## On the control of the spatial orientation of the transverse profile of a light beam

Optics Express, Vol. 14, Issue 3, pp. 1086-1093 (2006)

http://dx.doi.org/10.1364/OE.14.001086

Acrobat PDF (112 KB)

### Abstract

A first-order optical system (represented by its 4×4 ABCD matrix) is given in order to obtain a beam that preserves its spatial orientation of the transverse profile under free propagation from a beam with rotating irradiance distribution in free space. Within the formalism of the second-order irradiance moments, this transverse orientation is analyzed in terms of the evolution of the principal axes of the field irradiance distribution. It is shown that the spatial profile of the beam emerging from the proposed optical system does not rotate when light freely propagates. The improvement of the joint near-field and far-field beam spread product at the output of this optical system is also studied.

© 2006 Optical Society of America

## 1. Introduction

01. R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. **65**, 322–328 (1988). [CrossRef]

07. ISO 11146, Laser and laser related equipment-Test methods for laser beam widths, divergence angles and beam propagation ratios: ISO 11146-1:2005, Part 1: Stigmatic and simple astigmatic beams; ISO11146-2:2005, Part 2: General astigmatic beams; ISO/TR 11146-3:2004, Part 3: Intrinsic and geometrical laser beam classification, propagation, and details of test method; ISO/TR 11146-3:2004/Cor1:2005 (International Organization for Standardization, Geneva, Switzerland, 2005).

01. R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. **65**, 322–328 (1988). [CrossRef]

08. F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero, and P. M. Mejías, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. **34**, 1835–1838 (1998). [CrossRef]

09. P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. **26**, 65–130 (2002), and references therein. [CrossRef]

10. J. A. Arnaud and H. Kogelnik, “Light beams with general astigmatism,” Appl. Opt. **8**, 1687–1693 (1969). [CrossRef] [PubMed]

10. J. A. Arnaud and H. Kogelnik, “Light beams with general astigmatism,” Appl. Opt. **8**, 1687–1693 (1969). [CrossRef] [PubMed]

05. J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A **8**, 1094–1098 (1991). [CrossRef]

14. J. Serna, P. M. Mejías, and R. Martínez-Herrero, “Rotation of partially coherent beams through free space,” Opt. Quantum Electron. **24**, 873–880 (1992). [CrossRef]

## 2. Formalism and key definitions

*W*(

**r**_{1}

**r**_{2}), where

**r**_{j}, j = 1, 2, represent the two-dimensional position vectors at two points over the beam cross-section, transverse to the propagation direction

*z*. Since we will consider quasimonochromatic fields, explicit dependence on frequency ω will, for simplicity, be omitted in our expressions.

*W*, here we are interested on the global behaviour of the beam, described by certain overall parameters that propagate according to simple laws. Let us then introduce the Wigner distribution function (WDF) associated with the CSD function through a Fourier transform relationship: [5

05. J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A **8**, 1094–1098 (1991). [CrossRef]

18. M. J. Bastiaans, “Wigner distribution function and its applications to first-order optics,” J. Opt. Soc. Am. **69**, 1710–1716 (1979). [CrossRef]

*(*

**r***x*,

*y*) denotes again the two-dimensional position vector, the dot symbolizes the inner product, and

*k*= (

**η***ku*,

*kv*) = (

*k*,

_{x}*k*) provides the wavevector components along the x and y-axes (accordingly,

_{y}*u*and

*v*represent angles of propagation, without taking the evanescent waves into account). The WDF can be physically understood in Optics as the amplitude associated to a ray passing through a point along a certain direction. [19

19. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. **58**, 1256–1259 (1968). [CrossRef]

*I*= ∫

_{O}*h*d

*d*

**r***is proportional to the total beam power. As is well known, the four first-order moments, <*

**η***x*>, <

*y*>, <

*u*> and <

*v*>, characterize the centre of the beam and its mean direction. For simplicity, in what follows we assume that these moments vanish (this is not a restriction, since it is equivalent to a shift of the Cartesian coordinate system). On the other hand, the (squared) beam width at a plane z = constant and the (squared) far-field divergence are represented by <

