})

, and

ψ (- ρ_{d}) = - ψ (ρ_{d})

. If

μ

is real, then

a = μ

, and

μ

can be substituted directly into Eq. (14).

1.3.

New $M_{x}^{2}$ Vector Analysis

Substituting in Eq. (14) for the vector component beam quality factors in Eq. (10) and simplifying produce as follows:

Eq. (16)

M_{x}^{4} = \frac{1}{P_{x} + P_{y}} [\frac{P_{x}}{P_{x} + P_{y}} σ_{x, x}^{2} + \frac{P_{y}}{P_{x} + P_{y}} σ_{x, y}^{2}] \times {\frac{P_{x}}{σ_{x, x}^{2}} [{(M_{x, x}^{c})}^{4} - {4 σ_{x, x}^{2} \frac{\partial^{2} a_{x x} (ρ_{d})}{\partial x_{d}^{2}} |}_{ρ_{d} = 0}] + \frac{P_{x}}{σ_{x, y}^{2}} [{(M_{x, y}^{c})}^{4} - {4 σ_{x, y}^{2} \frac{\partial^{2} a_{y y} (ρ_{d})}{\partial x_{d}^{2}} |}_{ρ_{d} = 0}]} .

Because

M_{x}^{2}

in Eq. (16) is expressed in terms of vector component quantities (i.e.,

M_{x, α}^{c}

,

σ_{x, α}^{2}

,

P_{α}

, and

a_{α α}

), it is easy to use, in practice, to calculate the beam quality factor. It is not, however, the most simplified or physical form. Noting that the bracketed quantity on the first line of Eq. (16) is equal to

σ_{x}^{2}

via Eq. (7), expanding, and then simplifying Eq. (16) yields as follows:

Eq. (17)

M_{x}^{4} = σ_{x}^{2} [\frac{P_{x}}{P_{x} + P_{y}} \frac{{(M_{x, x}^{c})}^{4}}{σ_{x, x}^{2}} + \frac{P_{y}}{P_{x} + P_{y}} \frac{{(M_{x, y}^{c})}^{4}}{σ_{x, y}^{2}}] - 4 σ_{x}^{2} ({\frac{P_{x}}{P_{x} + P_{y}} \frac{\partial^{2} a_{x x} (ρ_{d})}{\partial x_{d}^{2}} |}_{ρ_{d} = 0} + {\frac{P_{y}}{P_{x} + P_{y}} \frac{\partial^{2} a_{y y} (ρ_{d})}{\partial x_{d}^{2}} |}_{ρ_{d} = 0}) .

Last, substituting in

{(M_{x, α}^{c})}^{2} = 4 π σ_{x, α} σ_{f_{x}, α}^{c}

, recalling Eq. (7), and simplifying, the desired result is obtained as follows:

Eq. (18)

M_{x}^{4} = {(4 π σ_{x})}^{2} [\frac{P_{x}}{P_{x} + P_{y}} {(σ_{f_{x}, x}^{c})}^{2} + \frac{P_{y}}{P_{x} + P_{y}} {(σ_{f_{x}, y}^{c})}^{2}] - 4 σ_{x}^{2} [{\frac{P_{x}}{P_{x} + P_{y}} \frac{\partial^{2} a_{x x} (ρ_{d})}{\partial x_{d}^{2}} |}_{ρ_{d} = 0} + {\frac{P_{y}}{P_{x} + P_{y}} \frac{\partial^{2} a_{y y} (ρ_{d})}{\partial x_{d}^{2}} |}_{ρ_{d} = 0}] = {(M_{x}^{c})}^{4} - 4 σ_{x}^{2} [{\frac{P_{x}}{P_{x} + P_{y}} \frac{\partial^{2} a_{x x} (ρ_{d})}{\partial x_{d}^{2}} |}_{ρ_{d} = 0} + {\frac{P_{y}}{P_{x} + P_{y}} \frac{\partial^{2} a_{y y} (ρ_{d})}{\partial x_{d}^{2}} |}_{ρ_{d} = 0}] .

Equation (18)—in particular, the last line of Eq. (18)—is the main analytical result of this paper and generalizes the scalar result presented in Ref. 6. Its form is very similar to the scalar result in that $M_{x}^{2}$ depends on the coherence of the source only through the second derivatives of the vector correlation functions evaluated at $ρ_{d} = 0$ . Rather intuitively, it differs in the fact that the “coherence contributions,” $σ_{x}$ and $σ_{f_{x}}^{c}$ , are weighted by the fraction of power in the associated vector component. This being the case, one is tempted to try to manipulate Eq. (18) into the sum of weighted, component, beam quality factors $M_{x, α}^{2}$ . Unfortunately, this simple, physical form for $M_{x}^{2}$ is spoiled by cross terms, which are evident in Eq. (16).

1.4.

Examples

Here, we calculate the $M_{x}^{2}$ for two example, vector Schell-model sources. We then simulate them in terms of $M_{x}^{2}$ in Sec. 2.

1.4.1.

EM Gaussian Schell-model source

The elements of the CSD matrix for an EGSM source are as follows:

Eq. (19)

W_{α β} (ρ_{1}, ρ_{2}) = A_{α} \exp (- \frac{ρ_{1}^{2}}{4 σ_{α}^{2}}) A_{β} \exp (- \frac{ρ_{2}^{2}}{4 σ_{β}^{2}}) B_{α β} \exp (- \frac{{| ρ_{1} - ρ_{2} |}^{2}}{2 δ_{α β}^{2}}),

where

α, β = x, y

,

A_{α}

is the amplitude of the

α

field component, and

σ_{α}

is the RMS width of the

α

component SD. Also in Eq. (19),

B_{α β}

is the complex correlation coefficient between the

α

and

β

field components and

δ_{α β}

is the RMS width of the cross-correlation function

μ_{α β}

, i.e., the cross-correlation function of the

α

and

β

field components.¹⁷^,²⁹ The EGSM source parameters—

A_{α}

,

σ_{α}

,

δ_{α β}

, and

B_{α β}

—must satisfy the realizability conditions derived in Refs. 17 and 30. Since only the diagonal elements of the CSD matrix are required to compute

M_{x}^{2}

, the

B_{α α} = 1

criterion is the most relevant here.

