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Abstract
We present modified scanning-type wavefront folding interferometers (WFIs), which allow spatial coherence measurements of non-uniformly correlated fields, where the degree of coherence is a function of two absolute spatial coordinates instead of coordinate separation only (Schell model). As an alternative to conventional prism-based WFI implementations, we introduce a scheme based on reflections by three mirrors. This setup allows us to avoid obstructions due to prism corners, and it is remarkably robust to polarization effects. Experimental results on measurement of fields that do not obey the Schell model are provided with the three-mirror WFI, and the results are compared to those obtained with a Young’s interferometer realized using a digital micromirror device.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Besides being one of the fundamental properties of optical fields, spatial coherence of light is gaining importance due to increasingly widespread emergence of partially coherent light sources in scientific instruments and consumer products. While many primary sources of spatially partially coherent light are of the Schell-model form, – meaning that the two-coordinate degree of spatial coherence is a function of the two-point separation only – this is not true for all relevant sources and especially not if light from a primary source passes through an optical system. Hence, it is of substantial interest to develop rapid and light-efficient spatial-coherence measurement techniques capable of characterizing fields that do not necessarily obey the Schell model.
Young’s two-pinhole interferometer [1] is the classical instrument for measuring the full two-point spatial coherence properties of light. Despite of several practical developments in the implementation of the two-pinhole interferometer [2–10], its main limitations persist: low light efficiency and long measurement time. Alternative techniques have been introduced to overcome these limitations, including the use of masks with multiple apertures [11], shadowing obstacles [12], Sagnac [13,14] and Mach–Zehder [15,16] interferometers, and gratings [17–19]. Despite these important developments, the wavefront-folding interferometer (WFI) [20–23] remains as one of the most important coherence characterization devices with several attractive features: it is a light-efficient and parallel instrument, and being based on reflective optics it is also suitable for characterizing polychromatic fields.
The conventional WFI consists of a 50 : 50 beam splitter in combination with one or two Porro prisms as shown in Fig. 1(a). The prisms may be either bare or metal-coated, or one may employ metal-coated corner retroreflectors. This conventional scheme involves a practical problem associated with the prims edges, which cause a central obstruction that prevents coherence measurements at small point-to-point separations. Even if ‘knife-edge’ prisms are used and the prism corners are imaged onto the detector, corner diffraction effects cause some blurring at small two-point separations.
Fig. 1 Basic configuration for a scanning WFI. The input is split by a 50 : 50 beamsplitter BS and retroreflected by prisms P1 and P2 towards the imaging system D, which captures the interference pattern. In (a) the beam is centered on the prism edge, whereas in (b) the beam has been shifted along the x-axis. Dashed lines show the paths through P1 and solid ones through P2.
In its traditional form, the WFI can be used to characterize only Schell-model sources correctly. In this paper we point out that the measurement of the full two-point coherence function is possible by simple scanning schemes, where the optical axes of the incident field and the WFI are sheared. In addition, we introduce a WFI setup that employs only planar mirrors instead of Porro prisms or corner retroreflectors, and is thereby free from the central obstruction problems. One may argue that such problems can be overcome expanding the incident beam sufficiently. However, even though a WFI is a light efficient instrument, this is not always a feasible option. Weak signals are encountered, e.g., when measuring coherence of fields generated in nonlinear media, in fluorescing experiments, and, most recently, in plasmonic Bose-Einstein condensates [24].
The theory of scanning-type WFIs in the space-frequency and space-time domains is given in Sects. 2 and 3, respectively. Some practical implementations, including the three-mirror setup, are discussed in Sect. 4. In Sect. 5 we compare the polarization-dependent responses of different WFI implementations. In Sect. 6 we apply the three-mirror WFI to characterize the spatial coherence of certain primary light sources that do not obey the Schell model, providing a comparison with results obtained by Young’s interferometer. Finally, conclusions are drawn in Sect. 7.
