Partially coherent sources with circular coherence: comment

Milo W. Hyde; Santasri R. Bose-Pillai

doi:10.1364/OL.42.003084

In Ref. [1], the authors present a new class of non-uniformly correlated (NUC) sources that are perfectly coherent for any two points lying on a circle centered on the source plane origin. The cross-spectral density function $W$ for such sources is

W (ρ_{1}, ρ_{2}) = τ (ρ_{1}) τ^{*} (ρ_{2}) g (\frac{ρ_{1}^{2} - ρ_{2}^{2}}{δ_{u}^{2}}),

where

ρ = | ρ | = | \hat{x} x + \hat{y} y |

,

τ

is a complex function,

g

is any function having a non-negative Fourier transform, and

δ_{u}

determines the spatial coherence properties of the source. The authors then describe an experimental setup for generating this class of sources, which uses the Van Cittert–Zernike theorem.

In this comment, we present a complementary way to derive and analyze the partially coherent sources discussed in Ref. [1]. From this analysis, we propose a new way to physically realize these sources using a laser, spatial light modulator (SLM), and deformable mirror (DM).

Let an instance of a random scalar optical field be

U (ρ) = τ (ρ) \exp [j f (ρ) \cdot S (ρ)],

where

S (ρ)

is a random vector phase function, and

f (ρ) = \hat{x} f_{x} (ρ) + \hat{y} f_{y} (ρ)

is a real vector function [2]. Equation (2) is initially presented in a very general form. In the analysis to follow, we specialize it to produce fields with

W

given in Eq. (1).

Taking the autocorrelation of $U$ and comparing it to Eq. (1) reveals the following equality:

g (f_{1} - f_{2}) = ⟨ \exp [j (f_{1} \cdot S_{1} - f_{2} \cdot S_{2})] ⟩,

where

f_{1} = f (ρ_{1})

and likewise for

f_{2}

,

S_{1}

, and

S_{2}

. The moment in Eq. (3) is the joint characteristic function of

S

. Since

g

is a function of

f_{1} - f_{2}

, Eq. (3) reduces to

g (f_{1} - f_{2}) = \iint_{- \infty}^{\infty} \exp [j (f_{1} - f_{2}) \cdot S] P (S_{x}, S_{y}) d S_{x} d S_{y},

where

S = \hat{x} S_{x} + \hat{y} S_{y}

is now a random vector, and

P

is the joint probability density function (PDF) of

S_{x}

and

S_{y}

.

Using the same example as in Ref. [1], let $τ = \exp (- a ρ^{2})$ , $g = \sin (x) / x$ , and $f = \hat{x} ρ^{2}$ . The field instance that produces this source is $U (ρ) = \exp (- a ρ^{2}) \exp (j S ρ^{2}),$ where $S$ is uniformly distributed on $[- 1 / δ_{u}^{2}, 1 / δ_{u}^{2}]$ . The PDF of $S$ is determined from the near Fourier transform relationship between $g$ and $P$ given in Eq. (4).

Equation (2) and the above example provide insight into how to physically realize these sources. For clarity, Eq. (2) is rewritten, taking into account the analysis presented in the preceding paragraphs:

U (ρ) = τ (ρ) \exp [j S h (ρ)] .

In this form,

h

is physically an optical aberration, and

S

is the aberration’s random weight. In the above example,

h = ρ^{2}

is obviously defocus; however, nothing prevents one from choosing astigmatism, coma, or any other real-valued

h

.

Field instances given by Eq. (5) can easily be synthesized using a SLM and DM [2]—the SLM applies the transmittance function $τ$ , while the DM applies the aberration $h$ and its random weight $S$ . The use of a SLM for beam shaping and a DM for phase control assumes that speed is paramount. If speed is not critical in the application, then only the SLM is required.

In conclusion, we presented an alternative way to analyze the NUC sources introduced in Ref. [1]. Also, we discussed another approach for synthesizing these sources, using a laser, SLM, and DM.

REFERENCES

1. M. Santarsiero, R. Martínez-Herrero, D. Maluenda, J. C. G. de Sande, G. Piquero, and F. Gori, Opt. Lett. 42, 1512 (2017). [CrossRef]

2. M. W. Hyde IV, S. Bose-Pillai, X. Xiao, and D. G. Voelz, J. Opt. 19, 025601 (2017). [CrossRef]

Partially coherent sources with circular coherence: comment

Abstract

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Optics Letters