*x*

^{2}+

*y*

^{2}> and <

*u*

^{2}+

*v*

^{2}> , respectively. In addition, the crossed moment <

*xu*+

*yv*> gives the position of the beam waist through the condition <

*xu*+

*yv*> = 0. Finally, it would also be useful to introduce an easy-to-measure figure of merit,

*Q*, that provides a joint information about the near and far-field behaviour of the beam. For brevity, we will refer to this parameter as the beam spread product, which has been defined for partially coherent fields as follows [5

05. J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A **8**, 1094–1098 (1991). [CrossRef]

*k*

^{2}factor apart, the beam propagation factor (

*M*

^{2})

^{2}first introduced by Siegman.[4

04. A. E. Siegman, “New developments in laser resonators” in *Laser Resonators*, Proc. SPIE **1224**, 2–14 (1990). [CrossRef]

*Q*as a 3D quality parameter relies on its invariance under propagation through rotationally-symmetric ABCD systems. In particular,

*Q*remains constant under free propagation. Note, however, that Q is not constant in the general case.[20–21

20. A. T. Friberg, E. Tervonen, and T. Turunen, “Interpretation and experimental demonstration of twisted Gauss-Schell-mode beams,” J. Opt. Soc. Am A **11**, 1818–1826 (1994). [CrossRef]

*Q*is a kind of “partial invariant”, which can be modified by using non-rotationally-symmetric systems. Consequently, this class of optical systems could then improve this parameter: a lower value of

*Q*would mean better simultaneous focusing and collimation capabilities for freely propagating light. It should also be noted that

*Q*has a lower limit,

*Q*≥ 1/

*k*

^{2}, where the equality is only reached by the idealized Gaussian beam.

**8**, 1094–1098 (1991). [CrossRef]

14. J. Serna, P. M. Mejías, and R. Martínez-Herrero, “Rotation of partially coherent beams through free space,” Opt. Quantum Electron. **24**, 873–880 (1992). [CrossRef]

*x*-

*y*moment vanishes, i.e. <

*xy*> = 0. It can also be shown that the beam widths <

*x*

^{2}>

^{1/2}and <

*y*

^{2}>

^{1/2}reach their extreme values along these axes. [14

14. J. Serna, P. M. Mejías, and R. Martínez-Herrero, “Rotation of partially coherent beams through free space,” Opt. Quantum Electron. **24**, 873–880 (1992). [CrossRef]

## 3. Optical system after which the beam profile is non-rotating

*as the product of*

**r***k*= 2π/λ by the conventional Cartesian coordinates of the point at which the field is evaluated. Consequently, from now on,

*x*and

*y*should be considered dimensionless variables, and all the irradiance moments (e.g., <

*x*

^{2}+

*y*

^{2}>, <

*u*

^{2}+

*v*

^{2}>, <

*xu*+

*yv*>, <

*xy*>), given by Eq. (2), will become dimensionless functions.

*Q*

_{x}=

*Q*

_{y}, where

*Q*

_{j}, j =

*x*,

*y*, would represent the 2D beam spread products associated with each transverse field component, namely,

*θ*between this new coordinate system and the former one (laboratory Cartesian axes) would be given by the formula

*f*denotes the irradiance moments referred to the former laboratory coordinate system. Remember that <

*x*

^{2}>, <

*y*

^{2}> and <

*xy*> are here dimensionless parameters.

*i*refers to the values of the beam irradiance moments at the input plane of this system. Note again that, according with the normalization introduced at the beginning of this section, the parameters

*a*and

*b*are dimensionless quantities.