2. Theory in the space-frequency domain
In the standard WFI configuration a collimated beam is split into two equal parts, which are folded in x and y directions by prisms P1 and P2 and recombined at the detection plane, as shown in Fig. 1(a). The axial optical paths through the two arms are made equal by z-scanning one of the prisms. In addition, the prisms are tilted slightly to produce lateral interference fringes, and the prism corners are imaged onto the detector in module D. An array detector can be used for parallel recording of the fringe visibility and position data, or an imaging spectrograph can be employed as module D to observe spectrally resolved interference patterns.
When the center of the input beam is aligned to the edges of the two prisms, as in Fig. 1(a), the WFI measures spatial coherence between two symmetrically located transverse spatial points, but as a function of their separation only. In this case the device is limited to spatial coherence measurements of fields that obey the Schell model (also sometimes referred to as uniformly correlated fields). In pure spatial-coherence measurements, the axial optical path length between the two arms is to be equalized with a z-scanning translation stage on which one of the prisms is mounted. Longitudinal spatial coherence (or, equivalently, temporal coherence) of the incident field can be studied by z-scanning such a stage, at the cost of losing perfect imaging of the plane of both prims edges on the detector. We note that, as an alternative to tilting the prims, the z-scan dithering method used in, e.g., Refs. [15,16] could be employed.
If one introduces a shear between the optical axes of the incident field E0(x, y; ω) and the optical axis of the interferometers as shown in Fig. 1(b), a shift by an amount (Δx, Δy) leads to interference of light fields between coordinates (x, y′) and (x′, y), where x′ = −x + 2Δx and y′ = −y + 2Δy are the folded and sheared transverse coordinates. Let us assume that prism P1 is tilted by an angle αx in the x direction and P2 by αy in the y direction. Further, if P2 is assumed to be shifted in the z direction by an amount Δz from the equal-path position, the spectral field at point (xd, yd) on the detector can be written in the form
where ϕ(ω) = 2Δzω/c, Cx(ω) = 2 sin αxω/c, and Cy(ω) = 2 sin αyω/c. Here we have neglected polarization effects – for the time being – and all losses apart from the recombination loss at the beam splitter. If the magnification of the imaging system in D is M, we have xd = −Mx and yd = −My.If we define the cross-spectral density (CSD) of the incident field as and its spectral density as S0(x, y; ω) = W0(x, y, x, y; ω), then the spectral density at the detector, S(xd, yd; ω) = 〈|E(xd, yd; ω|2〉, is given by
Using the Hermitian property of the CSD we obtain where ℜ denotes the real part. Introducing the complex degree of spectral coherence of the incident field, gives where Φ0(x, y′, x′, y; ω) = arg {μ0(x, y′, x′, y; ω)}. Therefore, the visibility of the interference fringes around point (xd, yd) is given by Measuring the spectral density from both arms of the interferometer together with the intensity of the interference fringes at frequency ω allows the determination of the absolute value of the complex degree of spectral coherence, |μ0(x, y′, x′, y; ω)|. The phase Φ0(x, y′, x′, y; ω) can be retrieved from the positions of the interference fringes.3. Theory in the space-time domain
The space-time domain coherence properties are characterized by the mutual coherence function (MCF) for stationary fields, which is connected to the CSD via the Wiener–Khintchine theorem
where τ is the time delay. The corresponding normalized quantity is the complex degree of coherence in the space-time domain, defined as where I0(x, y) = Γ0(x, y, x, y; 0) is the spatial intensity distribution of the incident field, g0(x1, y1, x2, y2; τ) is the envelope of γ0(x1, y1, x2, y2; τ), and ω0 is some suitable optical reference frequency (such as the center or peak frequency of the spectrum of the incident field). With τ = 0 (which is equivalent to ϕ(ω) = 0 in the experimental setting), we can determine the complex degree of spatial coherence γ0(x, y′, x′, y, 0) of the incident field by frequency integrating the spectrally resolved measurement results with the Friberg–Wolf theorem [33].If the detector unit D in Fig. 1 is not an imaging spectrograph but a combination of an imaging system and a square-law array detector, the WFI measures the time-integrated intensity, which is equal to
