Abstract

A numerical procedure that uses an explicit finite difference method to solve the wave equation is described. This technique results in a propagation algorithm that can accurately propagate an arbitrary electric field through a uniform medium or a medium that is nonuniform, transversely flowing, saturable, and contains index inhomogeneities. By using the propagation algorithm to propagate an arbitrary field back and forth between two resonator mirrors, the three-dimensional transverse mode and the output beam characteristics for a laser resonator can be determined. The advantage of the finite difference method is that unlike integral techniques the computational accuracy and efficiency improve as the resonator Fresnel number increases. The computational techniques are explained, and results for several specific empty cavity confocal unstable resonators are presented and compared to results obtained using an established calculation technique. The application of the finite difference method to inhomogeneous laser media is described, and computational results for an existing CO2 gas dynamic laser are presented and compared to measured data. The medium kinetics and shock wave models used in the calculations are described.

© 1974 Optical Society of America

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References

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  1. D. B. Rensch, A. N. Chester, Appl. Opt. 12, 997 (1973).
    [CrossRef] [PubMed]
  2. See, for example, Eqs. (8.2.1) and (8.3.20) in M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).
  3. E. V. Locke, R. Hella, L. Westra, G. Zeiders, IEEE J. Quantum Electron. QE-8, 389 (1972).
    [CrossRef]
  4. A. N. Chester, Appl. Opt. 12, 2353 (1973).
    [CrossRef] [PubMed]
  5. R. D. Richtmyer, K. W. Morton, Difference Methods for Initial Value Problems (Interscience, New York, 1967).
  6. D. B. Rensch, A. N. Chester, J. Opt. Soc. Am. 63, 502A (1973).
  7. V. E. Sherstobitov, G. N. Vinokurov, Sov. J. Quantum Electron. 2, 224 (1972).
    [CrossRef]
  8. Yu. A. Ananev, Sov. J. Quantum Electron. 1, 565 (1972).
    [CrossRef]
  9. G. L. McAllister, W. H. Steier, W. B. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974).
    [CrossRef]
  10. A. E. Siegman, Proc. IEEE 53, 277 (1965).
    [CrossRef]
  11. R. L. Taylor, S. Bitterman, Rev. Mod. Phy. 41, 26 (January1969).
    [CrossRef]
  12. T. K. McCubbin, T. R. Mooney, J. Quant. Spectrosc. Rad. Trans. 8, 1255 (1968).
    [CrossRef]
  13. E. T. Gerry, D. A. Leonard, Appl. Phy. Lett. 8, 227 (1966).
    [CrossRef]
  14. P. O. Clark, AIAA Paper 72-708, AIAA 5th Fluid and Plasma Dynamics Conference, Boston, Massachusetts, 26–28 June 1972.

1974 (1)

G. L. McAllister, W. H. Steier, W. B. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974).
[CrossRef]

1973 (3)

1972 (3)

V. E. Sherstobitov, G. N. Vinokurov, Sov. J. Quantum Electron. 2, 224 (1972).
[CrossRef]

Yu. A. Ananev, Sov. J. Quantum Electron. 1, 565 (1972).
[CrossRef]

E. V. Locke, R. Hella, L. Westra, G. Zeiders, IEEE J. Quantum Electron. QE-8, 389 (1972).
[CrossRef]

1969 (1)

R. L. Taylor, S. Bitterman, Rev. Mod. Phy. 41, 26 (January1969).
[CrossRef]

1968 (1)

T. K. McCubbin, T. R. Mooney, J. Quant. Spectrosc. Rad. Trans. 8, 1255 (1968).
[CrossRef]

1966 (1)

E. T. Gerry, D. A. Leonard, Appl. Phy. Lett. 8, 227 (1966).
[CrossRef]

1965 (1)

A. E. Siegman, Proc. IEEE 53, 277 (1965).
[CrossRef]

Ananev, Yu. A.

