## Generalized phase contrast matched to Gaussian illumination

Optics Express, Vol. 15, Issue 19, pp. 11971-11977 (2007)

http://dx.doi.org/10.1364/OE.15.011971

Acrobat PDF (193 KB)

### 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. 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

## 1. Introduction

1. F. M. Dickey and S. C. Holswade, *Laser Beam Shaping: Theory and Techniques* (Marcel Dekker, New York, 2000). [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]

13. S. Maruo, O. Nakamura, and S. Kawata, “Three-dimensional microfabrication with two-photon-absorbed photopolymerization,” Opt. Lett. **22**, 132–134 (1997). [CrossRef] [PubMed]

14. Y. Liu, S. Sun, S. Singha, M. R. Cho, and R. J. Gordon, “3D femtosecond laser patterning of collagen for directed cell attachment,” Biomaterials **26**, 4597–4605 (2005). [CrossRef] [PubMed]

15. D. Psaltis, “Coherent optical information systems,” Science **298**, 1359–1363 (2002). [CrossRef] [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]

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]

2. F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., *Laser. Beam Shaping Applications* (CRC Press,2005). [CrossRef]

14. Y. Liu, S. Sun, S. Singha, M. R. Cho, and R. J. Gordon, “3D femtosecond laser patterning of collagen for directed cell attachment,” Biomaterials **26**, 4597–4605 (2005). [CrossRef] [PubMed]

21. P.J. Rodrigo, V.R. Daria, and J. Glückstad, “Real-time three-dimensional optical micromanipulation of multiple particles and living cells,” Opt. Lett. **29**2270–2272 (2004). [CrossRef] [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]

11. J. Glückstad, “Phase contrast image synthesis,” Opt. Commun. **130**, 225–230 (1996). [CrossRef]

26. J. Glückstad and P.C. Mogensen, “Optimal phase contrast in common-path interferometry,” Appl. Opt. **40**, 268–282 (2001). [CrossRef]

27. F. Zernike, “How I discovered phase contrast,” Science **121**, 345–349 (1955). [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]

1. F. M. Dickey and S. C. Holswade, *Laser Beam Shaping: Theory and Techniques* (Marcel Dekker, New York, 2000). [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]

5. F. Wippermann, U. D. Zeitner, P. Dannberg, A. Bräuer, and S. Sinzinger, “Beam homogenizers based on chirped microlens arrays,” Opt. Express **15**, 6218–6231 (2007). [CrossRef] [PubMed]

1. F. M. Dickey and S. C. Holswade, *Laser Beam Shaping: Theory and Techniques* (Marcel Dekker, New York, 2000). [CrossRef]

6. M. T. Eismann, A. M. Tai, and J. N. Cederquist, “Iterative design of a holographic beamformer,” Appl. Opt. **28**, 2641–2650 (1989). [CrossRef] [PubMed]

8. A. J. Caley, M. Braun, A. J. Waddie, and M. R. Taghizadeh, “Analysis of multimask fabrication errors for diffractive optical elements,” Appl. Opt. **46**, 2180–2188 (2007). [CrossRef] [PubMed]

8. A. J. Caley, M. Braun, A. J. Waddie, and M. R. Taghizadeh, “Analysis of multimask fabrication errors for diffractive optical elements,” Appl. Opt. **46**, 2180–2188 (2007). [CrossRef] [PubMed]

*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 [29

29. J. Glückstad, L. Lading, H. Toyoda, and T. Hara, “Lossless light projection,” Opt. Lett. **22**, 1373–1375 (1997). [CrossRef]

30. D. Palima and V. R. Daria, “Effect of spurious diffraction orders in arbitrary multifoci patterns produced via phase-only holograms,” Appl. Opt. **45**, 6689–6693 (2006). [CrossRef] [PubMed]

31. D. Palima and V. R. Daria, “Holographic projection of arbitrary light patterns with a suppressed zero-order beam,” Appl. Opt. **46**, 4197–4201 (2007) [CrossRef] [PubMed]

19. R. L. Eriksen, V. R. Daria, and J. Glückstad, “Fully dynamic multiple-beam optical tweezers,” Opt. Express **10**, 597–602 (2002). [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]

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. **43**2223–2227 (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]

## 2. Mathematical analysis of GPC with Gaussian illumination

### 2.1 Optimal GPC parameters under uniform illumination

*f*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.

*θ*within an aperture region defined by

*S*(

*f*,

_{x}*f*) in the Fourier plane is mathematically described as

_{y}*I*(

*x*’,

*y*’), at the output plane is [26

26. J. Glückstad and P.C. Mogensen, “Optimal phase contrast in common-path interferometry,” Appl. Opt. **40**, 268–282 (2001). [CrossRef]

*a*(

*x′*,

*y′*)exp[

*iϕ*(

*x′*,

*y′*)], with the synthetic reference wave (SRW),

*[exp(*α ¯

*iθ*)-1]

*g*(

*x′*,

*y′*). The strength of the SRW in Eq. (2) depends on the normalized zero-order

*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:

*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

*w*

_{0}, is equivalent to having a Gaussian aperture function given by

^{2}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.

