## Intensity profiles and propagation of optical beams with bored helical phase

Optics Express, Vol. 17, Issue 18, pp. 16244-16254 (2009)

http://dx.doi.org/10.1364/OE.17.016244

Acrobat PDF (338 KB)

### Abstract

A modification of the helical phase profile obtained by eliminating the on-axis screw-dislocation is presented. Beams with this phase possess a variety of interesting properties different from optical vortex beams. Numerical simulations verify analytic predictions and reveal that beams with this phase have intensity patterns which vary as a function of the phase parameters, as well as the propagation distance. Calculations of the Poynting vector and orbital angular momentum are also performed. Experiments verify the intensity profiles obtained in simulation.

© 2009 Optical Society of America

## 1. Introduction

16. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A **56**, 4064–4075 (1997).
[CrossRef]

*φ*is the azimuthal angle while

*ℓ*determines the topological charge, the total number of 2π phase shifts around a closed loop along the azimuth. The phase of the wave changes continuously by

*ℓ*2π in one complete revolution around the optical axis with an indeterminate phase resulting along the dislocation center. Equiphase surfaces in the phase profile form a periodic helicoidal structure of pitch

*ℓλ*[1–7

7. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. **40**, 73–87 (1993).
[CrossRef]

4. M. V. Berry, “Optical Vortices Evolving from helicoidal integer and fractional phase steps,” J. Opt. A **6**, 259–268 (2004).
[CrossRef]

12. G.-H. Kim, J.-H. Jeon, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “Optical vortices produced with a nonspiral phase plate,” Appl. Opt. **36**, 8614–8621 (1997).
[CrossRef]

16. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A **56**, 4064–4075 (1997).
[CrossRef]

19. S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. **110**, 670–678 (1994).
[CrossRef]

*mħ*per photon on birefringent and partially-absorbing particles resulting in their translation around a circuit centered on the beam’s axis [20

20. J. E. Molloy and M. J. Padgett, “Lights, action: optical tweezers,” Contemp. Phys. **43**, 241–258 (2002).
[CrossRef]

22. J. B. Gotte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with orbital angular momentum and their vortex structure,” Opt. Express **16**, 993–1006 (2008).
[CrossRef] [PubMed]

23. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express **15**, 8619–8625 (2007).
[CrossRef] [PubMed]

24. K Volke-Sepulveda, V Garces-Chavez, S Chavez-Cerda, J Arlt, and K Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B **4**, S82–S89 (2002).
[CrossRef]

25. C. A. Alonzo, P. J. Rodrigo, and P. Gluckstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express **13**, 1749–1760 (2005).
[CrossRef] [PubMed]

26. N. P. Manaois and C. O. Hermosa II, “Phase structure of Helico-conical optical beams,” Opt. Commun. **271**, 178–183 (2007).
[CrossRef]

*z*. Within the bounds of the paraxial approximation of the scalar diffraction theory, an analysis is presented detailing the features and dynamics of this new class of beams. This preliminary treatment does not explore further the vortex dynamics which include birth and evolution of OV’s and beam propagation very near the focal plane.

## 2. Bored helical beams

*ρ*denotes the bore radius

_{REL}*ρ*in proportion to the outer radius

_{i}*ρ*as shown in Fig. 1. In this study, the topological charge

_{o}*ℓ*is limited to integer-order values. More specifically, |

*ℓ*|∈

**N**. The characteristic function

*χ*of a set

_{A}*A*is given by [27]

*I*and

*O*are as follows:

*O*\

*I*is thus

*ρ*replacing the screw dislocation previously found on-axis. In particular, a BH phase of charge

_{i}*ℓ*possesses |

*ℓ*| equally-spaced π – phase point discontinuities along the radius

*ρ*. This phase distribution will be shown to generate vortices despite the absence of the screw dislocation.

