Prony method revised
1
Prony method revised Edgardo Gerck Centro Tecnico Aeroespacial, Instituto de Atividades Es- paciais Divisao de Estudos Avancados, 12200-Sao Jose dos Campos, S.P., Brazil. Received 16 November 1978. 0003-6935/79/183075-01$00.50/0. © 1979 Optical Society of America. The Prony method described in its application to optical resonators by Siegman and Miller 1 has since then deserved some attention. 2,3 This Letter will show that the Prony method for symmetric matrix is all that is necessary for any scalar resonator calculation, regardless of misalignments or misfiguring of mirrors. This is accomplished by simple modifications of the original algorithm. In this way a simpler and faster procedure is presented only by considering the correct scalar product to be used. Our proposed method, however, could not possibly apply directly in the cases for which the polarization of the beam is changed as in "optical diodes" 4 or axicons. The integral equation for the optical resonator has the following general form in the scalar field approximation: where the kernel K may depend on l as for a cavity in cylin- drical coordinates. The A(s) factor is only A(s) = s or A(s) = 1. In the case of misalignments or radially unsymmetric mirror misfiguring, Eq. (1) must be in the bidimensional form. In such cases Eq. (1) follows the notation of Ref. 5, in which s is symbolic for s = (x,y) and r for r = (u,v). In the strip rectangular case the variables can be separated, and the effect of misalignments can be accounted for by con- sidering different upper and lower integration bounds on Eq. (1), as usual. 6 As in Ref. 7, we cast the eigenproblem [Eq. (1)] in a sym- metric matrix formulation using the scalar product with the same upper and lower integration bounds as Eq. (1), defining where M′ ij = K l (r i ,s j ). Hence M' will be symmetric because K l (r,s) is symmetric. 7 K l is symmetric for any resonator re- gardless of misalignments or misfiguring of mirrors. 7,8 Thus we see that the matrix problem is always symmetric under the correct scalar product [Eq. (2)]. Exactly as in the original procedure, for the generalized Prony method we form the propagation vectors u m : v m = Mv m–1 = (M',v m–1 ) and the coefficients F ij , but now defined as F ij = (v i ,v j ) with the scalar product [Eq. (2) J extended from the continuous to the discrete case using Wk, the integration weights, in F ij = Σ k v k i v k j w k A(s k ), where s k are the abscissa's values. The F ij obey, as in Ref. 1, the useful property: F ij = F i+j,0 = F i+j because of the symmetry of M'. This nice general property of F ij allows the direct application of the rest of the Siegman-Miller Prony methud. For the eigenvectors the simple algorithm described in Ref. 3 can be used to advantage. What changes should be made in the existing Prony pro- grams to conform to the general method proposed? There is one answer: in the chosen integration plane instead of calculating F mn by F mn = Σ i (v m ) i (v n ) i , we use F mn = Σ i - (v m ) i (v n ) i A(s i )w i , where A(s) = s in cylindrical coordinates or A(s) = 1 in rectangular coordinates. If the fast Fourier transform (FFT) is used for that integration plane, the w i factors are constant and can be considered as w i = 1, although losing the eigenfunction's normalization. The propagation vectors are calculated as before because (M',v m ) = Mv m . The propagation codes are not to be changed. The simple generalized method proposed in this Letter can be applied to any resonator as described by Eq. (1), aligned or misaligned, and to any quadrature formula including the FFT. With our method any scalar resenator described by an unsymmetric matrix equation can be formulated as a sym- metric matrix problem. References 1. A. E. Siegman and H. Y. Miller, Appl. Opt. 9, 2729 (1970). 2. J. F. Perkins and C. Canson, Appl. Phys. Lett. 31, 198 (1977). 3. W. D. Murphy and M. L. Bernabe, Appl. Opt. 17, 2358 (1978). 4. T. F. Johnston, Jr., and W. P. Proffitt, in Proceedings of the Fourth International Conference on Laser Spectroscopy, Rottach-Egern, Germany (1979). 5. A. G. Fox and T. Li, Proc. IEEE 51, 80 (1963). 6. P. Horwitz, Appl. Opt. 15, 167 (1976). 7. E. Gerck, and C. H. Brito Cruz, Appl. Opt. 18, 1341 (1979). 8. Ki(r,s) is the Green function of the problem and, as the light path is always reversible in an interferometer or laser, K l (r,s) = K l (s,r) for any resonator which does not change the polarization of the beam. This physical constraint on K l (r,s) applies for any reso- nator, whether misaligned or not. 15 September 1979 / Vol. 18, No. 18 / APPLIED OPTICS 3075
Transcript of Prony method revised
Prony method revised Edgardo Gerck
Centro Tecnico Aeroespacial, Instituto de Atividades Es-paciais Divisao de Estudos Avancados, 12200-Sao Jose dos Campos, S.P., Brazil. Received 16 November 1978. 0003-6935/79/183075-01$00.50/0. © 1979 Optical Society of America.
