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Abstract
We examine the spatial coherence properties of supercontinuum fields generated by illuminating rotating bulk media with intense pulsed beams. Theoretical models are presented, which indicate the possibility of generating a class of Bessel-correlated fields (in time-averaged sense) using tilted plane-parallel glass plates and wedges as media for generation of supercontinuum radiation. In special cases, the ensuing fields have a strictly identical functional form in the spatial and angular domains. Some of the main results are verified experimentally by measuring the spatial coherence properties of bulk-generated supercontinuum fields using a wavefront-folding interferometer.
© 2017 Optical Society of America
1. Introduction
The coherence properties of supercontinuum (SC) light are of interest in a broad range of applications [1–3]. The analysis of optical coherence of SC has thus far been limited mainly to fields generated in microstructured optical fibers, since in that case it is possible to simulate the spectral and temporal realizations of SC pulses by means of a nonlinear Schrödinder equation [2]. The two-time and two-frequency coherence properties of fiber generated SC light have recently been clarified in terms of second-order coherence theory of non-stationary light, on the basis of such simulations [4], and the main results have been verified experimentally [5]. Much less attention has been paid to SC generated in bulk media, even though this is a simple experimental task. In particular, the coherence properties of such radiation has received little attention. A notable exception is the work of Zeylikovich and Alfano [6], who studied some of the spatial and temporal coherence of SC radiation generated in bulk media using grating interferometry, and found both to be partial.
In this work we concentrate on the spatial coherence of SC light generated in bulk media and, in particular, on the possibility of generating highly intense (pulsed) broadband fields with partial but well-controlled spatial coherence properties. Such flexible coherence control is not (directly) possible with SC sources based on single-mode fibers, since in this case the spatial coherence is complete in the spatial-frequency domain and remains very high also in the space-time domain [3]. The use of multimode fibers could be an option that could be worth looking into, but in that case the choice of available spatial coherence functions would depend on the mode structure of the particular fiber. We therefore propose a scheme based on a rotating glass plate or a wedge to generate SC pulse trains with reduced and controlled spatial coherence properties if a sufficiently long time average (over at least one rotation period), is taken. We add a new tool to the existing arsenal of methods to generate partially coherent beams [7, 8] or, in fact, pulsed beams in our case. We concentrate on so-called Bessel-correlated fields [9–14] and on some of their variants, which are to be discussed below. We verify some of the main theoretical findings experimentally using a wavefront-folding interferometer (WFI) [15, 16] to measure the spatial coherence properties of bulk-generated SC fields.
In Sect. 2 we present the principle of generating partially coherent SC fields in rotating plane-parallel glass plates and wedges, and derive the coherence properties of the ensuing fields in the spatial domain. The angular coherence properties of such fields are discussed in Sect. 3, and a class of ‘self-transforming’ Bessel-correlated fields is introduced in Sect. 4. Numerical illustrations are provided in Sect. 5. Some theoretical considerations are given in Sect. 6 concerning the pulsed nature of the beams in our experiments. The experimental setup and the results that confirm our main theoretical predictions are presented in Sect. 6. In Sect. 7 we discuss some possible extensions of the work, and finally conclude the paper in Sect. 8.
2. Source models in the space–frequency domain
Let us consider the geometries in Fig. 1. Here a completely coherent Gaussian beam, centered at the red line (principal ray), propagates from the left towards a rotating glass plate or wedge. If the beam is sufficiently intense, SC light is generated in the medium, but the theoretical analysis applies to any incident Gaussian beam. In the geometry of Fig. 1(a), the principal ray is deflected laterally such that it draws a circle of radius r when the plate is rotated. In Fig. 1(b) the exit point of the principal ray is the same for all rotation angles of the wedge, but the exit angle with respect to the z-axis is β, and hence the output defines a cone when the wedge is rotated. In the final geometry of Fig. 1(c), where the orientation of the wedge is reversed, we have the same deflection angle β but the exit point also draws a circle of radius r.
Fig. 1 Different mechanisms that produce Bessel-correlated beams. (a) A rotating glass plate tilted at an angle α. (b) A rotating glass wedge with the front surface perpendicular to the propagation direction of the incident field. (c) The same glass wedge, but with the back surface perpendicular to the propagation direction. The geometry after a rotation of 180 degrees is shown shaded.
