## Abstract

We show that the generalized phase contrast method (GPC) can be used as a versatile tool for shaping an incident Gaussian illumination into arbitrary lateral beam profiles with uniform intensity. This energy-efficient technique shapes the beam by redistributing light from designated dark regions and homogenizes the beam by redistributing excess energy from the center to the edges. Results from numerical experiments show efficiencies around 84% for various profiles with minimal intensity inhomogeneities.

©2008 Optical Society of America

## 1. Introduction

The demand from the actively growing array of laser applications for beams with uniform intensities within specified transverse distributions fuels research interest on techniques to homogenize and shape the Gaussian profile emitted by most lasers [1,2]. One of the oldest tricks, where an expanded Gaussian beam is truncated to minimize inhomogeneity, remains a popular choice for its simplicity especially in applications where energy throughput is not of prime concern. These applications may also exploit other lossy approaches such as inhomogeneous absorptive filters that attenuate the central beam parts more than the peripheral portions to get a homogenized beam [3,4].

Expectedly, energy-efficient approaches are preferred over a wider spectrum of laser applications. Designs based on geometric optics, implemented via refractive or reflective systems, redirect portions of an incident Gaussian beam into a homogenized distribution [5–11]. Energy rerouting schemes may be implemented using customized lenses, mirrors, or lenslet arrays that initially split an incident Gaussian beam into discrete beams that are later recombined into homogeneous distributions [8]. Geometric solutions map each point on an incident beam to target locations on the output plane to homogenize the energy distribution. The designs obtained from geometric analysis may be adapted to phase-wrapped surfaces in diffractive elements through fabrication technologies that allow for continuous surface relief. Diffraction-based techniques [9–12] that work with limited phase levels are based on designs where every point on the homogenized output draws contributions from all points on the incident beam. Diffractive elements that execute this global mapping are commonly designed through iterative optimization procedures [10–12].

In this work, we use the generalized phase contrast (GPC) [13,14] to propose a system that generates arbitrary lateral beam profiles with uniform intensity from an incident Gaussian illumination. Building upon Zernike’s phase contrast [15], the GPC prescribes optimal parameters for generating desired intensity distributions using programmed spatial phase modulation. The GPC’s capacity for efficient pattern generation from an incident uniform illumination [16] has been applied with considerable success in dynamic, interactive and parallel optical micromanipulation [17–20], among others [21,22].

In a recent work [23], we showed that GPC can operate like a phase-only aperture that channels energy from intended dark regions in a Gaussian beam into designated transverse intensity distributions. However, like its amplitude mask counterparts, artifacts akin to the incident Gaussian rolloff remain evident in the illuminated regions of the output. In the present work we present an input phase compensation scheme to homogenize the output. Like the inhomogeneous absorptive filters, this scheme reduces the central intensity. However, this phase-only approach effectively redirects energy from the brighter central portion of the beam towards the dimmer edges to homogenize the output without suffering significant energy loss.

We present the GPC framework in section 2 and briefly review the principles for shaping a Gaussian input and develop prescriptions for generating arbitrary flattop profiles. Section 3 presents and discusses the numerical experiments that explore the performance of GPC-based flattop beam shaping. Finally, section 4 presents the conclusions and outlook.

## 2. GPC-based Gaussian-to-flattop conversion: Mathematical analysis

#### 2.1 GPC matched to Gaussian illumination

Let’s consider the 4*f* optical processing setup in Fig. 1, which is the typical implementation of the generalized phase contrast method. Following the discussion in [23], we define the field at the input plane as *p*(*x*, *y*) = *a*(*x*, *y*)exp[i*ϕ*(*x*, *y*), which is generated when an incident beam illuminates a phase-only spatial light modulator (SLM). The phase contrast filter (PCF) at the common focus between the Fourier lenses is described mathematically by

This filter transmits the original field and synthesizes a reference wave from the *θ*-phase-shifted diffraction-broadened zero-order beam that is transmitted through an aperture defined by *S*(*f _{x}*,

*f*). The intensity pattern at the output plane,

_{y}is formed by the interference of the input field image, *a*(*x*′,*y*′)exp[i*ϕ*(*x*′,*y*′)], with the synthetic reference wave (SRW), *$\overline{\alpha}$*[exp(i*θ*)-1]*g*(*x*′,*y*′).

