A general formulation is presented for spatial shaping of single-spatial-mode broadband pulsed beams with partial spectral and temporal coherence properties. The model is based on the second-order coherence theory of non-stationary fields, and examples are presented on Gaussian to flat-top transformations. Spatiotemporal intensity distributions are evaluated in the target plane of the shaping element for idealized Gaussian Schell-model pulse trains and for realistic supercontinuum pulses.
© 2015 Optical Society of America
Spatial shaping of spatially coherent, stationary, and quasimonochromatic light beams is possible using either free-form refractive optics or diffractive optics [1–3]. The beam shaping elements are often designed using geometrical map transformations  either to obtain the final result  or to obtain a starting point for further refinement using iterative wave-optical algorithms . The effects of partial spatial coherence  and broadband illumination [6–8] in the performance of such beam shaping elements have also been studied.
If pulsed illumination is applied to elements designed to convert Gaussian beams into flat-top beams, the spatiotemporal intensity profiles in the target plane have been found to bend [6, 7]. It is possible that the spatial profile observed in the target resembles the desired profile at no single instant of time, even though the time-integrated spatial profile would be of high quality. This, of course, can have serious consequences in time-resolved experiments in ultrafast optics . The effect has an intuitive geometrical explanation based on the character of the map-transform design method: the optical path of a ray from a point of origin in the input plane of the element to the point of arrival at the target plane varies with transverse position, and hence the ‘flight time’ of the pulse becomes position-dependent. If the geometrical time delay between two pairs of input and output points is of the order of pulse duration or larger, the pulse may have passed one point in the target plane before it arrives at another point, or vice versa.
In Refs. [6, 7] only spectrally fully coherent pulses were considered. However, ultrafast pulse trains can also be partially spectrally and temporally coherent in the sense of second-order coherence theory [10–12]. This is the case for, e.g., excimer and free-electron lasers , and for supercontinuum (SC) light generated in fibers . In the latter case the second-order coherence functions can be constructed directly from numerical simulations, and the spectral and temporal coherence properties may vary widely depending on the generation conditions of the SC pulse train [14, 15]. The purpose of this paper is to study the effects of partial spectral (and temporal) coherence in spatiotemporal target-plane distributions of beam shaping elements.
We begin by presenting a general theory for spatial shaping of partially coherent pulse trains without assuming any specific form for the beam-shape transformation or for the spectral correlation function. We assume a single-spatial-mode pulse train and make use of the coherent-mode expansion of the cross-spectral density function (CSD) to characterize the spectral coherence properties of the pulse train . We apply the method to Gaussian to flat-top transformation assuming first an idealized model, an isodiffracting Gaussian Schell-model pulse train , for which the coherent modes of the CSD are known analytically . We then consider realistic SC pulse trains, for which the coherent modes can be determined numerically by starting from simulated field realizations . These pulse trains are representative examples of non-stationary fields that can be adequately characterized only by considering their partial spectral and temporal coherence.
2. General theory of spatial shaping of pulsed beams
Referring to the geometry of z 1, we represent the space–frequency-domain field at the input plane of the beam shaping system in the form U0(x; ω) = a(ω)V0(x; ω), where a(ω) is a complex-valued random function of frequency ω and V0(x; ω) is a deterministic function of position x, which may depend on ω (we assume a y-invariant geometry for brevity of notation). The spectral second-order coherence properties of this field are fully determined by the two-frequency CSD, given byEq. (1) is separable in spatial coordinates x1 and x2, it represents a spatially fully coherent field at any single frequency. However, at this stage, we allow the field to have arbitrary spectral (and therefore also temporal) coherence properties.
In writing Eq. (1) we have assumed a certain degree of space-frequency separability of the optical field in the sense that the field is random only spectrally, not spatially. More generally, one could consider light fields that are both spectrally and spatially partially coherent. However, the assumption involved in Eq. (1) is valid for a number of important light sources including supercontinuum pulse trains generated in single-mode nonlinear optical fibers .
