## Effect of ABCD transformations on beam paraxiality |

Optics Express, Vol. 19, Issue 27, pp. 25944-25953 (2011)

http://dx.doi.org/10.1364/OE.19.025944

Acrobat PDF (1031 KB)

### Abstract

The limits of the paraxial approximation for a laser beam under ABCD transformations is established through the relationship between a parameter concerning the beam paraxiality, the paraxial estimator, and the beam second-order moments. The applicability of such an estimator is extended to an optical system composed by optical elements as mirrors and lenses and sections of free space, what completes the analysis early performed for free-space propagation solely. As an example, the paraxiality of a system composed by free space and a spherical thin lens under the propagation of Hermite-Gauss and Laguerre-Gauss modes is established. The results show that the the paraxial approximation fails for a certain feasible range of values of main parameters. In this sense, the paraxial estimator is an useful tool to monitor the limits of the paraxial optics theory under ABCD transformations.

© 2011 OSA

## 1. Introduction

11. P. Vaveliuk, “Quantifying the paraxiality for laser beams from the *M*^{2}-factor,” Opt. Lett. **34**, 340–342 (2009). [CrossRef] [PubMed]

## 2. Beam second-order moments and ABCD optical systems

*z*> 0 direction. Its complex amplitude in a transverse plane (fixed value of

*z*) is given by the function

*E*(

*x,y*), where we assume that the beam waist (minimum transverse size),

*w*, is located at the plane

_{m}*z*= 0. This plane is the input plane to a single or cascade ABCD system. The plane wave decomposition of such a beam around the

*z*axis is done by means of the bidimensional Fourier transform (FT) [12] where ℰ (

*u,v*) is the Fourier amplitude of the beam in the conjugate space, (

*u, v*), with 2

*πu*and 2

*πv*being the transverse wavenumbers associated with the

*x*and

*y*coordinates, respectively. We will assume the indefinite integrals to the whole real axis if not stated otherwise. Notice that

*u*and

*v*are dimensional coordinates with units of (length)

^{−1}. It is a common practice to describe the beams by means of the Wigner distribution (WD) for its characterization. The WD is a real function that represents the beam in the phase-space, (

*x,y,u, v*), i. e., it possesses the information from both the spatial and spectral amplitudes of the beam. Consequently, the global properties of the beam can easily be extracted calculating the moments of the WD, which are indirectly measurable [13

13. K. Sundar, N. Mukunda, and R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A **12**, 560–569 (1995). [CrossRef]

14. G. Nemes and A. E. Siegmam, “Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A **11**, 2257–2264 (1994). [CrossRef]

*E*(

*x,y*) in the plane

*z*= 0, its WD is defined as where

**r**= [

*x,y*]

*and*

^{t}**p**= [

*u,v*]

*are the spatial and spectral column vectors, correspondingly,*

^{t}*t*denotes the transpose operation and 〈·〉 the ensemble average. Its WD moments are defined in the same way the statistical moments are defined for statistical distributions. In particular the only zero-order moment is proportional to the irradiation of the beam, while its four first-order moments define the centroid and propagation direction of the beam. Without loss of generality one can always select such an axis for the transverse plane that the first-order moments of the beam are zero. Then, we can define the normalized second-order moments in this frame of reference as where

*s*and

*s*′ are placeholders for the spatial and spectral coordinates,

*x*,

*y*,

*u*, and

*v*. These moments are widely used in beam characterization as they contain important information, as for example, the beam size and beam spread in the

*x*and

*y*spatial coordinates and the quality factor of the beam [15]. All the ten independent second-order moments of a beam can conveniently be arranged into a 4 × 4 real symmetric positive-definite matrix [16

16. S. Ramee and R. Simon, “Effect of holes and vortices on beam quality,” J. Opt. Soc. Am. A **17**, 84–94 (2000). [CrossRef]

*is dimensionless, 𝕄*

_{qp}*has units of (length)*

_{qq}^{2}and 𝕄

*of (length)*

_{pp}^{−2}.