To compute $M_{x}^{2}$ , we use Eq. (16). The normalized beam widths of the $x$ and $y$ components of the EGSM source, computed using Eq. (8), are $σ_{x, α} = σ_{α}$ . The powers in the $x$ and $y$ components of the source are computed using the denominators in Eq. (3) and are $P_{α} = 2 π σ_{α}^{2} A_{α}^{2}$ . The corresponding coherent source to the EGSM source defined in Eq. (19) is as follows:

Eq. (20)

W_{α β}^{c} (ρ_{1}, ρ_{2}) = A_{α} A_{β} B_{α β} \exp (- \frac{ρ_{1}^{2}}{4 σ_{α}^{2}}) \exp (- \frac{ρ_{2}^{2}}{4 σ_{β}^{2}}) .

This CSD function describes a vector source composed of horizontally and vertically polarized, spatially coherent, correlated, Gaussian beams. Since the

x

and

y

vector components of the source are Gaussian beams, the component beam quality factors are

{(M_{x, α}^{c})}^{2} = 1

. Last, the component spectral correlation functions

μ_{α α}

are real; therefore,

a_{α α} = μ_{α α}

and the second derivatives in Eq. (16) evaluate to

Eq. (21)

{\frac{\partial^{2} a_{α α} (ρ_{d})}{\partial x_{d}^{2}} |}_{ρ_{d} = 0} = {\frac{\partial^{2}}{\partial x_{d}^{2}} \exp (- \frac{ρ_{d}^{2}}{2 δ_{α α}^{2}}) |}_{ρ_{d} = 0} = - \frac{1}{δ_{α α}^{2}} .

Substituting the quantities discussed in the previous paragraph into Eq. (16) and simplifying produce as follows:

Eq. (22)

M_{x}^{2} = \sqrt{\frac{A_{x}^{2} σ_{x}^{4} + A_{y}^{2} σ_{y}^{4}}{{(A_{x}^{2} σ_{x}^{2} + A_{y}^{2} σ_{y}^{2})}^{2}}} {[A_{x}^{2} (1 + 4 \frac{σ_{x}^{2}}{δ_{x x}^{2}}) + A_{y}^{2} (1 + 4 \frac{σ_{y}^{2}}{δ_{y y}^{2}})]}^{1 / 2} .

This expression is equal to the

M_{x}^{2}

for an EGSM source derived in Refs. 9 10.11.–12 using different methods. Note that letting either

A_{x} = 0

or

A_{y} = 0

yields the

M_{x}^{2}

for a scalar GSM source first derived in Ref. 6. We generate an EGSM source with a specified

M_{x}^{2}

in Sec. 2.

1.4.2.

Vector optical coherence lattices

Another popular and relatively new example of a vector partially coherent source is the so-called vector optical coherence lattice (VOCL).¹⁴ The CSD matrix elements for a VOCL take the form:

Eq. (23)

W_{α β} (ρ_{1}, ρ_{2}) = A_{α} \exp (- \frac{ρ_{1}^{2}}{4 σ_{α}^{2}}) A_{β} \exp (- \frac{ρ_{2}^{2}}{4 σ_{β}^{2}}) \times B_{α β} jinc (\frac{| ρ_{1} - ρ_{2} |}{\sqrt{2} δ_{α β}}) \frac{1}{N} \sum_{n = 1}^{N} \exp [- j v_{n} \cdot (ρ_{1} - ρ_{2})],

where

jinc (x) = 2 J_{1} (x) / x

,

J_{1}

is a first-order Bessel function of the first kind, and

v_{n} = \hat{x} v_{n x} + \hat{y} v_{n y}

is a vector that points from the origin to the

n

’th node of the “coherence lattice.” The other symbols have been defined previously and generally have the same physical interpretation as the EGSM source. One should immediately recognize that the spectral cross-correlation function in Eq. (23) is equivalent to the FZ pattern (spatial Fourier transform) of an array of circular transmitters.

To find the beam quality factor, we again use Eq. (16). The $σ_{x, α} = σ_{α}$ , $P_{α} = 2 π σ_{α}^{2} A_{α}^{2}$ , and ${(M_{x, α}^{c})}^{2} = 1$ are the same as in the EGSM source example. The second derivative of $a_{α α}$ is much more difficult. Here, for analytical convenience, we assume that the lattice nodes are symmetric about the origin. This assumption means that $μ_{α α}$ is real, and thus, $a_{α α} = μ_{α α}$ . After much calculus and algebra (details in Sec. 4 Appendix A), the second derivatives in Eq. (16) evaluate to

Eq. (24)

{\frac{\partial^{2} a_{α α} (ρ_{d})}{\partial x_{d}^{2}} |}_{ρ_{d} = 0} = \frac{\partial^{2}}{\partial x_{d}^{2}} {[jinc (\frac{ρ_{d}}{\sqrt{2} δ_{α α}}) \frac{1}{N} \sum_{n = 1}^{N} \exp (- j v_{n} \cdot ρ_{d})] |}_{ρ_{d} = 0} = - \frac{1}{8 δ_{α α}^{2}} - \frac{1}{N} \sum_{n = 1}^{N} v_{n x}^{2} .

Substituting $σ_{x, α} = σ_{α}$ , $P_{α} = 2 π σ_{α}^{2} A_{α}^{2}$ , ${(M_{x, α}^{c})}^{2} = 1$ ; Eq. (24) into Eq. (16); and simplifying produce as follows:

Eq. (25)

M_{x}^{2} = \sqrt{\frac{A_{x}^{2} σ_{x}^{4} + A_{y}^{2} σ_{y}^{4}}{{(A_{x}^{2} σ_{x}^{2} + A_{y}^{2} σ_{y}^{2})}^{2}}} \times {A_{x}^{2} [1 + \frac{1}{2} \frac{σ_{x}^{2}}{δ_{x x}^{2}} (1 + \frac{8}{N} \sum_{n = 1}^{N} δ_{x x}^{2} v_{n x}^{2})] + {A_{y}^{2} [1 + \frac{1}{2} \frac{σ_{y}^{2}}{δ_{y y}^{2}} (1 + \frac{8}{N} \sum_{n = 1}^{N} δ_{y y}^{2} v_{n x}^{2})]}}^{1 / 2} .

We generate and analyze a VOCL, in terms of

M_{x}^{2}

, in Sec. 2.

1.5.

Generating Stochastic Vector Fields in Terms of $M_{x}^{2}$

An instance of a Schell-model source can be generated by filtering a two-dimensional (2-D) array of circular complex Gaussian random numbers. For computational efficiency, it is best to perform the filtering in the spectral domain using the convolution theorem. This process has been described in the literature many times.³¹^–³⁷ Here, we present the necessary equations to implement the technique.