22. R. Simon and K. Bernardo Wolf, “Fractional Fourier transform in two dimensions,” J. Opt. Soc. Am. A **17**, 2368–2381 (2000). [CrossRef]

22. R. Simon and K. Bernardo Wolf, “Fractional Fourier transform in two dimensions,” J. Opt. Soc. Am. A **17**, 2368–2381 (2000). [CrossRef]

*z*-evolution of the spatial orientation of the beam profile, we will analyze the angle φ between the principal axes of the output field and our (fixed) Cartesian coordinate axes (for which

*Q*

_{x}=

*Q*

_{y}). At each transverse plane

*z*, the angle φ is given in terms of the intensity moments at such plane by the formula [5

**8**, 1094–1098 (1991). [CrossRef]

**24**, 873–880 (1992). [CrossRef]

*x*

^{2}>, <

*y*

^{2}> and <

*xy*> are evaluated at plane

*z*. Taking into account the free-propagation law of the irradiance moments, this expression can also be written in the form

*o*denotes the values at the output plane

*z*= 0 of the system S. Note that, in Eq. (11),

*kz*is a dimensionless product.

*x*

^{2}>

*and <*

_{o}*y*

^{2}>

*, are connected with their values at the input plane of the system by means of the following expressions:*

_{o}*xu*+

*yv*>

*= 0 (beam waist at the input plane) and <*

_{i}*x*

^{2}>

*<*

_{i}*u*

^{2}>

*= <*

_{i}*y*

^{2}>

*<*

_{i}*v*

^{2}>

*(since*

_{i}*Q*

_{x}=

*Q*y). Consequently, from Eqs. (12) and (13) it follows at once

*u*

^{2}>

*=<*

_{o}*v*

^{2}>

*. Moreover, <*

_{o}*xu*>

*=<*

_{o}*yv*>

*= 0. Therefore, after substitution of these expressions in Eq. (9), we obtain*

^{o}*z*) = constant = π/4. Accordingly, the spatial profile of the output beam does not rotate after travelling through the optical system described by matrix S, Q.E.D.

## 4. Beam spread product at the output of the system

*Q*, defined in Eq. (3), when the beam travels through the optical system represented by matrix S (cf Eq. (7)). More specifically, we will next show that the value of the parameter

*Q*decreases after propagation through the proposed optical system.

*xu*>

*=<*

_{o}*yv*>

*= 0 . Then*

_{o}*Q*denotes the beam spread product at the output plane of the system S. But, since the waist plane is placed at its input plane, we have <

_{o}*xu*>

*= - <*

_{i}*yv*>

*, and one gets*

_{i}*Q*

_{x}=

*Q*

_{y}, then <

*x*

^{2}>

*<*

_{i}*u*

^{2}>

*= <*

_{i}*y*

^{2}>

*<*

_{i}*v*

^{2}>

*, and the beam spread product*

_{i}*Q*at the input plane would read

_{i}*Q/*≤

_{o}*Q*, Q.E.D.

_{i}*Q*=

_{o}*Q*, in Eq. (21), the input beam should fulfil two conditions, namely, <

_{i}*u*

^{2}>

*,=<*

_{i}*v*

^{2}>

*and<*

_{i}*xu*>

*= 0. But, since*

_{i}*Q*=

_{x}*Q*, we would also have <

_{y}*x*

^{2}>

*=<*

_{i}*y*

^{2}>

*, and this input beam would not rotate under free propagation. In other words, use of system S would not be required for such beam. As a consequence, we finally get that*

_{i}*Q*<

_{o}*Q*for any input rotating beam.

_{i}## 5. Conclusions

15. G. Nemes and J. Serna, “Do not use spherical lenses and free spaces to characterize beams: a possible improvement of the ISO/DIS 11146 document,” in *Proceedings of the Fourth Workshop on Laser Beam and
Optics Characterization*,
A. Giesen and M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düseldorf, Germany, 1997), pp. 29–49.