**M**is defined only for a fixed transversal plane (in our particular case, for the input plane

*z*= 0), so it may change (and in general does) as the beam propagates through an optical system to the output plane

*z*=

*z*′. In this paper we consider the propagation through any two dimensional ABCD lossless optical system, including anisotropic (astigmatic) optical systems that are described by a 4 × 4 matrix called characteristic ray-transformation matrix: with 𝔸, 𝔹, ℂ, and 𝔻 being 2×2 sub-matrices. It has to satisfy the symplectic condition [17

17. M. Nazarathy and J. Shamir, “First-order optics–a canonical operator representation lossless systems,” J. Opt. Soc. Am. **72**, 356–364 (1982). [CrossRef]

18. E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta **32**, 855–872 (1985). [CrossRef]

**M**in the input plane

*z*= 0 to

**M**′ in the output plane

*z*=

*z*′ by the relation [1

1. R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical system and associated invariants,” Phys. Rev. A **31**, 2419–2434 (1985). [CrossRef] [PubMed]

**M**′ and

**M**, submatrices 𝔸 and 𝔻 have to be dimensionless, while 𝔹 has to have units of (length)

^{2}and ℂ of (length)

^{−2}. If we have a ABCD system as a cascade of

*n*optical elements (e.g. a sequence of spaces and lenses) whose single ray-transfer matrix are

**T**

_{1},

**T**, ...,

_{2}**T**

_{n}, it is well-known that such multi-element system is equivalent to a single optical component of ray-transfer matrix

**M**=

**T**

_{n}...

**T**

_{2}

**T**

_{1}. The above description of the second-order moments and rules of transformation through ABCD optical systems give us the machinery to quantitatively analyze the validity of the paraxial approximation. It must be emphasized that the ABCD matrices are real-valued for the cases treated here (lossless media). But complex-valued matrices are also possible that arise, for example, from optical systems with inherent apertures. The effects of finite-sized optical elements, tilt and random jitter, and distributed random inhomogeneities along the optical path were rigorously tackled in Ref. [19

19. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A **4**, 1931–1948 (1987). [CrossRef]

20. H. T. Yura, B. Rose, and S. G. Hanson, “Dynamic laser speckle in complex ABCD optical systems” J. Opt. Soc. Am. A **15**, 1160–1166 (1998). [CrossRef]

## 3. Beam paraxiality under ABCD transformations

6. P. Vaveliuk, B. Ruiz, and A. Lencina, “Limits of the paraxial aproximation in laser beams,” Opt. Lett. **32**, 927–929 (2007). [CrossRef] [PubMed]

*𝒫*, quantifying the validity of the paraxial approximation in free space propagation was introduced and applied in analyzing the paraxiality of Hermite-Gaussian, Laguerre-Gaussian, and Bessel-Gaussian modes. The definition of this parameter was based on the comparison between the propagation invariants of Helmholtz and paraxial equations. It was established that

*𝒫*is a real scalar value less than one. As

*𝒫*tends to unit, the PA fulfils more robustly for a given light beam. Working in the conjugate space,

*𝒫*is expressed in terms of the wavelength of the light,

*λ*, and the beam second-order moments in the conjugate space

*m*and

_{uu}*m*, that are associated with the spread of the beam in the transverse plane. Explicitly [11

_{vv}11. P. Vaveliuk, “Quantifying the paraxiality for laser beams from the *M*^{2}-factor,” Opt. Lett. **34**, 340–342 (2009). [CrossRef] [PubMed]

*𝒫*and at the output plane in which the paraxial estimator is denoted by

*𝒫*′. In this case, we are analyzing the validity of the PA for a given system between both planes. The paraxial estimator at the output plane

*z*=

*z*′ will be where the prime denotes evaluation in the output plane of the optical system. Using Eq. (7) we can relate the required output moments with the input moments as follows: It is important to be aware that the paraxial estimator in the exiting plane of an ABCD optical system depends on two sets of parameters: the beam characteristics, encoded in both, the wavelength and the second-order moments; and the optical system properties, encoded in the ray-transformation matrix.

*using Eq. (10) and introduce the results in Eq. (9). Also, it is obvious from Eqs. (9,10) that*

_{pp}*𝒫*′ does not depend on the submatrices 𝔸 and 𝔹 of the ABCD system. This fact is important because it directly yields that the single free-space propagation does not change this system paraxiality. On the contrary, the inclusion of linear optical elements as mirrors and lenses, in addition to the free space, could significatively alter the system paraxiality as it will showed below.