The strength and phase of the SRW in Eq. (2) depends on the normalized zero-order,

while the SRW spatial profile, *g*(*x*′,*y*′), arises from the diffraction of the zero-order beam through the PCF aperture

The input amplitude profile for Gaussian illumination with a beam waist, *w*
_{0}, is

The resulting SRW profile is given by

As we showed in ref [23], tweaking the PCF size can result in a good match between the SRW spatial profile and the Gaussian illumination. For matched profiles the output intensity becomes

Equation (7) prescribes a method for spatially modulating the output intensity by modulating the input phase to exploit interference effects. In ref [23], we used this to implement “phase-only apertures” that channel energy from the designated dark regions into the desired intensity distributions. Although much more efficient than truncation, the output patterns retained similar intensity rolloffs from the incident Gaussian illumination. In the next section, we will discuss how to minimize this rolloff to homogenize the output intensity.

#### 2.2 Homogenizing the output intensity for matched illumination and SRW profiles

We can see from the output intensity described by Eq. (7) that producing darkness in the output plane is contingent upon the condition

Under this condition, we can encode darkness by using zero-phase input and maximum intensity with π-phase input, subject to the Gaussian roll-off as described by Eq. (7). These binary phase inputs result in a real-valued *$\overline{\alpha}$*, which then requires a PCF that shifts the phase by *θ*=π. Until recently [24], most GPC applications have used this π–shift PCF.

Equation (7) shows the possibility to eliminate the Gaussian roll-off at the output by producing a reciprocal profile from the interference term:

where *A*(*x*′,*y*′) is the desired intensity profile with uniform intensity *I*
_{0} that is determined according to energy conservation constraints. Realizing the input phase modulation that satisfies Eq. (9) necessitates abandoning binary phase inputs and requires encoding over a continuous phase range instead. The principle for rolloff correction is graphically illustrated in Fig. 3(a), which shows the SRW phasors, ‒*a*
_{1} and ‒*a*
_{2}, and signal phasors,
${a}_{1}{e}^{i{\varphi}_{1}}$
and
${a}_{2}{\text{e}}^{i{\varphi}_{2}}$
, at two points in the output plane with mismatched amplitudes, *a*(*x*
_{1}, *y*
_{1})=*a*
_{1} and *a*(*x*
_{2}, *y*
_{2})=*a*
_{2}. To achieve homogenized interference at the amplitude mismatched points, we adopt a corrective phase encoding scheme where points having smaller amplitudes are encoded closer to π and points with larger amplitudes are encoded conversely.

The “darkness condition” specified by Eq. (8) leads, upon employing trigonometric identities, to the required normalized zero-order

This requirement specifies a practical range for *$\overline{\alpha}$* as illustrated in the phasor diagram of Fig. 3(b) for the upper semi-circle (the lower semi-circle has a symmetric set of solutions). The dotted lines indicate the requirement set by Eq. (10) that the real part of *$\overline{\alpha}$* be ½ and the maximal magnitude of the imaginary part is √3/2. Furthermore, Eq. (10) allows us to determine the required phase shift, *θ*, from the imaginary part of *$\overline{\alpha}$*:

Choosing the filter phase shift, *θ*, using Eq. (11) ensures that the SRW is π-shifted with respect to the projected image of the zero-phase encoded input, allowing us to designate dark regions and ensure that minimal energy is lost to these regions.

Applying the condition in Eq. (8) into Eq. (9) leads to the simplified relation

Substituting the trigonometric identity |exp(*iϕ*) ‒ 1|^{2}=2 ‒ 2cos(*ϕ*) into Eq. (12) yields

Solving this equation yields phase inputs with homogenized outputs.