On inserting from Eq. (2) and integrating, we get where is a position-dependent time delay due to tilted wavefronts and τ = 2Δz/c = ϕ(ω)/ω is the time delay caused by the optical path difference. By using the envelope form of the complex degree of temporal coherence defined in Eq. (8), we can cast Eq. (10) into the form where ψ0 [x, y′, x′, y; τ(x, y) − τ] = arg {g0 [x, y′, x′, y; τ(x, y) − τ]} is the envelope phase and we have used |g0 [x, y′, x′, y; τ(x, y) − τ]| = |γ0 [x, y′, x′, y; τ(x, y) − τ]|. The expression (12) represents a modulated interference pattern, where the fringe visibility is given by Because of the term τ(x, y) in Eq. (12), the WFI mixes lateral and longitudinal spatial coherence (or spatial and temporal coherence) to some extent. This is a significant concern for broadband fields only. If the coherence length of the incident field is large compared with the values of τ(x, y) that occur in the WFI measurement (which is true for quasi-monochromatic fields), the space-time domain coherence properties of the field are obtained directly by setting Δz = 0, since then τ(x, y) ≈ 0 in the argument of γ0. For broadband fields, a path-length compensation at each measurement point (xd, yd) is possible by setting the local path delay equal to zero by adjustment of τ0. This can be accomplished automatically using piezoelectric z-scanning. No such procedure was required for characterization of the relatively narrow-band sources considered in our experiments (to be reported Sect. 6). However, it would be required to get accurate spatial coherence measurements for fields with spectra wider than a few tens of nanometers.4. Alternative configurations
Instead of using two Porro prisms, a WFI can also be implemented using a plane mirror and a reflective axicon with a 45° cone angle. Such an arrangement has only a single ‘blind spot’ on the top of the cone of the axicon, but even that can be harmful. This problem is avoided with a modified Michelson interferometer in Fig. 2(a). Here a lens is inserted in one of the arms, one focal length away from the mirror [25], to fold the wavefront in two dimensions. An advantage of this setup over the conventional WFI in Fig. 1 is that the whole interference pattern is visible since no prisms are used. Beam shearing is also possible in this setup across the region where the lens L is diffraction-limited. This setup can be modeled along the lines presented in Sects. 2 and 3. If we assume that the mirror M in the lensless arm is tilted in x and y directions by angles αx and αy, and shifted in the z direction by an amount Δz from the equal-path position, the spectral field at the detector is
Then Eqs. (5) and (12) are replaced by and respectively. It is noteworthy that, because of the lack of prisms, this setup does not necessarily need an imaging system before the detector D. We note in passing that, without lateral shearing, this configuration is attractive for conversion of Schell-model incident beams into specular and anti-specular light beams [26,27]. In [27], diffraction by the prism edges in the setup of Fig. 1 prevented experimental studies of propagation of such beams away from the (secondary) source plane.Fig. 2 Some of the possible configurations for the WFI, (a) Michelson to WFI, and (b) triple mirror. The components use the same notations as before and F is the focal length of lens L.
The option we consider in our experiments is shown in Fig. 2(b). This type of WFI uses two beam splitters and three mirrors, two to fold the wavefront in one arm and one to reflect the original beam in the other arm. By shearing the optical axes of the input beam and the interferometer in the x direction, it is possible to measure the full two-coordinate (x1, x2) CSD or MCF. While the modified WFI shown in Fig. 2(b) can not measure spatial coherence between two arbitrary points in the (x, y) plane, it is possible to design extensions that are capable of doing so. One such implementation (to be discussed elsewhere) employs two beam splitters and six planar mirrors in a three-dimensional setup.
If we assume that the mirror M3 in Fig. 2(b) is tilted in x and y directions and shifted in the z direction, the spectral field at the detector is (using the same notation as above)
Here we have taken into account beam splitter losses arising from the fact that the beam reflected from mirror M3 is split three times, while the beam traversing through mirrors M1 and M2 is split only twice. In this setup the space-frequency and space-time interference patterns read as and respectively. Hence the fringe visibility in the spectral domain is and an expression with obvious replacements holds in the temporal domain. A more transparent form for the visibility of the interference fringes is obtained if we denote the spectral densities from the two arms as and . Substituting these into Eq. (20), yields the visibility in a form analogous to Eq. (6), Thus, by measuring the spectral densities or the temporal intensities from the two arms separately, it is always possible to use them to normalize the interference fringes in the manner of Eq. (21).5. Polarization effects
Thus far we have employed the scalar theory to model various WFI configurations. It is clear, however, that the state of polarization of the incident field should play a role in the observed visibility of the WFI interference pattern, as the passage of the beam through the system involves reflections from prisms or mirrors. Such devices are polarization-dependent, but the electromagnetic extension of the scalar results is quite straightforward with the complex Fresnel reflectance coefficients.
