Abstract

The Prony method is extended to handle the nonsymmetric algebraic eigenvalue problem and improved to search automatically for the number of dominant eigenvalues. A simple iterative algorithm is given to compute the associated eigenvectors. Resolution studies using the QR method are made in order to determine the accuracy of the matrix approximation. Numerical results are given for both simple well defined resonators and more complex advanced designs containing multiple propagation geometries and misaligned mirrors.

© 1978 Optical Society of America

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References

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  1. B. T. Smith, J. M. Boyle, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines (Springer-Verlag, New York, 1974).
  2. International Mathematical and Statistical Libraries, Inc., Houston, Texas, 1975.
  3. B. N. Parlett, Math. Comput. 28, 679 (1974).
    [CrossRef]
  4. J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, Oxford, 1965).
  5. A. E. Siegman, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
    [CrossRef] [PubMed]
  6. A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, New York, 1965).
  7. A. E. Siegman, “A Quasi-Fast Hankel Transform Using the Fast Fourier Transform,” Stanford University Technical Report (1976).

1974 (1)

B. N. Parlett, Math. Comput. 28, 679 (1974).
[CrossRef]

1970 (1)

Boyle, J. M.

B. T. Smith, J. M. Boyle, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines (Springer-Verlag, New York, 1974).

Garbow, B. S.

B. T. Smith, J. M. Boyle, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines (Springer-Verlag, New York, 1974).

Ikebe, Y.

B. T. Smith, J. M. Boyle, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines (Springer-Verlag, New York, 1974).

Klema, V. C.

B. T. Smith, J. M. Boyle, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines (Springer-Verlag, New York, 1974).

Miller, H. Y.

Moler, C. B.

B. T. Smith, J. M. Boyle, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines (Springer-Verlag, New York, 1974).

Parlett, B. N.

B. N. Parlett, Math. Comput. 28, 679 (1974).
[CrossRef]

Ralston, A.

A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, New York, 1965).

Siegman, A. E.

A. E. Siegman, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
[CrossRef] [PubMed]

A. E. Siegman, “A Quasi-Fast Hankel Transform Using the Fast Fourier Transform,” Stanford University Technical Report (1976).

Smith, B. T.

B. T. Smith, J. M. Boyle, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines (Springer-Verlag, New York, 1974).

Wilkinson, J. H.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, Oxford, 1965).

Appl. Opt. (1)

Math. Comput. (1)

B. N. Parlett, Math. Comput. 28, 679 (1974).
[CrossRef]

Other (5)

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, Oxford, 1965).

A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, New York, 1965).

A. E. Siegman, “A Quasi-Fast Hankel Transform Using the Fast Fourier Transform,” Stanford University Technical Report (1976).

B. T. Smith, J. M. Boyle, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines (Springer-Verlag, New York, 1974).

International Mathematical and Statistical Libraries, Inc., Houston, Texas, 1975.

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Figures (3)

Fig. 1
Fig. 1

Half symmetric unstable resonator with internal axicon (HSURIA): (a) an isometric of the resonator mirrors as they would be configured in a real system; (b) schematic of computational system geometries.

Fig. 2
Fig. 2

Mode discrimination.

Fig. 3
Fig. 3

(a) Intensity profile, 1.0-μm rad tilt; (b) intensity profile, 5.0-μm rad tilt; (c) intensity profile, 10-μm rad tilt.

Tables (2)

Tables Icon

Table I Resolution Studies

Tables Icon

Table II Prony Method with Search

Equations (46)