#### 2.3 Homogenizing the output intensity for mismatched illumination and SRW profiles

Encoding spatial phase information on an incident Gaussian beam can introduce frequency components close to the zero-order beam at the Fourier plane. This can set upper limits on practical PCF sizes to avoid SRW distortions. The Fourier relation between the PCF plane and the output plane implies that a smaller PCF broadens the SRW. In this case, we revise the GPC output described by Eq. (7) into

to account for the mismatched profiles.

The mismatched amplitude profiles mean that complete darkness can no longer be guaranteed at arbitrarily chosen points. However, GPC-based homogenized beam shaping exploits the high contrast that results from interference even with amplitude mismatch. For example, the interference of two beams with 10% amplitude mismatch yields a minimum intensity that is less than 0.28% of the maximum intensity. The conditions are even more favorable in beam shaping tasks such as Gaussian-to-circular flattop conversion that require darkness only in certain peripheral regions– these can be achieved with minor losses since these peripheral regions have minimal intensities to begin with.

Choosing, for convenience, *ϕ*=0 as the phase input for minimum output intensity sets the SRW phase to π, which requires the condition

where the constant *k* is not necessarily equal to unity. We can again rewrite the relationship between the normalized zero-order and the PCF phase shift as

For this we can see that *k*=2*α _{R}* and the matching phase shift is

*θ*=2 arccot (

*α*/

_{I}*α*).

_{R}Homogenizing the output for mismatched illumination and SRW profiles involves similar corrections to the encoded phase to compensate for the spatially varying amplitude. Substituting Eq.(15) into Eq. (14) and expanding the result yields

where *a*
_{0}(*x*′,*y*′) describes the Gaussian input profile. The phase input is then obtained using the image-plane phase

However, solving for the input phase in Eq. (18) is nontrivial since it requires knowledge of the profile *g*(*x*′,*y*′), and constant *k*, which both depend on the input phase. Moreover, the target homogeneous level for *I*(*x*′,*y*′) is likewise determined based on the constraints from the input and SRW profiles. However, we may treat the phase correction as perturbations to the binary phase inputs that generate inhomogeneous outputs. This allows us to use the *k* and *g*(*x*′,*y*′) obtained using binary inputs as suitable approximations. We can then use the phase inputs obtained to iteratively correct the SRW parameters and improve the phase input. We illustrate these design principles in the next section where we consider GPC-based projection of a circular flattop from a Gaussian input.

#### 2.4 Converting a Gaussian profile to a circular flattop

Let us consider a Gaussian beam, *a*
_{0}(*r*)=exp(-*r*
^{2}/*w*
^{2}
_{0}), as input for generating a circular flattop having intensity, *I*(*x*′,*y*′)=*I*
_{0}circ(*r*′/*r*
_{0}). The uniform intensity, *I*
_{0}, is determined based on the highest efficiency that can be achieved at the edge when the signal and SRW interference is optimized

To find the initial SRW profile, we start with a Gaussian beam that is encoded with a π-phase disc. We may treat this input as the superposition of an unmodulated Gaussian beam with a truncated, π-shifted Gaussian having twice the amplitude:

This gives a real-valued normalized zero order, which determines the scaling constant *k*

The π-shifting PCF is

and the SRW profile is

By using results obtained from Eqs. (19), (21), and (23) in Eq. (18) we are able to find a phase input, *ϕ*(*r*), that improves the output homogeneity. The phase input obtained is used to find the matching PCF phase shift, *θ*=2 arccot (*α _{I}*/

*α*), as

_{R}The highly symmetric case of generating a circular flattop from a Gaussian input allows for analytically finding the required initial parameters, such as *g*(*r*), when determining the phase input. For instance, the diffraction of truncated Gaussian beams, such as those encountered in Eqs. (20) and (23), are well-studied analytically and with various approximations proposed in the literature (see, e.g., Ref. [25] and references therein). When generating a circular flattop, we find that it is sufficient to approximate the far-field pattern of a truncated Gaussian by broadening the usual far-field pattern of an untruncated Gaussian. The amount of broadening is determined based on the expansion coefficients obtained from the Hankel transform of the truncated Gaussian that substitutes a series expansion for both the Gaussian input and the Bessel kernel

