## Calculating the fine structure of a Fabry-Perot resonator using spheroidal wave functions

Optics Express, Vol. 18, Issue 9, pp. 9580-9591 (2010)

http://dx.doi.org/10.1364/OE.18.009580

Acrobat PDF (847 KB)

### Abstract

A new set of vector solutions to Maxwell’s equations based on solutions to the wave equation in spheroidal coordinates allows laser beams to be described beyond the paraxial approximation. Using these solutions allows us to calculate the complete first-order corrections in the short-wavelength limit to eigenmodes and eigenfrequencies in a Fabry-Perot resonator with perfectly conducting mirrors. Experimentally relevant effects are predicted. Modes which are degenerate according to the paraxial approximation are split according to their total angular momentum. This includes a splitting due to coupling between orbital angular momentum and spin angular momentum.

© 2010 Optical Society of America

## 1. Introduction

1. I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. **4**, 382–385 (2008). [CrossRef]

2. A. Kuhn, M. Hennrich, and G. Rempe, “Deterministic single-photon source for distributed quantum networking,” Phys. Rev. Lett. **89**, 067901 (2002). [CrossRef] [PubMed]

_{0,0}mode to the regularly spaced eigenmode spectrum.

*et al*. [4

4. H. Laabs and A. T. Friberg, “Nonparaxial Eigenmodes of Stable Resonators,” IEEE J Quanum Electron. **35**, 198–207 (1999). [CrossRef]

*et al*. [7

7. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A **11**, 1365–1370 (1975). [CrossRef]

*et al*. [5

5. J. Visser and G. Nienhuis, “Spectrum of an optical resonator with spherical aberration,” J. Opt. Soc. Am. A **22**, 2490–2497 (2005). [CrossRef]

*et al*. [6

6. F. Zomer, V. Soskov, and A. Variola, “On the nonparaxial modes of two-dimensional nearly concentric resonators,” Appl. Opt. **46**, 6859–6866 (2007). [CrossRef] [PubMed]

*et al*. restricted their analysis to scalar fields in two-dimensional resonators. Visser

*et al*. focused solely on spherical aberrations, albeit leading to an analytical result reproduced here. The most rigorous approach so far by Laabs

*et al*. used an insufficient approximation by treating the mirror surfaces purely in terms of a position-dependent phase shift.

*ξ*as well as between the variation in beam diameter along a beam and surfaces of constant

*η*. With appropriate approximations, these coordinates have been applied in the past to resonators to obtain results in agreement with paraxial theory [8

8. R. W. Zimmerer, “Spherical Mirror Fabry-Perot Resonators,” IEEE Trans. Microwave Theory Tech. **11**, 371–379 (1963). [CrossRef]

9. W. A. Specht Jr., “Modes in Spherical-Mirror Resonators,” J. Appl. Phys. **36**, 1306–1313 (1965). [CrossRef]

## 2. Mathematical foundations

11. M. Zeppenfeld, “Solutions to Maxwell’s equations using spheroidal coordinates,” New J. Phys. **11**, 073007 (2009). [CrossRef]

11. M. Zeppenfeld, “Solutions to Maxwell’s equations using spheroidal coordinates,” New J. Phys. **11**, 073007 (2009). [CrossRef]

*ξ,η,ϕ*) to cylindrical coordinates (

*r,z,ϕ*) is given by

*ϕ*-coordinate being the same in both coordinate systems and

*d*being the interfocal distance as shown in Fig. 1. The scalar wave equation is separable in spheroidal coordinates, allowing the solutions to be written as the product of three functions depending only on

*ξ*,

*η*and

*ϕ*, respectively. This leads to so-called scalar spheroidal wave functions

*ψ*=

_{mν}*R*(

_{mν}*ξ*)

*S*(

_{mν}*η*)

*e*with

^{imϕ}*m*and

*ν*. Note that

*m*is simply the integer orbital angular momentum due to the

*e*

^{imϕ}*ϕ*-dependence. The variable

*x*= 2

*c*̄(1 -

*η*), not to be confused with the Cartesian coordinate, is introduced to simplify calculations. The parameter

*c*̄ =

*kd*/2, with wavevector

*k*, quantifies the scaling of the coordinate system relative to the wavelength. For short wavelengths compared to

*d*,

*c*̄ is a large number, and the functions

*r*(

_{mν}*ξ*) and

*s*(

_{mν}*x*) can be expanded as asymptotic series in

*r*(

_{mν}*ξ*) = 1 and

*s*(

_{mν}*x*) =

*L*

_{ν}^{(m)}(

*x*) where

*L*

_{ν}^{(m)}(

*x*) is a Laguerre polynomial.