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A u = γ u ,
v m = A v m 1 ,
v m = k 1 γ 1 m u 1 + k 2 γ 2 m u 2 + k 3 γ 3 m u 3 + .
v 0 = k 1 u 1 + k 2 u 2 + + k M u M + , v 1 = k 1 γ 1 u 1 + k 2 γ 2 2 M u 2 + + k M γ M u M + , · · · v 2 M = k 1 γ 1 2 M u 1 + k 2 γ 2 2 M u 2 + + k M γ M 2 M u M + , = c 1 u 1 + c 2 u 2 + + c M u M + ,
F 2 M + 1 = v 1 · v 2 M = A 1 γ 1 + A 2 γ 2 + + A M γ M + , F 2 M + 2 = v 2 · v 2 M = A 1 γ 1 2 + A 2 γ 2 2 + + A M γ M 2 + , · · · F 4 M = v 2 M · v 2 M = A 1 γ 1 2 M + A 2 γ 2 2 M + + A M γ M 2 M + .
γ M + Q 1 γ M 1 + + Q M = 0 ,
F 3 M + m + Q 1 F 3 M + m 1 + + Q M F 2 M + m = 0.
F 3 M Q 1 + F 3 M 1 Q 2 + + F 2 M + 1 Q M = F 3 M + 1 , F 3 M + 1 Q 1 + F 3 M Q 2 + + F 2 M + 2 Q M = F 3 M + 2 , · · · F 4 M 1 Q 1 + F 4 M 2 Q 2 + + F 3 M Q M = F 4 M .
F i + j = v i · v j = k 1 2 γ 1 i + j + k 2 2 γ 2 i + j + + k M 2 γ M i + j + i = 1,2 , , M and j = 1,2 , , M .
v m · A v m / v m · v m = { γ 1 + [ ( γ 2 / γ 1 ) 2 m ] if A is symmetric γ 1 + [ ( γ 2 / γ 1 ) m ] if A is nonsymmetric .
f i ( A ) u j = { 0 i j f i ( γ i ) u i i = j .
p k ( γ ) = ( γ γ 1 ) ( γ γ 2 ) ( γ γ k ) = γ k + s 1 γ k 1 + s 2 γ k 2 + + s k ,
p k + 1 ( γ ) = p k ( γ ) ( γ γ k + 1 ) = γ k + 1 + t 1 γ k + t 2 γ k 1 + + t k + 1 ,
t 1 = s 1 γ k + 1 , t 2 = s 2 s 1 γ k + 1 , t 3 = s 3 s 2 γ k + 1 , · · t k = s k s k 1 γ k + 1 , t k + 1 = s k γ k + 1 .
γ ϕ ( x ) = j l + 1 c 0 1 y J l ( c x y ) exp [ j ( c g / 2 ) ( x 2 + y 2 ) ] ϕ ( y ) d y ,
γ ψ ( s ) = j l + 1 ( c / 2 ) 0 1 J l [ c ( s t ) 1 / 2 ] × exp [ j ( c g / 2 ) ( s + t ) ] ψ ( t ) d t 0 1 K ( s , t ) ψ ( t ) d t ,
γ ψ ( s i ) = j = 1 n w j K ( s i , t j ) ψ ( t j ) , i = 1,2 , , n ,
( K W γ I ) ψ = 0 ,
lim n i = 1 n γ i = 0 1 K ( s , s ) d s ,
ERR = | i = 1 n γ i 0 1 K ( s , s ) d s | ,
ERRVEC max i | r i | ,
A = M 1 D 1 M 2 E 1 M 3 F 1 M 4 F 2 M 5 E 2 M 6 D 2 .
u 2 l ( r , ϕ ) = l = u 1 l ( r ) exp ( i l ϕ ) ,
u 2 l ( r 2 ) = j l + 1 L k exp ( j k r 2 2 / 2 L ) × 0 a u 1 l ( r 1 ) J l ( k r 1 r 2 L ) exp ( j k r 1 2 / 2 L ) r 1 d r 1 ,
ψ 2 l ( r 2 ) = r 2 u 2 l ( r 2 ) ( 1 j l + 1 ) exp ( j k r 2 2 / 2 L ) .
ψ 2 l ( r 2 ) = k L 0 a u 1 l ( r 1 ) J l ( k r 1 r 2 L ) exp ( j k r 1 2 / 2 L ) r 1 r 2 d r 1 .
r 1 = ρ 1 exp ( α x ) ,
r 2 = ρ 2 exp ( α y ) .
ψ 2 l [ ρ 2 exp ( α y ) ] = k L 1 / [ α ln ( a / ρ 1 ) ] × u 1 l [ ρ 1 exp ( α x ) ] J l { k ρ 1 ρ 2 exp [ α ( x + y ) ] L } × exp { j k [ ρ 1 exp ( α x ) ] 2 2 L } ρ 1 2 ρ 2 exp [ α ( x + y ) ] α exp ( α x ) d x .
r 1 = ρ 1 exp ( α n ) ,
( 18 a )
r 2 = ρ 2 exp ( α m ) ,
( 18 b )
f l ( n ) = ρ 1 exp ( α n ) u 1 l [ ρ 1 exp ( α n ) exp { j k [ ρ 1 exp ( α n ) ] 2 2 L } ,
g l ( n + m ) = α k ρ 1 ρ 2 exp [ α ( n + m ) ] L J l { k ρ 1 ρ 2 exp [ α ( n + m ) ] L } .
ψ 2 l ( m ) = n = 0 N G 1 f l ( n ) g l ( n + m ) = f l ( n ) * g l * ( m + n )
f l ( n ) FFT F l ( n ) , g l ( n ) FFT G l ( n ) , Ψ 2 l ( n ) = F l ( n ) · G l * ( n ) , Ψ 2 l ( n ) FFT ψ 2 l ( n ) .
ρ 1 = a 1 k 1 N F ρ 2 = a 2 k 1 N F Δ r max 1 = a 1 k 2 N F , Δ r max 2 = a 2 k 2 N F α = 1 N F k 2 N G = k 2 N F ln ( k 1 N F ) ,
u 2 l ( r 2 ) = j l + 1 L k exp ( j k r 2 2 / 2 L ) a 1 a 2 u 1 l ( r 1 ) J l ( k r 1 r 2 L ) × exp ( j k r 1 2 / 2 L ) r 1 d r 1 .
J l ( k r 1 r 2 L ) ( 2 L π k r 1 r 2 ) 1 / 2 cos ( k r 1 r 2 L 2 l + 1 4 π ) .
J l ( k r 1 r 2 L ) 1 2 ( 2 L π k r 1 r 2 ) 1 / 2 [ exp ( j θ ) + exp ( j θ ) ] ,
J l ( k r 1 r 2 L ) 1 2 ( 2 L π k r 1 r 2 ) 1 / 2 exp ( j k r 1 r 2 / L ) j l 1 / 2 .
( r 2 ) 1 / 2 u 2 l ( r 2 ) = ( j k 2 π L ) 1 / 2 a 1 a 2 u 1 l ( r 1 ) exp [ j k ( r 1 r 2 ) 2 / 2 L ] ( r 1 ) 1 / 2 d r 1 .
υ i l ( r i ) = ( r i ) 1 / 2 u i l ( r i ) ,
υ 2 l ( r 2 ) = ( j k 2 π L ) 1 / 2 a 1 a 2 υ 1 l ( r 1 ) exp [ j k ( r 1 r 2 ) 2 / 2 L ] d r 1 .
N e q = ( a 2 / λ z ) [ ( M 2 1 ) / 2 M ] ,

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