Combined with the input in Eq. (20), this enables us to describe the Fourier plane field and determine the maximum PCF size that maximizes the efficiency while minimizing distortions in the SRW profile. A similar technique is applied to analyze the truncation effects at the Fourier plane due to the PCF aperture and determine the SRW profile at the output plane.

Alternately, one can use numerical approaches to find the relevant parameters instead. The numerical approach is especially handy when generating arbitrary shapes that do not necessarily have particular symmetries. Moreover, a numerical approach is well-suited for possible iterations that may be needed to improve the output. In the next section, we apply the design principles described above and explore the performance of GPC-based conversion of an incident Gaussian illumination into arbitrary flattop profiles through numerical experiments.

## 3. GPC-based Gaussian-to-flattop conversion: Numerical experiments

To examine the expected performance of beamshaping systems based on the design principles outlined above, we performed numerical experiments using a Fourier optics-based model of the GPC optical system illustrated in Fig. 1. We considered various flattop profiles starting with the highly symmetric circular flattop and then advancing into various arbitrary shapes.

The results for Gaussian-to-circular flattop beam conversion are illustrated in Fig. 3. For comparison, we present in Fig. 3(d) the generated output when the binary (0 and π) phase input, shown in Fig. 3(a), is used with a π-phase shifting PCF. The GPC-based system is able to generate an output pattern that directly mimics the binary input shape, but with an intensity rolloff that is reminiscent of the Gaussian input. In effect, the GPC system operates like an energy-efficient “phase-only aperture” that redirects light from designated dark regions into the central spot instead of blocking light. The size of the PCF, chosen to optimize the output efficiency while minimizing output distortions, is 1.1 times the conjugate beam waist parameter of the zero-order beam in the Fourier plane.

The parameters obtained from this binary case are used in Eq. (18) to obtain the phase input shown in Fig. 3(b). This phase profile is π-valued at the edge and monotonically decreases towards the center. As a result, the image of the phase-encoded input is in phase with the SRW at the edges and gradually becomes out of phase towards the center. The output pattern that results from the interference is shown in Fig. 3(e). Comparing with the initial output, the new output illustrates that we can introduce corrections to the input phase to suppress the intensity in the central region while enhancing the intensity in the outer region to improve the homogeneity. In effect, it works similar to the inhomogeneous absorptive filters [3,4] that attenuate the central intensities but with the major difference that it redistributes the energy towards the outer regions instead of wasting it.

While it improves the homogeneity, using the parameters obtained from the binary case in Eq. (18) yields a phase input that overcompensates for the inhomogeneity. This reduces the efficiency due to the over-attenuated central region. By refining the input profile [see Fig. 3(c)] and using a matched PCF phase shift to set the correct SRW phase, we are able to obtain the homogenized output with improved efficiency as illustrated in Fig. 3(f). Prior to correction the output intensity profile monotonically rolls off away from the center and attenuates by as much as 25% at the edge. The corrected output exhibits a flattop profile whose maximum peak-to-peak fluctuation, Δ, is only 0.4% of the peak intensity within the target region and with minimal intensity loss compared to the initial inhomogeneous output.

We have previously shown [23] that a GPC-based system can be used to generate various lateral profiles from an incident Gaussian beam. This is accomplished with high efficiency using simple binary phase inputs that are shaped to mimic the desired output profiles. However the outputs suffer from an intensity roll off that could still be problematic for some applications. By applying compensation schemes on the input phase similar to that employed above for the circular flattop it is now possible to generate various profiles with homogenized intensity from an incident Gaussian beam. Some examples obtained from numerical experiments are illustrated in Fig. 4.