_{i}*u*(

*x*,

*y*,

*z*)is obtained by convolving the initial disturbance

*u*(

*x*,

*y*,

*z*=0) with the transfer function

*u*(

*x*,

*y*,

*z*) vanishes at the edges. The model simulates (linear) propagation of a He-Ne laser (

*λ*=632.8nm) in free space. The two-dimensional transverse wavefront and intensity profiles were observed as variations in the parameters

*ρ*values near unity, in particular for {

_{REL}*ρ*∈[0,1]: ~0.6≤

_{REL}*ρ*≤~0.9}, transverse intensity profiles viewed at a distance

_{REL}*z*are observed to form intense arms. These arms extend from the center, spiraling outwards perpendicular to the propagation axis. Alternatively, for

*ρ*values within the range [~0.1, ~0.3], BH beams instead form into symmetric patterns resembling polygons which encapsulate

_{REL}*ℓ*distinct OV’s distributed uniformly within the profile. The intermediate

*ρ*interval serves as a transition wherein beams are observed to form quasi-ring structures containing inner intense-arm formations. In all cases, the number of intense-arm or outer ring formations equal the integer-order topological charge

_{REL}*ℓ*of the BH phase.

*ρ*is decreased, interference of the wavefronts results in the intensity distribution of the beam gradually shifting from the center towards the outer ring structures. The limiting case of which being

_{REL}*ρ*=0, where the BH phase reverts to the corkscrew topology of a charge

_{REL}*ℓ*helical phase.

*ρ*value of 0.84 to 0.48, even though the encoded topological charges have the same sign and the intensity patterns were obtained at the same propagation distance. This is evident most specially in

_{REL}*ℓ*=1. It seems that the rotational sense is not determined by the sign of the topological charge alone but also of the value of

*ρ*. An analysis similar to Soskin [6

_{REL}6. I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A **6**, S166–S169 (2004).
[CrossRef]

16. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A **56**, 4064–4075 (1997).
[CrossRef]

4. M. V. Berry, “Optical Vortices Evolving from helicoidal integer and fractional phase steps,” J. Opt. A **6**, 259–268 (2004).
[CrossRef]

*ρ*from 0 to 2

_{i}*πℓ*.

*z*(Fig. 3). In contrast to the propagation invariant LG modes however, BH beams are, in general, structurally unstable upon propagation. Moreover, the variation in the intensity pattern as the beam propagates gives the impression of rotation. Such rotation is unlike the rigid rotation of intensity profiles discussed in Refs. [7

7. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. **40**, 73–87 (1993).
[CrossRef]

**56**, 4064–4075 (1997).
[CrossRef]

*ℓ*and

*ρ*as well. Beams with identical charges but differing

_{REL}*ρ*values exhibit dissimilar rotation rates. It is observed that beams with smaller

_{REL}*ρ*values rotate more quickly than beams with larger cavity radii (Fig. 4). In particular, it was found that beams with

_{REL}*ρ*values corresponding to the transition interval possess rotation rate greater than intense arms BH beams. Similarly, for beams of identical

_{REL}*ρ*values, it is observed that rotation rate monotonically decreases with the magnitude of charge |

_{REL}*ℓ*| (Fig. 5). In all cases, angular displacements asymptotically approach a maximum value as the wave propagates. This behavior suggests an association with the Gouy phase shift. Further details pertaining to the particular role of the Gouy effect as well as vortex dynamics in relation to beam profile evolution is outside the scope of this paper.

*et al*[5

5. V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. **39**, 985–990 (1992).
[CrossRef]

*Λ*denotes the spatial period of the plane wave in the transverse plane [26

26. N. P. Manaois and C. O. Hermosa II, “Phase structure of Helico-conical optical beams,” Opt. Commun. **271**, 178–183 (2007).
[CrossRef]

*ℓ*and

*ρ*were constructed. The incident beam was expanded and collimated so that it uniformly illuminates the CGH’s. Experiments verify the intensity profiles obtained in the numerical simulation.

_{REL}*f*) parallel to the input beam. This method simulates propagation by shifting the focal plane of the diffracting lens instead of mechanically translating the viewing plane (CCD). Figure 10 shows a sequence illustrating propagation of a

_{1}*ℓ*=3,

*ρ*=0.6 BH beam. This verifies qualitatively the unstable behavior of BH beams upon propagation as well as the decrease in rotation rate as the beam departs the focal plane.

