Abstract

We present a procedure for calculating the three-dimensional mode pattern, the output beam characteristics, and the power output of an oscillating high-power laser taking into account a nonuniform, transversely flowing, saturable gain medium; index inhomogeneities inside the laser resonator; and arbitrary mirror distortion and misalignment. The laser is divided into a number of axial segments. The saturated gain-and-index variation across each short segment is lumped into a complex gain profile across the midplane of that segment. The circulating optical wave within the resonator is propagated from midplane to midplane in free-space fashion and is multiplied by the lumped complex gain profile upon passing through each midplane. After each complete round trip of the optical wave inside the resonator, the saturated gain profiles are recalculated based upon the circulating fields in the cavity. The procedure when applied to typical unstable-resonator flowing-gain lasers shows convergence to a single distorted steady-state mode of oscillation. Typical near-field and far-field results are presented. Several empirical rules of thumb for finite truncated Hermite-Gaussian expansions, including an approximate sampling theorem, have been developed as part of the calculations.

© 1974 Optical Society of America

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  1. H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966); Appl. Opt. 5, 1550 (1966).
    [Crossref] [PubMed]
  2. Yu. A. Ananev, Sov. J. Quant. Elect. 1, 565 (May/June 1972).
    [Crossref]
  3. A. E. Siegman, Appl. Opt. 13, 353 (1974).
    [Crossref] [PubMed]
  4. D. B. Rensch, A. N. Chester, Appl. Opt. 12, 997 (1973).
    [Crossref] [PubMed]
  5. D. B. Rensch, A. N. Chester, J. Opt. Soc. Am. 63, 502 (abstract only) (April1973).
  6. M. Abramowitz, I. A. Stegum, Eds., Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 22, pp. 773–792.
  7. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), Sec. 21.7–1, p. 722; Sec. 21.7–6, p. 726.
  8. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part 1, pp. 786–787.
  9. A. E. Siegman, R. W. Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
    [Crossref]
  10. R. L. Sanderson, W. Streifer, Appl. Opt. 8, 2129 (1969).
    [Crossref] [PubMed]
  11. Yu. A. Ananev et al., Sov. Phys. JETP 31, 420 (1970).
  12. V. E. Sherstobitor, G. N. Vinokurov, Sov. J. Quantum Electron. 2, 224 (1972).
    [Crossref]
  13. P. Horowitz, J. Opt. Soc. Am. (to be published) 00, 000 (197x).
  14. G. H. McAllister, W. H. Steier, W. B. Lacina, IEEE J. Quantum Electron. (to be published) 00, 000 (197x).
  15. E. T. Gerry, IEEE Spectrum, 51 (November1970).
    [Crossref]
  16. A. E. Siegman, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
    [Crossref] [PubMed]
  17. M. N. Director, “Aerodynamic Parameters Affecting Practical Gas Dynamic Laser Design,” presented at the AIAA Sixth Fluid and Plasma Dynamics Conference, Palm Springs, California (July1973), AIAA Preprint 73-626.
  18. R. L. Taylor, S. Bitterman, Rev. Mod. Phys. 41, 26 (1969).
    [Crossref]
  19. R. J. Freiberg, P. Chenausky, C. J. Buczek, IEEE G. Quantum Electron. QE-8, 882 (1972).
    [Crossref]
  20. E. A. Sziklas, A. E. Siegman, Proc. IEEE 62, 410 (1974).
    [Crossref]

1974 (2)

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

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

1973 (2)

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

D. B. Rensch, A. N. Chester, J. Opt. Soc. Am. 63, 502 (abstract only) (April1973).

1972 (3)

Yu. A. Ananev, Sov. J. Quant. Elect. 1, 565 (May/June 1972).
[Crossref]

R. J. Freiberg, P. Chenausky, C. J. Buczek, IEEE G. Quantum Electron. QE-8, 882 (1972).
[Crossref]

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

1970 (3)

E. T. Gerry, IEEE Spectrum, 51 (November1970).
[Crossref]

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

Yu. A. Ananev et al., Sov. Phys. JETP 31, 420 (1970).

1969 (2)

R. L. Sanderson, W. Streifer, Appl. Opt. 8, 2129 (1969).
[Crossref] [PubMed]

R. L. Taylor, S. Bitterman, Rev. Mod. Phys. 41, 26 (1969).
[Crossref]

1967 (1)

A. E. Siegman, R. W. Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
[Crossref]

1966 (1)

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

Ananev, Yu. A.

Yu. A. Ananev, Sov. J. Quant. Elect. 1, 565 (May/June 1972).
[Crossref]

Yu. A. Ananev et al., Sov. Phys. JETP 31, 420 (1970).