**E**

^{±}

*propagating in the ±*

_{Jσν}*ξ*direction are defined which satisfy the wave equation as well as ∇ ·

**E**

^{±}

*= 0. The*

_{Jσν}**E**

^{±}

*can therefore be regarded as the electric field of a solution to Maxwell’s equations in free space. With spin angular momentum*

_{Jσν}*σ*=

*σ*

^{±}corresponding to left and right circular polarization,

*J*=

*m*± 1 denotes the total integer angular momentum of the field about the symmetry axis.

## 3. Satisfying the boundary conditions

**E**

^{±}

*. For perfectly conducting mirrors, the component of the electric field parallel to a mirror surface*

_{Jσν}**S**must vanish at

**S**, i.e.

**E**∣

_{S,∥}=0.

*L*with spherical mirrors with radius of curvature

*R*

_{-}and

*R*

_{+}, the spheroidal coordinate system is appropriately scaled by choosing the distance

*d*between the focal points to be [8

8. R. W. Zimmerer, “Spherical Mirror Fabry-Perot Resonators,” IEEE Trans. Microwave Theory Tech. **11**, 371–379 (1963). [CrossRef]

*ξ*=

*ξ*

_{±}with

*L*and have a radius of curvature of

*R*

_{±}on the resonator axis.

*ξ*is generally necessary. Switching briefly to cylindrical coordinates (

*z,r,ϕ*), the mirror surfaces are specified as the distance

*z*=

*z*̄

_{±}(

*r,ϕ*) above the plane

*z*= 0. For a smooth, cylindrically symmetric mirror,

*z*̄

_{±}(

*r,ϕ*) can be expanded in powers of

*r*

^{2}as

*z*

_{±}and

*c*

_{4±}are expansion coefficients and

*R*

_{±}is the radius of curvature of the surface at

*r*= 0. For a spherical surface,

*c*

_{4±}= 1/(8

*R*

^{3}

_{±}). Other values of

*c*

_{4±}can be chosen to describe cylindrically-symmetric mirror aberrations. For example, for a spheroidal surface of constant

*ξ*=

*ξ*

_{±}we have

*z*

_{±}=

*dξ*

_{±}/2 and

*c*

_{4±}= 1/(4

*dξ*

_{±}

*R*

^{2}

_{±}). For a parabolic mirror,

*c*

_{4±}= 0 by definition.

*ξ*as a function of

*x*= 2

*c*̄(1 -

*η*) and

*ϕ*. Eq. (7) is transformed into

**S**

_{±}being the mirror surface at

*ξ*

_{±}. Note that for small

*r*,

*x*is proportional to

*c*̄

*r*

^{2}. Additionally, due to the factor

*e*

^{-x/2}in the definition of

*S*(

_{mν}*η*), Eq. (3), the spheroidal wave functions vanish for

*x*≫ 1, motivating the use of

*x*in Eq. (8). The term linear in

*x*is missing due to the specific choice of

*d*and

*ξ*

_{±}in Eqs. (5) and (6). As will be seen, the

*x*

^{2}term in Eq. (8) is the highest-order term in

*x*which must be retained to calculate first-order corrections to the resonator eigenfrequencies.

*η*and

*ϕ*as

**x**(

*ξ*,

*η*,

*ϕ*) is the vector in ℝ

^{3}from the origin to the point denoted in spheroidal coordinates by (

*ξ*, η,

*ϕ*). A basis for the tangent space to the mirror surface at a point on the mirror surface, needed to calculate the component of

**E**parallel to the mirror surface, is given by

**e**̂

*. are unit vectors and the*

_{ui}*h*are scale factors in spheroidal coordinates [12]. The component of the electric field parallel to the mirror surface is proportional to the inner product of the electric field with these basis vectors.

_{ui}**e**̂

*,*

_{ϕ}**e**̂

*} as an approximation for the basis (11). For*

_{η}*x*= 𝒪(1), we have the following order of magnitudes,

11. M. Zeppenfeld, “Solutions to Maxwell’s equations using spheroidal coordinates,” New J. Phys. **11**, 073007 (2009). [CrossRef]

**e**̂

*·*

_{ξ}**E**term is of order

*c*̄

^{-2}compared to the contribution of the

**e**̂

*·*

_{η}**E**term and can therefore be neglected, justifying our simplified basis.