Any type of vector Schell-model source can be formed by the following optical field realization:³¹^,³³

Eq. (26)

E_{α} (ρ) = C_{α} \exp (- \frac{ρ^{2}}{4 σ_{α}^{2}}) T_{α} (ρ),

where

C_{α}

is the complex amplitude and

T_{α}

is the stochastic complex transmittance screen for the

α = x, y

component of the field, respectively.

T_{α}

is formed from correlated, circular complex Gaussian random numbers. Here, we have assumed a Gaussian shape for the source, considering that Sec. 1.4 examples also have Gaussian shapes. In general, the source can take any shape.

Hereafter, we specialize the mathematics to generate the sources discussed in Sec. 1.4. The procedure for synthesizing any other type of vector Schell-model source is the same as presented here, only the mathematical details change.

Taking the cross-correlation of Eq. (26) with $E_{β}$ and comparing the resulting expression to Eqs. (19) and (23) yields the following equalities: $| C_{α} | = A_{α}$ , $\arg (C_{α}) - \arg (C_{β}) = \arg (B_{α β})$ , and

Eq. (27)

⟨ T_{α} (ρ_{1}) T_{β}^{*} (ρ_{2}) ⟩ = {\begin{cases} | B_{α β} | \exp (- \frac{{| ρ_{1} - ρ_{2} |}^{2}}{2 δ_{α β}^{2}}) & EGSM source \\ | B_{α β} | jinc (\frac{| ρ_{1} - ρ_{2} |}{\sqrt{2} δ_{α β}}) \frac{1}{N} \sum_{n = 1}^{N} \exp [- j v_{n} \cdot (ρ_{1} - ρ_{2})] & VOCL \end{cases} .

Because potentially $| B_{x y} | > 0$ , $T_{x}$ and $T_{y}$ , in general, must be generated from correlated Gaussian random numbers. To see this and reveal the conditions on the values of $| B_{x y} |$ and $δ_{x y}$ , we note that an instance of $T_{α}$ is generated by³¹^,³²

Eq. (28)

T_{α} [i, j] = \sum_{m} \sum_{n} r_{α} [m, n] \sqrt{\frac{Φ_{T_{α}} [m, n]}{2 L_{x} L_{y}}} \exp (j \frac{2 π}{N_{x}} m i) \exp (j \frac{2 π}{N_{y}} n j),

where

N_{x}

,

N_{y}

are the numbers of grid points in the

x

,

y

directions,

L_{x} = N_{x} Δ

,

L_{y} = N_{y} Δ

are the lengths of the grid in the

x

,

y

directions in meters, and

Δ

is the grid spacing. In Eq. (28),

r_{α}

is an

N_{y} \times N_{x}

grid of zero-mean, unit-variance circular complex Gaussian random numbers and

Φ_{T_{α}}

is the spatial power spectrum of

T_{α}

, i.e., the Fourier transform of the autocorrelation of

T_{α}

. For the examples discussed in Sec. 1.4:

Eq. (29)

Φ_{T_{α}} (f) = {\begin{cases} 2 π δ_{α α}^{2} \exp (- 2 π^{2} δ_{α α}^{2} f^{2}) & EGSM source \\ 8 π δ_{α α}^{2} \frac{1}{N} \sum_{n = 1}^{N} circ (\sqrt{8} π δ_{α α} | f + \frac{v_{n}}{2 π} |) & VOCL \end{cases},

where

circ (x)

is the circle function defined in Ref. 38. We note that Eq. (28) is in the form of a discrete, inverse Fourier transform; thus,

T_{α}

can be generated quickly and efficiently using the fast Fourier transform algorithm.

Using Eqs. (28) and (29), a $T_{α}$ with a specified $δ_{α α}$ is produced. Physically, this leads to an EGSM source or VOCL with the correct, diagonal, CSD matrix elements. Recall that $M_{x}^{2}$ depends only on these elements, and therefore, one could ignore any correlation between the $x$ and $y$ vector components (i.e., produce instances of $T_{x}$ and $T_{y}$ from independent Gaussian random numbers) and still produce a beam with the desired $M_{x}^{2}$ . Doing this step results in a beam that is randomly, partially, or fully linearly polarized.

To generate a vector Schell-model source with a general polarization state, one must control the off-diagonal elements of the CSD matrix as well. This requires examination of the cross-correlation of $T_{x}$ with $T_{y}$ , namely,

Eq. (30)

⟨ T_{x} [i_{1}, j_{1}] T_{y}^{*} [i_{2}, j_{2}] ⟩ = \sum_{m_{1}} \sum_{n_{1}} \sum_{m_{2}} \sum_{n_{2}} \frac{⟨ r_{x} [m_{1}, n_{1}] r_{y}^{*} [m_{2}, n_{2}] ⟩}{2 L_{x} L_{y}} \sqrt{Φ_{T_{x}} [m_{1}, n_{1}] Φ_{T_{y}} [m_{2}, n_{2}]} \times \exp (j \frac{2 π}{N_{x}} m_{1} i_{1}) \exp (j \frac{2 π}{N_{y}} n_{1} j_{1}) \exp (- j \frac{2 π}{N_{x}} m_{2} i_{2}) \exp (- j \frac{2 π}{N_{y}} n_{2} j_{2}) .

The moment

⟨ r_{x} [m_{1}, n_{1}] r_{y}^{*} [m_{2}, n_{2}] ⟩ = 2 Γ δ [m_{1} - m_{2}] δ [n_{1} - n_{2}]

, where

Γ

is the correlation coefficient between the

r_{x}

and

r_{y}

random numbers and

δ [n]

is the discrete Dirac delta function. Substituting this into Eq. (30) and simplifying produce as follows:

Eq. (31)

⟨ T_{x} [i_{1}, j_{1}] T_{y}^{*} [i_{2}, j_{2}] ⟩ = \sum_{m} \sum_{n} Γ \sqrt{Φ_{T_{x}} [m, n] Φ_{T_{y}} [m, n]} \times \frac{1}{L_{x} L_{y}} \exp [j \frac{2 π}{N_{x}} m (i_{1} - i_{2})] \exp [j \frac{2 π}{N_{y}} n (j_{1} - j_{2})] .

