## Interference of a pair of symmetric partially coherent beams

Optics Express, Vol. 18, Issue 5, pp. 4816-4828 (2010)

http://dx.doi.org/10.1364/OE.18.004816

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### Abstract

We study theoretically and experimentally the interference of light produced by a pair of mutually correlated Schell-model sources. The spatial distributions of the fields produced by the two sources are inverted with respect to each other through their common center in the source plane. When the beams are in phase, a bright spot appears in the center of the spatial distribution of the beam intensity. When the beams have a phase shift *ϕ* = *π*, a dark spot appears in the center of the spatial distribution of the beam intensity. Experimental results that illustrate these results are included. Both bright and dark spots diverge more slowly with the increasing distance from the sources than the beam itself.

© 2010 Optical Society of America

## 1. Introduction

1. Zu-Han Gu, E. R. Mendez, M. Ciftan, T. A. Leskova, and A. A. Maradudin, “Interference of a pair of symmetric Collett-Wolf beams,” Opt. Lett. **30**, 1605–1607 (2005). [CrossRef] [PubMed]

*iϕ*). Second we assume a more general expression for the cross-spectral density in the source plane of each of the two sources producing the interfering beams than was used in [1

1. Zu-Han Gu, E. R. Mendez, M. Ciftan, T. A. Leskova, and A. A. Maradudin, “Interference of a pair of symmetric Collett-Wolf beams,” Opt. Lett. **30**, 1605–1607 (2005). [CrossRef] [PubMed]

## 2. The Radiated Intensity Distribution

*x*

_{3}= 0) at frequency ω by

*U*(

**x**

_{∥}, 0∣

*ω*), where

**x**

_{∥}= (

*x*

_{1},

*x*

_{2}, 0) is an arbitrary point in this plane. The cross-spectral density of this field in the source plane is then defined by [6]

*U*(

**x**

_{∥},0∣

*ω*)}. Since everything in this paper occurs at frequency

*ω*, we omit it in writing the cross-spectral density. The superscript (0) in Eq. (1

1. Zu-Han Gu, E. R. Mendez, M. Ciftan, T. A. Leskova, and A. A. Maradudin, “Interference of a pair of symmetric Collett-Wolf beams,” Opt. Lett. **30**, 1605–1607 (2005). [CrossRef] [PubMed]

*x*

_{3}= 0.

*W*

^{(0)}(

**x**

_{∥}, 0∣

**x**′

_{∥}, 0) that we consider in this paper has the Schell-model form [5]

*S*

^{(0)}(

**x**

_{∥}) = ⟨∣

*U*(

**x**

_{∥}, 0∣

*ω*)∣

^{2}⟩ is the spectral density (intensity) of the light at a typical point in the source plane, and

*g*

^{(0)}(

**x**

_{∥}-

**x**′

_{∥}) is the spectral degree of coherence of the source in the source plane. It is Hermitian,

*x*

_{3}direction can be expressed in terms of the field in the source plane as [7

7. J. J. Foley and M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. **26**, 297–300 (1978). [CrossRef]

*ϕ*is a phase independent of the coordinates. The spatial distributions of the fields are assumed to be symmetric with respect to inversion in the

*x*

_{1}

*x*

_{2}plane,

*x*

_{3}, they also obey the same conditions at

*x*

_{3}= 0, i.e. in the source plane.

*x*

_{3}from the source plane is given by

*U*

_{1}(

**x**′

_{∥}, 0 ∣

*ω*)

*U*

^{*}

_{2}(

**x**″

_{∥}, 0 ∣

*ω*)⟩ and ⟨

*U*

^{*}

_{1}(

**x**″

_{∥}, 0 ∣

*ω*)

*U*

_{2}(

**x**″

_{∥}, 0 ∣

*ω*)⟩ entering Eq. (9) can be written in the forms

*W*

^{(0)}(

**x**′

_{∥}, 0 ∣

**x**″

_{∥}, 0) given by Eq. (2) and obtain

**x**′

_{∥}= ±

**x**″∥ +

**u**

_{∥}followed by the change ±

**x**″∥ to

**x**″∥ transform Eq. (13)into

## 3. A Gaussian Spectral Density in the Source Plane

**x**″

_{∥}in Eq. (14) with the spectral density given by Eq. (15) we obtain

*x*

_{3}≫ 2(

*ω*/

*c*)

*σ*

^{2}

_{s}. In this far-field regime, and if we also assume that the width of the spectral coherence function is much smaller than that of the source, Eqs. (16) take the forms

*I*

_{+}(

**x**

_{∥},

*x*

_{3}∣

*ω*)⟩, depends on the spectral degree of coherence through what is essentially a Fourier transform operation.

*I*

_{-}(

**x**

_{∥},

*x*

_{3}∣

*ω*)⟩ is independent of the form of the spectral degree of coherence, and has a Gaussian shape ⟨

*I*

_{-}(

**x**

_{∥},

*x*

_{3}∣

*ω*)⟩ ~ exp[−(

*x*

_{∥}/

*x*

_{3})

^{2}/Δ

^{2}

_{i}], where Δ

_{i}is the angular divergence and is given by

*ϕ*between the initial beam and its inverted version and can be changed from 2 (

*ϕ*= 0) to 0 (

*ϕ*=

*π*) as the interference regime changes from constructive to destructive interference.