In the analysis of vectorial beamlike fields through any WFI – instead of the scalar CSD or MCF – we need to employ the corresponding 2 × 2 coherence matrices. The CSD matrix has the form
where the superscript T denotes transpose. The quantity observed by the detector is the trace (or the first Stokes parameter) of the spectral polarization matrix Φ(x, y; ω) = W0(x, y, x, y; ω), i.e., Correspondingly, in the space-time domain, one considers the mutual coherence matrix Γ(x1, y1, x2, y2; τ) = 〈E*(x1, y1; t)ET(x2, y2; t + τ)〉 and the quantity observed by the detector is the trace of the time-domain polarization matrix J(x, y, τ) = Γ(x, y, x, y; τ).Let us first consider the WFI configuration in Fig. 1. The x component of the electric field experiences two TE reflections at prism P1 and two TM reflections at prism P2. Similarly, the y component experiences two TM reflections at P1 and two TM reflections at P2. We assume an ideal non-polarizing beam splitter, which does not introduce any polarization-dependent phase changes, though such changes could be included in the analysis. Denoting the Fresnel reflection coefficients at 45° angle of incidence by rTE and rTM, we then have
where we have suppressed the ω dependence for brevity. Hence, by following the derivation in Sect. 2, we obtain where j = x, y, are the normalized spectral correlation coefficients of the incident field and φ0jj (x1, y1, x2, y2; ω) are their phases. The corresponding time-domain results are again obtained by obvious replacements.A similar vectorial analysis can be performed for the WFI configurations in Fig. 2. Considering the three-mirror setup, we denote the reflection coefficients at 45° angle by rTE and rTM, and the reflection coefficient at normal incident by r. Then the expressions for Φxx and Φyy at D read as
It is clear from Eqs. (26) and (27), as well as Eqs. (29) and (30), that if the incident light is linearly polarized in either x or y direction, we can straightforwardly obtain the complex correlation coefficients from the visibility and the positions of the fringes. In the general case of partially coherent and partially polarized light, these two elements can be obtained using polarizers in front of D. However, a full characterization of the coherence matrix also requires knowledge of μ0xy(x1, y1, x2, y2). This can be determined by inserting suitable polarizer-retarder combinations in front of the incident beam entering the WFI and by measuring all of the four so-called two-point Stokes parameters [28–31]; we intend to discuss the details of measuring them with the WFI elsewhere. However, it is clear from Eqs. (26) and (27), or from Eqs. (29) and (30), that the modulation contrast of 𝒮0 does not necessarily reach unity even for fully coherent fields because of the polarization effects in the WFI. The same is true for the modulation contrasts of the other Stokes parameters. Hence it is of interest to compare different implementations of the WFI in terms of polarization effects.Let us consider measurements with the WFI in Fig. 1 around the axial point (xd, yd) = (0, 0) by setting x = x′ = y′ = y = 0 in Eqs. (26) and (27). Since |μ0xx(0, 0, 0, 0)| = |μ0yy(0, 0, 0, 0)| = 1, we have
where ζj(x, y) = φ0jj (0, 0, 0, 0) + Cxx + Cyy − ϕ, j = x, y. Obviously the normalized visibilities of Φxx(0, 0), Φyy(0, 0), and 𝒮(0, 0) are generally less than unity. In particular, the two arms of the WFI rotate the polarization in opposite directions, which can have a large effect on the visibility. Correspondingly, for the three-mirror WFI in Fig. 2(b) we have We note that expressions of this type hold around an arbitrary point (xd, yd) for fully spatially coherent fields with |μ0xx(x1, y1, x2, y2)| = |μ0yy(x1, y1, x2, y2)| = 1.Figure 3 shows a comparison of the polarization modulation properties of various versions of the WFI. Here the normalized visibility of 𝒮0 around the axial point (or for fully coherent incident field) is plotted for various polarization states of the incident field defined by the Stokes vector 𝒮 = [𝒮0, 𝒮1, 𝒮2, 𝒮3]T. In the Porro-prism-based WFI |rTE| = |rTM| = 1 because of total internal reflection (TIR) but the phase delays at TIR cause a significant polarization dependence of the normalized visibility. If the prisms are metal-coated (here with Aluminium), the polarization dependence is reduced, and an even better option is to use metallic corner reflectors instead of prisms. However, the three-mirror setup of Fig. 2(b) is the most robust configuration in terms of polarization dependence. In fact, the polarization dependence of this setup is weak enough to be neglected in most measurements since its effects are usually masked by experimental errors in visibility measurements.
