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  1. H. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  2. H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).
  3. H. Kogelnik, “On the Propagation of Gaussian Beams of Light Through Lenslike Media Including those with a Loss or Gain Variation,” Appl. Opt. 4, 1562 (1965).
    [CrossRef]
  4. P. Baues, “Huygens’ Principle in Inhomogeneous Isotropic Media and a General Integral Equation Applicable to Optical Resonators,” Opto-electronics 1, 37 (1969).
    [CrossRef]
  5. P. Baues, “The Connection of Geometrical Optics with the Propagation of Gaussian Beams and the Theory of Optical Resonators,” Opto-electronics 1, 103 (1969).
    [CrossRef]
  6. S. A. Collins, “Lens-System Diffraction Integral Written in Terms of Matrix Optics,” J. Opt. Soc. Am. 60, 1168 (1970).
    [CrossRef]
  7. A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975), pp. 119–122.
  8. M. Nazarathy, J. Shamir, “First-Order Optics: Operator Representation for Systems with Loss or Gain,” J. Opt. Soc. Am. 72, 1398 (1982).
    [CrossRef]
  9. Examples of complex-valued matrices are given in L. W. Casperson, “Synthesis of Gaussian Beam Optical Systems,” Appl. Opt. 20, 2243 (1981) and references therein.
    [CrossRef] [PubMed]
  10. M. Nazarathy, A. Hardy, J. Shamir, “Generalized Mode Propagation in First-Order Optical Systems with Loss or Gain,” J. Opt. Soc. Am. 72, 1409 (1982).
    [CrossRef]
  11. A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).
  12. A. E. Siegman, “Hermite-Gaussian Functions of Complex Argument as Optical-Beam Eigenfunctions,” J. Opt. Soc. Am. 63, 1093 (1973).
    [CrossRef]
  13. J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 104–6, 286–93.
  14. P. Muys, H. Vanherzeele, “Huygens’ Integral for an Axially Symmetric Phase Conjugate Resonator,” IEEE J. Quantum Electron. QE-18, 1403 (1982).
    [CrossRef]
  15. A. Erdélyi, Ed., Tables of Integral Transforms, Vol. 2 (McGraw-Hill, New York, 1954).
  16. T. Li, “Diffraction Loss and Selection of Modes in Maser Resonators with Circular Mirrors,” Bell Syst. Tech. 44, 917 (1965), Eq. (4).
  17. J. Arnaud, “Representation of Gaussian beams by Complex Rays,” Appl. Opt. 24, 538 (1985).
    [CrossRef] [PubMed]
  18. L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci., Paris 110, 1251 (1890); “Sur la propagation anomale des ondes,” Ann. Chim. Phys. 24, 145 (1891).
  19. A. E. Siegman, “A Canonical Formulation for Analyzing Multi-element Unstable Resonators,” IEEE J. Quantum Electron. QE-12, 35 (1976); correction in IEEE J. Quantum Electron. QE-12, 315 (1976).
    [CrossRef]

1985 (1)

1982 (3)

1981 (1)

1976 (1)

A. E. Siegman, “A Canonical Formulation for Analyzing Multi-element Unstable Resonators,” IEEE J. Quantum Electron. QE-12, 35 (1976); correction in IEEE J. Quantum Electron. QE-12, 315 (1976).
[CrossRef]

1973 (1)

1970 (1)

1969 (2)

P. Baues, “Huygens’ Principle in Inhomogeneous Isotropic Media and a General Integral Equation Applicable to Optical Resonators,” Opto-electronics 1, 37 (1969).
[CrossRef]

P. Baues, “The Connection of Geometrical Optics with the Propagation of Gaussian Beams and the Theory of Optical Resonators,” Opto-electronics 1, 103 (1969).
[CrossRef]

1966 (1)

1965 (3)

H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).

H. Kogelnik, “On the Propagation of Gaussian Beams of Light Through Lenslike Media Including those with a Loss or Gain Variation,” Appl. Opt. 4, 1562 (1965).
[CrossRef]

T. Li, “Diffraction Loss and Selection of Modes in Maser Resonators with Circular Mirrors,” Bell Syst. Tech. 44, 917 (1965), Eq. (4).

1890 (1)

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci., Paris 110, 1251 (1890); “Sur la propagation anomale des ondes,” Ann. Chim. Phys. 24, 145 (1891).