At any individual instant of time the output field is a single Gaussian beam, which is deflected spatially or angularly (or both) with respect to the incident beam. This implies that the spatial and angular coherence properties of the output field become partial whether the illuminating field is a continuous-wave or pulsed beam. With a constant rotation speed of the plate/wedge, all realizations with different rotation angles are equally probable (in the case of pulsed illumination we assume a train of pulses with a constant repetition rate). Hence we may characterize the spatial (and angular) correlations by evaluating ensemble averages of mutually uncorrelated realizations of the output beam, taken over a single rotation period. Let us first consider such averaging processes in the space-frequency domain, i.e., determine the resulting forms of the cross-spectral density function (CSD) [17]. For brevity of notation, we drop the frequency dependence of all quantities until Sect. 6.
2.1. Plane-parallel plates
In the case of Fig. 1(a), it is convenient to model the spatial coherence of the output field at the exit plane of the element by using the shifted-elementary-beam theory [18–21], which expresses the CSD in the form
where p(x′, y′) is the weight function of mutually identical elementary fields e(x, y) centered at positions (x′, y′). In what follows, it will be convenient to introduce cylindrical coordinates (ρ′, ϕ) by writing x′= ρ′ cos ϕ, y′ = ρ′ sin ϕ, which leads to In view of the model for plane-parallel plates we take the weight function to be a delta ring of radius r, which gives Since the incident field is a Gaussian beam, the elementary field can be written in the form and we arrive at where S0 = |e0|2. The integral can be easily evaluated and the result is where I0 denotes the modified Bessel function of the first kind and order zero. Hence the spectral density (intensity) of the source field is of the form and the complex degree of spectral (spatial) coherence is given by This is a real-valued function, and its nominator depends on the average spatial position of the observation points as well as on the ratio r/w0.2.2. Wedges
In the geometry in Fig. 1(b), the field exiting the wedge can be modeled as an incoherent superposition of elementary contributions of the form
where ϕ is the azimuthal rotation angle of the wedge. These are Gaussian beams originating from the optical axis and propagating at an angle of inclination β according to the model. If the speed of rotation is constant, all of these contributions have an equal weight and the resulting CSD is The integral can be evaluated straightforwardly and the result is which represents a Bessel-correlated Schell-model source [9] with a Gaussian distribution of spectral density.Considering finally the geometry in Fig. 1(c), in which the elementary contributions form an angle β with respect to the optical axis and originate from a ring with radius r, we can express the elementary contributions in the form
On inserting this expression in Eq. (11) and integrating, we obtain with and we have introduced average spatial coordinates and . The spatial intensity distribution takes the form of Eq. (8) regardless of the value of β. As is also physically clear by consideration of the geometries in Fig. 1, Eq. (14) reduces formally to Eq. (12) in the limit r → 0 and to Eq. (7) in the limit β → 0. The source-plane intensity distribution reduces to that given by Eq. (8) for any value of β, but the complex degree of spatial coherence is a complex-valued function that depends on both r and β.3. Fields in the angular domain
The angular coherence properties of a spatially partially coherent field are characterized by the angular correlation function (ACF)
where (kx1 + ky1) and (kx2, ky2) are the spatial frequencies that define the observation directions. If the source-plane CSD is of the form of Eq. (11), the ACF is given by where is the angular spectrum of the elementary-field contribution e(x, y, ϕ).Let us consider an elementary field of the form of Eq. (13) with the understanding that the cases pertaining to Figs. 1(a) and 1(b) can be dealt with by inserting β = 0 and r = 0, respectively, in the final results. The Fourier transform of this field is
where . On performing the integration in Eq. (17) we obtain where and In writing Eq. (21) we have introduced the Rayleigh range of the Gaussian elementary field, and denoted the center and difference spatial-frequency coordinates by , and Δkx = kx2 − kx1, Δky = ky2 − ky1, respectively. Furthermore, the angular intensity distribution takes the form regardless of the value of r.If we set β = 0 to consider the case of a rotating plane-parallel plane in Fig. 1(a), we find that the angular intensity takes the Gaussian form