**e**̂ ·

**E**

_{±}

*and*

_{Jσν}**e**̂

_{ϕ}·

**E**

^{±}

*we additionally need the following expressions,*

_{Jσν}**e**̂

*are not needed, but have been included for completeness. Using previous expressions and with a significant amount of algebra, one obtains*

_{ξ}*L*

_{ν}^{(J)}are Laguerre polynomials

*L*

_{ν}^{(J)}(

*x*) with the dependence on

*x*being implicit. Heavy use of recursion relations among the Laguerre polynomials has been made. Note that we have evaluated the spheroidal wave functions at the mirror surfaces

**S**

_{±}. For

**E**

_{Jσν}^{-}, the field running in the other direction, the results are the same except that

*ξ*

_{±}is replaced by -

*ξ*

_{±}and

*f*

_{4±}is replaced by -

*f*

_{4±}.

**E**for the mirror at

*ξ*

_{±}can be reformulated as a matrix equation for the coefficients

*b*

_{Jσν}^{∓}of the outgoing wave in terms of the coefficients

*b*

_{Jσν}^{±}of the incoming wave. We begin by defining the set of vector functions

**v**

*as*

_{Jσν}**u**

_{1Jσν}and

**u**

_{2Jσν}according to

**E**

^{+}

*at the mirror surfaces can then be written in terms of the*

_{Jσν}**v**

*as*

_{Jσν}**E**

^{-}

*we must again replace*

_{Jσν}*ξ*

_{±}everywhere by -

*ξ*

_{±}and

*ξ*

_{4±}by -

*ξ*

_{4±}. The

*A*are expansion coefficients for the angular spheroidal functions [11

^{s}_{Jν}**11**, 073007 (2009). [CrossRef]

**E**

^{±}

*∣*

_{Jσν}_{s±,∥}is simply proportional to

**v**

*, motivating the choice of*

_{Jσν}**v**

*.*

_{Jσν}*a*

^{±1±2,J,σ′,σ}

*. The subscripts on the ± signs denote two independent choices of + or -. Eqs. (24) and (25) contain factors of*

_{ν′ν}*x*as part of the coefficients of the functions

*v*

*. These can be removed using the relations*

_{Jσν}*a*

^{±1±2,J,σ′,σ}

*independent of*

_{ν′ν}*x*and

*ϕ*.

**E**can now be written as

*J*do not couple, so we have restricted our attention to a single

*J*. The

**v**

*, considered as vector functions of*

_{Jσν}*x*and

*ϕ*, are linearly independent. As a result, the expression in the square bracket in Eq. (29) must be zero for each value of

*σ*′ and

*ν*′.

*A*

^{±1,±2}maps the traveling wave described by the vector

**b**

_{σ}^{±1}of coefficients of wave-functions

**E**

^{±1}

*onto a set of coefficients of surface functions*

_{Jσν}**v**

*describing the electric field component parallel to the surface at*

_{Jσν}**S**

_{±2}. This finally allows us to write the boundary conditions as matrix equations,

**b**

_{σ}^{∓}in terms

**b**

_{σ}^{±}using the boundary condition imposed by the mirror at

*ξ*

_{±}, we find the round-trip matrix for the resonator to be given by

*A*are vectors of coefficients

*b*

^{+}

*of resonator eigenmodes, the corresponding eigenvalues are the round-trip phase shifts. To lowest order in*

_{Jσν}*A*is diagonal. The resonator eigenmodes are therefore of the form

*b*

^{+}

_{Jσ±ν}

**E**

^{+}

_{Jσ±ν}+

*b*

^{-}

_{Jσ±ν}

**E**

^{-}

_{Jσ±ν}and the corresponding round-trip phase shift is equal to

## 4. First-order corrections to the round-trip phase shift

*A*which are degenerate to lowest order in

**E**

_{Jσ+ν}and

*E*

_{Jσ-,ν-1}are the same for arbitrary

*J*and

*ν*. This reflects the fact that, within the paraxial approximation, the resonator eigenfrequency is independent of the polarization. Additionally, for a cavity geometry such that

*p*and

*n*, the lowest-order eigenvalues corresponding to the modes

**E**

**E**

*and*

_{Jσν}**E**

_{Jσ,ν+n}are the same for arbitrary

*J*,

*ν*and

*σ*.