_{REL}## 3. Calculations of Poynting vector and the orbital angular momentum

*BH*is the BH phase (Eq.(2)) with cavity radius

_{ℓ}*ρ*,

_{i}*σ*denotes the initial waist of the truncated Gaussian amplitude profile, and

*C*the normalization constant. The initial beam waist

*σ*is chosen to be larger than the outer radius

*ρ*to ensure the beam varies only slightly within the region

_{o}*O*. Outside this region, the transmission is zero. From Eq.(8), the propagated field can be obtained, within the paraxial limit, using the Huygens-Fresnel integral [28, 29]

*I*and

*O*\

*I*are disjoint, the two integrals

*(*

^{I}u*ρ*,

*z*) and

*(*

^{O}u*ρ*,

*φ*,

*z*) in Eq. (9) do not overlap. The separation of the integral occurs when the separation in the radial coordinate of the phase term of the integrand factor

*u*(

^{BH}*ρ*,

*φ*,

*z*=0) is considered. Expressed in this manner, the field

*u*(

^{BH}*ρ*,

*φ*,

*z*) at some

*z*>0 may be interpreted as the interference of two co-propagating wavefronts, with only the

*(*

^{O}u*ρ*,

*φ*,

*z*) component possessing a helical phase structure. Note that the

*(*

^{O}u*ρ*,

*φ*,

*z*) term includes

*χ*

_{O/I}. The presence of the plane-phase component in the wavefront is responsible for the decomposition of a single

*ℓ*-charged (|

*ℓ*|>1) vortex into nearly-situated |

*ℓ*| unit-charged vortices of the same sign (identical helicity) [12

12. G.-H. Kim, J.-H. Jeon, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “Optical vortices produced with a nonspiral phase plate,” Appl. Opt. **36**, 8614–8621 (1997).
[CrossRef]

14. I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. **103**, 422–428 (1993).
[CrossRef]

**56**, 4064–4075 (1997).
[CrossRef]

*(*

^{I}u*ρ*,

*z*) and

*(*

^{O}u*ρ*,

*φ*,

*z*) is dictated by the cavity radius

*ρ*. Larger

_{i}*ρ*values would imply greater contribution of the plane-phase wave

_{i}*(*

^{I}u*ρ*,

*z*) giving rise to intensity arms, while as

*ρ*approaches zero, the intensity profiles of BH beams revert to those of OV beam patterns.

_{i}**〉·**

*S**φ*̂ is attributed to the helical phase structure of the beam. This implies a spiraling Poynting vector trajectory similar to OV beams.

**also gives rise to a nonzero component of the angular momentum density of**

*S**u*(

^{BH}*ρ*,

*φ*,

*z*) associated with the transverse plane. This is given by [17

17. M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. **121**, 36–40 (1995).
[CrossRef]

**Ξ**is thus

*ρ*>0. Eq. (13) also exhibits that a plane-phase wavefront such as

_{i}*does not contribute to the OAM expression of the beam [14*

^{I}u14. I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. **103**, 422–428 (1993).
[CrossRef]

13. K. Staliunas, “Dynamics of optical vortices in a laser beam,” Opt. Commun. **90**, 123–127 (1992).
[CrossRef]

**56**, 4064–4075 (1997).
[CrossRef]

## 4. Conclusion

*I*, thereby replacing the on-axis screw-dislocation with several edge-dislocations. Beams with this phase possess a variety of interesting properties different from OV beams.