and the complex degree of angular coherence becomes Hence the angular field produced by a rotating plane-parallel plate is of pure Bessel-correlated Schell-model form with a Gaussian intensity profile, and therefore complementary with the field produced by the rotating wedge in Fig. 1(b). On the other hand, setting r = 0 in Eq. (20) leads to the result which has a functional form similar to Eq. (7).4. Self-transforming fields
The CSDs defined in Eqs. (14) and (15) have a remarkable resemblance with the ACFs defined in Eqs. (20) and (21). To investigate this similarity further, we begin by comparing the expressions (8) and (22) for the field intensities in the spatial and angular domains, which hold for any combination of the parameters w0, r, and β. Normalizing the spatial coordinates to the ring radius r by writing x̃ = x/r and ỹ = y/r, we may cast Eq. (8) into the form
The angular domain radius of the cone defined by the output rays in Fig. 1(c) is k0 sin β. If we normalize the spatial frequencies to this radius by writing kx̃ = kx/k0 sin β and k̃y = ky/k0 sin β, Eq. (22) takes the form The similarity of Eqs. (26) and (27) is striking. Let us next suppose that the parameters w, r, and β are chosen such that the condition or, equivalently, holds. In this case the field has identical functional forms of intensity distributions in the spatial and angular domains. Fields of this type can be realized by choosing the thickness and the wedge angle of the wedge in Fig. 1(c) appropriately.Using the normalized coordinates introduced above and introducing dimensionless parameters r̃ = r/w0 and c = (zR/r) sin β, we can cast Eq. (14) in the form
where we have denoted Δx̃ = Δx/r, Δỹ = Δy/r, x̃ = x̄/r, and ỹ = ȳ/r in accordance with previously introduced notation. Correspondingly, Eq. (20) can be written in the form where Δk̃x = Δkx/k0 sin β, Δk̃y = Δky/k0 sin β, k̃x = k̄x/k0 sin β, and k̃y = k̄y/k0 sin β. In view of Eqs. (30) and (31), it is obvious that if the condition implied by Eq. (29), i.e., c = 1, is satisfied, not only the intensity distributions but also the coherence properties of the field have the same functional form in both spatial and angular domains. Hence, it seems appropriate to talk about self-Fourier-transforming fields in the present context.5. Numerical illustrations
To illustrate the character of the type of fields introduced above, it is useful to consider cross-sections y1 = y2 = ky1 = ky2 = 0. Figure 2 exhibits some source-plane and angular intensity distributions with different values of the parameters r and sin β. In these intensity plots the normalization factors S0 and T0 have been chosen equal to unity for clarity of illustration, although the requirement that the beam energy is equal to unity for all values of r would require and . The intensity distributions at the source plane depend (in the functional form) only on the ratio r/w0. It is evident from Fig. 2(a) that a ring-like source profile begins to be formed when this ratio exceeds unity, as one may expect on geometrical grounds. Correspondingly, an increase of the value of sin β leads to the formation of a ring pattern in the spatial-frequency domain. With the value w0/λ = 5 chosen here, this begins to occur at around sin θ = 0.05. However, if w0/λ is increased, the elementary contributions become more directional and the ring starts to form at smaller values of sin β. In general, this occurs when sin β > λ/πw0, which is (the sine of) the far-field divergence angle of the elementary field.
Fig. 2 (a) Cross sections of the intensity at the source plane with r/w0 = 1 (black), r/w0 = 0.75 (red), r/w0 = 1.5 (green), and r/w0 = 3 (blue) and (b) the corresponding cross sections of the degree of spatial coherence. (c) Cross sections of the angular intensity distribution with the same values of r/w0, with c = 0.75, and (d) the corresponding cross sections of the degree of angular coherence.
Figure 3 shows some selected distributions of the complex degrees of spatial and angular coherence. The parameters can be chosen in such a way that a distinct ring-like intensity distribution is observed both at the source plane and in the spatial-frequency domain. The source-plane distribution of the complex degree of spatial coherence shows an hour-glass pattern, which is tilted at 45° in the (x1, x2) coordinates but would be upright if plotted in (Δx, x̄) coordinates. Thus, the field at the source plane has highly non-uniform correlations, i.e., it does not obey the Schell-model even closely. At small values of x̄ the coherence width (the effective width of |μ(x̄, Δx) in the Δx direction) has a minimum, and it grows steadily when |x̄| increases. In the space-frequency domain, the absolute values of the complex degree of angular coherence also exhibits an hour-glass pattern.
Fig. 3 (a) The absolute value and (b) the phase of the complex degree of spatial coherence at the source plane. (c) The absolute value and (d) the phase of the complex degree of angular coherence. The parameters are chosen as c = 0.75 and r̃ = 1.
