Abstract

We present further calculations of the three-dimensional mode patterns and power outputs from a high-power gas-dynamic laser, including a nonuniform flowing, saturable gain medium plus index inhomogeneities (shocks) inside the laser. The calculations are carried out using a plane-wave or k-space expansion together with the fast Fourier transform. A new expanding-beam coordinate transform converts all diverging or converging sections of the resonator mode into equivalent collimated beam sections. The resulting FFT propagation code is significantly faster than earlier propagation codes using other eigenmode expansions.

© 1975 Optical Society of America

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References

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  1. A. E. Siegman, E. A. Sziklas, Appl Opt. 13, 2775 (1974).
    [CrossRef] [PubMed]
  2. D. B. Rensch, Appl. Opt. 13, 2546 (1974).
    [CrossRef] [PubMed]
  3. D. B. Rensch, A. N. Chester, Appl. Opt. 12, 997 (1973).
    [CrossRef] [PubMed]
  4. A. N. Chester, Appl. Opt. 12, 2353 (1973).
    [CrossRef] [PubMed]
  5. G. D. Bergland, IEEE Spectrum41 (July1969).
    [CrossRef]
  6. L. R. Rabiner, C. M. Rader, Eds., Digital Signal Processing, IEEE Press Selected Reprint Series (Wiley/IEEE, New York, 1972).
  7. J. A. Ratcliffe, Rep. Prog. Phys.19, 188 (1956).
    [CrossRef]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  9. H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966);Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  10. E. A. Sziklas, A. E. Siegman, Proc. IEEE 62, 410 (1974).
    [CrossRef]
  11. A. E. Siegman, Appl. Opt. 13, 353 (1974).
    [CrossRef] [PubMed]
  12. R. B. Blackman, J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1958), pp. 14, 15,68–70, 90–100.
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 460, 461.

1974 (4)

A. E. Siegman, E. A. Sziklas, Appl Opt. 13, 2775 (1974).
[CrossRef] [PubMed]

E. A. Sziklas, A. E. Siegman, Proc. IEEE 62, 410 (1974).
[CrossRef]

A. E. Siegman, Appl. Opt. 13, 353 (1974).
[CrossRef] [PubMed]

D. B. Rensch, Appl. Opt. 13, 2546 (1974).
[CrossRef] [PubMed]

1973 (2)

1969 (1)

G. D. Bergland, IEEE Spectrum41 (July1969).
[CrossRef]

1966 (1)

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966);Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

Bergland, G. D.

G. D. Bergland, IEEE Spectrum41 (July1969).
[CrossRef]

Blackman, R. B.

R. B. Blackman, J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1958), pp. 14, 15,68–70, 90–100.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 460, 461.

Chester, A. N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Kogelnik, H.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966);Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

Li, T.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966);Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

Ratcliffe, J. A.

J. A. Ratcliffe, Rep. Prog. Phys.19, 188 (1956).
[CrossRef]

Rensch, D. B.

Siegman, A. E.

A. E. Siegman, Appl. Opt. 13, 353 (1974).
[CrossRef] [PubMed]

E. A. Sziklas, A. E. Siegman, Proc. IEEE 62, 410 (1974).
[CrossRef]

A. E. Siegman, E. A. Sziklas, Appl Opt. 13, 2775 (1974).
[CrossRef] [PubMed]

Sziklas, E. A.

A. E. Siegman, E. A. Sziklas, Appl Opt. 13, 2775 (1974).
[CrossRef] [PubMed]

E. A. Sziklas, A. E. Siegman, Proc. IEEE 62, 410 (1974).
[CrossRef]

Tukey, J. W.

R. B. Blackman, J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1958), pp. 14, 15,68–70, 90–100.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 460, 461.

Appl Opt. (1)

A. E. Siegman, E. A. Sziklas, Appl Opt. 13, 2775 (1974).
[CrossRef] [PubMed]

Appl. Opt. (4)

IEEE Spectrum (1)

G. D. Bergland, IEEE Spectrum41 (July1969).
[CrossRef]

Proc. IEEE (2)

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966);Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

E. A. Sziklas, A. E. Siegman, Proc. IEEE 62, 410 (1974).
[CrossRef]

Other (5)

R. B. Blackman, J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1958), pp. 14, 15,68–70, 90–100.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 460, 461.