Yu. A. Ananev, Sov. J. Quantum Electron. 1, 565 (1972).
[CrossRef]

Bitterman, S.

R. L. Taylor, S. Bitterman, Rev. Mod. Phy. 41, 26 (January1969).
[CrossRef]

Born, M.

See, for example, Eqs. (8.2.1) and (8.3.20) in M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Chester, A. N.

Clark, P. O.

P. O. Clark, AIAA Paper 72-708, AIAA 5th Fluid and Plasma Dynamics Conference, Boston, Massachusetts, 26–28 June 1972.

Gerry, E. T.

E. T. Gerry, D. A. Leonard, Appl. Phy. Lett. 8, 227 (1966).
[CrossRef]

Hella, R.

E. V. Locke, R. Hella, L. Westra, G. Zeiders, IEEE J. Quantum Electron. QE-8, 389 (1972).
[CrossRef]

Lacina, W. B.

G. L. McAllister, W. H. Steier, W. B. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974).
[CrossRef]

Leonard, D. A.

E. T. Gerry, D. A. Leonard, Appl. Phy. Lett. 8, 227 (1966).
[CrossRef]

Locke, E. V.

E. V. Locke, R. Hella, L. Westra, G. Zeiders, IEEE J. Quantum Electron. QE-8, 389 (1972).
[CrossRef]

McAllister, G. L.

G. L. McAllister, W. H. Steier, W. B. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974).
[CrossRef]

McCubbin, T. K.

T. K. McCubbin, T. R. Mooney, J. Quant. Spectrosc. Rad. Trans. 8, 1255 (1968).
[CrossRef]

Mooney, T. R.

T. K. McCubbin, T. R. Mooney, J. Quant. Spectrosc. Rad. Trans. 8, 1255 (1968).
[CrossRef]

Morton, K. W.

R. D. Richtmyer, K. W. Morton, Difference Methods for Initial Value Problems (Interscience, New York, 1967).

Rensch, D. B.

D. B. Rensch, A. N. Chester, J. Opt. Soc. Am. 63, 502A (1973).

D. B. Rensch, A. N. Chester, Appl. Opt. 12, 997 (1973).
[CrossRef] [PubMed]

Richtmyer, R. D.

R. D. Richtmyer, K. W. Morton, Difference Methods for Initial Value Problems (Interscience, New York, 1967).

Sherstobitov, V. E.

V. E. Sherstobitov, G. N. Vinokurov, Sov. J. Quantum Electron. 2, 224 (1972).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Proc. IEEE 53, 277 (1965).
[CrossRef]

Steier, W. H.

G. L. McAllister, W. H. Steier, W. B. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974).
[CrossRef]

Taylor, R. L.

R. L. Taylor, S. Bitterman, Rev. Mod. Phy. 41, 26 (January1969).
[CrossRef]

Vinokurov, G. N.

V. E. Sherstobitov, G. N. Vinokurov, Sov. J. Quantum Electron. 2, 224 (1972).
[CrossRef]

Westra, L.

E. V. Locke, R. Hella, L. Westra, G. Zeiders, IEEE J. Quantum Electron. QE-8, 389 (1972).
[CrossRef]

Wolf, E.

See, for example, Eqs. (8.2.1) and (8.3.20) in M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Zeiders, G.

E. V. Locke, R. Hella, L. Westra, G. Zeiders, IEEE J. Quantum Electron. QE-8, 389 (1972).
[CrossRef]

Appl. Opt. (2)

Appl. Phy. Lett. (1)

E. T. Gerry, D. A. Leonard, Appl. Phy. Lett. 8, 227 (1966).
[CrossRef]

IEEE J. Quantum Electron. (2)