*E*

_{Jσ+ν}in general couples to the modes

**E**

_{Jσ+, ν±1}as well as to the modes

*E*

_{Jσ-ν′}with

*ν′*=

*ν*- 2,

*ν*- 1 and

*ν*. A mode

**E**

_{Jσ-ν}couples to the modes

*E*

_{Jσ-,ν±1}as well as to the modes

*E*

_{Jσ+ν}with

*ν*′ =

*ν*,

*ν*+ 1 and

*ν*+ 2. For nonzero

*f*

_{4±}coefficient, the modes

**E**

_{Jσ+ν}and

**E**

_{Jσ-ν}additionally couple to the modes

**E**

_{Jσ+ν±2}and

**E**

_{Jσ-ν±2}, respectively.

**E**

_{Jσ+ν}and

**E**

_{Jσ-,ν-1}must be taken into account. On the other hand, a degeneracy between the modes

**E**

_{Jσ+ν}and

**E**

_{Jσ+,ν+n}due to resonator geometry is only relevant for

*n*= 1 and for

*n*= 2,

*f*

_{4±}≠ 0. For these two special cases, an infinite set of degenerate modes are coupled by first-order off-diagonal matrix elements, and finding the first-order corrections to the eigenvalues of

*A*is significantly more difficult. We therefore exclude these two cases from further analysis.

*n*= 1 and

*n*= 2,

*f*

_{4±}≠ 0, the first-order corrections to the eigenvalues of

*A*can be obtained from the two-by-two submatrix of

*A*which corresponds to the subspace spanned by

**E**

_{Jσ+ν}and

**E**

_{Jσ-,ν-1}. We denote this matrix as the two-by-two round-trip matrix. Since the four matrices

*A*

^{±1,±2}are all diagonal to lowest order in , the two-by-two round-trip matrix is given to first order in by a product as in Eq. (32), except that each of the matrices

*A*

^{±1,±2}is replaced by an appropriate two-by-two submatrix. Specifically, these four submatrices are made up of the coefficients

*a*

_{ij}^{±1,±2}with

*i*,

*j*∈ {0,1}, given by

*a*

_{ij}^{+,±}, obtained by applying Eqs. (27) and (28) to Eqs. (24) and (25) and comparing with Eq. (26), are given explicitly by

*a*

_{ij}^{-,±}one can check that

*a*

_{ij}^{-,±}= (

*a*

_{ij}^{+,±})* with * denoting complex conjugation.

*ξ*

_{±}is given by

*i*(

*R*

_{J,ν-1}(

*ξ*)-

*R*(

_{Jν}*ξ*)),

*R*

_{J+1,ν-1}(

*ξ*) and

*R*

_{J-1,ν}(

*ξ*) is equal to lowest order in

*a*

_{ij}^{±1,±2}with fixed ±

_{1}and ±

_{2}. Since

*a*

_{ij}^{-,±}is the complex conjugate of

*a*

_{ij}^{+,±}the lowest-order term of

*a*

_{ij}^{-,±}

*a*

_{i′j′}^{+,±}is real for arbitrary

*i*,

*j*,

*i′*and

*j′*. Since

*a*

_{ij}^{+,±}= 𝒪(1) for

*i*=

*j*and for

*i*≠

*j*, we have

*λ*

^{±}

_{Jσ-,ν-1}, is given by

*a*

_{0,0}

^{±,±}/

*a*

_{0,0}

^{∓,±}, and the bottom right element, which we denote

*λ*

^{±}

_{Jσ-,ν-1}is given by

*a*

_{1,1}

^{±,±}/

*a*

_{1,1}

^{∓,±}.

**E**

_{Jσ+ν}and

**E**

_{Jσ-,ν-1}to first order. As a result,

**E**

_{Jσ+ν}and

**E**

_{Jσ-ν-1}remain lowest-order resonator eigenmodes when taking into account first-order corrections, despite being degenerate at lowest order. The round-trip phase shifts for the two modes are given by the products

*λ*

_{Jσ+ν}=

*λ*

^{+}

_{Jσ+ν}

*λ*

^{-}

**E**

_{Jσ+ν}and

*λ*

_{Jσ-,ν-1}=

*λ*

^{+}

_{Jσ-,ν-1}

*λ*

^{-}

_{Jσ-,ν-1}. With a bit of algebra we obtain the central result,

*f*

_{4±}has been calculated previously [5

5. J. Visser and G. Nienhuis, “Spectrum of an optical resonator with spherical aberration,” J. Opt. Soc. Am. A **22**, 2490–2497 (2005). [CrossRef]

*f*

_{4±}.