*g*

^{(0)}(

**u**

_{∥}) = 1, the initial beam and its inverted version interfere over the entire beam area, so that the total intensity of the beam is

*g*

^{(0)}(

**u**

_{∥}) is essentially a delta function and far field conditions occur very rapidly. As a result, the interference feature appears on a uniform background,

## 4. Examples

*g*

^{(0)}(

*u*

_{∥}). To make comparisons among the results meaningful, we will normalize each expression for

*g*

^{(0)}(

*u*_{∥}) in such a way that as

*u*

_{∥}→ 0 it has the form

*σ*,

_{s}*σ*, and the wavelength

_{g}*λ*are

*σ*= 2.432 mm,

_{s}*σ*= 179.6

_{g}*μ*m, and

*λ*= 632.8 nm. They correspond to the values characterizing the Collett-Wolf source studied experimentally in Ref. [1

**30**, 1605–1607 (2005). [CrossRef] [PubMed]

### 4.1. Gaussian spectral degree of coherence: *g*^{(0)} (*u*_{∥}) = exp(-*u*^{2}_{∥}/2*σ*^{2}_{g}).

*g*

^{(0)}(

*u*

_{∥}) we use Eqs. (11) and (16) to evaluate ⟨

*I*

_{±}(

**x**

_{∥},

*x*

_{3}∣

*ω*)⟩ analytically, with the result

*I*(

**x**

_{∥},

*x*

_{3}) = ⟨

*I*(

**x**

_{∥},

*x*

_{3}∣

*ω*)⟩/⟨

*I*(0,

*x*

^{3}∣

*ω*)) of the beams that are inverted with respect to their common center at two values of the distance from the source planes

*x*

^{3}= 1m (Figs. 1(a) and (b)), and

*x*

^{3}= 100m (Figs. 1(c) and (d)).

*x*

_{3}≫ 2(

*ω*/

*c*)

*σ*

^{2}

_{s}and when

*σ*≫

_{s}*σ*as a Collett-Wolf source [1

_{g}**30**, 1605–1607 (2005). [CrossRef] [PubMed]

4. P. DeSantis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. **29**, 256–260 (1979). [CrossRef]

_{b}is the angular divergences of the beam and the angular divergence of the interference feature Δ

_{i}is given by Eq. (18). Therefore, in the case considered (

*σ*≫

_{s}*σ*) the bright or dark feature diverges significantly more slowly than the beam itself. The evolution of the beams that results from the interference of circularly symmetric beams as they propagate along the

_{g}*x*

_{3}axis is shown in Fig. 2. It is seen that in both cases, when

*ϕ*= 0 and

*ϕ*=

*π*, the central interference peak or dip diverges considerably more slowly that the beams themselves.

### 4.2. Lorentzian spectral degree of coherence: *g*^{(0)} (*u*_{∥}) = 2*σ*^{2}_{g}/(*σ*^{2}_{g} + *u*^{2}_{∥})

*g*

^{(0)}(

*u*

_{∥}) we use Eqs. (11) and (16) to evaluate ⟨

*I*

_{±}(

**x**

_{∥},

*x*

^{3}∣

*ω*)⟩. To simplify the calculations we use the fact that the spectral degree of coherence

*g*

^{(0)}(

**u**

_{∥}) →

*g*

^{(0)}(

*u*

_{∥}) is circularly symmetric. In this case the angular integrations in Eqs. (16) can be carried out analytically, leaving only one-dimensional integrals to evaluate numerically. Thus, if we denote the

**x**

_{∥}and

**u**

_{∥}by

*ϕ*and

*ϕ*, respectively, we can rewrite Eq. (16a) as

_{u}*J*

_{0}(

*x*) is the Bessel function of the first kind and zero order.

*I*

_{−}(

**x**

_{∥},

*x*

^{3}∣

*ω*)) is only a bit more subtle. Thus, the integral over

**u**

_{∥}in Eq. (16b) can be written as

*I*

_{0}(

*z*) is the modified Bessel function of the first kind and zero order.

*I*

_{±}(

**x**

_{∥},

*x*

^{3}∣

*ω*)⟩, Eqs. (27) and (28), are shown in Fig. 3. The striking result that can be easily seen when comparing Figs. 3(a) and 3(c) is that in the far field [Fig. 3(c)] the beam narrows down to the interference peak. This is easy to understand on the basis of Eq. (16a). Indeed, as

*u*

_{∥}→ ∞ the decay of the integrand is determined by the Gaussian function exp[-

*u*

^{2}

_{∥}/(8

*σ*

^{2}

_{s})], rather than by the asymptotic behavior of the spectral degree of coherence

*g*

^{(0)}(

**u**

_{∥}), as is the case for the Gauss-Schell model source. Therefore, the angle of divergence of the beam is determined by its initial width