6. Experiments with supercontinuum pulse trains
In the experiments we consider the case of a rotating plane-parallel plate illuminated with a pulse train from a femtosecond laser. The pulses are focused weakly inside the plate such that the beam diameter is nearly constant inside it, as illustrated in Fig. 4. The pulses are strong enough to generate supercontinuum radiation inside the medium, which therefore acts as a source of broadband light with a nearly Gaussian intensity distribution at each frequency. When considering SC light, we must take into account the possibility that the beam width w0 becomes a function of temporal frequency ω, a fact that has been left implicit thus far. To measure the spatial coherence properties of the SC field we employ a wavefront-folding interferometer and consider a time integral over many pulses in the train.
Fig. 4 Experimental setup for supercontinuum generation inside a rotating glass plate. Intensity distributions were measured both in the far field of the rotating glass plate and in its (magnified) image plane. The bottom row shows images of the source plane ring, with (a) α ≈ 2.0° and r ≈ 36 μm, (b) α ≈ 1.7° and r ≈ 29 μm, (c) α ≈ 1.2°1 and r ≈ 22 μm, (d) α ≈ 0.5° and r ≈ 9 μm. Here WP is a half-wave plate, PBS is a polarizing beam splitter, S is an aperture stop, L1 and L2 are achromatic lenses and F is a low-pass filter. Black & white pictures are shown to illustrate frequency-integrated intensity distributions.
We used 3 mm thick and 1″ diameter SiO2 plates as the bulk medium and focused highly coherent, near-Gaussian pulses of 120 femtosecond FWHM duration at 1 kHz repetition rate to generate supercontinuum radiation. The silica plate was attached to a rotating holder. This rotation served two purposes: it modulated the spatial coherence and increased the longevity of the plates. Although the pump pulse train had quite a low average power of about 5 mW, the peak power was more than enough to ablate the surface of the silica glass when it was exposed to a continuous train of pulses on a single focused spot. The ablated surface scatters the pump pulses, which also decreases the efficiency of continuum generation. To avoid this, we did not focus the beam at the rotation axis of the plate, but somewhat off-axis and rotated the plate so that the focused spot drew a ring on the plate. Hence the actual setup differed slightly from the geometry of Fig. 1(a). However, because of the small value of α, this does not have a significant effect. The output beam was filtered to remove the spectral peak due to the pump pulses and subsequently collimated (in the far field) to observe the angular correlation function.
6.1. Spectral intensity measurements
Figure 5 illustrates the spectral measurements in the far field, which were carried out using a fiber spectrometer and an integrating sphere. From these figures, we can see that the majority of the supercontinuum is in the visible range and cutting the longer wavelengths with a low-pass filter at a cut-off wavelength of 750 nm does not remove a significant portion from it. Apart from Fig. 5(b), all of the experiments reported below were performed with the low-pass filter in place, so that we would measure the newly generated light, excluding the pump pulses.
Fig. 5 (a) A color photo of the far field radiation pattern. A supercontinuum spectrum measured with an integrating sphere without (b) and with (c) a low-pass filter to remove the residual pump beam. (d) Filtered radial spectra measured by scanning the spectrometer fiber across the far-field pattern (without the integrating sphere). In (b) the peak at λ ∼ 800 nm, caused by residual pump pulses, is truncated.
We believe that the distinct concentric rings in Fig. 5(a) are caused by the conservation of momentum in Stimulated Raman Scattering (SRS) [22]. The wave vectors of the Stokes and Anti-Stokes scattered light are coupled to each other and to the vibrational modes of the molecules, such that the total wave vectors for the Stokes scattered light points in the direction of the pump, and the wave vectors associated with Anti-Stokes scattered light point at a slight angle relative to the pump. Therefore, we observe distinct concentric rings in the output emission and the peculiar spatial intensity distribution gives motivation to measure the radial spectra. We see from Fig. 5 that most of the white light is in the center of the emission, which is not caused by SRS. Additionally, by changing the diameter and average power of the pump beam, we can observe changes in the concentric rings, as well as other effects that we will not discuss in detail here. This would suggest that there are several distinct continuum generation mechanisms (as is the case in SC generated in fibers [2]). By engineering the excited volume in the bulk medium, we can observe transitions between different regimes.
In view of Eq. (23), the simplest way to investigate the frequency dependence of w0 appears to be spectral measurement of the width of the angular intensity distribution, which is done in Figure 6. Let us introduce a diffraction angle θ with respect to the optical axis by writing the spatial frequencies in the form kx = (ω/c) sin θ cos φ, ky = (ω/c) cos θ sin φ, where φ is the azimuthal diffraction angle. Then Eq. (23) takes the form
where S0(ω) is the on-axis power spectrum at the source plane and the angle θ0 defined by is the characteristic far-field diffraction angle of the SC field at frequency ω. Hence we can determine w0(ω) by fitting a Gaussian curve in this distribution.Fig. 6 (a) Color photo of the near field radiation pattern and (b) the corresponding RGB channel cross sections. (c) An example of a Gaussian fit to experimental data at a single wavelength. (d) Dependence of θ0 on λ and the resulting dependence of w0 on λ according to Eq. (33).