E. V. Locke, R. Hella, L. Westra, G. Zeiders, IEEE J. Quantum Electron. QE-8, 389 (1972).
[CrossRef]

G. L. McAllister, W. H. Steier, W. B. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974).
[CrossRef]

J. Opt. Soc. Am. (1)

D. B. Rensch, A. N. Chester, J. Opt. Soc. Am. 63, 502A (1973).

J. Quant. Spectrosc. Rad. Trans. (1)

T. K. McCubbin, T. R. Mooney, J. Quant. Spectrosc. Rad. Trans. 8, 1255 (1968).
[CrossRef]

Proc. IEEE (1)

A. E. Siegman, Proc. IEEE 53, 277 (1965).
[CrossRef]

Rev. Mod. Phy. (1)

R. L. Taylor, S. Bitterman, Rev. Mod. Phy. 41, 26 (January1969).
[CrossRef]

Sov. J. Quantum Electron. (2)

V. E. Sherstobitov, G. N. Vinokurov, Sov. J. Quantum Electron. 2, 224 (1972).
[CrossRef]

Yu. A. Ananev, Sov. J. Quantum Electron. 1, 565 (1972).
[CrossRef]

Other (3)

R. D. Richtmyer, K. W. Morton, Difference Methods for Initial Value Problems (Interscience, New York, 1967).

See, for example, Eqs. (8.2.1) and (8.3.20) in M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

P. O. Clark, AIAA Paper 72-708, AIAA 5th Fluid and Plasma Dynamics Conference, Boston, Massachusetts, 26–28 June 1972.

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Figures (16)

Fig. 1
Fig. 1

Schematic of confocal unstable resonator with laser medium divided into Nz segments of length Δz. Each segment is considered to be axially uniform in gain and index.

Fig. 2
Fig. 2

Two-dimensional mirror showing edge tapering with Gaussian profile.

Fig. 3
Fig. 3

Transformation of plane P into image plane P′ using lens with focal length f.

Fig. 4
Fig. 4

Geometry used to analyze unstable resonators with variable coordinate system.

Fig. 5
Fig. 5

(a) Near-field intensity distribution, (b) near-field phase distribution, and (c) far-field intensity distribution for empty cavity confocal unstable resonator N0 = 10, M = 2.

Fig. 6
Fig. 6

Comparison of near-field intensity distributions for confocal unstable resonator N0 = 10, M = 2. Fresnel integral distribution is for two-dimensional resonator with sharp mirror edges. Finite difference distribution is a cross section of three-dimensional distribution in Fig. 5(a).

Fig. 7
Fig. 7

(a) Near-field intensity distribution, (b) near-field phase distribution, and (c) far-field intensity distribution for empty cavity confocal unstable resonator N0 = 60, M = 2.5.

Fig. 8
Fig. 8

Comparison of near-field intensity distributions for confocal unstable resonator N0 = 60, M = 2.5. Fresnel integral distribution is for two-dimensional resonator with sharp mirror edges. Finite difference distribution is a cross section of three-dimensional distribution in Fig. 7(a).

Fig. 9
Fig. 9

Vibrational energy level diagram for CO2-N2 molecular system.

Fig. 10
Fig. 10

Computed small-signal gain for CO2-N2 GDL with rotational quantum number J = 18.

Fig. 11
Fig. 11

Schematic of GDL unstable resonator configuration used for experiment and computer modeling.

Fig. 12
Fig. 12

(a) Near-field intensity distribution, (b) near-field phase distribution, and (c) far-field intensity distribution for CO2 GDL with confocal unstable resonator N0 = 78, M = 1.58.

Fig. 13
Fig. 13

Measured far-field intensity distribution for CO2 GDL with unstable resonator (see Ref. 3).

Fig. 14
Fig. 14

Schematic showing position of shock waves used for modeling CO2 GDL with unstable resonator.

Fig. 15
Fig. 15

(a) Near-field intensity distribution, (b) near-field phase distribution, and (c) far-field intensity distribution for CO2 GDL with shock waves N0 = 78, M = 1.58.