*Q*, and the same near- and far-field behaviour under free propagation, but the differences would appear after travelling, for example, through a cylindrical lens following by a free space. We do not proceed further into these properties because the present work does not concern with the full detwisting problem. These aspects, however, deserve attention in the future.

## Acknowledgments

## Reference and links

01. | R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. |

02. | S. Lavi, R. Prochaska, and E. Keren, “Generalized beam parameters and transformation law for partially coherent light,” Appl. Opt. |

03. | M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik |

04. | A. E. Siegman, “New developments in laser resonators” in |

05. | J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A |

06. | H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. |

07. | ISO 11146, Laser and laser related equipment-Test methods for laser beam widths, divergence angles and beam propagation ratios: ISO 11146-1:2005, Part 1: Stigmatic and simple astigmatic beams; ISO11146-2:2005, Part 2: General astigmatic beams; ISO/TR 11146-3:2004, Part 3: Intrinsic and geometrical laser beam classification, propagation, and details of test method; ISO/TR 11146-3:2004/Cor1:2005 (International Organization for Standardization, Geneva, Switzerland, 2005). |

08. | F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero, and P. M. Mejías, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. |

09. | P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. |

10. | J. A. Arnaud and H. Kogelnik, “Light beams with general astigmatism,” Appl. Opt. |

11. | J. Serna and G. Nemes, “Decoupling of coherent Gaussian beams with general astimatism,” Opt. Lett. |

12. | G. Nemes and A. E. Siegman, “Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am A |

13. | G. Nemes, “Synthesis of general astigmatic optical systems, the detwisting procedure and the beam quality factors for general astigmatic laser beams,” in |

14. | J. Serna, P. M. Mejías, and R. Martínez-Herrero, “Rotation of partially coherent beams through free space,” Opt. Quantum Electron. |

15. | G. Nemes and J. Serna, “Do not use spherical lenses and free spaces to characterize beams: a possible improvement of the ISO/DIS 11146 document,” in |

16. | G. Nemes and J. Serna, “Laser beam characterization with use of second order moments: an overview,” in |

17. | J. Serna, F. Encinas, and G. Nemes, “Complete spatial characterization of a pulsed doughnut-type beam by use of spherical optics and a cylindrical lens,” J. Opt. Soc. Am. A |

18. | M. J. Bastiaans, “Wigner distribution function and its applications to first-order optics,” J. Opt. Soc. Am. |

19. | A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. |

20. | A. T. Friberg, E. Tervonen, and T. Turunen, “Interpretation and experimental demonstration of twisted Gauss-Schell-mode beams,” J. Opt. Soc. Am A |

21. | A. T. Friberg, C. Gao, B. Eppich, and H. Weber, “Generation of partially coherent fields with twist,” Proc. SPIE |

22. | R. Simon and K. Bernardo Wolf, “Fractional Fourier transform in two dimensions,” J. Opt. Soc. Am. A |

**OCIS Codes**

(140.3300) Lasers and laser optics : Laser beam shaping

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: October 31, 2005

Revised Manuscript: December 27, 2005

Manuscript Accepted: January 27, 2006

Published: February 6, 2006

**Citation**

R. Martínez-Herrero and P. Mejías, "On the control of the spatial orientation of the transverse profile of a light beam," Opt. Express **14**, 1086-1093 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-3-1086