Arnaud, J.

J. Arnaud, “Representation of Gaussian beams by Complex Rays,” Appl. Opt. 24, 538 (1985).
[CrossRef] [PubMed]

J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 104–6, 286–93.

Baues, P.

P. Baues, “Huygens’ Principle in Inhomogeneous Isotropic Media and a General Integral Equation Applicable to Optical Resonators,” Opto-electronics 1, 37 (1969).
[CrossRef]

P. Baues, “The Connection of Geometrical Optics with the Propagation of Gaussian Beams and the Theory of Optical Resonators,” Opto-electronics 1, 103 (1969).
[CrossRef]

Burch, J. M.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975), pp. 119–122.

Casperson, L. W.

Collins, S. A.

Gerrard, A.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975), pp. 119–122.

Gouy, L. G.

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci., Paris 110, 1251 (1890); “Sur la propagation anomale des ondes,” Ann. Chim. Phys. 24, 145 (1891).

Hardy, A.

Kogelnik, H.

Li, T.

H. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

T. Li, “Diffraction Loss and Selection of Modes in Maser Resonators with Circular Mirrors,” Bell Syst. Tech. 44, 917 (1965), Eq. (4).

Muys, P.

P. Muys, H. Vanherzeele, “Huygens’ Integral for an Axially Symmetric Phase Conjugate Resonator,” IEEE J. Quantum Electron. QE-18, 1403 (1982).
[CrossRef]

Nazarathy, M.

Shamir, J.

Siegman, A. E.

A. E. Siegman, “A Canonical Formulation for Analyzing Multi-element Unstable Resonators,” IEEE J. Quantum Electron. QE-12, 35 (1976); correction in IEEE J. Quantum Electron. QE-12, 315 (1976).
[CrossRef]

A. E. Siegman, “Hermite-Gaussian Functions of Complex Argument as Optical-Beam Eigenfunctions,” J. Opt. Soc. Am. 63, 1093 (1973).
[CrossRef]

A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).

Vanherzeele, H.

P. Muys, H. Vanherzeele, “Huygens’ Integral for an Axially Symmetric Phase Conjugate Resonator,” IEEE J. Quantum Electron. QE-18, 1403 (1982).
[CrossRef]

Appl. Opt. (4)

Bell Syst. Tech. (1)

T. Li, “Diffraction Loss and Selection of Modes in Maser Resonators with Circular Mirrors,” Bell Syst. Tech. 44, 917 (1965), Eq. (4).

Bell Syst. Tech. J. (1)

H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).

C. R. Acad. Sci., Paris (1)

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci., Paris 110, 1251 (1890); “Sur la propagation anomale des ondes,” Ann. Chim. Phys. 24, 145 (1891).

IEEE J. Quantum Electron. (2)

A. E. Siegman, “A Canonical Formulation for Analyzing Multi-element Unstable Resonators,” IEEE J. Quantum Electron. QE-12, 35 (1976); correction in IEEE J. Quantum Electron. QE-12, 315 (1976).
[CrossRef]

P. Muys, H. Vanherzeele, “Huygens’ Integral for an Axially Symmetric Phase Conjugate Resonator,” IEEE J. Quantum Electron. QE-18, 1403 (1982).
[CrossRef]

J. Opt. Soc. Am. (4)

Opto-electronics (2)

P. Baues, “Huygens’ Principle in Inhomogeneous Isotropic Media and a General Integral Equation Applicable to Optical Resonators,” Opto-electronics 1, 37 (1969).
[CrossRef]

P. Baues, “The Connection of Geometrical Optics with the Propagation of Gaussian Beams and the Theory of Optical Resonators,” Opto-electronics 1, 103 (1969).
[CrossRef]

Other (4)

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975), pp. 119–122.

A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).

J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 104–6, 286–93.

A. Erdélyi, Ed., Tables of Integral Transforms, Vol. 2 (McGraw-Hill, New York, 1954).

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Figures (1)

Fig. 1
Fig. 1

Optical paraxial system illuminated by a beam propagating along the z axis. The wave amplitude distribution is u(r,ϕ) with r and ϕ being polar coordinates. The ABCD matrix elements are real.