Fig. 3 Normalized fringe visibility, 𝒮0, with different WFI implementations when the input Stokes vector traces a path along the surface of the Poincaré sphere. (a) 𝒮3 = 0 (b) 𝒮1 = 0, where cyan ellipses correspond to right handed and magenta to left handed polarization.
6. Experimental considerations and results
To extract the spatial coherence information with the three-mirror WFI, we recorded three images: the first with both paths open and the other two with only one of the paths open at a time. The latter two measurements were used to normalize the measured visibility in accordance with Eq. (21). What is then left is the cosine-modulated absolute value of the degree of coherence. To remove the cosine term, we employ simple Fourier signal processing techniques [32]. First, we take one line along the direction we are measuring and Fourier transform it. Then we remove the negative frequency components from the signal and inverse Fourier transform back to the original domain. This yields both the phase and the amplitude of the complex degree of spatial coherence.
The coherence of light radiated by the considered sources was measured with a Young’s interferometer and the modified WFI in Fig. 2(b). The Young’s interferometer was realized by using a digital micromirror device (DMD) shown in Fig. 4, which can produce arbitrary mirror formations, allowing the full two-coordinate degree of spatial coherence to be measured (see [8] for details). The measurement speed of this interferometer was ∼ 4 data points per second.
Fig. 4 Young’s interferometer used for reference measurements. The slits are realized using a digital micromirror device (DMD).
A noteworthy feature of the employed setups is the difference in this speed. If we measure a n × n data matrix, the measurement time with the Young’s interferometer scales proportionally to n2, whereas for the WFI it scales to , when the two devices have similar spatial resolution. This allows for a remarkable reduction in the measurement time, which is due to the WFI being able to measure a slice of the correlation function along the antidiagonal at every value of z-scan, instead of a single point. With the components we employed, the interferometer was able to measure a set of points in about 3 seconds. For Schell-model sources the increase in speed is the most dramatic, since no scanning is necessary and all relevant points can be measured at once. For non-Schell model sources the measuring time is longer due to the lateral shearing.
First, we characterized the spatial coherence (at zero path length difference) of a high-power broad-area edge-emitting laser diode (Thorlabs L808P500MM) with center wavelength λ = 808 nm and a bandwidth of ∼ 2 nm. This laser radiates a large number of transverse (and longitudinal) modes, but the radiation is linearly polarized in the direction of the junction (x direction). The intensity profile of the collimated beam has a more or less flat-top (but noisy) profile of width ∼ 2 mm. The measurement results are shown in Fig. 5.
Fig. 5 Absolute value of the complex degree of coherence, |μ0(x1, x2)|, for the emission from a multimode laser diode, measured with the Young’s interferometer (left) and the modified WFI (right). The beam size was ∼ 2 mm.