The Prony method described in its application to optical resonators by Siegman and Miller1 has since then deserved some attention.2,3 This Letter will show that the Prony method for symmetric matrix is all that is necessary for any scalar resonator calculation, regardless of misalignments or misfiguring of mirrors. This is accomplished by simple modifications of the original algorithm. In this way a simpler and faster procedure is presented only by considering the correct scalar product to be used. Our proposed method, however, could not possibly apply directly in the cases for which the polarization of the beam is changed as in "optical diodes"4 or axicons.
The integral equation for the optical resonator has the following general form in the scalar field approximation:
where the kernel K may depend on l as for a cavity in cylindrical coordinates. The A(s) factor is only A(s) = s or A(s) = 1.
In the case of misalignments or radially unsymmetric mirror misfiguring, Eq. (1) must be in the bidimensional form. In such cases Eq. (1) follows the notation of Ref. 5, in which s is symbolic for s = (x,y) and r for r = (u,v).
In the strip rectangular case the variables can be separated, and the effect of misalignments can be accounted for by considering different upper and lower integration bounds on Eq. (1), as usual.6
As in Ref. 7, we cast the eigenproblem [Eq. (1)] in a symmetric matrix formulation using the scalar product
with the same upper and lower integration bounds as Eq. (1), defining
where M′ij = Kl(ri,sj). Hence M' will be symmetric because Kl(r,s) is symmetric.7 Kl is symmetric for any resonator regardless of misalignments or misfiguring of mirrors.7,8 Thus we see that the matrix problem is always symmetric under the correct scalar product [Eq. (2)].
Exactly as in the original procedure, for the generalized Prony method we form the propagation vectors um: vm = Mvm – 1 = (M',vm – 1) and the coefficients Fij, but now defined as Fij = (vi,vj) with the scalar product [Eq. (2) J extended from the continuous to the discrete case using Wk, the integration weights, in Fij = Σkvk
ivkjwkA(sk), where sk are the abscissa's
values. The Fij obey, as in Ref. 1, the useful property: Fij = Fi+j,0
= Fi+j because of the symmetry of M'. This nice general property of Fij allows the direct application of the rest of the Siegman-Miller Prony methud.
For the eigenvectors the simple algorithm described in Ref. 3 can be used to advantage.
What changes should be made in the existing Prony programs to conform to the general method proposed? There is one answer: in the chosen integration plane instead of calculating Fmn by Fmn = Σ i(vm) i(vn) i , we use Fmn = Σi-
(vm)i(vn)iA(si)wi, where A(s) = s in cylindrical coordinates or A(s) = 1 in rectangular coordinates. If the fast Fourier transform (FFT) is used for that integration plane, the wi factors are constant and can be considered as wi = 1, although losing the eigenfunction's normalization. The propagation vectors are calculated as before because (M',vm) = Mvm. The propagation codes are not to be changed.
The simple generalized method proposed in this Letter can be applied to any resonator as described by Eq. (1), aligned or misaligned, and to any quadrature formula including the FFT. With our method any scalar resenator described by an unsymmetric matrix equation can be formulated as a symmetric matrix problem.
References 1. A. E. Siegman and H. Y. Miller, Appl. Opt. 9, 2729 (1970). 2. J. F. Perkins and C. Canson, Appl. Phys. Lett. 31, 198 (1977). 3. W. D. Murphy and M. L. Bernabe, Appl. Opt. 17, 2358 (1978). 4. T. F. Johnston, Jr., and W. P. Proffitt, in Proceedings of the Fourth
International Conference on Laser Spectroscopy, Rottach-Egern, Germany (1979).
5. A. G. Fox and T. Li, Proc. IEEE 51, 80 (1963). 6. P. Horwitz, Appl. Opt. 15, 167 (1976). 7. E. Gerck, and C. H. Brito Cruz, Appl. Opt. 18, 1341 (1979). 8. Ki(r,s) is the Green function of the problem and, as the light path
is always reversible in an interferometer or laser, Kl(r,s) = Kl(s,r) for any resonator which does not change the polarization of the beam. This physical constraint on Kl(r,s) applies for any resonator, whether misaligned or not.
15 September 1979 / Vol. 18, No. 18 / APPLIED OPTICS 3075
