## Propagation law for the generating function of Hermite-Gaussian-type modes in first-order optical systems

Optics Express, Vol. 13, Issue 4, pp. 1107-1112 (2005)

http://dx.doi.org/10.1364/OPEX.13.001107

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### Abstract

Based on the common Hermite-Gaussian modes, a general class of orthonormal sets of Hermite-Gaussian-type modes is introduced. Such modes can most easily be defined by means of their generating function. It is shown that these modes remain in their class of orthonormal Hermite-Gaussian-type modes, when they propagate through first-order optical systems. A propagation law for the generating function is formulated.

© 2005 Optical Society of America

## 1. Introduction

## 2. Hermite-Gaussian-type modes

*H*(°) are the Hermite polynomials [1, Section 22] and where the column vector

_{n}**r**=(

*x,y*)

^{t}is a short-hand notation for the spatial variables

*x*and

*y*, with the superscript

^{t}denoting transposition. Note that

*𝓗*(

_{n}*x;w*) has been defined such that we have the orthonormality relationship

*δ*the Kronecker delta. (All integrals in this paper extend from -∞ to +∞.) From the generating function of the Hermite polynomials [1, Eq. (22.9.17)],

_{nl}*𝓗*(

_{n,m}**r**;

*w*):

_{x},w_{y}*𝓗*(

_{n,m}**r;K,L**) that we propose, is most easily defined by the generating function

**s**=(

*s*)t and three (possibly complex) 2×2-matrices

_{x}, s_{y}**K, L=L**

^{t}, and

**M=M**

^{t}. For the common Hermite-Gaussian modes

*𝓗*(

_{n,m}**r**;

*w*) we have, see Eq. (5),

_{x},w_{y}**M**is completely determined by

**K**, see Eq. (19), and therefore does not have to be included as a parameter in

*𝓗*(

_{n,m}**r;K,L**).

## 3. Propagation through first-order optical systems

*𝓗*(

_{n,m}**r;K,L**) propagate through a lossless, first-order optical system – also called an

**ABCD**-system – and determine the generating function of the set of modes to which the beam that appears at the output of this system belongs. Any lossless, first-order optical system can be described by its ray transformation matrix [2], which relates the position ri and direction qi of an incoming ray to the position

**r**

_{o}and direction q

_{o}of the outgoing ray:

**A, B**, and

**D**, and assuming that

**B**is a non-singular matrix, we can represent the first-order optical system by the Collins integral [3

3. S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. **60**, 1168–1177 (1970). [CrossRef]

*f*(

_{o}**r**) is expressed in terms of the input amplitude f i(r). The phase factor exp(

*iϕ*) in Eq. (10) is rather irrelevant and can often be chosen arbitrarily. We remark that the signal transformation

*f*(

_{i}**r**)→

*f*(

_{o}**r**) that corresponds to a lossless, first-order optical system, is unitary, i.e.

*𝓗*(

_{n,m}**r;K**

_{i},

**L**

_{i}) at the input of an

**ABCD**-system, we denote the output beam by

*𝓗*(

_{n,m}**r;K**

_{o},

**L**

_{o}). To find the generating function of the set of modes to which this output beam belongs, we write

4. M. J. Bastiaans and T. Alieva, “Generating function for Hermite-Gaussian modes propagating through first-order optical systems,” J. Phys. A: Math. Gen **38**, L73–L78 (2005). [CrossRef]

**K**

_{i},

**L**

_{i}, and

**M**

_{i}by the output matrices

**K**

_{o},

**L**

_{o}, and

**M**

_{o}, respectively, in accordance with the input-output relationships (12–14). Note that Eq. (13) is in fact the wellknown

**ABCD**-law, and that Eqs. (12) and (13) can be combined into

## 4. Conditions resulting from orthonormality

**K, L**, and

**M**. To derive these, we consider the expression

**t**=(

*t*)

_{x},t_{y}^{t}on the analogy of s=(

*s*)

_{x}, s_{y}^{t}, and where we have substituted from the generating function (6). We note that the integral in this expression equals

**L**and

**M**once again. Note that

**M=KK***

^{-1}is completely determined by

**K**, see Eq. (19), which is the reason why we did not include

**M**as a parameter in

*𝓗*(

_{n,m}**r;K,L**).

**K**

^{-1}in its real and imaginary parts,

**K**

^{-1}=

**a**+

*i*

**b**, we immediately get from the realness of (

**K***

^{t}

**K**)-1, see Eq. (20), that the matrix

**ab**

^{t}is symmetric. If we then express

**L**as

**L**=(

**d**-

*i*c)

**K**=(

**d**-

*i*

**c**)(

**a**+

*i*

**b**)-1, the symmetry of

**L**leads to

5. A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. A: Math Gen. **33**, 1603–1629 (2000). [CrossRef]

*π*

**r**

^{t}

**Lr**), with a matrix

**L**that can be chosen freely if we would only require Eq. (17) and not necessarily Eq. (18), whereasWünsche uses a fixed expression of the form exp(-