In the thin-element approximation we can describe the optical function of the beam shaping element by a deterministic complex-amplitude transmission function
The linear system with response function K(u, x; ω) illustrated in Fig. 1 transforms the incident field U0(x; ω) into an output field U(u; ω) = a(ω)V(u; ω), where
Hence the CSD in the output (target) plane is given by
In this paper we employ the geometrical map-transform approach  to design the beam shaping element. This analytical method is applicable only in special cases, such as the Gaussian to flat-top transformation considered here. The design of beam shaping elements for target patterns of arbitrary shape can be performed using the Iterative Fourier Transform Algorithm (IFTA), which is discussed in detail in . The most often-used system in beam shaping is the 2F Fourier-transforming system with an achromatic lens of focal length F, for which
The spatiotemporal properties of the field in the target plane are fully specified by the two-time mutual coherence function (MCF), which may be calculated from the two-frequency CSD using the generalized Wiener–Khintchine theorem for non-stationary light,
The space–time intensity distribution is then given by I(u, t) = Γ(u, u; t, t), the spatial coherence in the time domain is characterized by the function Γ(u1, u2; t, t), and the temporal coherence properties of the field at any spatial position u are defined by the function Γ(u, u; t1, t2).Eq. (9) leads to
Hence, e.g., the space–time intensity profile in the target plane is
Using Eq. (13), we could also study the time-domain spatial coherence in the target plane by examining the function Γ(u1, u2; t, t), or the spatial variations of the two-time temporal coherence by studying the function Γ(u, u; t1, t2).
Some symmetry properties of the general beam shaping problem in the 2F geometry are presented in Appendix A. In the following examples, however, we consider the free-space beam shaping problem with K(u, x; ω) given by Eq. (8).
3. Gaussian Schell-model pulses
Let us first examine the general character of the influence of partial spectral coherence of the incident field in the spatiotemporal properties of the field at the target plane by considering a convenient mathematical model for the two-frequency CSD, known as the Gaussian Schell model (see, e.g., Refs. [12, 19, 20]). In this model the spectral CSD is expressed in the form10, 12]
The spatial profile of the incident field is assumed to be the waist of the fundamental Gaussian mode of a spherical-mirror resonator, which is of the form 8]
Here a is a parameter that characterizes the intended half-width of the flat-top profile in the geometrical-optics framework used to derive Eq. (23). We use this phase function exclusively in the present paper, noting that it can be realized physically as either a diffractive or refractive phase plate. However, the main results given below, concerning the effects of partial spectral and temporal coherence, can be directly applied to more complicated beam shaping problems.Eq. (2) into the form
The first exponential term in Eq. (26) describes a thin dispersive lens with focal length F(ω) = Δz/D(ω) and the second term is responsible for beam shaping. Defining also a normalized frequency , Eq. (4) may be written in the form
Hence, if dispersion can be ignored, the third exponential term inside the integral vanishes. However, this assumption is not valid for wideband pulses.
In Fig. 2 we illustrate the space–frequency and space–time profiles of Gaussian Schell-model pulse trains in the target plane. Here the central wavelength of the pulse train is chosen as 800 nm (i.e., ω0 ≈ 2.36 × 1015 Hz) and the dispersion data of Polycarbonate  is used for D(ω). The propagation distance is Δz = 0.5 m and the incident beam diameter is w0 = 0.3 mm, which gives a Rayleigh range zR ≈ 0.353 m. Hence, from Eq. (24), the diffraction-limited spot size in target plane is w ≈ 0.21 mm. In all cases we choose Q = 20, which implies a target spot half-width Qw ≈ 4.2 mm, ensuring that we are well within the paraxial domain. We keep the spectral width of the pulse train constant at Ω = 6 × 1013 Hz, which implies that in the fully coherent case we have an incident Gaussian pulse with axial temporal half-width of T = 2/Ω ≈ 33 fs. Figure 2(a) illustrates the space–frequency profile in the target plane. In Fig. 2(b) and Media 1 we vary the ratio Σ/Ω to reveal the effects of partial spectral coherence in the spatiotemporal profiles.
Figure 2(a) shows that good-quality spatial flat-top profiles (with smoothed-out edges and some frequency-dependent width variation) are obtained throughout the effective spectral extent of the pulse trains. In view of Fig. 2(b), the space–time distributions become bent, implying that the axial part of the pulse is seen first at the target plane and the on-axis pulse has in fact passed the target plane before any significant contributions arrive at the edges of the flat-top profile. Reduction of the spectral coherence implies a widening of the temporal profile throughout the flat-top region, as one would expect since also the temporal width of the incident pulse increases when the degree of spectral coherence is reduced.
The temporal evolution of the space–time profile is investigated more quantitatively in Fig. 3 and Media 2. Here the parameters are as described above but we have fixed the ratio Σ/Ω = 0.5. We see that the spatial profiles at different instants of time are very different, and never resemble of flat-top. At first we see light only in the central part of the intended flat-top profile, whereas a double-peaked profile is observed towards the end. This general behavior is in agreement with the results of . However, we stress that the nature of the temporal evolution depends critically on the choice of the parameters of the beam shaping geometry. In Appendix A we show that a simple change of the sign of the phase function would cause perfect target-plane time-reversal for real-valued incident pulses in a 2F geometry, i.e., one could also first observe contributions at the edges of the flat-top region. Similar behavior can also be observed in free-space beam shaping, although no simple rule has been found as to when this happens. Generally, however, such reversal takes place when the intended flat-top region is substantially narrower than the incident beam size, in which case the frequency-domain field V(u; ω) exhibits converging (aberrated) phase fronts in the target plane.
