OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 5 — Mar. 1, 2010
  • pp: 4816–4828
« Show journal navigation

Interference of a pair of symmetric partially coherent beams

E. E. García-Guerrero, E. R. Méndez, Zu-Han Gu, T. A. Leskova, and A. A. Maradudin  »View Author Affiliations


Optics Express, Vol. 18, Issue 5, pp. 4816-4828 (2010)
http://dx.doi.org/10.1364/OE.18.004816


View Full Text Article

Acrobat PDF (1474 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

In a recent paper [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]

] the interference of light produced by a pair of mutually correlated Gaussian Schell-model sources [2

2. 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]

,3

3. 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]

] was investigated. The spatial distributions of the fields produced by these sources were assumed to be symmetric with respect to a plane through their common center, and to differ by a phase factor exp(). One of the results of this investigation was that when ϕ = 0 the resulting radiation is a beam with an intensity distribution that displays a narrow bright line at its center. When the parameters characterizing the Gaussian Schell-model source are chosen in such a way that it becomes a Collett-Wolf source [4

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

], the resulting bright line diverges much more slowly than the radiated beam itself. These theoretical predictions were confirmed experimentally, and suggested that the interference of a pair of correlated Collett-Wolf beams can be used to produce a pseudo-nondiffracting beam.

In this paper we extend the investigation 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]

] in several ways. First of all we assume that 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 and differ by a phase factor exp(). 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]

], namely a Schell–model source [5

5. A. C. Schell, The Multiple Plate Antenna, Ph. D. Dissertation, Massachusetts Institute of Technology, 1961, Section 7.5.

] instead of a Gaussian Schell–model source. On the basis of scalar diffraction theory, we obtain an expression for the mean intensity of the field produced by the interference of these two beams in terms of the spectral density and spectral degree of coherence of each source. This result is illustrated by applying it to situations in which the cross–spectral density function in the source plane has a non-Gaussian form.

It is found that the angular divergence of the bright spot in the intensity distribution of the field produced by the interference of the fields produced by two sources with these properties is determined by the width of the initial beams and, as a result, is considerably smaller than that of the initial beams, which is determined by the spectral degree of coherence. Since the divergence of the interference feature can be much smaller than that of a beam of the same size, this result supports the suggestion that the interference of two mutually correlated beams produced by Schell–model sources can be used to produce a pseudo-nondiffracting beam.

2. The Radiated Intensity Distribution

We denote a component of the radiated field in the source plane (x3 = 0) at frequency ω by U(x, 0∣ ω), where x = (x1, x2, 0) is an arbitrary point in this plane. The cross-spectral density of this field in the source plane is then defined by [6

6. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995), Section 4.3.2.

]

W(0)x0x0=U(x,0ω)U*x0ω,
(1)

where the angle brackets denote an average over the ensemble of realizations of the functions {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]

) and in subsequent expressions emphasizes that the corresponding quantity refers to the source plane x3 = 0.

The class of cross-spectral density functions W(0) (x, 0∣x, 0) that we consider in this paper has the Schell-model form [5

5. A. C. Schell, The Multiple Plate Antenna, Ph. D. Dissertation, Massachusetts Institute of Technology, 1961, Section 7.5.

]

W(0)x0x0=[S(0)(x)]12g(0)(xx)[S(0)(x)]12.
(2)

In this expression 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,

g(0)(xx)=g(0)(xx)*,
(3)

and has the additional properties [6

6. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995), Section 4.3.2.

]

0g(0)(xx)1
(4)

and

g(0)(0)=1.
(5)

In the Fresnel approximation, the field that propagates in the x3 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]

]

Uxx3ω=(ω2πicx3)exp[iωcx3]d2xexp[iω2cx3(xx)2]Ux0ω.
(6)

Let us consider a quasi-monochromatic beam in free space that is a superposition of two beams produced by mutually correlated Schell-model sources characterized by identical cross-spectral density functions of the form given by Eqs. (1), (2), so that

U(x,x3ω)=U1xx3ω+U2xx3ωexp(),
(7)

where 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 x1 x2 plane,

U2x1x2x3|ω=U1x1x2x3|ω.
(8)

Since the spatial distributions of the fields of the two beams obey the symmetry conditions expressed by Eq. (8) at a distance x3, they also obey the same conditions at x3 = 0, i.e. in the source plane.

