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We consider a class of spatially partially coherent light beams, which are generated by passing a Gaussian Schell-model beam though a wavefront-folding interferometer. In certain cases these beams are shape-invariant on propagation and can exhibit sharp internal structure with a central peak (specular beam) or a central dip (antispecular beam) whose dimensions depend on the spatial coherence area. Such beams are demonstrated experimentally and their cross-like distributions of the complex degree of spatial coherence are measured with a digital micromirror device.
© 2015 Optical Society of America
1. Introduction
Model sources and fields play a key role in partially coherent optics since they often allow the description of coherence and propagation phenomena in simple analytical terms [1]. The most commonly used model is the Schell model, in which the complex degree of spatial coherence at a given plane depends only on coordinate differences. However, this model excludes certain interesting properties of partially coherent light; several decades ago Gori, Guattari, Palma, and Padovani [2] introduced the concept of specular cross-spectral density functions, and discussed some general features of such specular fields. By definition, a specular cross-spectral density function W(x1, x2) satisfies (in one dimension) the condition W(−x1, x2) = W(x1, x2), which implies that W(x1, −x2) = W(x1, x2). Gori, Guattari, Palma, and Padovani also proposed a method of generating specular fields by means of a Porro-prism interferometer.
Despite of its peculiar nature, the phenomenon of specularity has received little attention since its introduction. An important exception is the paper of Ponomarenko and Agrawal [3], who studied theoretically the specular properties of vectorial partially coherent solitons propagating in non-instantaneous nonlinear media. However, to our best knowledge, specular fields have not been studied experimentally.
In this paper we study and demonstrate a class of light fields that can be generated by passing a partially coherent field through a perfectly aligned wavefront-folding interferometer (WFI), which employs Porro prisms instead of plane mirrors in both arms of a classical Michelson’s interferometer [4–7]. We begin (Sect. 2) with a brief description of the spatial coherence modulation properties of such an interferometer when there is a short optical path delay between the two arms. We show that specular fields are generated for certain values of the phase delay (such as zero). We also introduce another related class of fields, which we call anti-specular, and show that these fields can be generated with the WFI with certain other choices of the phase delay. One example of such transformations, where the interferometer is illuminated with a Bessel-correlated field [8], has already been considered [9] and it was shown that the WFI then provides propagation-invariant fields with sharp intensity variations [9, 10]. In Sect. 3 we first briefly recall this case and consider also the case of Gaussian-correlated illumination. We then proceed by assuming that the WFI is illuminated by a Gaussian Schell-model beam (Ref. [1], Sect. 5.6.4), and present (in Sect. 4) a detailed analysis of the resulting field and its propagation characteristics. Some properties of specular and antispecular fields generated in a WFI are demonstrated experimentally in Sect. 5. Concluding remarks are provided in Sect. 6.
2. Coherence modulation by a wavefront-folding interferometer
Figure 1 illustrates the wavefront-folding interferometer, which is a device primarily used to measure the spatial coherence properties of partially coherent light fields [4–7]. The operation of this device is based on the two perpendicularly oriented right-angle prisms in the interferometer arms. One of the prisms retroreflects the incident field in the x direction and the other in the y direction. In spatial coherence measurements, one (or both) of the prims is tilted slightly in a direction of the prims edge to generate spatial interference fringes, but we assume a perfectly aligned device.
Fig. 1 The wavefront-folding interferometer: S is the source, BS is a non-polarizing beam splittter, P1 and P2 are right-angle prisms, and D is a detector.