Arrathoon, R. W.

A. E. Siegman, R. W. Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
[Crossref]

Bitterman, S.

R. L. Taylor, S. Bitterman, Rev. Mod. Phys. 41, 26 (1969).
[Crossref]

Buczek, C. J.

R. J. Freiberg, P. Chenausky, C. J. Buczek, IEEE G. Quantum Electron. QE-8, 882 (1972).
[Crossref]

Chenausky, P.

R. J. Freiberg, P. Chenausky, C. J. Buczek, IEEE G. Quantum Electron. QE-8, 882 (1972).
[Crossref]

Chester, A. N.

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

D. B. Rensch, A. N. Chester, J. Opt. Soc. Am. 63, 502 (abstract only) (April1973).

Director, M. N.

M. N. Director, “Aerodynamic Parameters Affecting Practical Gas Dynamic Laser Design,” presented at the AIAA Sixth Fluid and Plasma Dynamics Conference, Palm Springs, California (July1973), AIAA Preprint 73-626.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part 1, pp. 786–787.

Freiberg, R. J.

R. J. Freiberg, P. Chenausky, C. J. Buczek, IEEE G. Quantum Electron. QE-8, 882 (1972).
[Crossref]

Gerry, E. T.

E. T. Gerry, IEEE Spectrum, 51 (November1970).
[Crossref]

Horowitz, P.

P. Horowitz, J. Opt. Soc. Am. (to be published) 00, 000 (197x).

Kogelnik, H.

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

Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), Sec. 21.7–1, p. 722; Sec. 21.7–6, p. 726.

Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), Sec. 21.7–1, p. 722; Sec. 21.7–6, p. 726.

Lacina, W. B.

G. H. McAllister, W. H. Steier, W. B. Lacina, IEEE J. Quantum Electron. (to be published) 00, 000 (197x).

Li, T.

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

McAllister, G. H.

G. H. McAllister, W. H. Steier, W. B. Lacina, IEEE J. Quantum Electron. (to be published) 00, 000 (197x).

Miller, H. Y.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part 1, pp. 786–787.

Rensch, D. B.

D. B. Rensch, A. N. Chester, J. Opt. Soc. Am. 63, 502 (abstract only) (April1973).

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

Sanderson, R. L.

Sherstobitor, V. E.

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

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, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
[Crossref] [PubMed]

A. E. Siegman, R. W. Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
[Crossref]

Steier, W. H.

G. H. McAllister, W. H. Steier, W. B. Lacina, IEEE J. Quantum Electron. (to be published) 00, 000 (197x).

Streifer, W.

Sziklas, E. A.

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

Taylor, R. L.

R. L. Taylor, S. Bitterman, Rev. Mod. Phys. 41, 26 (1969).
[Crossref]

Vinokurov, G. N.

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

Appl. Opt. (4)

IEEE G. Quantum Electron. (1)

R. J. Freiberg, P. Chenausky, C. J. Buczek, IEEE G. Quantum Electron. QE-8, 882 (1972).
[Crossref]

IEEE J. Quantum Electron. (1)

A. E. Siegman, R. W. Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
[Crossref]

IEEE Spectrum (1)

E. T. Gerry, IEEE Spectrum, 51 (November1970).
[Crossref]

J. Opt. Soc. Am. (1)

D. B. Rensch, A. N. Chester, J. Opt. Soc. Am. 63, 502 (abstract only) (April1973).

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]

Rev. Mod. Phys. (1)

R. L. Taylor, S. Bitterman, Rev. Mod. Phys. 41, 26 (1969).
[Crossref]

Sov. J. Quant. Elect. (1)

Yu. A. Ananev, Sov. J. Quant. Elect. 1, 565 (May/June 1972).
[Crossref]

Sov. J. Quantum Electron. (1)

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

Sov. Phys. JETP (1)

Yu. A. Ananev et al., Sov. Phys. JETP 31, 420 (1970).

Other (6)

M. Abramowitz, I. A. Stegum, Eds., Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 22, pp. 773–792.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), Sec. 21.7–1, p. 722; Sec. 21.7–6, p. 726.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part 1, pp. 786–787.

P. Horowitz, J. Opt. Soc. Am. (to be published) 00, 000 (197x).

G. H. McAllister, W. H. Steier, W. B. Lacina, IEEE J. Quantum Electron. (to be published) 00, 000 (197x).

M. N. Director, “Aerodynamic Parameters Affecting Practical Gas Dynamic Laser Design,” presented at the AIAA Sixth Fluid and Plasma Dynamics Conference, Palm Springs, California (July1973), AIAA Preprint 73-626.