The mean intensity at a distance x3 from the source plane is given by

I(x,x3ω)=U(x,x3ω)2
=(ω2πcx3)2d2xd2xexp[iω2cx3(xx)2]
×exp[iω2cx3(xx)2]Ux0ωU*x0ω
=(ω2πcx3)2d2xd2xexp[iω2cx3(xx)2]
×exp[iω2cx3(xx)2]{2W(0)x0x0
+U1x0ωU2*x0ωexp()+U1*x0ωU2x0ωexp()}.
(9)

In view of the spatial symmetry of the fields the ensemble averages ⟨U1(x, 0 ∣ω)U*2(x, 0 ∣ ω)⟩ and ⟨U*1(x, 0 ∣ω)U2(x, 0 ∣ ω)⟩ entering Eq. (9) can be written in the forms

U1(x,0ω)U2*x0ω=U1x1x20ωU1*x1x20ω
=W(0)(x,0x,0)
(10a)

and

U1*(x,0ω)U2x0ω=U2*x1x20ωU2x1x20ω
=W(0)(x,0x,0).
(10b)

With these results the mean intensity of the beam, Eq. (9), becomes

I(x,x3ω)=I+xx3ω+cosϕI(x,x3ω),
(11)

where

I±xx3ω=2(ω2πcx3)2d2xd2xexp[iω2cx3(xx)2]
×exp[iω2cx3(xx)2]W(0)(x,0|±x",0).
(12)

We substitute into Eq. (12) the expression for W(0) (x, 0 ∣x, 0) given by Eq. (2) and obtain

I±(x,x3ω)=2(ω2πcx3)2d2xd2xg(0)(xx)[S(0)(x)]12
×[S(0)(±x")]12exp[iω2cx3(x2x2)]exp[iωcx3(xx)·x].
(13)

The changes of variables x = ±x″∥ + u followed by the change ±x″∥ to x″∥ transform Eq. (13)into

I+xx3ω=2(ω2πcx3)2d2ug(0)(u)exp[iω2cx3u2]exp[iωcx3x·u]
×d2xexp{iωcx3[u·x]}[S(0)(|x+u|)]12[S(0)(x)]12
(14a)
Ixx3ω=2(ω2πcx3)2d2ug(0)(u)exp[iω2cx3u2]exp[iωcx3x·u]
×d2xexp{iωcx3[(u2x)·x]}[S(0)(|x+u|)]12[S(0)(x)]12.
(14b)

Equations (11) and (14), are the main results of this section.

We now turn to several applications of these results.

3. A Gaussian Spectral Density in the Source Plane

We begin by assuming that the spectral density of the radiated field in the source plane has the Gaussian form

S(0)(x)=exp(x2/2σs2).
(15)

On carrying out the integration over x in Eq. (14) with the spectral density given by Eq. (15) we obtain

I+xx3ω=1π(ω2σs2c2x32)d2ug(0)(u)exp[iωcx3x·u]
×exp{u2[18σs2+ω2σs22c2x32]},
(16a)
Ixx3ω=1π(ω2σs2c2x32)exp[2ω2σs2c2x32x2]d2ug(0)(u)exp[2ω2σs2c2x32u·x]
×exp{u2[18σs2+ω2σs22c2x32]}.
(16b)

Equations (16) describe the evolution of the beam as it propagates away from the source plane. The angular divergence of the beam, however, is determined by the behavior of these integrals in the far-field zone where x3 ≫ 2(ω/c)σ2s. 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,x3ω)=1π(ω2σs2c2x32)d2ug(0)(u)exp(iωcx3x·u),
(17a)
I(x,x3ω)=1π(ω2σs2c2x32)exp{2(ωσscx3)2x2}d2ug(0)(u).
(17b)

Two important conclusions can be drawn from Eqs. (17). First of all, it follows from Eq. (17a) that the angle of divergence of the primary beam, associated with ⟨I+(x, x3ω)⟩, depends on the spectral degree of coherence through what is essentially a Fourier transform operation.