## 5. Discussion of the results

*ν*and equal orbital angular momentum

*m*=

*J*∓ 1 but opposite spin

*σ*

^{±}, we find a difference in round-trip phase shift equal to

*ν*+

*m*are degenerate, independent of polarization. This allows significant arbitrariness in the choice of eigenmode basis for a resonator, i.e. Hermite-Gaussian modes, Laguerre-Gaussian modes and many other linear combinations are equally valid. For a resonator geometry such that

*f*

_{4±}term in the round-trip phase shift opens completely new possibilities via the capability to manufacture aspherical mirrors. For a choice of

*ν*and mainly quadratic in

*J*. This allows for the construction of resonators with a degeneracy of a large number of modes reestablished. Note however that this possibility does not include confocal resonators. Such resonators fall in the class of resonators with

*n*= 1 as defined above, causing the derivation of Eqs. (40) and (41) in section 4 to fail. Together with the fact that they lie at the edge of the zone of stability, confocal resonators turn out to be among the worst possible choices for a degenerate resonator.

*J*and

*ν*should allow the correction term to be relatively easily observed for higher-order modes. Verification of the present results in such a system would constitute a precision test of diffraction and propagation in a resonator.

## Acknowledgments

## References and links

1. | I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. |

2. | A. Kuhn, M. Hennrich, and G. Rempe, “Deterministic single-photon source for distributed quantum networking,” Phys. Rev. Lett. |

3. | V. F. Lazutkin, “An equation for the natural frequencies of a nonconfocal resonator with cylindrical mirrors which takes mirror aberration into account,” Opt. Spectr. |

4. | H. Laabs and A. T. Friberg, “Nonparaxial Eigenmodes of Stable Resonators,” IEEE J Quanum Electron. |

5. | J. Visser and G. Nienhuis, “Spectrum of an optical resonator with spherical aberration,” J. Opt. Soc. Am. A |

6. | F. Zomer, V. Soskov, and A. Variola, “On the nonparaxial modes of two-dimensional nearly concentric resonators,” Appl. Opt. |

7. | M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A |

8. | R. W. Zimmerer, “Spherical Mirror Fabry-Perot Resonators,” IEEE Trans. Microwave Theory Tech. |

9. | W. A. Specht Jr., “Modes in Spherical-Mirror Resonators,” J. Appl. Phys. |

10. | C. Flammer |

11. | M. Zeppenfeld, “Solutions to Maxwell’s equations using spheroidal coordinates,” New J. Phys. |

12. | M. Abramowitz and I. A. StegunHandbook of Mathematical Functions (Dover, New York, 1964). |

**OCIS Codes**

(220.2560) Optical design and fabrication : Propagating methods

(230.5750) Optical devices : Resonators

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: December 16, 2009

Revised Manuscript: February 25, 2010

Manuscript Accepted: March 8, 2010

Published: April 23, 2010

**Citation**

M. Zeppenfeld and P. W. H. Pinkse, "Calculating the fine structure of a Fabry-Perot resonator using spheroidal wave functions," Opt. Express **18**, 9580-9591 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-9-9580

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### References

- I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008). [CrossRef]
- A. Kuhn, M. Hennrich, and G. Rempe, “Deterministic single-photon source for distributed quantum networking,” Phys. Rev. Lett. 89, 067901 (2002). [CrossRef] [PubMed]
- V. F. Lazutkin, “An equation for the natural frequencies of a nonconfocal resonator with cylindrical mirrors which takes mirror aberration into account,” Opt. Spectrosc. 24, 236 (1968).
- H. Laabs, and A. T. Friberg, “Nonparaxial Eigenmodes of Stable Resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999). [CrossRef]
- J. Visser, and G. Nienhuis, “Spectrum of an optical resonator with spherical aberration,” J. Opt. Soc. Am. A 22, 2490–2497 (2005). [CrossRef]
- F. Zomer, V. Soskov, and A. Variola, “On the nonparaxial modes of two-dimensional nearly concentric resonators,” Appl. Opt. 46, 6859–6866 (2007). [CrossRef] [PubMed]
- M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975). [CrossRef]
- R. W. Zimmerer, “Spherical Mirror Fabry-Perot Resonators,” IEEE Trans. Microw. Theory Tech. 11, 371–379 (1963). [CrossRef]
- W. A. Specht, Jr., “Modes in Spherical-Mirror Resonators,” J. Appl. Phys. 36, 1306–1313 (1965). [CrossRef]
- C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, Calif., 1957).
- M. Zeppenfeld, “Solutions to Maxwell’s equations using spheroidal coordinates,” N. J. Phys. 11, 073007 (2009). [CrossRef]
- M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

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