Let us assume that a (linearly polarized) field E0(x, y) with an essentially planar wavefront is incident on the interferometer and that the wavefront remains essentially planar in propagation through the system. The field at the detector may be written (up to a constant phase factor) as
where ϕ denotes the phase difference between the two beam paths. Considering E0(x, y) as a single realization of a spatially partially coherence field, we may characterize the coherence properties of the incident field by a cross-spectral density function (CSD) where the sharp brackets denote ensemble averaging and we have ignored the frequency dependence for brevity. Defining the CSD W(x1, y1, x2, y2) of the output field analogously and considering the retroreflections, we find that This field satisfies the condition W(−x1, −y1, x2, y2) = W(x1, y1, x2, y2) if ϕ = 2πn, where n in an integer, hence being specular in two dimensions. If, on the other hand, ϕ = π/2 + 2πn, we have W(−x1, −y1, x2, y2) = −W(x1, y1, x2, y2). We call this kind of fields anti-specular.The far-zone properties of the field defined in Eq. (3) are determined entirely by the angular correlation function (ACF) [1], which is related to the CSD by a transformation
Explicitly, by inserting from Eq. (3) into Eq. (4), we have The ACF satisfies the condition T(−kx1, −ky1, kx2, ky2) = T(kx1, ky1, kx2, ky2) if ϕ = 2πn and the condition T(−kx1, −ky1, kx2, ky2) = −T(kx1, ky1, kx2, ky2) if ϕ = π/2 + 2πn. Hence, if the CSD is specular, so is the ACF, and a similar conclusion applies also to the antispecular case.3. Transformation of Bessel- and Gaussian-correlated input fields
If the WFI is illuminated by a fundamental Bessel-correlated field with a CSD of the form
where α is a constant, the output field is given by [9] This field is indeed specular when ϕ = 2πn and anti-specular when ϕ =π/2 + 2πn. The spectral density S(x, y) = W(x, y, x, y) of the field has the form for all values of ϕ. In the specular case the field thus has a sharp central peak on a uniform background, and in the anti-specular case it has an axial dip. The spectral density has the same form, given by Eq. (6), for all values of the propagation distance (if we ignore the finite aperture of the interferometer). Hence the fields defined by Eq. (5) have been called dark and antidark diffraction-free beams [10].Let us proceed to consider another example by assuming that the incident field is of unit-amplitude Gaussian-correlated form
where the parameter σ0 is a measure of the spatial coherence width of the field. In view of Eq. (3), the CSD of the output field is readily seen to be and its spectral density is given by As with a Bessel-correlated incident field, we have a central peak in the specular case and a central dip in the anti-specular case. The width of the peak or dip is determined directly by the spatial coherence area of the incident field. We will demonstrate fields that obey Eqs. (10) and (11) experimentally in Sect. 5. It should be noted, on comparison with the previous example, that these Gaussian-correlated fields are not propagation-invariant but experience diffractive spreading under propagation. Analytical characterization of such spreading is, however, not possible and we therefore consider in the following section a case in which simple closed-form expressions exist.4. Transformation of Gaussian Schell-model incident fields
Let us replace the rotation-symmetric uniform-intensity CSD defined in Eq. (9) with that of an anisotropic Gaussian Schell-model source [11], i.e.,
where the parameters w0x, w0y represent the beam widths and σ0x, σ0y the coherence widths in the x and y directions. Now Eq. (10) is replaced with and the spectral density is given byFigure 2 illustrates S(x, y) and the complex degree of spatial coherence μ(x1, y1, x2, y2) = W(x1, y1, x2, y2)/[S(x1, y1)S(x2, y2)]1/2 for some chosen parameter values in the specular case ϕ = 0. The profiles S(x, y) show a central dip, which is elliptical in anisotropic cases. The distributions of μ(x1, 0, x2, 0) feature a characteristic cross-like shape.
Fig. 2 Distributions of the spectral density S(x, y) and the complex degree of spatial coherence μ(x1, 0, x2, 0) of some specular output fields when the WFI is illuminated with Gaussian Schell-model fields. Left: an isotropic case with w0y = w0x and σ0y = σ0x = w0x/4. Center: an anisotropic case with w0y = w0x, σ0y = w0x/4, and σ0x = w0x/2. Right: another anisotropic case with w0y = w0x/2, σ0y = w0x/4, and .