*σ*, Δ

_{s}_{b}=

*c*/(√2

*ω*

*σ*), rather than by the correlation length

_{s}*σ*. The analogous effect was pointed out in Ref. [9

_{g}9. R. M. Fitzgerald, T. A. Leskova, and A. A. Maradudin, “Control of coherence of light scattered from a one-dimensional randomly rough surface that acts as a Schell-model source,” J. Lumin. **125**, 147–155 (2007). [CrossRef]

*x*

_{2}= 0 of the far field intensity distribution of the beam produced by a pair of Schell-model sources whose spectral degree of coherence has a Lorentzian form, which produce fields inverted with respect to each other through their common center in the source plane and are in phase. The distance from the source plane is

*x*

_{3}= 500 m. For comparison, in the same Figure we plot the cross-section of the far field intensity distribution of the beam produced by the interference of the two Collett–Wolf beams (red lines). For clarity, we present the absolute intensities in Fig. 4(a), and the normalized intensities in Fig. 4(b). We also point out that the integrated intensities of the two beams coincide, as they should.

*g*

^{(0)}(

**u**

_{∥}) for which this is the case, but conjecture that they exist.

### 4.3. Spectral degree of coherence *g*^{(0)}(**u**_{∥}) = sinc(√3*u*_{1}/(*σ*_{g})sin(√3*u*_{2}/*σ*_{g})

_{g}

_{g}

*g*

^{(0)}(u

_{∥}) we use Eqs. (11) and (16) to evaluate ⟨

*I*

_{±}(

**x**

_{∥},

*x*

_{3}∣

*ω*)⟩ numerically. The results are shown in Fig. 5.

*b*= √3

*c*/(

*ω*

*σ*).

_{g}## 5. A General Circularly Symmetric Spectral Density in the Source Plane

*π*.

## 6. Experimental Details and Results

*λ*= 633 nm). After passing through a spatial filter, the diverging Gaussian beam passes through a rotating ground glass. The light emerging from the diffuser is allowed to propagate a small distance and illuminate a circular aperture, behind which we placed a collimating lens. The light emerging from the lens constitutes a secondary light source that can be regarded as a partially coherent source with a Gaussian spectral degree of coherence and a uniform intensity distribution within a circular domain. The radius of the beam is approximately 1 cm and the parameter

*σ*≈ 1 mm. The value of

_{g}*σ*is simply the size

_{g}*λ*/2, the output is a pair of Collet-Wolf beams that have a phase shift

*ϕ*=

*π*. The resulting pattern displays a dark spot in the spatial distribution of the intensity, as in Fig. 8(b).

## 7. Summary and conclusion

## Acknowledgements

## References and links

1. | Zu-Han Gu, E. R. Mendez, M. Ciftan, T. A. Leskova, and A. A. Maradudin, “Interference of a pair of symmetric Collett-Wolf beams,” Opt. Lett. |

2. | E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?”, Opt. Lett. |

3. | E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. |

4. | P. DeSantis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. |

5. | A. C. Schell, |

6. | L. Mandel and E. Wolf, |

7. | J. J. Foley and M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. |

8. | 6, Section 5.2.1. |

9. | R. M. Fitzgerald, T. A. Leskova, and A. A. Maradudin, “Control of coherence of light scattered from a one-dimensional randomly rough surface that acts as a Schell-model source,” J. Lumin. |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(140.3300) Lasers and laser optics : Laser beam shaping

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: December 21, 2009

Revised Manuscript: February 2, 2010

Manuscript Accepted: February 12, 2010

Published: February 23, 2010

**Citation**

E. E. García-Guerrero, E. R. Méndez, Zu-Han Gu, T. A. Leskova, and A. A. Maradudin, "Interference of a pair of symmetric partially coherent beams," Opt. Express **18**, 4816-4828 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-4816

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### References

- Zu-Han Gu, E. R. Mendez, M. Ciftan, T. A. Leskova, and A. A. Maradudin, "Interference of a pair of symmetric Collett-Wolf beams," Opt. Lett. 30, 1605-1607 (2005). [CrossRef] [PubMed]
- E. Collett and E. Wolf, "Is complete spatial coherence necessary for the generation of highly directional light beams?," Opt. Lett. 2, 27-29 (1978). [CrossRef] [PubMed]
- E. Wolf and E. Collett, "Partially coherent sources which produce the same far-field intensity distribution as a laser," Opt. Commun. 25, 293-296 (1978). [CrossRef]
- P. DeSantis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979). [CrossRef]
- A. C. Schell, The Multiple Plate Antenna, Ph. D. Dissertation, Massachusetts Institute of Technology, 1961, Section 7.5.
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995), Section 4.3.2.
- J. J. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297-300 (1978). [CrossRef]
- Ref. 6, Section 5.2.1.
- R. M. Fitzgerald, T. A. Leskova, and A. A. Maradudin, "Control of coherence of light scattered from a onedimensional randomly rough surface that acts as a Schell-model source," J. Lumin. 125, 147-155 (2007). [CrossRef]

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