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Figures (17)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

(a) The Hermite-Gaussian parameters w2 and R2 at the output plane can be calculated for any choice of values w1 and R1 at the input plane. (b) The ratio a2/w2 at the output plane vs the ratio a1/w1 at the input plane under self-consistent Gaussian radius conditions for various magnifications.

Fig. 3
Fig. 3

(a) Analytical model of the laser energy levels for the flowing saturable gain calculation. (b) The step approximation to the transverse intensity profile along the transverse flow direction.

Fig. 4
Fig. 4

Geometry of the 10.6-μm laser system used for trial calculations with the Hermite-Gaussian computational procedure.

Fig. 5
Fig. 5

Density flow field assumed for trial calculations (see text).

Fig. 6
Fig. 6

(a) Intensity distribution, and (b) phase distribution, incident upon the output coupling mirror for the bare-resonator case with Neq = 0.5 (cf. Table I).

Fig. 7
Fig. 7

(a) Far-field isointensity contours, and (b) integrated percent power within a far-field radius r, for the bare-resonator case with Neq = 0.5 and D = 1.2 × 2a2 (cf. Table I).

Fig. 8
Fig. 8

(a) Intensity distribution, and (b) phase distribution, incident upon the output coupling mirror for the bare-resonator case with Neq = 1.5 (cf. Table I).

Fig. 9
Fig. 9

(a) Far-field isointensity contours, and (b) integrated percent power within a far-field radius r, for the bare-resonator case with Neq = 1.5 and D = 1.05 × 2a2 (cf. Table I).

Fig. 10
Fig. 10

(a) Intensity distribution, and (b) phase distribution, incident upon the output coupling mirror for the loaded-resonator case with Neq = 0.5 (cf. Table I).

Fig. 11
Fig. 11

(a) Far-field isointensity contours, and (b) integrated percent power within a far-field radius r, for the loaded-resonator case with Neq = 0.5 and D = 1.2 × 2a2 (cf. Table I).

Fig. 12
Fig. 12

(a) Intensity distribution, and (b) phase distribution, incident upon the output coupling mirror for the loaded-resonator case with Neq = 1.5 (cf. Table I).

Fig. 13
Fig. 13

(a) Far-field isointensity contours, and (b) integrated percent power within a far-field radius r, for the loaded-resonator case with Neq = 1.5 and D = 1.05 × 2a2 (cf. Table I).

Fig. 14
Fig. 14

The Hermite-Gaussian fn(x) for n = 20.

Fig. 15
Fig. 15

The outermost maximum ζmax of the Hermite-Gaussian functions such as Fig. 14 plotted against the function index n. The resulting curve is asymptotic to ζmax = n1/2.

Fig. 16
Fig. 16

(a) The expansion coefficients |cn| for the Hermite-Gaussian expansion of a symmetric square pulse of width 2a, as shown in the inset, for various choices of the ratio a/w. (b) The mean-square error due to truncation of the Hermite-Gaussian series expansion vs the truncation index N, for the case of a half-symmetric square pulse u(x) as shown in the inset, for various choices of the ratio a/w.

Fig. 17
Fig. 17

The first-null index n0, as obtained from plots like Fig. 16(a), vs the epot-size ratio a/w, with the asymptotic curve (a/w)2.

Tables (2)

Tables Icon

Table I Numerical Parameters for the Unstable-Resonator Gas-Dynamic Lasers Used in the Test Calculations

Tables Icon

Table II Power Outputs from the Flowing Unstable Resonator Laser Models

Equations (56)