The method we have presented improves the output homogeneity by introducing intuitive multilevel corrections to the input phase. Alternately, one can consider a multilevel phase implementation of the PCF to utilize the higher degrees of freedom. However, GPC uses the PCF to modify a single diffraction order – the zero-order – by introducing a well-defined phase shift upon it, relative to the other orders. A well-defined phase shift requires a binary phase PCF and so does not benefit from using a multiphase element. A multilevel PCF may be explored for altering the SRW profile to match the illumination profile better. However, the severely limited space bandwidth within the PCF prevents the SRW from producing sharp transitions. We determined that it is better matched with Gaussian illumination and can be used for beam shaping but with a residual intensity roll off [23]. With the limited space bandwidth at the PCF, the current paper suggests how to improve the output by introducing multilevel phase corrections at the input plane instead.

Achieving a high degree of output uniformity requires analog input phase encoding over a continuous range. However, dynamic applications require spatial light modulation (SLM) devices that usually quantize the encoded phase into discrete levels. To assess the impact of phase quantization on the output uniformity, we implemented various phase quantization levels in the numerical experiments. Figure 5 shows the maximum fluctuation, Δ, for various phase quantization levels when generating a circular pattern from a Gaussian incident beam. The inset shows the projected pattern when encoding the phase with 16 quantization levels, where the biggest fluctuation is 10% of the peak intensity. The fluctuations decrease monotonically with an increasing number of phase quantization levels and is reduced to less than 3% when encoded with only 64 phase quantization levels. Thus, the required phase quantization requirements for generating outputs with acceptable intensity fluctuations can be sufficiently met by commercially available devices. However one must exercise caution with regard to the dead spaces between SLM pixels as these can map to the output pattern and degrade uniformity. This can be avoided by using so-called “non-pixelated SLMs” such as optically addressed devices. If using pixilated devices, one can block the tiled higher-order replicas at the PCF plane to improve the output uniformity by avoiding pixilation at the output with some efficiency tradeoff.

The 4*f* optical processing setup illustrated in Fig. 1 can be modified to suit the requirements of particular applications. For example, GPC has been implemented using planar integrated micro-optics for a compact optical decryption device [22]. It is also possible to reduce the number of optical components if required by the application. One can incorporate the lens phase into the phase input to eliminate the first Fourier lens and, similarly, combine the phase-only PCF and the second Fourier lens into a single phase element. This reduces the GPC system into just two optical elements. Other combinations with diffractive implementations can also be considered. However, one has to cautiously examine whether the advantages of having fewer components outweigh the side effects. When merging the Fourier lens with the input phase, one must keep in mind that SLM-based diffractive lenses can generate spurious secondary lenses [26]. Incorporating the lens phase into the PCF must be weighed against the simplicity of fabricating the standard and simple binary-phase PCF. A binary diffractive lens has a much lower efficiency while a multilevel implementation must contend with fabrication issues [27]. Furthermore, GPC achieves high reconfiguration rates by exploiting the degrees of freedom afforded by working with two conjugate planes. Consequently, the input phase simply mimics the spatial features of the desired output patterns and do not require the computationally expensive algorithms used in single plane designs such as computer generated holograms.

## 4. Conclusions and outlook

We have shown that the generalized phase contrast method (GPC) serves as a useful framework for developing a versatile tool that shapes a Gaussian beam into flattop beams having arbitrary lateral profiles. GPC efficiently diverts energy from designated dark regions into desired intensity distributions where they can be homogenized by intuitive corrections on the input phase. This helps expanding the repertoire of techniques for getting more out of the fundamental Gaussian laser output through beam homogenization. Future work will generalize to arbitrary inhomogeneous input beams such as the elliptical outputs of laser diodes and the rectangular emission from excimer lasers. The chromatic response of GPCbased beam shaping and homogenization technique, as well as effects of illumination coherence is also worthy of investigation in order to reach an even wider array of applications.

## Acknowledgments

We thank the support from the Danish Technical Scientific Research Council (FTP).

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