The

Γ \sqrt{Φ_{T_{x}} Φ_{T_{y}}}

must equal the cross-power spectra, or the Fourier transforms of the expressions in Eq. (27), viz.,

Eq. (32)

Φ_{T_{x} T_{y}} (f) = {\begin{cases} | B_{x y} | 2 π δ_{x y}^{2} \exp (- 2 π^{2} δ_{x y}^{2} f^{2}) & EGSM source \\ | B_{x y} | 8 π δ_{x y}^{2} \frac{1}{N} \sum_{n = 1}^{N} circ (\sqrt{8} π δ_{x y} | f + \frac{v_{n}}{2 π} |) & VOCL \end{cases} .

This step produces the following relation for EGSM sources:

Eq. (33)

Γ δ_{x x} δ_{y y} \exp [- 2 π^{2} (\frac{δ_{x x}^{2} + δ_{y y}^{2}}{2}) f^{2}] = | B_{x y} | δ_{x y}^{2} \exp (- 2 π^{2} δ_{x y}^{2} f^{2}) .

From here, it is quite clear that

Eq. (34)

δ_{x y} = \sqrt{\frac{δ_{x x}^{2} + δ_{y y}^{2}}{2}} | B_{x y} | = Γ \frac{2 δ_{x x} δ_{y y}}{δ_{x x}^{2} + δ_{y y}^{2}},

where

0 \leq Γ \leq 1

. These conditions were first derived in Ref. 31.

The VOCL conditions on $| B_{x y} |$ and $δ_{x y}$ are derived from

Eq. (35)

Γ \prod_{α = x, y} {[δ_{α α}^{2} \sum_{n = 1}^{N} circ (\sqrt{8} π δ_{α α} | f - \frac{v_{n}}{2 π} |)]}^{1 / 2} = | B_{x y} | δ_{x y}^{2} \sum_{n = 1}^{N} circ (\sqrt{8} π δ_{x y} | f - \frac{v_{n}}{2 π} |) .

Because of this expression, it is not likely that conditions similar to Eq. (34) can be derived for a VOCL. Two rather trivial, but physically important conditions can be derived by letting

Γ = | B_{x y} |

:

1. If $B_{x y} = 0$ , $δ_{x y}$ is physically meaningless and its value is irrelevant, $T_{x}$ and $T_{y}$ are statistically independent, and $δ_{α α}$ can be chosen freely. This produces a VOCL with a diagonal CSD matrix and the beam is randomly, partially, or fully linearly polarized.
2. If $| B_{x y} | \neq 0$ , then $δ_{x x} = δ_{y y} = δ_{x y}$ , and $T_{x}$ and $T_{y}$ are correlated to some degree. This produces a VOCL with a full CSD matrix and the beam is, most generally, elliptically partially polarized.

These same conditions apply to many other vector Schell-model sources, e.g., EM multi-Gaussian Schell-model¹⁷^,²¹ and EM Bessel-Gaussian Schell-model sources.²²^,³²

The last step is to express the EGSM source and VOCL conditions derived above in terms of $M_{x}^{2}$ . The EGSM source and VOCL component beam quality factors are as follows:

Eq. (36)

M_{x, α}^{4} = {\begin{cases} 1 + 4 \frac{σ_{α}^{2}}{δ_{α α}^{2}} & EGSM source \\ 1 + \frac{1}{2} \frac{σ_{α}^{2}}{δ_{α α}^{2}} (1 + \frac{8}{N} \sum_{n = 1}^{N} δ_{α α}^{2} v_{n x}^{2}) & VOCL \end{cases} .

Referring back to Eqs. (22) and (25), substituting Eq. (36) into those expressions and inverting produces as follows:

Eq. (37)

M_{x}^{4} \frac{{(A_{x}^{2} σ_{x}^{2} + A_{y}^{2} σ_{y}^{2})}^{2}}{A_{x}^{2} σ_{x}^{4} + A_{y}^{2} σ_{y}^{4}} = A_{x}^{2} M_{x, x}^{4} + A_{y}^{2} M_{x, y}^{4} .

We assume that $A_{α}$ and $σ_{α}$ are given. This makes sense as we would expect the on-axis intensity $A_{α}^{2}$ and size of the source to be known. In addition, since generally $M_{x, x}^{2} \neq M_{x, y}^{2}$ (or equivalently, $δ_{x x} \neq δ_{y y}$ ), one of those must be given. Without loss of generality, we assume $M_{x, y}^{2}$ is known. Simplifying Eq. (37) further produces as follows:

Eq. (38)

M_{x, x}^{4} = \frac{1}{A_{x}^{2}} [M_{x}^{4} \frac{{(A_{x}^{2} σ_{x}^{2} + A_{y}^{2} σ_{y}^{2})}^{2}}{A_{x}^{2} σ_{x}^{4} + A_{y}^{2} σ_{y}^{4}} - A_{y}^{2} M_{x, y}^{4}] .

Last, substituting Eq. (36) into Eq. (38) and simplifying yields

Eq. (39)

δ_{x x}^{2} = 4 A_{x}^{2} σ_{x}^{2} (A_{x}^{2} σ_{x}^{4} + A_{y}^{2} σ_{y}^{4}) {{(A_{x}^{2} σ_{x}^{2} + A_{y}^{2} σ_{y}^{2})}^{2} M_{x}^{4} {- (A_{x}^{2} σ_{x}^{4} + A_{y}^{2} σ_{y}^{4}) [A_{x}^{2} + A_{y}^{2} (1 + 4 \frac{σ_{y}^{2}}{δ_{y y}^{2}})]}}^{- 1}

for an EGSM source, and

Eq. (40)

δ_{x x}^{2} = \frac{1}{2} A_{x}^{2} σ_{x}^{2} (A_{x}^{2} σ_{x}^{4} + A_{y}^{2} σ_{y}^{4}) {{(A_{x}^{2} σ_{x}^{2} + A_{y}^{2} σ_{y}^{2})}^{2} M_{x}^{4} - (A_{x}^{2} σ_{x}^{4} + A_{y}^{2} σ_{y}^{4}) [A_{x}^{2} (1 + 4 \frac{σ_{x}^{2}}{N} \sum_{n = 1}^{N} v_{n x}^{2}) {+ A_{y}^{2} (1 + 4 \frac{σ_{y}^{2}}{N} \sum_{n = 1}^{N} v_{n x}^{2} + \frac{1}{2} \frac{σ_{y}^{2}}{δ_{y y}^{2}})]}}^{- 1}

for a VOCL. We note that the locations of the coherence lattice nodes

v_{n}

must be known to find

δ_{x x}

.