Fig. 16
Fig. 16

(a) Near-field intensity distribution, (b) near-field phase distribution, and (c) far-field intensity distribution for CO2 GDL with shock waves N0 = 78, M = 1.35.

Tables (2)

Tables Icon

Table I Resonator and Computer Model Parameters for Empty Cavity Unstable Resonator Calculations

Tables Icon

Table II Resonator and Computer Model Parameters for CO2 GDL Unstable Resonator Calculations

Equations (36)

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U ˆ ( x , y , z , t ) = i ˆ Re [ U ˆ ( x , y , z ) exp ( i ω t ) ,
( 2 + k 2 n 2 ) U ˆ ( x , y , z ) = 0 ,
U ˆ ( x , y , z + Δ z ) = U ˆ f ( x , y , z + Δ z ) exp { [ g ( x , y ) / 2 i k Δ n ( x , y ) ] Δ z + 0 ( Δ z 2 / l 2 ) } ,
[ i 2 k ( / z ) + T 2 ] U ˆ f ( x , y , z ) = 0 ,
U / z = ( 1 / 2 Δ z ) ( U j , l m + 1 U j , l m 1 ) ,
2 U / x 2 = [ 1 / ( Δ x ) 2 ] ( U j + 1 , l m U j , l m + 1 U j , l m 1 + U j 1 , l m ) ;
2 U / y 2 = [ 1 / ( Δ y ) 2 ] ( U j , l + 1 m U j , l m + 1 U j , l m 1 + U j , l 1 m ) ,
U j , l m + 1 = 1 1 + δ x 2 + δ y 2 + 2 δ x δ y [ U j , l m 1 + i δ x ( U j + 1 , l m 2 u j , l m 1 + u j 1 , l m ) + δ x 2 ( u j + 1 , l m u j , l m 1 + u j 1 , l m ) + δ x δ y ( u j + 1 , l m u j , l m 1 + u j 1 , l m ) + i δ y ( u j , l + 1 m 2 u j , l m 1 + u j , l 1 m ) + δ y 2 ( u j , l + 1 m u j , l m 1 + u j , l 1 m ) + δ x δ y ( u j , l + 1 m u j , l m 1 + u j , l 1 m ) ] + truncation error terms ,
U j , l 1 = U j , l 0 + i [ δ x ( U j 1 , l 0 2 U j , l 0 + U j + 1 , l 0 ) + δ y ( U j , l 1 0 2 U j , l 0 + U j , l + 1 0 ) ] ,
Truncation error = i k [ ( 2 U z 2 ) ( Δ z 2 Δ x 2 + Δ z 2 Δ y 2 ) ] + 1 3 ( 3 U z 3 ) Δ z 3 i 12 k [ ( 4 U x 4 ) Δ x 2 + ( 4 U y 4 ) Δ y 2 ] + .
Δ z < k Δ x 2 and Δ z < k Δ y 2 .
R = exp [ ( x x 0 ) 2 + ( y y 0 ) 2 ] / τ 2 | x | x 0 , a y a , | y | y 0 , a x a ,
2 a = 2 [ x 0 + ( 0.7 ) 1 / 2 τ ] ,
number of computer operations 3 ( N x N y ) N z .
y = f y z f , x = f x z f , and z = f z z f .
V p = c n p | Δ z Δ t | , V p = c n p | Δ z Δ t | ,
n p = f 2 / [ ( z f ) 2 ] n p = { [ ( z f ) 2 / f 2 ] } n p .
z = n p d z = [ f z / ( z f ) ] ,
y = y , x = x .