31. L. Kelemen, S. Valkai, and P. Ormos, “Parallel photopolymerisation with complex light patterns generated by diffractive optical elements,” Opt. Express **15**, 14488–14497 (2007).
[CrossRef] [PubMed]

## Acknowledgments

## References and Links

1. | M. V. Berry, “Much ado about nothing: optical dislocation lines (phase singularities, zeros, vortices,” in. Intl. Conf. on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 1–15 (1998). |

2. | G. V. Soskin and M. S. Borgatiryova, “Detection and metrology of optical vortex helical wavefronts,” SPQEO |

3. | A. Dreischuh, D. Neshev, G. G. Paulus, and H. Walther, “Experimental generation of steering odd dark beams of finite length,” J. Opt. Soc. Am. B |

4. | M. V. Berry, “Optical Vortices Evolving from helicoidal integer and fractional phase steps,” J. Opt. A |

5. | V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. |

6. | I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A |

7. | G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. |

8. | W.M. Lee, X.-C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. |

9. | J. Curtis and D. G. Grier, “Structure of Optical Vortices,” Phys. Rev. Lett. |

10. | S. Ramee and R. Simon, “Effects of holes and vortices on beam quality,” J. Opt. Soc. Am. A |

11. | X. Yuan, B. S. Ahluwalia, W. C. Cheong, L. Zhang, J. Bu, S. Tao, K. J. Moh, and J. Lin, “Micro-optical elements for optical manipulation,” Opt. Photon. News |

12. | G.-H. Kim, J.-H. Jeon, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “Optical vortices produced with a nonspiral phase plate,” Appl. Opt. |

13. | K. Staliunas, “Dynamics of optical vortices in a laser beam,” Opt. Commun. |

14. | I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. |

15. | I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. |

16. | M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A |

17. | M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. |

18. | L. Allen, M.J. Padgett, and M. Babiker, “The Orbital Angular Momentum of Light,” in Progress in Optics XXXIX, edited by E. Wolf, Elsevier Science B.V., New York, 294–372 (1999). |

19. | S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. |

20. | J. E. Molloy and M. J. Padgett, “Lights, action: optical tweezers,” Contemp. Phys. |

21. | M. J. Padgett and L. Allen, “Optical tweezers and spanners,” Phys. World |

22. | J. B. Gotte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with orbital angular momentum and their vortex structure,” Opt. Express |

23. | S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express |

24. | K Volke-Sepulveda, V Garces-Chavez, S Chavez-Cerda, J Arlt, and K Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B |

25. | C. A. Alonzo, P. J. Rodrigo, and P. Gluckstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express |

26. | N. P. Manaois and C. O. Hermosa II, “Phase structure of Helico-conical optical beams,” Opt. Commun. |

27. | H. L. Royden, |

28. | A. E. Siegman, |

29. | J. Goodman, |

30. | H. A. Haus, |

31. | L. Kelemen, S. Valkai, and P. Ormos, “Parallel photopolymerisation with complex light patterns generated by diffractive optical elements,” Opt. Express |

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(050.4865) Diffraction and gratings : Optical vortices

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: March 2, 2009

Revised Manuscript: April 9, 2009

Manuscript Accepted: May 1, 2009

Published: August 28, 2009

**Citation**

Stein Alec C. Baluyot and Nathaniel P. Hermosa II, "Intensity profiles and propagation of optical beams with bored helical phase," Opt. Express **17**, 16244-16254 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-18-16244