In summary, to produce an EGSM source or VOCL with a desired $M_{x}^{2}$ :

1. Specify $A_{x}$ , $σ_{x}$ , $A_{y}$ , $σ_{y}$ , $M_{x}^{2}$ , $M_{x, y}^{2}$ and for a VOCL, $v_{n}$ . Recall that $| C_{α} | = A_{α}$ . The values of $M_{x, y}^{2}$ , $σ_{y}$ , and $v_{n}$ (for a VOCL) determine $δ_{y y}$ [see Eq. (36)].
2. Use Eq. (39) for an EGSM source or Eq. (40) for a VOCL to find $δ_{x x}$ . For both sources, this effectively sets $δ_{x y}$ [see Eq. (34) and VOCL conditions immediately following Eq. (35)].
3. Specify $\arg (C_{x})$ , $\arg (C_{y})$ , and $| B_{x y} |$ . Recall that $\arg (C_{x}) - \arg (C_{y}) = \arg (B_{x y})$ . For both sources, the value of $| B_{x y} |$ determines $Γ$ [see Eq. (34) and for a VOCL, $Γ = | B_{x y} |$ ].
4. Use a multivariate Gaussian random number generator to produce correlated $r_{x}$ and $r_{y}$ . The means and covariance matrix are as follows:
Eq. (41)
$⟨ r_{x}^{r} ⟩ = ⟨ r_{x}^{i} ⟩ = ⟨ r_{y}^{r} ⟩ = ⟨ r_{y}^{i} ⟩ = 0 Σ = [\begin{matrix} ⟨ {(r_{x}^{r})}^{2} ⟩ & ⟨ r_{x}^{r} r_{x}^{i} ⟩ & ⟨ r_{x}^{r} r_{y}^{r} ⟩ & ⟨ r_{x}^{r} r_{y}^{i} ⟩ \\ ⟨ r_{x}^{i} r_{x}^{r} ⟩ & ⟨ {(r_{x}^{i})}^{2} ⟩ & ⟨ r_{x}^{i} r_{y}^{r} ⟩ & ⟨ r_{x}^{i} r_{y}^{i} ⟩ \\ ⟨ r_{y}^{r} r_{x}^{r} ⟩ & ⟨ r_{y}^{r} r_{x}^{i} ⟩ & ⟨ {(r_{y}^{r})}^{2} ⟩ & ⟨ r_{y}^{r} r_{y}^{i} ⟩ \\ ⟨ r_{y}^{i} r_{x}^{r} ⟩ & ⟨ r_{y}^{i} r_{x}^{i} ⟩ & ⟨ r_{y}^{i} r_{y}^{r} ⟩ & ⟨ {(r_{y}^{i})}^{2} ⟩ \end{matrix}] = [\begin{matrix} 1 & 0 & Γ & 0 \\ 0 & 1 & 0 & Γ \\ Γ & 0 & 1 & 0 \\ 0 & Γ & 0 & 1 \end{matrix}], r_{x} = r_{x}^{r} + j r_{x}^{i} r_{y} = r_{y}^{r} + j r_{y}^{i}$
where the superscripts “r” and “ $i$ ” stand for real and imaginary parts, respectively.
5. Use Eqs. (28) and (29), the values of $δ_{x x}$ , $δ_{y y}$ , and $v_{n}$ (for a VOCL), and the $r_{x}$ and $r_{y}$ from step 4 to generate instances of $T_{x}$ and $T_{y}$ .
6. Use Eq. (26), the values of $C_{x}$ , $C_{y}$ , $σ_{x}$ , $σ_{y}$ , and the $T_{x}$ and $T_{y}$ generated in step 5 to create an EGSM or VOCL field instance.

Since the generated fields are stochastic, this procedure will produce an EGSM source or VOCL with a desired, “on-average” $M_{x}^{2}$ .

2. Simulation

Here, we present simulation results to validate the analysis in the previous section. We also examine the convergence of the stochastic vector fields to the desired $M_{x}^{2}$ . Before presenting the results, we discuss the details of the setup.

2.1.

Setup

For these Monte Carlo simulations, we used computational grids with $N_{y} = N_{x} = 512$ points per side. The grid spacings were chosen such that $Δ = \min {σ_{x}, σ_{y}} / 10$ resulting in 3 and 0.97 mm for the EGSM source and VOCL simulations, respectively. These spacings easily satisfied the Nyquist sampling criterion derived for Gaussian signals in Ref. 39. The EGSM source and VOCL parameters are given in Table 1. For the VOCL, the coherence lattice was rectangular with 5 rows and 4 columns of nodes spaced $500 m^{- 1}$ apart.

Table 1

EGSM source and VOCL parameters.

	EGSM	VOCL
$A_{x}$	1.3	1
$A_{y}$	1	1.7
$σ_{x}$	5 cm	1 cm
$σ_{y}$	3 cm	0.97 cm
$B_{x y}$	$0.15 \exp (- j π / 6)$	0
$M_{x}^{2}$	10	11.4
$M_{y}^{2}$	10	14.2028
$M_{x, x}^{2}$	11	12.5
$M_{x, y}^{2}$	5.1446	10.9880
$δ_{x x}$	0.9129 cm	0.1286 cm
$δ_{y y}$	1.1889 cm	0.4707 cm
$δ_{x y}$	1.0599 cm	N/A

We generated 20,000 EGSM source and VOCL field instances. From these, we computed the near-zone (NZ) (i.e., source plane) and FZ Stokes parameters, $M_{x}^{2}$ , and $M_{y}^{2}$ . The Stokes parameters in terms of the CSD matrix elements are as follows:¹⁷^,²⁹

Eq. (42)

S_{0} (ρ) = W_{x x} (ρ, ρ) + W_{y y} (ρ, ρ) S_{1} (ρ) = W_{x x} (ρ, ρ) - W_{y y} (ρ, ρ) S_{2} (ρ) = W_{x y} (ρ, ρ) + W_{y x} (ρ, ρ) S_{3} (ρ) = j [W_{y x} (ρ, ρ) - W_{x y} (ρ, ρ)] .

For

M_{x}^{2}

and

M_{y}^{2}

, the

σ_{x}

,

σ_{f_{x}}

,

σ_{y}

, and

σ_{f_{y}}

[see Eq. (3)] were computed using trapezoidal numerical integration.