## 4. Numerical examples: paraxiality for free-space plus a thin lens under Hermite-Gauss and Laguerre-Gauss propagation

*λ*and the transverse size parameter

*w*

_{0}, this last encoded in its second moments. The HG modes in the waist transverse plane

*z*= 0 are given in rectangular coordinates (

*x,y*) by where

*E*

_{0}is a constant with electric field dimensions and

*H*is the Hermite Polynomial of degree

_{m,n}*m, n*. The LG

*mode profiles can be written in cylindrical coordinates (*

_{l,p}*r*,

*ϕ*,

*z*) at the beam waist as where

*ϕ*is the azimuthal coordinate and

*N*, being

*N*=

*m*+

*n*for HG and

*N*= 2

*p*+ |

*ℓ*| for LG. The mode

*N*= 0, common to both families, is the fundamental Gaussian beam. This mode fulfills

*w*=

_{m}*w*

_{0}, so that

*w*

_{0}really represents the minimum spot size or beam waist size. But this is not the case for higher order modes. All

*N*≠ 0 modes have the same transverse scale length

*w*

_{0}but higher order modes use a large transverse area and the minimum spot size is approximately

*z*= 0,

**M**

*, by using Eq. (3) giving The matrix*

_{HG}**M**

*is like Eq. (13) by replacing all (2*

_{LG}*m*+ 1) and (2

*n*+ 1) factors by the factor 2

*p*+ |

*ℓ*|. With explicit values for the second moments, the paraxial estimator can be then explicitly calculated from Eq. (9) for determining the validity of the paraxial approximation in systems composed by free space and thin lenses.

### 4.1. Single systems: Free-space and Generalized thin lens

*z*= 0) and output (

*z*=

*z*′) planes is then given by where

*d*is the propagation distance. Applying Eq. (7) one finds that It is clear that 𝕄′

*= 𝕄*

_{pp}*having no changes in the second moments of the conjugate space under the free-space transformation. Consequently, the beam propagation along*

_{pp}*z*leaves the paraxial estimator invariant between

*z*= 0 to

*z*=

*z*′ for HG and LG modes such that where

*w̃*

_{0}=

*w*

_{0}/

*λ*is the normalized size parameter to the wavelength. Equation (16) coincides with the result given in Ref. [6

6. P. Vaveliuk, B. Ruiz, and A. Lencina, “Limits of the paraxial aproximation in laser beams,” Opt. Lett. **32**, 927–929 (2007). [CrossRef] [PubMed]

*𝒫*follows the criterion for the validity limits of the paraxial approximation for the fundamental Gaussian mode (

*N*= 0) established in most standard textbooks. For instance, after a judicious analysis, Siegman [3] concluded that the paraxial optical beams can be focused or can diverge at semi-cone angles up to 28°(0.5 rad) before significant corrections to the paraxial wave approximation become necessary. Because

*θ*=

*λ*/(

*πw*

_{0}), the use of Eq. (16) gives

*𝒫*= 0.94 corresponding to

*w̃*

_{0}∼ 0.64. From this, the scale setting for the paraxial estimator is established. The paraxial approximation may be a questionable hypotheses for

*𝒫*-values of the order of and lower than 0.94. In the opposite limit,

*w̃*

_{0}≫ 1, the paraxial estimator quickly tends to unit and the Gaussian beam is fully paraxial. These limits were rigorously investigated in the interesting paper of Seshadri[10

10. S. R. Seshadri, “Quality of paraxial electromagnetic beams,” Appl. Opt. **45**, 5335–5345 (2006). [CrossRef] [PubMed]

*w̃*

_{0}since

*𝒫*decreases when

*N*increases. For example, for

*N*= 10, the PA begins to be critical from

*w̃*

_{0}∼ 2.15, or for a real value of the minimum spot size of

*w̃*≈ 7.13. Although the free space solely does not affect the beam paraxiality, it will affect if it is part of a cascade of optical systems as we will see in further examples.