These results illustrate the build-up of the final time-integrated profile in the target plane, which would be recorded by a ‘slow’ detector once the pulses has passed by completely. The black line in Fig. 3, which represents I(U), is indeed a high-quality flat-top profile.
The effect of varying the expansion parameter Q in the space–time profiles is illustrated in Fig. 4 and Media 3. Here the parameters are the same as above, and we consider a partially coherent case Σ/Ω = 0.5. We see that the temporal bending of the space–time profile increases rapidly with Q. Finally, Fig. 5 and Media 4 show how the pulse shape changes upon propagation.
4. Supercontinuum pulse trains
As a second example we consider the shaping of supercontinuum (SC) pulse trains generated in microstructured fibers. Individual realizations A(ω) of such SC pulses can be constructed by numerical simulations [13, 23], and the two-frequency CSD14, 15]. The coherent modes ϕm(ω) and their weights αm in Eq. (10) can then be found by numerical solution of the Fredholm equation (11) as described in . Depending on the excitation conditions and the length of the fiber, the spectral coherence properties of the SC pulse train can vary over a wide range. Under certain conditions one can obtain a train of virtually identical pulses, and one may speak of a quasi-coherent pulse train. In other conditions, trains of pulses of widely varying properties are observed, leading to low degrees of spectral and temporal coherence. In this case individual pulses are nearly statistically independent and we may speak of a quasi-stationary pulse train.
Since the realizations An(ω) are known for SC pulse trains, an alternative (more direct) approach is also available. We may first evaluate the space–time realizations in the target plane using a formula analogous to Eq. (14):
Then the space–time intensity distribution in the target plane is obtained as an ensemble average
For sufficiently large N, the result must be same as with the eigenmode method. This approach was adopted in the example that follows.
We consider a pulse train with ‘intermediate’ coherence properties, which exhibits simultaneously quasi-coherent (qc) and quasi-stationary (qs) contributions. These two can be roughly separated in both spectral and temporal domains as shown in . The qc contribution typically occupies more limited spectral and temporal regions than the qs contribution, as shown in Fig. 6 for picosecond pulses in the near-infrared region.
Some results for space–frequency and space–time profiles of SC pulses in the target plane are shown in Fig. 7. We assume that the incident field is of the isodiffracting form. This is not exactly valid for microstructured fibers, but the consequences of the exact choice of V(x; ω) are not too significant if we expand the beam well beyond the diffraction limit in the target plane (Q ≫ 1). In Fig. 7 we assume the geometrical parameters w0 = 0.3 mm and Δz = 0.5 m, and consider the case Q = 10. Despite of the wide bandwidth, the spectral profiles are still good approximations of flat-top profiles over the entire spectral range. The spatiotemporal profiles show the same type of bending that those of Gaussian Schell-model pulses. However, since we consider here picosecond rather than femtosecond pulses, the bending (relative to the total pulse length) is less prominent.
A general theory was presented for spatial shaping of spectrally and temporally partially coherent single-spatial-mode pulse trains. The general character of space–frequency and space–time distributions of the shaped pulses was discussed with the aid of the Gaussian Schell model. The theory was finally applied to realistic partially coherent supercontinuum pulse trains, which can be described adequately only within the framework of second-order coherence theory of non-stationary light, taking fully into account the partial spectral and temporal coherence properties of the pulse train.
Appendix A: Some symmetry considerations
In this Appendix we consider certain symmetries involved in general beam shaping problems in the 2F Fourier-transform geometry, with K(u, x; ω) given by Eq. (7). If we introduce the retarded time tr = t − 2F/c, Eq. (14) gives
Suppose that we make the transformation
If V0(x; ω) = |V0(x; ω)|, i.e., the incident field has a constant spatial and spectral phase, this transformation reduces to reversing the sign of the phase function of the beam shaping element at the design frequency: ϕ (x) → − ϕ (x). The introduction of the replacement (34) into Eq. (33) implies that
Hence each modal field in the space–time domain experiences spatial inversion, time reversal, complex conjugation, and a phase shift of π/2 radians. Further, using Eq. (13) and writing tr1 = t1 − 2F/c and tr2 = t2 − 2F/c, we see that
Finally, Eq. (15) implies that
Hence the space–time target-plane profile is inverted spatially around the optical axis and time-reversed. In the case of spatial inversion symmetry I(−u; tr) = I(u, tr), the replacement (34) only results in arrive-time reversal of the pulses in the target plane.
This work was supported by the Academy of Finland (project 252910). We are grateful to G. Genty for providing the supercontinuum field realizations.
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