By applying the standard Fresnel propagation formula for spatially partially coherent fields [1], we find that the CSD at any distance z from the output beam waist is given by
where k = 2π/λ is the wave number. The spectral density is The parameters wj(z), σj(z), and Rj(z) (j = x, y) obey the same propagation laws as for usual anisotropic Gaussian Schell-model beams: where are the Rayleigh ranges of the beam in the two orthogonal distances and are parameters that characterize the state of coherence of the beam in the x and y directions.Eqs. (15)–(20) show that in the isotropic case the beam is shape-invariant in propagation since w(z) and σ(z) grow at the same rate, just as they do for usual Gaussian Schell-model fields. In the anisotropic case the ellipticity of both the entire beam and the central peak or dip generally chance on propagation. However, if the condition zRx = zRy, or
is satisfied, we again obtain a shape-invariant beam. This is the case in the right-hand-side example presented in Fig. 2.5. Measurement results
We used the experimental setup described in Fig. 3 to demonstrate the main properties of fields introduced in Sect. 3. First a rotating diffuser modifies the fully coherent Gaussian beam from a HeNe laser into a partially coherent field with Gaussian spatial coherence properties [12]. Next a WFI transforms the beam into the form described in Sect. 3. Finally, the coherence properties of the output beam are measured with a double-pinhole interferometer setup realized with a digital micromirror device (DMD) [13].
The beam from a HeNe laser (λ0 = 633 nm) is focused by a lens f1 on a rotating diffuser plate that scatters the light. Lens f2 collimates the beam and an iris limits the beam diameter to about 1 mm. The beam splitter BS1 redirects the beam to two right-angle retroreflector prisms mounted 90° with respect to each other. The prisms flip the beam in y and x directions. The angles of the prisms are tuned so that the overlapping beams arrive on the detectors (cameras 1 and 2) in parallel direction, i.e., no interference fringes are seen. Prism 2 is mounted on a piezo translation table to fine tune the optical path length and phase difference between the interferometer arms. The lens f3 images the prism corner planes on the inspection plane (camera 2 and DMD). We used camera 2 to monitor the beam intensity profile simultaneously with the coherence measurement.
To measure the coherence function we placed the DMD into the image plane of the f3 lens. The DMD has a 608 × 684 array of tiltable rectangular mirrors with 10.8 μm diagonal size, arranged in 45° rotated orientation. We used Matlab software to control the whole setup. We measured the two-dimensional CSD W(x1, 0, x2, 0) = W(x1, x2) of the beam on one line by creating two virtual pinholes into coordinates x1 and x2, and scanning over all coordinate combinations. The pinholes consisted of three DMD mirrors oriented in line perpendicular to the scanning direction. The mirrors were tilted towards camera 1, which captured the interference pattern. We also measured the intensities in single pinhole coordinates by having only one pinhole open at the time. The intensity fringes were normalized as we describe in [13]. The absolute value |μ(x1, x2)| was determined from the visibility of the fringes and the phase of μ(x1, x2) from their lateral position on the camera plane. Measurement of a single coordinate pair took approximately one second. The path length between the arms did not stay constant over a long time, probably because of the drifting of the piezo driving current and thermal expansion of the components. Therefore we compensated the position of the piezo translator during the measurement cycle. In the beginning of the measurement we captured the intensity profile with camera 2, and every 20 coordinate-pair measurement we checked whether the intensity on the camera 2 matched with the original, and moved the piezo if necessary.
The horizontal and vertical lines in the center of the measured intensity figures and corresponding ripples are caused by the corners of the prisms. The intensity profile of the measured beam is not Gaussian because of the hard aperture used in the system, but it corresponds to the model presented in Sect. 3. An apodizing Gaussian spatial filter would allow the realization of beams discussed in Sect. 4. The coherence width of the simulated beam was taken as σ0 = 120 μm.