L. R. Rabiner, C. M. Rader, Eds., Digital Signal Processing, IEEE Press Selected Reprint Series (Wiley/IEEE, New York, 1972).

J. A. Ratcliffe, Rep. Prog. Phys.19, 188 (1956).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (20)

Fig. 1
Fig. 1

(a) Model of the unstable laser resonator and saturable gain medium used in the calculations. The saturated gain profile and index inhomogeneities across each shaded gain-phase segment are lumped into a complex gain profile across the segment's mid-plane or station, (b) Grid pattern across one transverse half-plane of the resonator (assumed to be symmetric about the x axis). The resonator axis is located at bottom center of the grid.

Fig. 2
Fig. 2

(a) Illustration of a diverging quasi-spherical-wave optical beam, (b) Expanding coordinate system for one complete unfolded round trip around a confocal unstable resonator, (c) Equivalent collimated coordinate system in the primed coordinate space.

Fig. 3
Fig. 3

Illustration of the periodic iteration inherent in any discrete Fourier transform calculation and the guard bands necessary to minimize spillover from adjoining cells in the real coordinate space.

Fig. 4
Fig. 4

Plots of (a) wave intensity and (b) phase just inside the output mirror of a bare unstable resonator with Neq = 1.5 and M = 2.5, as calculated by the HG method.

Fig. 5
Fig. 5

Plots of (a) wave intensity and (b) phase just inside the output mirror of a bare unstable resonator with Neq = 1.5 and M = 2.5, as calculated by the PFT method.

Fig. 6
Fig. 6

Plots of (a) wave intensity and (b) phase just inside the output mirror of a loaded unstable resonator with Neq = 0.5 and M = 2.5, as calculated by the HG method.

Fig. 7
Fig. 7

Plots of (a) wave intensity and (b) phase just inside the output mirror of a loaded unstable resonator with Neq = 0.5 and M = 2.5 as calculated by the FFT method.

Fig. 8
Fig. 8

Phase and intensity distributions just inside the output mirror for a loaded unstable resonator with iVeq = 1.5 and M = 2.5, calculated by the FFT method, with the normalized phase disturbance or density strength set equal to zero.

Fig. 9
Fig. 9

Phase and intensity distributions just inside the output mirror for a loaded unstable resonator with Neq = 1.5 and M = 2.5, calculated by the FFT method, with the normalized phase disturbance or density strength set equal to 0.25.

Fig. 10
Fig. 10

Phase and intensity distributions just inside the output mirror for a loaded unstable resonator with Neq = 1.5 and M = 2.5, calculated by the FFT method, with the normalized phase disturbance or density strength set equal to 1.0.

Fig. 11
Fig. 11

Phase and intensity distributions just inside the output mirror for a loaded unstable resonator with Neq = 1.5 and M = 2.5, calculated by the FFT method, with the normalized phase disturbance or density strength set equal to 1.5.

Fig. 12
Fig. 12

Far-field intensity distributions corresponding (except for one case) to the resonator mode patterns of Figs. 811. Normalized density strengths are (a) 0, (b) 0.5, (c) 1.0, and (d) 1.5.

Fig. 13
Fig. 13

Integrated far-field power within a given spot diameter in the far field for the different values of normalized density strength shown in Figs. 812.

Fig. 14
Fig. 14

Phase and intensity distributions just inside the output mirror for a bare unstable resonator with Neq = 1.5, M = 2.5, and a tilt θ2 = 50 μrad of the back mirror (compare Fig. 5).

Fig. 15
Fig. 15

Phase and intensity distributions just inside the output mirror for a loaded unstable resonator with Neq = 1.5, M = 2.5, and a tilt θ2 = 50 μrad of the back mirror (compare Fig. 8).

Fig. 16
Fig. 16

Phase and intensity distributions just inside the output mirror for a loaded unstable resonator with Neq = 1.5, M = 2.5, and a tilt θ2 = 100 μrad of the back mirror (compare Figs. 8 and 15).