To verify that, we produced a source with the correct NZ and FZ polarization properties, we compared the simulated Stokes parameters with the corresponding theoretical quantities. The theoretical expressions for the EGSM source and VOCL NZ Stokes parameters were computed using Eqs. (19) and (23) and the values in Table 1, respectively. We computed the theoretical FZ Stokes parameters using those same equations and the Fourier transform in Eq. (4) resulting in

Eq. (43)

{\tilde{W}}_{α β} (f, f) = {\begin{cases} A_{α} A_{β} B_{α β} \frac{16 π^{2} σ_{α}^{2} σ_{β}^{2} δ_{α β}^{2}}{2 σ_{α}^{2} + 2 σ_{β}^{2} + δ_{α β}^{2}} \exp [- \frac{4 π^{2} δ_{α β}^{2} (σ_{α}^{2} + σ_{β}^{2})}{2 σ_{α}^{2} + 2 σ_{β}^{2} + δ_{α β}^{2}} f^{2}] & EGSM source \\ A_{α} A_{β} B_{α β} \frac{32 π^{2} σ_{α}^{2} σ_{β}^{2} δ_{α β}^{2}}{σ_{α}^{2} + σ_{β}^{2}} \int_{0}^{\infty} \exp [- \frac{δ_{α β}^{2}}{2 (σ_{α}^{2} + σ_{β}^{2})} u^{2}] \\ \times J_{1} (u) \frac{1}{N} \sum_{n = 1}^{N} J_{0} (\sqrt{8} π δ_{α β} | f + \frac{v_{n}}{2 π} | u) d u & VOCL \end{cases} .

The integral in the VOCL

{\tilde{W}}_{α β}

can be evaluated in closed form using Mellin transform techniques.⁴⁰ The resulting expression is an infinite series of hypergeometric functions.⁴¹ Computing the result using this analytical relation is very slow; therefore, we evaluated the integral directly using numerical quadrature.

2.2.

Results

Figures 1 and 2 show the EGSM source and VOCL results, respectively. The figures are organized as follows: the Stokes parameters are displayed in the first four rows— $S_{0}$ , $S_{1}$ , $S_{2}$ , and $S_{3}$ , respectively. The first two columns show the NZ results, with column 1 (the left column) showing the theoretical (Thy) results and column 2 (the right column) showing the simulated (Sim) results. Columns 3 and 4 show the FZ results, with an identical left-right arrangement of Thy (column 3) and Sim (column 4) results. Note that we have added row and column headings to Figs. 1 and 2 to aid the reader. Last, the fifth rows [Figs. 1(q) and 2(q)] show the convergence of the simulated $M_{x}^{2}$ and $M_{y}^{2}$ to the corresponding theoretical values versus Monte Carlo trial number.

Fig. 1

EGSM source results—(a) $S_{0}^{NZ}$ Thy, (b) $S_{0}^{NZ}$ Sim, (c) $S_{0}^{FZ}$ Thy, (d) $S_{0}^{FZ}$ Sim, (e) $S_{1}^{NZ}$ Thy, (f) $S_{1}^{NZ}$ Sim, (g) $S_{1}^{FZ}$ Thy, (h) $S_{1}^{FZ}$ Sim, (i) $S_{2}^{NZ}$ Thy, (j) $S_{2}^{NZ}$ Sim, (k) $S_{2}^{FZ}$ Thy, (l) $S_{2}^{FZ}$ Sim, (m) $S_{3}^{NZ}$ Thy (n) $S_{3}^{NZ}$ Sim, (o) $S_{3}^{FZ}$ Thy, (p) $S_{3}^{FZ}$ Sim, and (q) convergence of simulated $M_{x}^{2}$ and $M_{y}^{2}$ to the corresponding theoretical values versus trial number.

Fig. 2

VOCL results—(a) $S_{0}^{NZ}$ Thy, (b) $S_{0}^{NZ}$ Sim, (c) $S_{0}^{FZ}$ Thy, (d) $S_{0}^{FZ}$ Sim, (e) $S_{1}^{NZ}$ Thy, (f) $S_{1}^{NZ}$ Sim, (g) $S_{1}^{FZ}$ Thy, (h) $S_{1}^{FZ}$ Sim, (i) $S_{2}^{NZ}$ Thy, (j) $S_{2}^{NZ}$ Sim, (k) $S_{2}^{FZ}$ Thy, (l) $S_{2}^{FZ}$ Sim, (m) $S_{3}^{NZ}$ Thy (n) $S_{3}^{NZ}$ Sim, (o) $S_{3}^{FZ}$ Thy, (p) $S_{3}^{FZ}$ Sim, and (q) convergence of simulated $M_{x}^{2}$ and $M_{y}^{2}$ to the corresponding theoretical values versus trial number.

Overall, the agreement between simulation and theory is excellent. We note that the visually conspicuous differences between the $S_{2}$ and $S_{3}$ Thy and Sim VOCL results in Fig. 2 are in fact quantitatively small (see the associated color bars above the subfigures). These discrepancies arise because of our choice of color scale and Thy $S_{2} = S_{3} = 0$ . Running more Monte Carlo trials would reduce these errors; however, $S_{2}$ and $S_{3}$ Sim will never be identically zero for that would require an infinite number of trials.

The simulated $M_{x}^{2}$ and $M_{y}^{2}$ converge to the desired values in approximately 1,000 trials. This finding is consistent with the scalar Schell-model beam results in Ref. 27. Figures 1 and 2 validate the theoretical analysis presented in Sec. 1.

3. Conclusion

Here, we derived the $M^{2}$ factor for a general vector Schell-model beam. Starting with Siegman’s $M^{2}$ definition, we found an expression for the $M^{2}$ factor of a vector partially coherent beam in terms of its vector component beam quality factors. Then, applying the prior scalar analysis, we derived a physical expression for the $M^{2}$ factor of a general vector Schell-model source. As an example, we computed the beam quality factors for two EM Schell-model beams found in the literature using our new $M^{2}$ relation and described how to synthesize vector Schell-model beams in terms of specified, desired $M^{2}$ .

To validate our analysis, we performed Monte Carlo simulations, where we generated two partially coherent sources. The simulated results were found to be in excellent agreement with the corresponding theory, and convergence to the specified, desired $M^{2}$ occurred within approximately 1000 vector field realizations. Although not performed here because of equipment availability, experimental synthesis and subsequent measurement of the $M^{2}$ factor for vector Schell-model sources can be performed using optical setups described in Refs. 15 16.17.–18, 32, and 42 43.–44.

The analysis presented in this paper will be useful in the design of vector Schell-model sources for applications from optical communications and directed energy to atomic optics and optical tweezers.