By moving the piezo we were able control the phase difference ϕ approximately with a resolution of 0.1π radians. Figure 4 depicts the simulated and measured beam with thee different optical path lengths ϕ = 0, ϕ = π/2 and ϕ = π. The left column displays the intensity profile of the beam captured with camera 2. The center column shows the absolute value of the degree of spatial coherence, |μ(x1, x2)|. The cross-shaped specular arms predicted in Fig. 2 are clearly visible, as is their disappearance when a pure Gaussian Shell-model beam is obtained at ϕ = π/2. The anti-specular arms with negative phase appear when ϕ = π. The right column in Fig. 4 shows the phase arg[μ(x1, x2)]. Theoretically μ(x1, x2) is real-valued, but the measurements show a somewhat uneven phase profile even after the spherical phase caused by the imaging system and beam spreading has been removed as described in [13].
Fig. 4 Theoretical (subscript t) and measured (subscript m) spectral densities S(x, y), corresponding absolute values |μ(x1, x2)| and phase arg[μ(x1, x2)] of the degree of coherence, with three different values of phase difference ϕ. The maximum and minimum values of the coordinate axes are ±324 μm. A corresponding animation with 21 measured values of ϕ is presented in Visualization 1.
Probably because of alignment errors in the system, the maximum peak and dip in the intensity profile do not appear at the same values of ϕ as the maximum specular arms in the degree of spatial coherence. This is better visible in the corresponding animation ( Visualization 1). One source of alignment errors is that we could not place the prism exactly perpendicularly to the beams because in that case the light reflected from front surface of the prism and its corners would propagate in same direction, thus interfering with each other and causing confusion with interference from the other WFI arm. Because of the Hermiticity W(x1, x2) = W*(x2, x1), and to save the measurement time, we only measured the upper half-diagonal of the data matrices.
Figure 5 shows the measured distribution μ(x1, 0, x2, 0) with x1 = 324 μm and x2 = −x1. The circles mark the measured real part of μ(x1, 0, x2, 0) and the crosses indicate the imaginary part (which should theoretically be zero, as it almost is). The measured values match well the sinusoidal shape of the theoretical values (solid line). The data is from a different shorter measurement cycle than Fig. 4, to prevent errors caused by long-time piezo drifting.
Figure 6 shows another example of a measured coherence function, with a higher resolution and larger detection area. Figure 6(a) shows the absolute value |W(x1, x2)| of the CSD, which also contains intensity information, (b) depicts normalized degree of coherence, (c) shows the directly measured phase before the spherical phase front is removed, and (d) after this have been done (it shows phase structures further away from the center of the beam). Finally, Fig. 6(e) shows the intensity profile measured by scanning a single pinhole in the Young interferometer.
Fig. 6 Measured coherence function. (a) Cross-spectral density, (b) degree of coherence, (c) originally measured phase, (d) spherical phase removed, and (e) intensity.
6. Final remarks
We introduced and demonstrated a class of partially spatially coherent light fields that can be generated by inserting a Shell-model beam into a wavefront-folding interferometer. One of the more interesting conclusions of the present work is that the intensity profile of the output field can vary in a scale that depends on the coherence properties of the incident field in addition to its intensity profile. Particularly, if the incident field is quasihomogeneous (coherence area is small compared to beam diameter), the dimensions of the central internal structure (dip or peak) of the output beam are entirely determined by the spatial coherence area of the input beam.
In this paper we have considered the field generated by the WFI within the framework of scalar coherence theory. Interesting effects might be revealed by extending the treatment to the electromagnetic case, especially by considering fields with spatially nonuniform states of polarization. Furthermore, our experimental demonstrations are restricted to the case of rotationally symmetric uniform-intensity incident fields and in the experiments presented here we considered only the fields at the output plane of the interferometer. Experimental demonstrations of fields considered in Sect. 4 are yet to be performed, but appear straightforward. The critical factor is the quality of the right-angle prisms. Our initial numerical simulations indicate that if the beam diameter and the coherence area are of the order of millimeters, the precision of the 90° angle should be of the order of milliradians.
Acknowledgments
This work was partially supported by a grant from Tekniikan edistämissäätiö (Finland). The present address of J. Tervo is Microsoft, Keilalahdentie 2–4, 02150 Espoo, Finland.
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