Figure 6(a) shows a color photograph of a single elementary field at the source plane without rotating the plate, and Fig. 6(b) shows the beam profiles at the RGB channels of the camera. In view of Figs. 6(c) and 6(d), the fitted values of w0 qualitatively agree with the widths of the RGB channel responses. The decrease of the beam width at shorter wavelengths can be attributed to the pump beam spatial intensity distribution. Wavelengths close to the pump are generated with relatively low pump intensities, whereas shorter wavelengths require higher power. Thus, the lower-intensity regions at the edges of the pump beam produce almost exclusively light in the red region, while the center with high intensity produces the majority of the supercontinuum.
6.2. Spatial coherence measurements
We employed a wavefront-folding interferometer illustrated in Fig. 7 for spatial coherence measurements. The WFI is a light-efficient interferometric instrument for measuring the coherence of light fields that obey the Schell model and certain symmetry conditions [23]. By employing retroreflector prisms, this device allows all of the spatial points along the difference coordinate to interfere with each other at the observation plane. The visibility of the resulting interference fringes then provides information about the spatial coherence of incident light in the measurement direction. Since the theoretical treatment presented earlier for rotating plane-parallel glass plates predicts that the desired output is of the Schell-model form in the far zone, the use of a WFI in the present form is justified in our experiments.
Fig. 7 Experimental setup for spatial coherence measurements, showing the WFI with a collimated input. Light is incident from the top to a 50 : 50 beam splitter BS, which divides the input between a knife-edge prism (KEP) and knife-edge prism on a piezo stage (P+KEP). Two electronic shutters, ES1 and ES2 were used to block one arm at a time and an imaging lens L was employed to image the edges of the prisms on the detector.
The instrument is implemented in much the same way as in our previous work [24]. Often it is sufficient to use only one right-angle prism as a retroreflector and a plane mirror in the second arm of the interferometer. However, since we are dealing with short pulses and the glass medium introduces group velocity dispersion, we use prisms in both arms to introduce the same amount of dispersion in both copies of the pulse. Additionally, because of the short duration of the individual pump pulses, it was necessary to equalize the optical paths through the two arms of the WFI at a high accuracy to ensure that the recombined SC pulses overlap temporally. In addition to using a manual translation stage, one of the prims was attached on a piezoelectric stage with a maximum movement of 100 μm and the smallest step size on the order of tens of nanometers. By moving this stage, the visibility of the fringes in the central region of the pattern was maximized (by equalization of the axial optical paths) before carrying out the experiments. Nevertheless, it proved difficult to keep the setup perfectly stable over a sufficient time period for the experiments, which resulted in a decrease of the fringe visibility due to path length imbalance in the interferometer. However, this only causes a systematic overall effect, which could be compensated by scaling the visibility to unity at zero spatial separation of the measurement points.