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### References

- M. V. Berry, "Much ado about nothing: optical dislocation lines (phase singularities, zeros, vortices," in. Intl. Conf. on Singular Optics, M. S. Soskin, ed., Proc. SPIE 3487, 1-15 (1998).
- G. V. Borgatiryova and M. S. Soskin, "Detection and metrology of optical vortex helical wavefronts," SPQEO 6, 254 - 258 (2003).
- A. Dreischuh, D. Neshev, G. G. Paulus, and H. Walther, "Experimental generation of steering odd dark beams of finite length," J. Opt. Soc. Am. B 17, 2011 - 2017 (2000). [CrossRef]
- M. V. Berry, "Optical Vortices Evolving from helicoidal integer and fractional phase steps," J. Opt. A 6, 259 - 268 (2004). [CrossRef]
- V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, "Screw dislocations in light wavefronts," J. Mod. Opt. 39, 985 - 990 (1992). [CrossRef]
- I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, "Synthesis and analysis of optical vortices with fractional topological charges," J. Opt. A 6, S166 - S169 (2004). [CrossRef]
- G. Indebetouw, "Optical vortices and their propagation," J. Mod. Opt. 40, 73 - 87 (1993). [CrossRef]
- W.M. Lee, X.-C. Yuan, K. Dholakia, "Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step," Opt. Commun. 239, 129-135 (2004). [CrossRef]
- J. Curtis and D. G. Grier, "Structure of Optical Vortices," Phys. Rev. Lett. 90, 133901-1 1 133901-4 (2003). [CrossRef]
- S. Ramee and R. Simon, "Effects of holes and vortices on beam quality," J. Opt. Soc. Am. A 17, 84 - 94 (2000). [CrossRef]
- X. Yuan, B. S. Ahluwalia, W. C. Cheong, L. Zhang, J. Bu, S. Tao, K. J. Moh, and J. Lin, "Micro-optical elements for optical manipulation," Opt. Photon. News 17, 36 - 41 (2006). [CrossRef]
- G.-H. Kim, J.-H. Jeon, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, "Optical vortices produced with a nonspiral phase plate," Appl. Opt. 36, 8614 - 8621 (1997). [CrossRef]
- K. Staliunas, "Dynamics of optical vortices in a laser beam," Opt. Commun. 90, 123 - 127 (1992). [CrossRef]
- I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, "Optics of light beams with screw dislocations," Opt. Commun. 103, 422 - 428 (1993). [CrossRef]
- I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, "Optical wavefront dislocations and their properties," Opt. Commun. 119, 604 - 612 (1995). [CrossRef]
- M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, "Topological charge and angular momentum of light beams carrying optical vortices," Phys. Rev. A 56, 4064 - 4075 (1997). [CrossRef]
- M. J. Padgett, and L. Allen, "The Poynting vector in Laguerre-Gaussian laser modes," Opt. Commun. 121, 36 - 40 (1995). [CrossRef]
- L. Allen, M.J. Padgett, and M. Babiker, "The Orbital Angular Momentum of Light," in Progress in Optics XXXIX, edited by E. Wolf, Elsevier Science B.V., New York, 294 - 372 (1999).
- S. M. Barnett, and L. Allen, "Orbital angular momentum and nonparaxial light beams," Opt. Commun. 110, 670 - 678 (1994). [CrossRef]
- J. E. Molloy and M. J. Padgett, "Lights, action: optical tweezers," Contemp. Phys. 43, 241 - 258 (2002). [CrossRef]
- M. J. Padgett and L. Allen, "Optical tweezers and spanners," Phys. World 10, 35 - 38 (1997).
- J. B. Gotte, K. O'Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, "Light beams with orbital angular momentum and their vortex structure," Opt. Express 16, 993 - 1006 (2008). [CrossRef] [PubMed]
- S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, "Optical ferris wheel for ultracold atoms," Opt. Express 15, 8619 - 8625 (2007). [CrossRef] [PubMed]
- K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, "Orbital angular momentum of a high-order Bessel light beam," J. Opt. B 4, S82-S89 (2002). [CrossRef]
- C. A. Alonzo, P. J. Rodrigo, and P. Gluckstad, "Helico-conical optical beams: a product of helical and conical phase fronts," Opt. Express 13, 1749 - 1760 (2005). [CrossRef] [PubMed]
- N. P. HermosaII and C. O. Manaois, "Phase structure of Helico-conical optical beams," Opt. Commun. 271, 178 - 183 (2007). [CrossRef]
- H. L. Royden, Real Analysis, 3rd ed. (Macmillan Publishing, USA 1988).
- A. E. Siegman, LASERS, 1st ed. (University Science Books, USA 1986).
- J. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, Inc., USA, 1968).
- H. A. Haus, Waves and fields in optoelectronics, 1st ed. (Prentice Hall, Englewood Cliffs, New Jersey 1984).
- L. Kelemen, S. Valkai, and P. Ormos, "Parallel photopolymerisation with complex light patterns generated by diffractive optical elements," Opt. Express 15, 14488-14497 (2007). [CrossRef] [PubMed]

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