_{m}### 4.2. A cascade system: free space portion plus spherical thin lens

*f*with the (beam waist) input plane placed at

*z*= 0 and the output plane at the exit lens plane

*z*=

*z*′ so that the distance among both planes is

*d*+

*δ*

*∼ d*as illustrated by Fig. 1 (c). First of all, we derive the paraxial estimator. The ray transformation matrix of this composite system is

**T**

*=*

_{FL}**T**

_{L}**T**

*, where*

_{F}**T**

*and*

_{F}**T**

*are defined in the Eq. (14) and Eq. (17), respectively. That way, the second-order matrix transformation law under this system between input and output planes is*

_{L}*𝒫*′ accounts the curvature phase of the beam at the lens entrance and its transformation features. If

*d*= 0, the beam waist plane coincides with the lens plane and Eq. (23) reduces to Eq. (20). If one has a collimated beam exiting from the lens (

*d*=

*f*) then

*w*′

_{0}=

*fλ*/

*πw*

_{0}(see for example Ref. [4

4. B. E. A. Saleh and M. C. Teich, *Fundamentals of Photonics*, (John Wiley, 1991), Chaps. 1–4,7–9,14. [CrossRef]

*𝒫*′, reduces to

*w*′

_{0}. Hence,

*𝒫*′ really estimates the paraxiality of the exiting beam supporting the relationship between the paraxial estimator and the beam second order moments. On the other hand, the paraxial estimator will give rise identical value if the input plane is symmetrically shifted from the focal plane: a exiting beam will acquire the same phase curvature if

*d*=

*f*+

*ɛ*or if

*d*=

*f*–

*ɛ*, being

*ɛ*<

*f*. Both output beams thus corresponding to the same paraxiality.

*𝒫*′ for the free space plus spherical thin lens system crossed by a Gaussian beam (

*N*= 0) in terms of both the normalized beam waist parameter

*w̃*

_{0}and the ratio

*d*/

*f*for a fixed focal length

*f*= 2 × 10

^{4}

*λ*. The dotted line is a reference for the paraxial-nonparaxial limit at the input plane,

*𝒫*= 0.94 that was included for comparative purposes. It is clear that the PA validity at the output plane is strongly dependant on the distance among the beam waist plane and the lens entrance plane. For instance, a beam with waist size of

*w̃*

_{0}= 2.0 that is paraxial at the input plane,

*𝒫*= 0.99, and symbolized by “+” in the figure, becomes nonparaxial at the output plane (

*𝒫*′ = 0.89) if

*d*/

*f*= 5. This same beam, now placed at

*d*/

*f*= 2 (symbolized by “*” in the figure), becomes paraxial at the output plane:

*𝒫*′ = 0.99. Hence, the distance between input and output planes is critical for evaluating the PA validity. Finally, notice that around the collimation zone

*d*≈

*f*, there would be no restrictions on the system paraxiality in the limit

*w̃*

_{0}→ 0 since

*𝒫*′ ≃ 1. This would imply that an ultra-focalized beam (for instance

*w̃*

_{0}≈ 0.4) whose paraxial estimator at the input is

*𝒫*= 0.84 would give paraxial at the output plane. This apparent inconsistency is due to the failure of the paraxial approximation of the incident beam. In such case, the ABCD theory is not suitable to predict the beam characteristics at the lens exit. The fulfillment of the PA at the entrance plane is a mandatory condition in order to apply the results given by Eq. (23).

## 5. Conclusion

## Acknowledgments

## References and links

1. | R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical system and associated invariants,” Phys. Rev. A |

2. | P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. |

3. | A. E. Siegman, |

4. | B. E. A. Saleh and M. C. Teich, |

5. | H. Kogelnik, “Imaging of Optical Modes -Resonators and Internal Lenses”, Bell Syst. Opt. Tech. J. |

6. | P. Vaveliuk, B. Ruiz, and A. Lencina, “Limits of the paraxial aproximation in laser beams,” Opt. Lett. |

7. | P. Vaveliuk, “Comment on degree of paraxiality for monochromatic light beams,” Opt. Lett. |

8. | P. Vaveliuk, G. F. Zebende, M. A. Moret, and B. Ruiz, “Propagating free-space nonparaxial beams,” J. Opt. Soc. Am. A |