In contrast, from Eq. (17b), we see that the interference feature associated with ⟨I-(x, x3ω)⟩ is independent of the form of the spectral degree of coherence, and has a Gaussian shape ⟨I-(x, x3ω)⟩ ~ exp[−(x/x3)22i], where Δi is the angular divergence and is given by

Δi=12(ω/c)σs.
(18)

We note, that the superposition of the initial beam and its inverted version leads to an interference feature only within a coherence area around the center of inversion. Outside this area the beams add incoherently. Thus, it is not surprising that in the far field the divergence properties of this interference feature coincide with those of a speckle. The relative intensity of the interference peak is determined by the phase shift 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.

Before proceeding, it is of interest to consider two extreme limits of the secondary source. In the case where the radiated field is completely coherent, i.e. the spectral degree of coherence is 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

Ixx3ω=2(1+cosϕ)1+(cx32ωσs2)2exp{x22σs2[1+(cx32ωσs2)2]}.
(19)

In contrast, if the field is incoherent, 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,

Ixx3ω=Acπ(ωσscx3)2{1+cosϕexp[2(ωσscx3)2x2]},
(20)

where

Ac=d2ug(0)(u)
(21)

is a measure of the area under the coherence function.

We now consider different forms of the spectral degree of coherence, illustrating the results through some examples.

4. Examples

In this section we apply the results of the preceding section to several choices for the spectral degree of coherence 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

g(0)(u)=1u22σg2+o(u2).
(22)

In all calculations carried out here the values of the parameters σs, σg, and the wavelength λ are σs = 2.432 mm, σg = 179.6μm, and λ = 632.8 nm. They correspond to the values characterizing the Collett-Wolf source studied experimentally in Ref. [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]

].

4.1. Gaussian spectral degree of coherence: g(0) (u) = exp(-u2/2σ2g).

For this circularly symmetric form for g(0)(u) we use Eqs. (11) and (16) to evaluate ⟨I±(x, x3ω)⟩ analytically, with the result

Ixx3ω=4σs2σeff2(x3)exp(x2σeff2(x3)){1+cosϕexp[4σs2σg2x2σeff2(x3)]},
(23)

where

σeff2(x3)=2σs2+c2x32ω2(12σs2+2σg2).
(24)

In Fig. 1 we present plots of the normalized intensities I(x, x3) = ⟨I(x, x3ω)⟩/⟨I(0, x3ω)) of the beams that are inverted with respect to their common center at two values of the distance from the source planes x3 = 1m (Figs. 1(a) and (b)), and x3 = 100m (Figs. 1(c) and (d)).

In the far field where x3 ≫ 2(ω/c)σ2s and when σsσg as a Collett-Wolf source [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]

,4

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

], Eq. (24) becomes

σeff(x3)=2(ω/c)σgx3=Δbx3,
(25)

so that Eq. (23) takes the form

Ixx3ω=(2ωcσsσgx3)2exp{(x/x3)2Δb2}[1+cosϕexp]{(x/x3)2Δi2},
(26)

where Δb is the angular divergences of the beam and the angular divergence of the interference feature Δi is given by Eq. (18). Therefore, in the case considered (σsσg) 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 x3 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) = 2σ2g/(σ2g + u2)

For this form for g(0)(u) we use Eqs. (11) and (16) to evaluate ⟨I±(x, x3ω)⟩. 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 ϕu, respectively, we can rewrite Eq. (16a) as