Fig. 17
Fig. 17

Far-field intensity distributions for the loaded resonators with mirror tilt shown in Figs. 15 and 16. Back mirror tilt angles are (a) θ2 = 50 μrad and (b) θ2 = 100 μrad. The beam deflects as predicted by geometrical optics, with very little beam distortion at these tilt angles.

Fig. 18
Fig. 18

Plots of (a) wave intensity and (b) phase just inside the output mirror of a bare unstable resonator with Neq = 5.5 and M = 2.5, as calculated by the FFT method.

Fig. 19
Fig. 19

Plots of (a) wave intensity and (b) phase just inside the output mirror of a loaded unstable resonator with Neq = 5.5 and M = 2.5, as calculated by the FFT method.

Fig. 20
Fig. 20

Far-field intensity distribution for the loaded large-Fresnel-number unstable resonator of Fig. 19, with Neq = 5.5, M = 2.5, and normalized density strength = 4.

Tables (4)

Tables Icon

Table I Comparison of Various Loaded-Resonator and Bare-Resonator Parameters Calculated for the same Cases Neq = 0.5 and Neq = 1.5 Using both the Fast Fourier Transform (FFT) and Hermite-Gaussian (HG) Methods

Tables Icon

Table II Calculated Results Using the FFT Method for a Loaded Neq = 1.5, M = 2.5 Resonator in Which the Normalized Strength of the Index Perturbation due to Sidewall Shocks was Systematically Varied

Tables Icon

Table III Mirror Tilt Results for the Neq = 1.5 Bare Resonator

Tables Icon

Table IV Mirror Tilt Results for the Neq = 1.5 Loaded Resonator

Equations (22)

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2 u x 2 + 2 u y 2 2 j k u z = 0 .
u 0 ( x , y ) = P 0 ( ν x , ν y ) exp ( j 2 π ν x x j 2 π ν y y ) d ν x d ν y ,
P 0 ( ν x , ν y ) = u 0 ( x , y ) exp ( j 2 π ν x x + j 2 π ν y y ) dxdy .
exp ( j k · r ) = exp ( j 2 π ν x x j 2 π ν y y j k z z ) ,
k z ( ν x , ν y ) = [ k 2 ( 2 π ν x ) 2 ( 2 π ν y ) 2 ] 1 / 2 k π λ ( ν x 2 + ν y 2 ) .
P 1 ( ν x , ν y ) = P 0 ( ν x , ν y ) exp [ j π L λ ( ν x 2 + ν y 2 ) ] ,
u 1 ( x , y ) = P 1 ( ν x , ν y ) exp ( j 2 π ν x x j 2 π ν y y ) d ν x d ν y ,
u ( x , y , z ) = z 1 exp [ j ( π / z λ ) ( x 2 + y 2 ) ] υ ( x , y , z ) ,
x ( x , z ) = ( α x / z ) , y ( y , z ) = ( α y / z ) , z = α 2 ( z z 0 ) z z 0
2 υ x 2 + 2 υ y 2 2 j k υ z = 0 ,
N c a 2 L λ = a 0 2 R 0 λ R 0 + L L .
N c = a 2 2 ( M L + L ) λ = N T M + 1 ,
N c = 2 M 2 M 2 1 N eq
u ( y ) j ( j ) 1 / 2 2 π ( N c ) 1 / 2 exp [ j π N c ( y 1 ) 2 ] y 1 × [ 1 y 1 y + 1 exp ( j 4 π N c y ) ] ,
( G ) = G | u ( y ) | 2 d y 1 2 π 2 N c 1 G 1 .
G 1 + 1 2 π 2 N c 1 ,
P ( ν ) = 2 a sinc 2 π a ν ,
= 1 a ν max | P ( ν ) | 2 d ν 2 G π 2 N P ,
N P 2 G π 2 2
N P 4 G ( G + 1 ) N c .
N P 2 G π 2 1 + 8 G N c = N p + 8 G N c .
θ = 2 M θ 2 2 θ 1 M 1

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