*π*

**r**

^{t}

**r**). Wünsche’s results arise indeed from ours for the special choice

**L**=I, in which case Eq. (17) leads to

**M=KK**

^{t}, yielding the generating function

5. A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. A: Math Gen. **33**, 1603–1629 (2000). [CrossRef]

**KK***

^{t}=

**I**.

*x*and

*y*directions determined by

*γ*and

_{x}*γ*, respectively), for which the matrices

_{y}**a, b, c**, and

**d**are given by

*γ*=0. Note that for Hermite-Gaussian modes the matrix

_{x}=γ_{y}=γ_{1}=γ_{2}*diagonal*matrix, and that the

**ABCD**-law (13) is useful when such modes propagate through separable systems (for which

**A, B, C**, and

**D**are

*diagonal*matrices). For (rotationally symmetric) Laguerre-Gaussian modes (with its curvature determined by

*γ*) we have

*γ=γ*=0 has been reported, for instance, in [6

_{1}=γ_{2}6. M.W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. **96**, 123–132 (1993). [CrossRef]

**L**=(1+

*i*tan

*γ*)

*w*

^{-2}

**I**is a

*scalar*matrix, and that the

**ABCD**-law (13) is useful when such modes propagate through isotropic systems (for which

**A, B, C**, and

**D**are

*scalar*matrices). We remark that the discriminating parameters in the above expressions are the widths (

*w*) and the curvatures (

_{x}, w_{y}, w*γ*) of the modes; the parameters γ

_{x}, γ_{y}, γ_{1}and γ

_{2}lead to a mere multiplication of the complex field amplitude by a phase factor that depends on the mode-number (

*n,m*) but not on the space variables

**r**.

## 5. An alternate propagation law

**K**and the output matrices

_{i}, L_{i}, M_{i}**K**

_{o},

**L**

_{o},

**M**

_{o}can be expressed in the special forms

**a**

_{i},

**b**

_{i},

**c**

_{i},

**d**

_{i}, and the real matrices

**a**

_{o},

**b**

_{o},

**c**

_{o},

**d**

_{o}constitute two real symplectic matrices, we can – after some straightforward but rather lengthy calculations, in which we make extensive use of the symplecticity properties, cf. Eqs. (9) – bring the input-output relationships (12–14) into a different form and formulate the elegant propagation law

7. J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. **61**, 751–758 (1971). [CrossRef]

*i*(

**a+**),

*i*b**d**-

*i*c correspond to the ‘matricial rays’

**Q,P**[7

7. J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. **61**, 751–758 (1971). [CrossRef]

**Q**√

*π*=

*i*

**K**

^{-1}=

*i*(

**a**+

*i*

**b**) and

**P**√

*π*=λ

**LK**

^{-1}=λ(

**d**-

*i*c), with λ the wavelength of the light. The treatment in [7

7. J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. **61**, 751–758 (1971). [CrossRef]

**Q,P**that are associated with the mode that is to be generated. The modes in [7

**61**, 751–758 (1971). [CrossRef]

**a,b**and

**c,d**that characterize this generating function correspond to the real and imaginary parts of

**Q**and

**P**, and constitute a real, symplectic matrix. With

**r=x**and λ

**s**=

*π*

**y**√2, there is indeed a one-to-one correspondence between our generating function (6) and the one used in [7

**61**, 751–758 (1971). [CrossRef]

**61**, 751–758 (1971). [CrossRef]

**61**, 751–758 (1971). [CrossRef]

## 6. Conclusion

## Acknowledgments

## References and links

1. | M. Abramowitz and I. A. Stegun, eds., |

2. | R. K. Luneburg, |

3. | S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. |

4. | M. J. Bastiaans and T. Alieva, “Generating function for Hermite-Gaussian modes propagating through first-order optical systems,” J. Phys. A: Math. Gen |

5. | A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. A: Math Gen. |

6. | M.W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. |

7. | J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. |

8. | T. Alieva and M. J. Bastiaans, “Mode mapping in paraxial lossless optics,” submitted to Opt. Lett. (2005). |

**OCIS Codes**

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

(070.4690) Fourier optics and signal processing : Morphological transformations

(080.2730) Geometric optics : Matrix methods in paraxial optics

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

(140.3300) Lasers and laser optics : Laser beam shaping

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 30, 2004

Revised Manuscript: November 26, 2004

Published: February 21, 2005

**Citation**

Martin Bastiaans and Tatiana Alieva, "Propagation law for the generating function of Hermite-Gaussian-type modes in first-order optical systems," Opt. Express **13**, 1107-1112 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-4-1107

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### References

- M. Abramowitz and I. A. Stegun, eds., Pocketbook of Mathematical Functions (Deutsch, Frankfurt am Main, Germany, 1984).
- R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, CA, USA, 1966).
- S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970). [CrossRef]
- M. J. Bastiaans and T. Alieva, “Generating function for Hermite-Gaussian modes propagating through first-order optical systems,” J. Phys. A: Math. Gen 38, L73–L78 (2005). [CrossRef]
- A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. A: Math Gen. 33, 1603–1629 (2000). [CrossRef]
- M.W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P.Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]
- J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. 61, 751–758 (1971). [CrossRef]
- T. Alieva and M. J. Bastiaans, “Mode mapping in paraxial lossless optics,” submitted to Opt. Lett. (2005).

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