4. Appendix A: Derivation of Eq. (24)

Starting with

Eq. (44)

{\frac{\partial^{2} a_{α α} (ρ_{d})}{\partial x_{d}^{2}} |}_{ρ_{d} = 0} = {\frac{\partial^{2}}{\partial x_{d}^{2}} [jinc (\frac{ρ_{d}}{\sqrt{2} δ_{α α}}) \frac{1}{N} \sum_{n = 1}^{N} \exp (- j v_{n} \cdot ρ_{d})] |}_{ρ_{d} = 0},

we apply the derivative multiplication rule twice to yield as follows:

Eq. (45)

{\frac{\partial^{2} a_{α α} (ρ_{d})}{\partial x_{d}^{2}} |}_{ρ_{d} = 0} = {\frac{\partial}{\partial x_{d}} [\frac{1}{N} \sum_{n = 1}^{N} \exp (- j v_{n} \cdot ρ_{d})] |}_{ρ_{d} = 0} {\frac{\partial}{\partial x_{d}} [jinc (\frac{ρ_{d}}{\sqrt{2} δ_{α α}})] |}_{ρ_{d} = 0} + {\frac{\partial}{\partial x_{d}} [jinc (\frac{ρ_{d}}{\sqrt{2} δ_{α α}})] |}_{ρ_{d} = 0} {\frac{\partial}{\partial x_{d}} [\frac{1}{N} \sum_{n = 1}^{N} \exp (- j v_{n} \cdot ρ_{d})] |}_{ρ_{d} = 0} + {\frac{1}{N} \sum_{n = 1}^{N} \exp (- j v_{n} \cdot ρ_{d}) |}_{ρ_{d} = 0} {\frac{\partial^{2}}{\partial x_{d}^{2}} [jinc (\frac{ρ_{d}}{\sqrt{2} δ_{α α}})] |}_{ρ_{d} = 0} + {jinc (\frac{ρ_{d}}{\sqrt{2} δ_{α α}}) |}_{ρ_{d} = 0} \frac{\partial^{2}}{\partial x_{d}^{2}} [\frac{1}{N} \sum_{n = 1}^{N} \exp (- j v_{n} \cdot ρ_{d})] |_{ρ_{d} = 0} .

The first derivatives in Eq. (45) are odd functions of

x_{d}

; therefore, evaluating them at

ρ_{d} = 0

is trivially zero. Simplifying Eq. (45) produces as follows:

Eq. (46)

{\frac{\partial^{2} a_{α α} (ρ_{d})}{\partial x_{d}^{2}} |}_{ρ_{d} = 0} = {\frac{\partial^{2}}{\partial x_{d}^{2}} [jinc (\frac{ρ_{d}}{\sqrt{2} δ_{α α}})] |}_{ρ_{d} = 0} {+ \frac{\partial^{2}}{\partial x_{d}^{2}} [\frac{1}{N} \sum_{n = 1}^{N} \exp (- j v_{n} \cdot ρ_{d})] |}_{ρ_{d} = 0} .

We now evaluate the second derivatives in Eq. (46) separately. We begin with the second derivative of the coherence lattice term. Bringing the derivatives inside the summation and evaluating the second derivative of the exponential yields as follows:

Eq. (47)

{\frac{\partial^{2}}{\partial x_{d}^{2}} [\frac{1}{N} \sum_{n = 1}^{N} \exp (- j v_{n} \cdot ρ_{d})] |}_{ρ_{d} = 0} = {\frac{1}{N} \sum_{n = 1}^{N} {(- j v_{n x})}^{2} \exp (- j v_{n} \cdot ρ_{d}) |}_{ρ_{d} = 0} = - \frac{1}{N} \sum_{n = 1}^{N} v_{n x}^{2} .

To evaluate the second derivative of the jinc function and argument, we first let the argument

ρ_{d} / (\sqrt{2} δ_{α α}) = u

. The associated second derivative becomes

Eq. (48)

\frac{\partial^{2}}{\partial x_{d}^{2}} [jinc (\frac{ρ_{d}}{\sqrt{2} δ_{α α}})] = \frac{\partial}{\partial x_{d}} {\frac{\partial u}{\partial x_{d}} \frac{\partial}{\partial u} [\frac{2 J_{1} (u)}{u}]} .

We now use the Bessel function identity:

Eq. (49)

\frac{2 n}{x} J_{n} (x) = J_{n - 1} (x) + J_{n + 1} (x)

to simplify Eq. (48) to

Eq. (50)

\frac{\partial^{2}}{\partial x_{d}^{2}} [jinc (\frac{ρ_{d}}{\sqrt{2} δ_{α α}})] = \frac{\partial}{\partial x_{d}} {\frac{\partial u}{\partial x_{d}} [J_{0^{'}} (u) + J_{2^{'}} (u)]} .

Equation (50) can be further simplified by using the Bessel function identity:

Eq. (51)

2 J_{n}^{'} (x) = J_{n - 1} (x) - J_{n + 1} (x),

yielding

Eq. (52)

\frac{\partial^{2}}{\partial x_{d}^{2}} [jinc (\frac{ρ_{d}}{\sqrt{2} δ_{α α}})] = \frac{\partial}{\partial x_{d}} {\frac{\partial u}{\partial x_{d}} [\frac{1}{2} J_{- 1} (u) - \frac{1}{2} J_{3} (u)]} = - \frac{\partial}{\partial x_{d}} {\frac{\partial u}{\partial x_{d}} [\frac{1}{2} J_{1} (u) + \frac{1}{2} J_{3} (u)]} = - \frac{\partial}{\partial x_{d}} {\frac{\partial u}{\partial x_{d}} \frac{2 J_{2} (u)}{u}} .

In going from line 1 to 2 in Eq. (52), we used the Bessel function identity

J_{- n} (x) = {(- 1)}^{n} J_{n} (x)

, and in going from line 2 to 3, we used Eq. (49).

Continuing, we apply the derivative multiplication and chain rules producing

Eq. (53)

\frac{\partial^{2}}{\partial x_{d}^{2}} [jinc (\frac{ρ_{d}}{\sqrt{2} δ_{α α}})] = - \frac{2 J_{2} (u)}{u} \frac{\partial^{2} u}{\partial x_{d}^{2}} - {(\frac{\partial u}{\partial x_{d}})}^{2} \frac{\partial}{\partial u} [\frac{2 J_{2} (u)}{u}] .