## 3. Numerical experiments: GPC-generated optical patterns from Gaussian beams

### 3.1 GPC-based phase-only circular aperture

*A*

^{2}/

*w*

^{2}

_{0}) when using a circular aperture of radius

*A*to truncate a Gaussian beam having a 1/e

^{2 }radius of

*w*

_{0}. This is equal to the relative center-to-edge intensity difference of the transmitted beam.

*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 0.34

*w*

_{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

## 4. Summary and conclusions

## Acknowledgments

## References and links

1. | F. M. Dickey and S. C. Holswade, |

2. | F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., |

3. | B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. |

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. |

5. | F. Wippermann, U. D. Zeitner, P. Dannberg, A. Bräuer, and S. Sinzinger, “Beam homogenizers based on chirped microlens arrays,” Opt. Express |

6. | M. T. Eismann, A. M. Tai, and J. N. Cederquist, “Iterative design of a holographic beamformer,” Appl. Opt. |

7. | V. A. Soifer, |

8. | A. J. Caley, M. Braun, A. J. Waddie, and M. R. Taghizadeh, “Analysis of multimask fabrication errors for diffractive optical elements,” Appl. Opt. |

9. | C. O. Weiss and Y. Larionova, “Pattern formation in optical resonators,” Rep. Prog. Phys. |

10. | R. L. Eriksen, P. C. Mogensen, and J. Glückstad, “Elliptical polarisation encoding in two dimensions using phase-only spatial light modulators,” Opt. Commun. |

11. | J. Glückstad, “Phase contrast image synthesis,” Opt. Commun. |

12. | C. Blanca and S. Hell, “Axial superresolution with ultrahigh aperture lenses,” Opt. Express |

13. | S. Maruo, O. Nakamura, and S. Kawata, “Three-dimensional microfabrication with two-photon-absorbed photopolymerization,” Opt. Lett. |

14. | Y. Liu, S. Sun, S. Singha, M. R. Cho, and R. J. Gordon, “3D femtosecond laser patterning of collagen for directed cell attachment,” Biomaterials |

15. | D. Psaltis, “Coherent optical information systems,” Science |

16. | S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-dimensional optically bound arrays of microscopic particles,” Phys. Rev. Lett |

17. | G. Milne, D. Rhodes, M. MacDonald, and K. Dholakia, “Fractionation of polydisperse colloid with acousto-optically generated potential energy landscapes,” Opt. Lett. |

18. | S. Maruo and H. Inoue, “Optically driven micropump produced by three-dimensional two-photon microfabrication,” Appl. Phys. Lett. |

19. | R. L. Eriksen, V. R. Daria, and J. Glückstad, “Fully dynamic multiple-beam optical tweezers,” Opt. Express |

20. | P. J. Rodrigo, R. L. Eriksen, V. R. Daria, and J. Glückstad, “Interactive light-driven and parallel manipulation of inhomogeneous particles,” Opt. Express |

21. | P.J. Rodrigo, V.R. Daria, and J. Glückstad, “Real-time three-dimensional optical micromanipulation of multiple particles and living cells,” Opt. Lett. |

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. |

23. | P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Four-dimensional optical manipulation of colloidal particles,” Appl. Phys. Lett. |

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 |

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 |

26. | J. Glückstad and P.C. Mogensen, “Optimal phase contrast in common-path interferometry,” Appl. Opt. |

27. | F. Zernike, “How I discovered phase contrast,” Science |

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. |

29. | J. Glückstad, L. Lading, H. Toyoda, and T. Hara, “Lossless light projection,” Opt. Lett. |

30. | D. Palima and V. R. Daria, “Effect of spurious diffraction orders in arbitrary multifoci patterns produced via phase-only holograms,” Appl. Opt. |

31. | D. Palima and V. R. Daria, “Holographic projection of arbitrary light patterns with a suppressed zero-order beam,” Appl. Opt. |

32. | P. C. Mogensen and J. Glückstad, “Phase-only optical encryption,” Opt. Lett. |

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. |

34. | J. Glückstad, “Adaptive array illumination and structured light generated by spatial zero-order self-phase modulation in a Kerr medium,” Opt. Commun. |

**OCIS Codes**

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(070.6110) Fourier optics and signal processing : Spatial filtering