Recent simulations and experiments [3–5] on SC generation in optical fibers have revealed that the CSD of a supercontinuum pulse train is generally a complicated function of two frequencies ω1 and ω2, and hence the corresponding mutual coherence function (MCF) depends in a sophisticated way on two instants of time t1 and t2. In general, the CSD and the MCF are related by the generalized Wiener-Khintchine theorem for non-stationary light, which reads as
In spatial coherence measurements we are interested in the equal-time MCF, i.e., Γ(x1, y1, x2, y2; t, t), and in practice we perform a time integrated measurement over numerous pulses in the train. Hence the result of our measurement is the time-integrated MCF which is independent on the two-frequency or two-time correlation properties of the field. In the experiments to be presented below we measure coherence in the far field of the source, thus effectively considering the time-domain angular correlation function In particular, we measure the amplitude and phase of the time-integrated degree of angular coherence by observation of the visibility and spatial position of the interference fringes in the wavefront-folding interferogram. Such fringes are seen by tilting one of the prisms (or both) slightly, so that the reflected wavefronts are at a small angle with respect to each other.As illustrated in Fig. 7, the far-field radiation pattern is collimated before it enters the WFI. By measuring also the intensity profile in the far field (by blocking one arm of the WFI), we then construct the normalized quantity
i.e., the time-integrated complex degree of angular coherence of the field. We also point out that the spectral response of the detector is not uniform over the entire SC spectrum. Since we do not compensate for this spectral response in any way, we do not measure the genuine complex degree of coherence as defined in Eq. (38), but instead the “physical degree of coherence,” as seen by the detector.Figure 8 illustrates the measured coherence of the SC generated when the glass plate was not rotating. We assumed in Sect. 2 that the each elementary field in the superposition is completely coherent, and the results illustrated here confirm this assumption for SC generated in bulk medium. Since we employ retroreflectors in both arms of the interferometer, we can measure spatial coherence along both x and y axes. In Fig. 8(b) we show the absolute degree of coherence determined directly from fringe visibility in the two perpendicular directions. The results are slightly different because of the short pulse duration. In the vertical direction, the small tilt of the interfering wave fronts was introduced to observe the fringe pattern, which results also in a (linear) position-dependent time difference. This results in an apparent decrease of the degree of spatial coherence since the mutually delayed pulses do not perfectly overlap temporally. Hence the measurement in the horizontal direction gives a more reliable result. However, because of short pulse duration, the measured degree of coherence does not reach unity even at zero spatial separation (although theoretically it should). This occurs because of the difficulty to maintain the path lengths along the two arms of the WFI precisely equal over the time period required for carrying out the measurements.
Fig. 8 Normalized interference-fringe visibility with a glass plate at a fixed position. (a) The normalized interference pattern and (b) the absolute value of the degree of coherence determined along the vertical and horizontal axes.
Figure 9 shows measured (normalized and time-integrated) interference patterns when the glass plate rotates at different tilt angles. It shows that the output field is (at least approximately) Bessel-correlated, and that the tilt angle changes the width of the correlation function, as the theory predicts. By using Hilbert transform [25], we extracted the absolute value of the degree of coherence from the interferogram images as fringe pattern envelopes. This method would in principle also reveal the phase of the coherence function, but the imperfections of the retroreflecting prisms causes large distortions. Compensation of the phase measurement errors requires more fine-tuning, which we will not cover here. To get a more quantitative understanding of how well the measured values agree with the theory, we take cross sections from the measured values along the horizontal axis, and fit Bessel functions to the measured points. The fitted curves in Fig. 10 correspond remarkably well to the measured points.
Fig. 9 Measured interference patterns (a)–(d) and the corresponding absolute degrees of spatial coherence (e)–(h) with a glass plate rotating at tilt angles corresponding to Figs. 4(a)–(d).
Fig. 10 Measured (non-scaled) values of the degree of spatial coherence, with (a)–(d) corresponding to the interference patterns in Figs. 9(a)–(d). The solid curves are fitted Bessel functions to the measured points. The measured values are shown for (a) and (c), with others omitted for clarity.
7. Discussion
Supercontinuum generation in optical fibers has gained considerable attention in recent years and it is now relatively well understood. Although the cost of the required equipment is lower and the overall setup is simpler when generating a continuum in bulk media, the coherence of such radiation has been poorly studied both theoretically and experimentally. This dilemma can be attributed to the favorable properties of photonic crystal fibers (PCFs) and the possibility of constructing and (numerically) solving an analytical equation to describe the non-linear dynamics of SC generated in them. Experiments and simulations in different studies have shown that with suitable pump pulses, it is possible to produce spatially, temporally, and spectrally highly coherent supercontinuum radiation in such a non-linear fiber. Especially the spatial coherence of radiation generated in single-mode PCFs is always high. However, as we have demonstrated in this paper, SC generation in bulk medium offers new possibilities for spatial coherence control. It is possible to extend the coherence modulation technique described here, e.g., by using rotating diffractive elements in the spirit of Ref. [26]. The temporal and spectral coherence properties of bulk-generated SC remain to be studied in detail.
8. Conclusions
In conclusion, we have introduced simple model sources for experimentally generating Bessel-correlated fields, as well as a class of self-Fourier-transforming fields, which have the same functional form in the spatial and angular domains. The experimental studies of supercontinuum pulse trains generated in rotating plane-parallel fused silica plates confirm the main theoretical predictions. We stress that the method is not restricted to supercontinuum fields, but is applicable to any spatially coherent light beam with a Gaussian intensity profile.
Funding information
Academy of Finland (285880); Strategic funding of the University of Eastern Finland (930350).
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