Fig. 1. The spatial distribution of the normalized intensity of the beams that are inverted with respect to their common center with (a) and (c) ϕ = 0 and (b) and (d) ϕ = π. The distance from the source plane is x3 = 1 m (a) and (b), and x3 = 100m (c) and (d).
Fig. 2. Theoretical plot of the evolution along the x3 axis of the cross section x2 = 0 of the normalized intensity of the beams that are inverted with respect to their common center with ϕ = 0 (a) and ϕ = π (b).
I+xx3ω=1π(ω2σs2c2x32)0duug(0)(u)exp[(ω2σs22c2x32+18σs2)u2]
×ππdϕuexp[iωcx3xucos(ϕϕu)]
=2(ω2σs2c2x32)0duug(0)(u)J0(ωcx3xu)exp[(ω2σs22c2x32+18σs2)u2],
(27)

where J0(x) is the Bessel function of the first kind and zero order.

The evaluation of the angular integral in the expression for ⟨I(x, x3ω)) is only a bit more subtle. Thus, the integral over u in Eq. (16b) can be written as

Fig. 3. The spatial distribution of the normalized intensity of the beams that have the spectral degree of coherence of the Lorentzian form and are inverted with respect to their common center with (a) and (c) ϕ = 0 and (b) and (d) ϕ = π. The distance from the source plane is x3 = 1 m (a) and (b), and x3 = 100m (c) and (d).
Fig. 4. The intensity (a) and the normalized intensity (b) of the beam as a function of x1 at x2 = 0 at a distance from the source plane x3 = 500 m. The spectral degree of coherence of in the source plane has a Lorentzian form (black curve) or a Gaussian form (red curve).
Ixx3ω=1π(ω2σs2c2x32)exp(2ω2σs2c2x32x2)0duug(0)(u)
×exp[(ω2σs22c2x32+18σ32)u2]ππdϕuexp[2ω2σs2c2x32xucos(ϕϕu)]
=2(ω2σs2c2x32)exp(2ω2σs2c2x32x2)0duug(0)(u)I0(2(ωσscx3)2xu)
×exp[(ω2σs22c2x32+18σs2)u2],
(28)

where I0(z) is the modified Bessel function of the first kind and zero order.

The results of a numerical evaluation of I±(x, x3ω)⟩, 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[-u2/(8σ2s)], 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 σs, Δb = c/(√2ωσs), rather than by the correlation length σg. The analogous effect was pointed out in Ref. [9

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]

] for a one dimensional partially coherent beam whose spectral degree of coherence has a Lorentzian form.

To demonstrate this more clearly, in Fig. 4 we present the cross-sections x2 = 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 x3 = 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.

This example shows that by choosing the spectral degree of coherence suitably, e.g. with a Lorentzian form, the beam can be narrowed to the point where it is as narrow as the interference feature. We have not sought other functional forms for g(0)(u) for which this is the case, but conjecture that they exist.

4.3. Spectral degree of coherence g(0)(u) = sinc(√3u1/(σg)sin(√3u2/σg)

Fig. 5. Plots of the normalized intensity of the beams that have the spectral degree of coherence of the form g(0)(u) = sinc(√3u1/σg)sinc(√3u2/σg), and are inverted with respect to their common center with ϕ = 0 (a) and (c) and ϕ = π (b) and (d). The distance from the source plane is x3 = 1 m (a) and (b), and x3 = 100m (c) and (d).

For this non-circularly symmetric form for the spectral degree of coherence g(0)(u) we use Eqs. (11) and (16) to evaluate ⟨I±(x, x3ω)⟩ numerically. The results are shown in Fig. 5.

The beam characterized by such a form of the spectral degree of coherence evolves on propagation into a flat top beam in a square domain. The angle of divergence of this beam is Δb = √3c/(ωσg).

5. A General Circularly Symmetric Spectral Density in the Source Plane

Although the Gaussian form [Eq. (15)] is perhaps the most common form of the spectral density in the source plane, it is not the only one that has been used. As an example in this section we consider the interference of the beams produced by the sources whose spectral densities although circularly symmetric are not Gaussian, namely, they are constant in the circular domain (“flat top” beam). The beams are inverted with respect to their common center and can acquire an additional phase shift π.