Equations (26)

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u 2 ( r 2 , ϕ 2 ) = j λ B 0 0 2 π u 1 ( r 1 , ϕ 1 ) × exp [ - j k S ( r 1 , r 2 , ϕ 3 , ϕ 2 ) ] r 1 d r 1 d ϕ 1 ,
S ( r 1 , r 2 , ϕ 1 , ϕ 2 ) = L + 1 2 B ( Ar 1 2 - 2 r 1 r 2 cos ( ϕ 1 - ϕ 2 ) + D r 2 2 ) .
u 1 ( r 1 , ϕ 1 ) = a 1 ( 2 r 1 w 1 ) 1 L p l ( 2 r 1 2 w 1 2 ) × exp ( - j l ϕ 1 ) exp ( - j k 2 q 1 r 1 2 ) ,
1 q = 1 R - j λ π w 2 ,
0 0 2 π u 1 ( r 1 , ϕ 1 ) 2 r 1 d r 1 d ϕ 1 = 1.
0 x l [ L p l ( x ) ] 2 exp ( - x ) d x = ( p + l ) ! p ! ,
a 1 = [ π ( 1 + δ 0 l ) ] - 1 / 2 ( p ! ( p + l ) ! ) 1 / 2 2 w 1 ,
0 2 π exp { j [ k B r 1 r 2 cos ( ϕ 1 - ϕ 2 ) - l ϕ 1 ] } d ϕ 1 = 2 π j l exp ( - j l ϕ 2 ) J l ( k r 1 r 2 B ) ,
u 2 ( r 2 , ϕ 2 ) = a 1 j l + 1 k B exp ( j l ϕ 2 ) exp ( - j k L ) exp ( - j k 2 B D r 2 2 ) I ,
I = 0 ( 2 r 1 w 1 ) 1 L p l ( 2 r 1 2 w 1 2 ) × exp [ - j k 2 B ( A + B q 1 ) r 1 2 ] J l ( k r 1 r 2 B ) r 1 d r 1 .
ρ i = ( k B ) 1 / 2 r i ,             i = 1 , 2.
β = j 2 ( A + B q 1 ) ,
α = 2 B k w 1 2 .
I = 2 l / 2 ( B k ) l / 2 + 5 / 4 w 1 - l r 2 - 1 / 2 I .
I = 0 ρ 1 l + 1 / 2 L ρ l ( α ρ 1 2 ) exp ( - β ρ 1 2 ) J l ( ρ 1 ρ 2 ) ρ 1 ρ 2 d ρ 1 .
I = 2 - l - 1 β - 1 - ρ - 1 ( β - α ) p ρ 2 l + 1 / 2 L p l [ α ρ 2 2 4 β ( α - β ) ] exp ( - ρ 2 2 4 β ) .
exp ( - ρ 2 2 4 β ) = exp ( j k 2 B q 1 A q 1 + B r 2 2 ) .
exp ( j k 2 B q 1 A q 1 + B r 2 2 ) exp ( - j k 2 B D r 2 2 ) = exp ( - j k 2 C q 1 + D A q 1 + B r 2 2 ) .
q 2 = A q 1 + B C q 1 + D ,
( β - α ) = j 2 ( A + B q 1 * ) ,
α ρ 2 2 4 β ( α - β ) = 2 r 2 2 w 1 2 [ ( A + B q 1 ) ( A + B q 1 * ) ] ,
A + B q 1 = w 2 w 1 exp ( - j Φ ) ,
Φ = tan - 1 [ λ B π w 1 2 ( A + B R 1 ) ] .
α ρ 2 2 4 β ( α - β ) = 2 r 2 2 w 2 2 ,
I = 2 - l / 2 j - l - 1 ( B k ) - l / 2 - 1 / 4 r 2 1 / 2 w 1 l + 1 w 2 ( 2 r 2 w 2 ) l L p l ( 2 r 2 2 w 2 2 ) × exp ( j k 2 q 2 r 2 2 ) exp [ j ( 2 p + l + 1 ) Φ ] .
u 2 ( r 2 , ϕ 2 ) = a 2 ( 2 r 2 w 2 ) l L p l ( 2 r 2 2 w 2 2 ) exp [ - j ( k r 2 2 2 q 2 + l ϕ 2 ) ] × exp { - j [ k L - ( 2 p + l + 1 ) Φ ] } ,

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