9. | M. A. Bandres and M. Guizar-Sicairos, “Paraxial group,” Opt. Lett. |

10. | S. R. Seshadri, “Quality of paraxial electromagnetic beams,” Appl. Opt. |

11. | P. Vaveliuk, “Quantifying the paraxiality for laser beams from the |

12. | J. W. Goodman, |

13. | K. Sundar, N. Mukunda, and R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A |

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

15. | A. E. Siegman, G. Nemes, and J. Serna, in Proceedings of DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1. |

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

17. | M. Nazarathy and J. Shamir, “First-order optics–a canonical operator representation lossless systems,” J. Opt. Soc. Am. |

18. | E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta |

19. | H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A |

20. | H. T. Yura, B. Rose, and S. G. Hanson, “Dynamic laser speckle in complex ABCD optical systems” J. Opt. Soc. Am. A |

**OCIS Codes**

(070.2590) Fourier optics and signal processing : ABCD transforms

(120.4820) Instrumentation, measurement, and metrology : Optical systems

(200.4740) Optics in computing : Optical processing

**ToC Category:**

Fourier Optics and Signal Processing

**History**

Original Manuscript: August 1, 2011

Revised Manuscript: October 30, 2011

Manuscript Accepted: October 31, 2011

Published: December 6, 2011

**Citation**

Pablo Vaveliuk and Oscar Martinez-Matos, "Effect of ABCD transformations on beam paraxiality," Opt. Express **19**, 25944-25953 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-25944

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### References

- R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical system and associated invariants,” Phys. Rev. A31, 2419–2434 (1985). [CrossRef] [PubMed]
- P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett.16, 196–198 (1991). [CrossRef] [PubMed]
- A. E. Siegman, Lasers, (University Science Books, 1986), Chaps. 15–21.
- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (John Wiley, 1991), Chaps. 1–4,7–9,14. [CrossRef]
- H. Kogelnik, “Imaging of Optical Modes -Resonators and Internal Lenses”, Bell Syst. Opt. Tech. J.44, 455–494 (1965).
- P. Vaveliuk, B. Ruiz, and A. Lencina, “Limits of the paraxial aproximation in laser beams,” Opt. Lett.32, 927–929 (2007). [CrossRef] [PubMed]
- P. Vaveliuk, “Comment on degree of paraxiality for monochromatic light beams,” Opt. Lett.33, 3004–3005 (2008). [CrossRef] [PubMed]
- P. Vaveliuk, G. F. Zebende, M. A. Moret, and B. Ruiz, “Propagating free-space nonparaxial beams,” J. Opt. Soc. Am. A24, 3297–3302 (2007). [CrossRef]
- M. A. Bandres and M. Guizar-Sicairos, “Paraxial group,” Opt. Lett.34, 13–15 (2009). [CrossRef]
- S. R. Seshadri, “Quality of paraxial electromagnetic beams,” Appl. Opt.45, 5335–5345 (2006). [CrossRef] [PubMed]
- P. Vaveliuk, “Quantifying the paraxiality for laser beams from the M2-factor,” Opt. Lett.34, 340–342 (2009). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
- K. Sundar, N. Mukunda, and R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A12, 560–569 (1995). [CrossRef]
- G. Nemes and A. E. Siegmam, “Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A11, 2257–2264 (1994). [CrossRef]
- A. E. Siegman, G. Nemes, and J. Serna, in Proceedings of DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.
- S. Ramee and R. Simon, “Effect of holes and vortices on beam quality,” J. Opt. Soc. Am. A17, 84–94 (2000). [CrossRef]
- M. Nazarathy and J. Shamir, “First-order optics–a canonical operator representation lossless systems,” J. Opt. Soc. Am.72, 356–364 (1982). [CrossRef]
- E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta32, 855–872 (1985). [CrossRef]
- H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A4, 1931–1948 (1987). [CrossRef]
- H. T. Yura, B. Rose, and S. G. Hanson, “Dynamic laser speckle in complex ABCD optical systems” J. Opt. Soc. Am. A15, 1160–1166 (1998). [CrossRef]

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