2 U x 2 + 2 U y 2 + 2 U z 2 + n p 2 k 2 U = 0 ,
2 U x 2 + 2 U y 2 + f 4 ( z f ) 4 2 U z 2 + 2 f 2 ( z f ) 3 U z + n p 2 k 2 U = 0.
| U | 2 d x d y = | U | 2 d x d y ,
U = [ ( x y ) / ( x y ) ] 1 / 2 U = [ f / ( z f ) ] U = { [ ( z f ) U ] / f } .
( z ¯ 2 f 2 + x 2 z ¯ 2 ) 2 U x 2 + ( z ¯ 2 f 2 + y 2 z ¯ 2 ) 2 U y 2 + 2 U z 2 + 2 x z ¯ 2 U x + 2 y z ¯ 2 U y + 2 z ¯ U z + 2 x y z ¯ 2 2 U x y + 2 y z ¯ 2 U y z + 2 x z ¯ 2 U x z + z ¯ 2 f 2 n p 2 k 2 U = 0 ,
1 p 2 p ( p 2 U p ) + 1 p 2 sin ϕ 1 ϕ ( sin ϕ ϕ ) + 1 p 2 sin 2 ϕ 2 U θ 2 + n p 2 k 2 U = 0 ,
x = p ϕ cos θ , y = p ϕ sin θ , z ¯ = p .
( 1 + x 2 z ¯ 2 ) 2 U x 2 + ( 1 + y 2 z ¯ 2 ) 2 U y 2 + 2 U z ¯ 2 + 2 x z ¯ 2 U x + 2 y z ¯ 2 U y + 2 z ¯ U z + 2 x y z ¯ 2 2 U x y + 2 y z ¯ 2 U y z + 2 x z ¯ 2 U x z + n p 2 k 2 U = 0.
U j , l m + 1 = A 1 + δ x 2 + δ y 2 + 2 δ x δ y [ A U j , l m 1 + i δ x ( U j + 1 , l m 2 A U j , l m 1 + U j 1 , l m ) + δ x 2 ( U j + 1 , l m A U j , l m 1 + U j 1 , l m ) + δ x δ y ( U j + 1 , l m A U j , l m 1 + U j 1 , l m ) + i δ y ( U j , l + 1 m 2 A U j , l m 1 + U j , l 1 m ) + δ y 2 ( U j , l + 1 m A U j , l m 1 + U j , l 1 m ) + δ x δ y ( U j , l + 1 m A U j , l m 1 + U j , l 1 m ) ] + truncation error terms ,
f = z + V s , Δ x = Δ x i ( 1 + z / V s ) , Δ y = Δ y i ( 1 + z / V s ) ,
Resonator 1 : N 0 = 10 , M = 2 ( N eq = 1.25 ) ; Resonator 2 : N 0 = 60 , M = 2.5 ( N eq = 7.20 ) .
( 1 ) CO 2 * ( ν 2 ) + A CO 2 + A , ( 2 ) N 2 * + A N 2 + A .
( 3 ) CO 2 * ( ν 3 ) + N 2 CO 2 + N 2 * , ( 4 ) CO 2 * ( ν 3 ) + A CO 2 * * * ( ν 2 ) + A , ( 5 ) CO 2 * ( ν 1 ) + A CO 2 * * ( ν 2 ) + A ,
d E C 3 d x = 1 V [ ( d E C 3 d t ) C 2 + ( d E C 3 d t ) N + ( d E C 3 d t ) l ] , d E C 2 d x = 1 V [ ( d E C 2 d t ) C 1 + ( d E C 2 d t ) C 3 + ( d E C 2 d t ) V T ] , d E C 1 d x = 1 V [ ( d E C 1 d t ) C 1 + ( d E C 1 d t ) l ] , d E N d x = 1 V [ ( d E N d t ) C 3 + ( d E N d t ) V T ] ,
g ( ν ) = ( λ 2 / 8 π ) A u l [ N u ( G u / G l ) N l ] α ( ν ν 0 ) ,
α ( 0 ) = 2 / ( π Δ ν H )
Δ ν H = 5.72 P atm ( 300 / T ) 1 / 2 × ( ψ CO 2 + 0.71 ψ N 2 + 0.36 ψ H 2 O ) GHz ,

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