The derivative of

2 J_{2} (u) / u

can be found by applying Eqs. (49) and (51), such that

Eq. (54)

\frac{\partial^{2}}{\partial x_{d}^{2}} [jinc (\frac{ρ_{d}}{\sqrt{2} δ_{α α}})] = - \frac{2 J_{2} (u)}{u} \frac{\partial^{2} u}{\partial x_{d}^{2}} - {(\frac{\partial u}{\partial x_{d}})}^{2} [\frac{1}{4} J_{0} (u) - \frac{1}{4} J_{4} (u)] .

We now evaluate the first and second derivatives of $u$ . The first derivative of $u$ is found by applying the chain rule, namely,

Eq. (55)

\frac{\partial u}{\partial x_{d}} = \frac{1}{\sqrt{2} δ_{α α}} \frac{\partial ρ_{d}}{\partial x_{d}} = \frac{1}{\sqrt{2} δ_{α α}} \frac{\partial}{\partial x_{d}} {(x_{d}^{2} + y_{d}^{2})}^{1 / 2} = \frac{1}{\sqrt{2} δ_{α α}} x_{d} ρ_{d}^{- 1} .

The second derivative of

u

is found by applying the multiplication and chain rules:

Eq. (56)

\frac{\partial^{2} u}{\partial x_{d}^{2}} = \frac{\partial}{\partial x_{d}} (\frac{1}{\sqrt{2} δ_{α α}} x_{d} ρ_{d}^{- 1}) = \frac{1}{\sqrt{2} δ_{α α}} ρ_{d}^{- 1} - \frac{1}{\sqrt{2} δ_{α α}} x_{d}^{2} ρ_{d}^{- 3} .

Substituting Eqs. (55) and (56) into Eq. (54) and simplifying produce as follows:

Eq. (57)

\frac{\partial^{2}}{\partial x_{d}^{2}} [jinc (\frac{ρ_{d}}{\sqrt{2} δ_{α α}})] = \frac{2}{ρ_{d}^{2}} (1 - \frac{x_{d}^{2}}{ρ_{d}^{2}}) J_{2} (\frac{ρ_{d}}{\sqrt{2} δ_{α α}}) - \frac{1}{8 δ_{α α}^{2}} \frac{x_{d}^{2}}{ρ_{d}^{2}} [J_{0} (\frac{ρ_{d}}{\sqrt{2} δ_{α α}}) - J_{4} (\frac{ρ_{d}}{\sqrt{2} δ_{α α}})] .

Last, evaluating Eq. (57) at $ρ_{d} = 0$ yields as follows:

Eq. (58)

\frac{\partial^{2}}{\partial x_{d}^{2}} {[jinc (\frac{ρ_{d}}{\sqrt{2} δ_{α α}})] |}_{ρ_{d} = 0} = - \frac{1}{8 δ_{α α}^{2}} .

We obtain the second line of Eq. (24)—the desired result—by substituting Eqs. (58) and (47) into Eq. (46), i.e.:

Eq. (59)

{\frac{\partial^{2} a_{α α} (ρ_{d})}{\partial x_{d}^{2}} |}_{ρ_{d} = 0} = - \frac{1}{8 δ_{α α}^{2}} - \frac{1}{N} \sum_{n = 1}^{N} v_{n x}^{2} .

Acknowledgments

The views expressed in this paper are those of the authors and do not reflect the official policy or position of the US Air Force, the Department of Defense, or the US government. The authors have no conflicts of interest.

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Biography

Milo W. Hyde IV received his BS degree in computer engineering from Georgia Tech in 2001 and his MS degree and PhD in electrical engineering from AFIT in 2006 and 2010, respectively. Currently, he is an adjunct professor in the Department of Electrical and Computer Engineering at AFIT. He is a senior member of IEEE and SPIE. He is also a member of OSA and DEPS.

Mark F. Spencer is the principal investigator for the Aero Effects and Beam Control Program at the Air Force Research Laboratory, Directed Energy Directorate. He is also an adjunct assistant professor of optical sciences and engineering within the Department of Engineering Physics at the Air Force Institute of Technology. In addition to being a senior member of SPIE, he is a regular member of the Optical Society and the Directed Energy Professional Society.

CC BY: © The Authors. Published by SPIE under a Creative Commons Attribution 4.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

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Milo W. Hyde and Mark F. Spencer "M² factor of a vector Schell-model beam," Optical Engineering 58(7), 074101 (10 July 2019). https://doi.org/10.1117/1.OE.58.7.074101

Received: 25 April 2019; Accepted: 7 June 2019; Published: 10 July 2019

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KEYWORDS

Monte Carlo methods

Optical engineering

Fourier transforms

Stochastic processes

Bessel functions

Coherence (optics)

Directed energy weapons

Eq. (1)

1.

Theory

1.1.

Siegman’s M2

Eq. (2)

Eq. (3)

Eq. (4)

Eq. (5)

Eq. (6)

Eq. (7)

Eq. (8)

Eq. (9)

Eq. (10)

1.2.

Prior Mx2 Scalar Analysis

Eq. (11)

Eq. (12)

Eq. (13)

Eq. (14)

Eq. (15)

1.3.

New Mx2 Vector Analysis

Eq. (16)

Eq. (17)

Eq. (18)

1.4.

Examples

1.4.1.

EM Gaussian Schell-model source

Eq. (19)

Eq. (20)

Eq. (21)

Eq. (22)

1.4.2.

Vector optical coherence lattices

Eq. (23)

Eq. (24)

Eq. (25)

1.5.

Generating Stochastic Vector Fields in Terms of Mx2

Eq. (26)

Eq. (27)

Eq. (28)

Eq. (29)

Eq. (30)

Eq. (31)

Eq. (32)

Eq. (33)

Eq. (34)

Eq. (35)

Eq. (36)

Eq. (37)

Eq. (38)

Eq. (39)

Eq. (40)

Eq. (41)

2.

Simulation

2.1.

Setup

Table 1

Eq. (42)

Eq. (43)

2.2.

Results

Fig. 1

Fig. 2

3.

Conclusion

4.

Appendix A: Derivation of Eq. (24)

Eq. (44)

Eq. (45)

Eq. (46)

Eq. (47)

Eq. (48)

Eq. (49)

Eq. (50)

Eq. (51)

Siegman’s $M^{2}$

Prior $M_{x}^{2}$ Scalar Analysis

New $M_{x}^{2}$ Vector Analysis

Generating Stochastic Vector Fields in Terms of $M_{x}^{2}$