On comparing the results obtained with the different methods, we see clear similarities. However, the two degrees of coherence are not exactly the same, partly because they were not measured at precisely the same plane of propagation. Because of the complexity of the mode structure and the noisy intensity profile, even a small change in the propagation distance modified the measured coherence function. Still, the source is not exactly of the Schell-model type, as the spatial coherence profile in the antidiagonal (x2 − x1) direction varies with the mean position (x1 + x2)/2. The resolution obtained with the laterally scanning WFI is clearly superior to that obtained with the two-pinhole setup: in the WFI the resolution is limited mainly by the step size of the lateral scan, which was 5 μm, whereas in Young’s interferometer it is limited by the size of the micromirrors, that is 10.8 μm.
The number of measurement points with Young’s interferometer was 151 × 151, and hence the measurement time would be ∼ 1.6 hours, when the whole correlation function is measured. However, the process was sped up by measuring only one half of the data points, which is possible due to the Hermiticity of the correlation function, reducing the required time to ∼ 50 min. With the WFI, the scan was done over 400 lateral points, and the time was ∼ 20 min, constituting a factor of 2.5 increase in the measurement speed with a greater resolution. However, the mechanical parts we employed limited the speed greatly, and thus our measurement setup does not fully utilize the potential of the WFI. It would be quite straightforward to speed up the measurement by using more sophisticated mechanics. Similarly, the DMD could have been faster as well, since it can refresh in the kilohertz range. However, the limiting factor in our setup is the camera exposure time needed to capture the faint interference fringes. Additionally, the resolution of the Young’s interferometer system could be improved by using a DMD with smaller mirrors, but this would in turn increase the exposure time and slow down the measurement.
The second source we consider is a multimode He-Ne laser, which radiates a spatially partially coherent and unpolarized beam. This laser is rather similar to that considered in Ref. [2], with a nearly flat-top intensity profile (exhibiting only smooth fluctuations). The measurement results are presented in Fig. 6. Evidently, the two measurement systems produce essentially the same results, though again the resolution of the WFI is better. Slight differences in the results can again be attributed to the fact that the measurements were not done at exactly the same plane of propagation, although these planes were close to each other. Now the deviation from the Schell model is quite evident, in particular towards the edges of the beam.
Fig. 6 Absolute value of the complex degree of coherence, |μ0(x1, x2)|, for emission from a multimode He-Ne laser, measured with the Young’s interferometer (left) and the modified WFI (right). The beam size was ∼ 2.5 mm.
7. Conclusions
We have introduced modifications to the wavefront-folding interferometer, which remove its main limitations, and demonstrated one variation experimentally by measuring the spatial coherence properties of light fields of different nature. At a fixed lateral scan position, Δx, this modified WFI measures the spatial coherence between all pairs of points along the x direction. The scanning can be realized by either moving the input light or the entire interferometer, so that the beams are sheared over each other. This allows for spatial coherence characterization between any pairs of points (x1, x2) along the wavefront. A notable feature of this type of devices is the speed of measurement, which scales linearly with the number of measurement points along one dimension. If z-scanning is added to one of the interferometer arms, the device can measure the three-dimensional spatial coherence function. The device is, in particular, capable of dealing with non-homogeneous fields that do not obey the Schell model. By adding an imaging spectrograph, it allows for the measurement of space-frequency domain spatial coherence.
In this paper we have considered a prism-less three-mirror version of a WFI, which folds the spatial coordinates along one spatial axis. The disadvantages of retroreflecting Porro prisms include unwanted reflections from prism front surface, the difficulty to manufacture a prism with a precisely 90° angle corner and a sharp enough edge, as well as the polarization modulation effects caused by total internal reflection. These issues are avoided by using planar mirrors. It is also possible to design WFIs that fold the field in two dimensions, thus allowing (with xy scanning) space-time-domain coherence measurements of arbitrary two-dimensional fields. This, however, requires the construction of interferometers with more mirrors, in three-dimensionally arranged settings. One such extension of the presently used setup, involving two beam splitters and six mirrors, is currently under construction. The WFI demonstrated here allows rapid and high-resolution spatial coherence measurements, especially in comparison with standard Young’s two-pinhole techniques. In addition, the different WFI configurations considered here are expected to allow the generation of new varieties of partially coherent light fields with non-uniform spatial coherence properties.
Funding
Academy of Finland (project 285880); Graduate School of the University of Eastern Finland; KAUTE foundation.
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