Fig. 6. Plots of the normalized intensity of the beams whose spectral density in the source plane is a constant within a circular domain and their spectral degree of coherence has a Gaussian form. The beams fields are inverted with respect to their common center with ϕ = 0 (a) and (c) and ϕ = π (b) and (d). The distance from the source plane is x3 = 1 m (a) and (b), and x3 = 1Km (c) and (d).

We assume that

S(0)(x)={1,x<σs0,x>σs,
(29a)
g(0)(x)=exp(x2/2σg2).
(29b)

For this form for S(0)(x) we use Eqs. (11) and (14) to evaluate ⟨I±(x, x3ω)⟩ numerically. The results of this numerical evaluation are presented in Fig. 6.

6. Experimental Details and Results

The schematic diagram of the optical system employed is shown in Fig. 7. The illumination is provided by a HeNe laser beam (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 σg ≈ 1 mm. The value of σg is simply the size

To study the interference of the field produced by a pair of such partially coherent beams, the original beam is sent to a modified Michelson interferometer. The modification consists of the insertion of a spherical lens in one of the arms of the interferometer in such a way that the wave front is reversed about the axis of the lens. For this, the mirror of that arm of the interferometer must be placed on the focal plane of the lens. The output of the interferometer then consists of the superposition of a pair of partially coherent beams, where the first beam is symmetric with respect to the second about their common center.

Fig. 7. Schematic diagram of the experimental arrangement used.

When the beams are in phase, a bright spot appears in the center of the beam spot. The experimental interference pattern showing constructive interference is shown in Fig. 8(a). When the mirrors of the modified Michelson interferometer are arranged to induce a path difference of 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

We have shown theoretically and experimentally that the interference of the radiation produced by a pair of mutually correlated Schell-model sources produces a beam in the far field with an intensity distribution that displays a narrow bright or dark dot of small radius at its center, both of which diverge much more slowly with the increasing distance from the source plane than the beam itself. The bright dot arises when the fields produced by the two sources are inverted with respect to each other through their common center in the source plane, and are in phase. In the case where the inverted fields produced by the two sources are out of phase their interference produces the dark dot.

These results support the suggestion that the interference of the beams produced by two mutually correlated Schell-model sources can be used for the creation of a pseudo-nondiffracting beam.

We have also shown that when the spectral density of each source has a Gaussian form, while the spectral degree of coherence of each source has a Lorentzian form, the initial beams evolve upon propagation in such a way that in the far field the angular divergence of the initial beams coincides with the angular divergence of the interference peak.

Fig. 8. Experimental gray-level images of the normalized intensity of the symmetric beams for (a) constructive interference and (b) destructive interference. The horizontal lines in the images show the positions where the intensity scans in the lower graphs were taken.

Acknowledgements

A. A. M. would like to thank Professors R. Righini and S. Califano for the hospitality of the European Laboratory for Nonlinear Spectroscopy (LENS) where his contribution to this work was carried out.

The research of E.E.G-G. was supported in part by projects PROMEP/103.5/07/2599 and the internal project 471, the research of E.R.M was supported by the Consejo Nacional de Ciencia y Tecnología (grant 47712-F), and the research of Z.-H.G., T.A.L., and A.A.M. was supported in part by AFRL contract FA9453-08-C-0230.

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

2.

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]

3.

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]

4.

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

5.

A. C. Schell, The Multiple Plate Antenna, Ph. D. Dissertation, Massachusetts Institute of Technology, 1961, Section 7.5.

6.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995), Section 4.3.2.

7.

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

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. 125, 147–155 (2007). [CrossRef]

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  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. 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]
  3. 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]
  4. P. DeSantis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979). [CrossRef]
  5. A. C. Schell, The Multiple Plate Antenna, Ph. D. Dissertation, Massachusetts Institute of Technology, 1961, Section 7.5.
  6. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995), Section 4.3.2.
  7. J. J. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297-300 (1978). [CrossRef]
  8. Ref. 6, Section 5.2.1.
  9. 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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited