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. For illustration, we use GPC in an energy-efficient phase-only implementation of various apertures that do not block light but instead effectively redirect the available photons from a bell-shaped light distribution. GPC-based generation of lateral beam profiles can thus be achieved using a simplified optical implementation as it eliminates the need for a potentially lossy initial beam shaping. The required binary phase input is simple to fabricate for static applications and can be easily reconfigured up to device frame refresh rates for dynamic applications.
©2007 Optical Society of America
Control over the spatial distribution of photon energy, momentum and angular momentum can be achieved by appropriate laser beam shaping techniques [1–11]. Aside from fundamental interest, this research receives considerable attention for its promise of enabling tools that exploit light-matter interaction in applications that include microscopy , materials processing [13,14], information processing , and optical micromanipulation [16–25], among others . Of multidisciplinary interest is controlled light-matter interaction in biological materials [14,21,22].
Energy-efficient optical systems that achieve spatial control over the properties of light have been designed based on geometric optics for irradiance redistribution but physical optics opens doors to more sophisticated applications. Refractive effects are commonly used in geometric designs while diffractive effects are popular for a more general wavefront engineering approach. Considerable success has also been achieved by the generalized phase contrast method (GPC) , which employs spatial filtering and belongs to the class of non-absorbing common path interferometers .
The generalized phase contrast method is a versatile and energy-efficient beam shaping tool. On the theory side, GPC extends Zernike’s phase contrast  to accommodate larger phase modulations. Detailed mathematical analyses are available for circular  and rectangular  aperture geometries. In experiments, the GPC blends the dynamic spatial phase control afforded by spatial light modulators with the phase contrast ability to visualize phase variations in order to produce arbitrary intensity patterns.
Refractive beam shaping solutions [1,3,4] remap available energy with promise of lossless conversion over a wide range of illumination wavelengths. However, fabrication of the required refractive elements can be problematic, especially when aimed at generating arbitrary intensity profiles that lack circular symmetry. Microlens arrays, which divide an incident beam into beamlets that are later recombined in beam integrators, may be easier to fabricate but the resulting beams can have very poor homogeneity, especially under coherent illumination . These integrators are likewise limited in terms of achievable intensity patterns. Diffractive optical approaches [1,6–8] offer capacity for producing a variety of beam shapes, with some compromise on phase homogeneity, and are rich in design algorithms that promise theoretical conversion efficiencies in the upper 90% range. However, fabrication errors can degrade the efficiency and uniformity of the generated patterns . Since phase errors can easily give rise to a spurious zero-order beam, diffractive designs commonly avoid the optical axis, which is the more attractive reconstruction region in terms of optimizing efficiency and minimizing aberrations.
The GPC method achieves high efficiency using a very straightforward design of the needed optical element. The input simply requires an easy-to-fabricate binary phase mask (0 and π) that is patterned after the desired intensity distribution. Thus, static applications of GPC-based beam-shaping are less susceptible to fabrication errors and can as well provide excellent output phase homogeneity unlike that of diffractive approaches. Compared to diffractive optical elements, the GPC phase mask generally contains fewer locations with phase jumps and, hence, suffers less from scattering losses. Additionally, the GPC intensity projections can be centered on the optical axis to minimize aberration effects. The GPC method has been successfully implemented for lossless pattern projection under uniform illumination .
The simplicity of designing binary phase inputs in the GPC-based approach lends itself to dynamic pattern reconfiguration that is limited only by the frame-rate of the encoding device (e.g., this can reach up to kilohertz in ferroelectric liquid crystals). This high refresh rate is achieved without compromising issues associated with speckle, spurious higher orders and zero-order effects that are expected in computer generated phase holograms [30,31], especially when the number of iterations is compromised for faster computation. Coupled to microscope objectives, GPC-generated patterns have demonstrated capacity for fully dynamic and interactive optical micromanipulation [19–25]. The GPC method has also been exploited for phase-only optical encryption and decryption [32,33].
The combination of GPC with Gaussian illumination represents a natural step in GPC theory development. All previous theoretical and experimental GPC studies used uniform illumination except from the analysis of self-induced phase-filtering in a non-linear Kerr medium . Experimentally, uniform illumination requires beam shaping optics to convert the Gaussian laser beam to an appropriate profile and finding efficient conversion techniques continues to attract research attention. Thus, this work addresses the question of whether we can utilize GPC to efficiently generate optical beam patterns directly from an unshaped incident Gaussian beam. This contributes to the available techniques for reshaping Gaussian beams, considering the advantages of the GPC method over the other approaches. The GPC capacity for generating exotic shapes at rapid reconfiguration rates is particularly attractive, since the current literature is focused on static and simple patterns, owing to practical constraints in the other methods.
Section 2 presents a mathematical analysis of GPC with Gaussian illumination that builds upon the standard GPC analysis for uniform illumination. Section 3 discusses the results of numerical experiments that use GPC-based phase-only apertures to reshape an incident Gaussian beam. Finally, section 4 provides summary and concluding remarks.
2. Mathematical analysis of GPC with Gaussian illumination
2.1 Optimal GPC parameters under uniform illumination
The optical implementation of the generalized phase contrast method typically involves the 4f optical processing setup illustrated in Fig. 1. An incident Gaussian beam is first transformed into a truncated plane-wave, and then illuminates a phase-only spatial light modulator (SLM) that generates a field p(x,y)=a(x,y)exp[i ϕ(x,y)] at the input plane. The phase contrast filter (PCF), a small phase shifter at the common focus between the Fourier lenses, synthesizes a phase-shifted reference wave from the zero-order beam. The encoded phase information in the input field is then converted to intensity variations when the image of the input field interferes with the synthesized reference wave (SRW) at the output plane.
A PCF that phase shifts the diffraction-limited zero-order beam by θ within an aperture region defined by S(fx,fy) in the Fourier plane is mathematically described as
where the terms are grouped to explicitly show the dual action of the PCF: (1) transmitting the signal, and (2) synthesizing a phase-shifted reference wave. The resulting intensity pattern, I(x’,y’), at the output plane is 
This is formed by the interference of the input field image, a(x′,y′)exp[iϕ(x′,y′)], with the synthetic reference wave (SRW), [exp(iθ)-1]g(x′,y′). The strength of the SRW in Eq. (2) depends on the normalized zero-order
The SRW spatial profile,g(x′,y′), is governed by diffraction effects from the finite-sized PCF aperture of the diffraction-broadened zero-order beam at the Fourier plane:
As a result, GPC is able to transcend the default choice for a nominal PCF size. Instead, GPC prescribes a performance optimization based on the PCF size, the phase shift, and the strength of the zero-order beam to exploit phase contrast in high-efficiency pattern generation:
where K=g(0,0) is the central value of the SRW profile. This condition ensures that the signal and the SRW are amplitude-matched in the central region of the output plane.
2.2 Optimal GPC parameters under Gaussian illumination
Gaussian illumination with a beam waist, w 0, is equivalent to having a Gaussian aperture function given by
In this case the zero-order beam at the Fourier plane also assumes a Gaussian profile:
The GPC-output still follows the same functional form as in Eq. (2) with the SRW profile consequently adjusted based on the new aperture function,
Figure 2(a) shows the radial SRW profile obtained from numerical integration of Eq. (8) for different PCF sizes, measured relative to the 1/e2 intensity beam waist of the Gaussian zero-order beam. The SRW profile approaches the amplitude of the Gaussian illumination as the PCF size is increased. For comparison, Fig. 2(b) shows the corresponding variations in the SRW profile with PCF size for top-hat illumination. Better matching between the input and SRW amplitude profiles for Gaussian illumination is expected since the PCF captures the relevant Gaussian Fourier components, which are close to the zero order, while a flattop illumination contains higher spatial frequencies.
For perfectly matched signal and SRW profiles, and applying the optimal conditions of Eq. (5), the output intensity becomes
Achieving matched signal and SRW profiles, a requirement embodied in Eq. (5), allows us to generate a dark outer region by manipulating the signal phase to attain destructive interference. This leads to high energy efficiency as the energy channels into the central region, which is an attractive working area because of its relatively flat amplitude profile.
In the next section, we illustrate some features of GPC under Gaussian illumination in numerical experiments that use the method to implement various phase-only apertures.
3. Numerical experiments: GPC-generated optical patterns from Gaussian beams
3.1 GPC-based phase-only circular aperture
The simplest way to get a fairly flat illumination from a Gaussian beam is to expand the beam and use a truncating aperture to select the relatively flatter central region. This comes at the cost of energy efficiency. Optical throughput is only 1-exp(-2A 2/w 2 0) when using a circular aperture of radius A to truncate a Gaussian beam having a 1/e2 radius of w 0. This is equal to the relative center-to-edge intensity difference of the transmitted beam.
Using a π-phase shifting PCF, a GPC-based phase-only aperture can be implemented by encoding the signal beam with a π-phase at the intended bright regions and encoding a zero-phase where darkness is desired. This phase-only aperture promises a higher efficiency since the energy from the truncated portions of the Gaussian beam can be diverted into the transmitted region and will not be lost. This is confirmed in Fig. 3(a), which shows the efficiency of a GPC-based phase-only aperture relative to a simple truncation for a Gaussian input beam. The 86% efficiency obtained for an aperture radius of 0.34w 0 is 2.33 times better than the throughput of an equally-sized hard aperture. While the efficiencies of GPC apertures decrease with size, this actually represents an increasing gain when referenced to the energy throughput of truncating apertures of corresponding sizes. Residual light that is not diverted into the main spot can be easily blocked by an exit aperture in applications that cannot tolerate stray light. Additionally, some improvement in relative flatness is gained using a GPC aperture. For example, the center-to-edge intensity difference is 23% for aperture truncation but only 17% for GPC-based truncation for the output illustrated in Fig. 3(b). Also, a flat phase profile is maintained throughout the illuminated region.
3.2 GPC-based arbitrary phase-only apertures
An additional advantage of GPC over simple truncating apertures is that more complicated aperture functions can be implemented using appropriate phase-only spatial light modulators. The outputs of several GPC-based phase-only apertures are illustrated in Fig. 4 with their corresponding efficiencies. The efficiencies when using amplitude masks to accomplish the task are also presented for comparison. As in the case of circular apertures, GPC-based phase-only apertures show superior energy efficiency over their amplitude-based counterparts. In addition, a flat phase profile is maintained throughout the illuminated region.
4. Summary and conclusions
We have shown that the generalized phase contrast method can directly generate arbitrary lateral beam patterns fully utilizing an incident Gaussian illumination. The close match between the signal and SRW profiles enables efficient diversion of energy from designated dark regions into desired intensity distributions. Our results indicate that GPC can implement various aperture functions with high energy efficiency. This can be accomplished using a simple binary phase mask that is patterned after the desired intensity distribution. Thus, GPC offers a straightforward process for designing phase masks that are likewise simpler to fabricate. The sharp bright-to-dark transition in patterns obtained using GPC-designed apertures can be highly suitable in applications where a soft intensity gradient can produce undesirable effects. With most lasers naturally emitting a Gaussian beam profile, GPC’s capacity for efficiently handling Gaussian beam illumination can be beneficial for various applications requiring reconfigurable lateral beam shaping.
We thank the support from the EU-FP6-NEST program (ATOM3D), the ESF-Eurocores-SONS program (SPANAS), and the Danish Technical Scientific Research Council (FTP).
References and links
1. F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel Dekker, New York, 2000). [CrossRef]
2. F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., Laser. Beam Shaping Applications (CRC Press,2005). [CrossRef]
3. B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. 4, 1400–1403 (1965). [CrossRef]
4. J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a gaussian to a flattop beam,” Appl. Opt. 39, 5488–5499 (2000). [CrossRef]
7. V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements (John Wiley & Sons, New York, 2002).
9. C. O. Weiss and Y. Larionova, “Pattern formation in optical resonators,” Rep. Prog. Phys. 70, 255–335 (2007). [CrossRef]
10. R. L. Eriksen, P. C. Mogensen, and J. Glückstad, “Elliptical polarisation encoding in two dimensions using phase-only spatial light modulators,” Opt. Commun. 187, 325–336 (2001). [CrossRef]
11. J. Glückstad, “Phase contrast image synthesis,” Opt. Commun. 130, 225–230 (1996). [CrossRef]
12. C. Blanca and S. Hell, “Axial superresolution with ultrahigh aperture lenses,” Opt. Express 10, 893–898 (2002). [PubMed]
16. S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-dimensional optically bound arrays of microscopic particles,” Phys. Rev. Lett 89, 283901 (2002). [CrossRef]
17. G. Milne, D. Rhodes, M. MacDonald, and K. Dholakia, “Fractionation of polydisperse colloid with acousto-optically generated potential energy landscapes,” Opt. Lett. 32, 1144–1146 (2007). [CrossRef] [PubMed]
18. S. Maruo and H. Inoue, “Optically driven micropump produced by three-dimensional two-photon microfabrication,” Appl. Phys. Lett. 89, 144101 (2006). [CrossRef]
19. R. L. Eriksen, V. R. Daria, and J. Glückstad, “Fully dynamic multiple-beam optical tweezers,” Opt. Express 10, 597–602 (2002). [PubMed]
20. P. J. Rodrigo, R. L. Eriksen, V. R. Daria, and J. Glückstad, “Interactive light-driven and parallel manipulation of inhomogeneous particles,” Opt. Express 10, 1550–1556 (2002). [PubMed]
22. N. Arneborg, H. Siegumfeldt, G. H. Andersen, P. Nissen, V. R. Daria, P. J. Rodrigo, and J. Glückstad, “Interactive optical trapping shows that confinement is a determinant of growth in a mixed yeast culture,” FEMS Microbiol. Lett. 245, 155–159 (2005). [CrossRef] [PubMed]
23. P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Four-dimensional optical manipulation of colloidal particles,” Appl. Phys. Lett. 86, 074103 (2005). [CrossRef]
24. I. R. Perch-Nielsen, P. J. Rodrigo, C. A. Alonzo, and J. Glückstad, “Autonomous and 3D real-time multi-beam manipulation in a microfluidic environment,” Opt. Express 14, 12199–12205 (2006). [CrossRef] [PubMed]
25. P. Rodrigo, L. Gammelgaard, P. Bøggild, I. Perch-Nielsen, and J. Glückstad, “Actuation of microfabricated tools using multiple GPC-based counterpropagating beam traps,” Opt. Express 13, 6899–6904 (2005). [CrossRef] [PubMed]
26. J. Glückstad and P.C. Mogensen, “Optimal phase contrast in common-path interferometry,” Appl. Opt. 40, 268–282 (2001). [CrossRef]
28. C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Photon-efficient grey-level image projection by the generalized phase contrast method,” New J. Phys. 9, 132 (2007). [CrossRef]
29. J. Glückstad, L. Lading, H. Toyoda, and T. Hara, “Lossless light projection,” Opt. Lett. 22, 1373–1375 (1997). [CrossRef]
32. P. C. Mogensen and J. Glückstad, “Phase-only optical encryption,” Opt. Lett. 25, 566–568 (2000). [CrossRef]
33. V. R. Daria, P. J. Rodrigo, S. Sinzinger, and J. Glückstad, “Phase-only optical decryption in a planar-integrated micro-optics system,” Opt. Eng. 432223–2227 (2004). [CrossRef]
34. J. Glückstad, “Adaptive array illumination and structured light generated by spatial zero-order self-phase modulation in a Kerr medium,” Opt. Commun. 120, 194–203 (1995). [CrossRef]