(120.5060) Instrumentation, measurement, and metrology : Phase modulation

(140.3300) Lasers and laser optics : Laser beam shaping

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: July 5, 2007

Revised Manuscript: August 31, 2007

Manuscript Accepted: August 31, 2007

Published: September 5, 2007

**Citation**

Darwin Palima, Carlo A. Alonzo, Peter J. Rodrigo, and Jesper Glückstad, "Generalized phase contrast matched to Gaussian illumination," Opt. Express **15**, 11971-11977 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-19-11971

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### References

- F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel Dekker, New York, 2000). [CrossRef]
- F. M. Dickey, S. C. Holswade, & D. L. Shealy, eds., Laser. Beam Shaping Applications (CRC Press, 2005). [CrossRef]
- B. R. Frieden, "Lossless conversion of a plane laser wave to a plane wave of uniform irradiance," Appl. Opt. 4, 1400-1403 (1965). [CrossRef]
- 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]
- F. Wippermann, U. D. Zeitner, P. Dannberg, A. Bräuer, and S. Sinzinger, "Beam homogenizers based on chirped microlens arrays," Opt. Express 15, 6218-6231 (2007). [CrossRef] [PubMed]
- M. T. Eismann, A. M. Tai, and J. N. Cederquist, "Iterative design of a holographic beamformer," Appl. Opt. 28, 2641-2650 (1989). [CrossRef] [PubMed]
- V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements (John Wiley & Sons, New York, 2002).
- A. J. Caley, M. Braun, A. J. Waddie, and M. R. Taghizadeh, "Analysis of multimask fabrication errors for diffractive optical elements," Appl. Opt. 46, 2180-2188 (2007). [CrossRef] [PubMed]
- C. O. Weiss and Y. Larionova, "Pattern formation in optical resonators," Rep. Prog. Phys. 70,255-335 (2007). [CrossRef]
- 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]
- J. Glückstad, "Phase contrast image synthesis," Opt. Commun. 130, 225-230 (1996). [CrossRef]
- C. Blanca and S. Hell, "Axial superresolution with ultrahigh aperture lenses," Opt. Express 10, 893-898 (2002). [PubMed]
- S. Maruo, O. Nakamura, and S. Kawata, "Three-dimensional microfabrication with two-photon-absorbed photopolymerization," Opt. Lett. 22, 132-134 (1997). [CrossRef] [PubMed]
- Y. Liu, S. Sun, S. Singha, M. R. Cho, and R. J. Gordon, "3D femtosecond laser patterning of collagen for directed cell attachment," Biomaterials 26, 4597-4605 (2005). [CrossRef] [PubMed]
- D. Psaltis, "Coherent optical information systems," Science 298, 1359-1363 (2002). [CrossRef] [PubMed]
- S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, "One-dimensional optically bound arrays of microscopic particles," Phys. Rev. Lett 89, 283901 (2002). [CrossRef]
- 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]
- S. Maruo and H. Inoue, "Optically driven micropump produced by three-dimensional two-photon microfabrication," Appl. Phys. Lett. 89, 144101 (2006). [CrossRef]
- R. L. Eriksen, V. R. Daria, and J. Glückstad, "Fully dynamic multiple-beam optical tweezers," Opt. Express 10,597-602 (2002). [PubMed]
- 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]
- P.J. Rodrigo, V.R. Daria, and J. Glückstad, "Real-time three-dimensional optical micromanipulation of multiple particles and living cells," Opt. Lett. 29 2270-2272 (2004). [CrossRef] [PubMed]
- 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]
- P. J. Rodrigo, V. R. Daria, and J. Glückstad, "Four-dimensional optical manipulation of colloidal particles," Appl. Phys. Lett. 86, 074103 (2005). [CrossRef]
- 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]
- 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]
- J. Glückstad and P.C. Mogensen, "Optimal phase contrast in common-path interferometry," Appl. Opt. 40, 268-282 (2001). [CrossRef]
- F. Zernike, "How I discovered phase contrast," Science 121, 345-349 (1955). [CrossRef] [PubMed]
- 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]
- J. Glückstad, L. Lading, H. Toyoda, and T. Hara, "Lossless light projection," Opt. Lett. 22, 1373-1375 (1997). [CrossRef]
- D. Palima and V. R. Daria, "Effect of spurious diffraction orders in arbitrary multifoci patterns produced via phase-only holograms," Appl. Opt. 45, 6689-6693 (2006). [CrossRef] [PubMed]
- D. Palima and V. R. Daria, "Holographic projection of arbitrary light patterns with a suppressed zero-order beam," Appl. Opt. 46, 4197-4201 (2007) [CrossRef] [PubMed]
- P. C. Mogensen and J. Glückstad, "Phase-only optical encryption," Opt. Lett. 25, 566-568 (2000). [CrossRef]
- V. R. Daria, P. J. Rodrigo, S. Sinzinger, and J. Glückstad, "Phase-only optical decryption in a planar-integrated micro-optics system," Opt. Eng. 43 2223-2227 (2004). [CrossRef]
- 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]

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