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### References

- R. Simon, N. Mukunda and E. C. G. Sudarshan, "Partially coherent beams and a generalized ABCD-law," Opt. Commun. 65, 322-328 (1988). [CrossRef]
- S. Lavi, R. Prochaska and E. Keren, "Generalized beam parameters and transformation law for partially coherent light," Appl. Opt. 27, 3696-3703 (1988). [CrossRef] [PubMed]
- M. J. Bastiaans, "Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems," Optik 82, 173-181 (1989).
- A. E. Siegman, "New developments in laser resonators" in Laser Resonators, Proc. SPIE 1224, 2-14 (1990). [CrossRef]
- J. Serna, R. Martínez-Herrero and P. M. Mejías, "Parametric characterization of general partially coherent beams propagating through ABCD optical systems," J. Opt. Soc. Am. A 8, 1094-1098 (1991). [CrossRef]
- H. Weber, "Propagation of higher-order intensity moments in quadratic-index media," Opt. Quantum Electron. 24, 1027-1049 (1992). [CrossRef]
- ISO 11146, Laser and laser related equipment-Test methods for laser beam widths, divergence angles and beam propagation ratios: ISO 11146-1:2005, Part 1: Stigmatic and simple astigmatic beams; ISO11146-2:2005, Part 2: General astigmatic beams; ISO/TR 11146-3:2004, Part 3: Intrinsic and geometrical laser beam classification, propagation, and details of test method; ISO/TR 11146-3:2004/Cor1:2005 (International Organization for Standardization, Geneva, Switzerland, 2005).
- F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero and P. M. Mejías, "Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses," IEEE J. Quantum Electron. 34, 1835-1838 (1998). [CrossRef]
- P. M. Mejías, R. Martínez-Herrero, G. Piquero and J. M. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarized laser beams," Prog. Quantum Electron. 26, 65-130 (2002), and references therein. [CrossRef]
- J. A. Arnaud and H. Kogelnik, "Light beams with general astigmatism," Appl. Opt. 8, 1687-1693 (1969). [CrossRef] [PubMed]
- J. Serna and G. Nemes, "Decoupling of coherent Gaussian beams with general astimatism," Opt. Lett. 18, 1174-1176 (1993). [CrossRef]
- G. Nemes and A. E. Siegman, "Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics," J. Opt. Soc. Am A 11, 2257-2264 (1994). [CrossRef]
- G. Nemes, "Synthesis of general astigmatic optical systems, the detwisting procedure and the beam quality factors for general astigmatic laser beams," in Proceedings of the Second Workshop on Laser Beam Characterization, H. Weber, N. Reng, J. Ludtke and P. M. Mejías, eds., Festkorper-Laser Institut Berlin GmbH, Berlin, Germany, 1994, pp. 93-104.
- J. Serna, P. M. Mejías and R. Martínez-Herrero, "Rotation of partially coherent beams through free space," Opt. Quantum Electron. 24, 873-880 (1992). [CrossRef]
- G. Nemes and J. Serna, "Do not use spherical lenses and free spaces to characterize beams: a possible improvement of the ISO/DIS 11146 document," in Proceedings of the Fourth Workshop on Laser Beam and Optics Characterization, A. Giesen and M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düseldorf, Germany, 1997), pp. 29-49.
- G. Nemes and J. Serna, "Laser beam characterization with use of second order moments: an overview," in Diode Pumped Solid State Lasers: Applications and Issues, M. W. Dowley, ed., OSA TOPS 17, 200-207 (1998).
- J. Serna, F. Encinas and G. Nemes, "Complete spatial characterization of a pulsed doughnut-type beam by use of spherical optics and a cylindrical lens," J. Opt. Soc. Am. A 18, 1726-1733 (2001). [CrossRef]
- M. J. Bastiaans, "Wigner distribution function and its applications to first-order optics," J. Opt. Soc. Am. 69, 1710-1716 (1979). [CrossRef]
- A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 58, 1256-1259 (1968). [CrossRef]
- A. T. Friberg, E. Tervonen and T. Turunen, "Interpretation and experimental demonstration of twisted Gauss-Schell-mode beams," J. Opt. Soc. Am A 11, 1818-1826 (1994). [CrossRef]
- A. T. Friberg, C. Gao, B. Eppich and H. Weber, "Generation of partially coherent fields with twist," Proc. SPIE 3110, 317-318 (1997). [CrossRef]
- R. Simon and K. Bernardo Wolf, "Fractional Fourier transform in two dimensions," J. Opt. Soc. Am. A 17, 2368-2381 (2000). [CrossRef]

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