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K ( x 1 , y 1 , x 2 , y 2 ) = ( j / z λ ) exp { j ( π / z λ ) [ ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 ] } ,
N points = 2 a / Δ x = ( 8 π / δ φ ) N F .
u ( x , y , z ) = n m c n m ( z ) f n ( x , z ) f m ( y , z ) ,
f n ( x ) H n ( 2 x / w ) exp ( x 2 / w 2 )
u ( x , y , z ) = n = 0 m = 0 c n m ( z ) f n ( x , z ) f m ( y , z ) .
f n ( x , z ) = ( 2 π ) 1 / 4 [ 1 2 n n ! w 1 Q ( z ) ] 1 / 2 ( Q * ( z ) Q ( z ) ) n / 2 × H n [ 2 1 / 2 x w ( z ) ] exp [ j k x 2 2 q ( z ) ] ,
1 q 1 = ( 1 / R 1 ) j ( λ / π w 1 2 ) , q ( z ) = q 1 + z z 1 , 1 / q ( z ) = [ 1 / R ( z ) ] j { λ / [ π w 2 ( z ) ] } , Q ( z ) q ( z ) / q 1 .
f n ( x , z ) f m * ( x , z ) d x = δ n m .
c n m ( z ) = u ( x , y , z ) f n * ( x , z ) f m * ( y , z ) d x d y .
N c [ 2 M 2 / ( M 2 1 ) ] N eq .
( 1 / π N c ) ( a 1 / w 1 ) 2 1 ( 1 / 8 M 4 ) , ( 1 / π N c ) ( a 2 / w 2 ) 2 ( 1 / 2 ) ( 1 / 4 M 2 ) ,
a 1 / w 1 = ( π N c ) 1 / 2
( a 2 / w 2 ) ( π N c / 2 ) 1 / 2 for M 1 .
N terms ( π N c ) 1 / 2 { [ M 2 / ( M 2 1 ) ] 2 π N e q } 1 / 2 .
N ripple ( π N c / 2 ) [ M 2 / ( M 2 1 ) ] π N e q .
n 0 / N 0 χ CO 2 / χ N 2 = constant ,
χ CO 2 and χ N 2
υ ( n a / x ) = Λ N ( α + Γ ) n a ( σ I / h ν ) ( n a n b ) ,
υ ( n b / x ) = β n b + ( σ I / h ν ) ( n a n b ) ,
υ ( N / x ) = Γ n a Λ N .
Λ / Γ = χ CO 2 / χ N 2 .
χ CO 2 χ N 2 .
α β , Λ Γ .
g ( x , I ) = ( n a n b ) σ .
n a ( x ) = A exp [ r 1 ( x x n ) / υ ] + B exp [ r 2 ( x x n ) / υ ] + C exp [ r 3 ( x x n ) / υ ] ,
r 1 β ( α + W n ) β ( Λ + Γ ) + W n ( 2 Λ + Γ + β ) , r 2 1 2 { Λ + Γ + β + 2 W n [ ( Λ + Γ β ) 2 + 4 W n ( W n Λ ) ] 1 / 2 } , r 3 1 2 { Λ + Γ + β + 2 W n + [ ( Λ + Γ β ) 2 + 4 W n ( W n Λ ) ] 1 / 2 } ,
n a ( x ) = Λ Λ + Γ [ n a ( x n ) + N ( x n ) ] exp [ r 1 0 ( x x n ) / υ ] + [ Γ n ( x ) Λ N ( x ) Λ + Γ ] exp [ r 3 0 ( x x n ) / υ ] , n b ( x ) = n b ( x n ) exp [ r 2 0 ( x x n ) / υ ] , N ( x ) = Γ Λ + Γ [ n a ( x n ) + N ( x n ) ] exp [ r 1 0 ( x x n ) / υ ] [ Γ n a ( x n ) Λ N ( x n ) Λ + Γ ] exp [ r 3 0 ( x x n ) / υ ] ,
r 1 0 = α Λ / ( Λ + Γ ) , r 2 0 = β , r 3 0 = Λ + Γ .
Γ n a ( x ) Λ N ( x ) .
n a ( x n + 1 ) n b ( x n + 1 ) β ( Λ + Γ ) n a ( x n ) β ( Λ + Γ ) + W n ( 2 Λ + Γ + β ) exp [ r 1 ( W n ) Δ x n υ ] β β + W n n a ( x n ) exp [ r ( W n ) Δ x n υ ] ,
n a ( x n ) n a ( x n 1 ) exp [ r 1 ( W n 1 ) Δ x n 1 / υ ] .
n a ( x n + 1 ) n b ( x n + 1 ) β n a ( x 1 ) β + W n { exp [ r 1 ( W n ) Δ x n + r 1 ( W n 1 ) Δ x n 1 + + r 1 ( W 0 ) Δ x 0 ] / υ } .
n a ( x ) n b ( x ) n a ( x 0 ) 1 + W ( x ) exp [ 1 υ x 0 x d x r 1 ] = n a ( x 0 ) exp [ r 1 0 ( x x 0 ) / υ ] 1 + W ( x ) exp [ 1 υ x 0 x d x ( r 1 r 1 0 ) ] ,
W ( x ) = σ I ( x ) / h ν β .
g ( x ) = [ g 0 ( x ) 1 + W ( x ) ] exp [ χ CO 2 β χ N 2 υ x 0 x d x W ( x ) 1 + W ( x ) ] ,
g 0 ( x ) = g 0 ( x 0 ) exp [ χ CO 2 α ( x x 0 ) / χ N 2 υ ] .
χ CO 2 = 0.13 χ N 2 = 0.85 υ = 1.55 × 1 0 5 cm / sec 1 / α = 47 μ sec 1 / β = 5.2 μ sec σ = 10 18 cm 2 g 0 ( at x = 6 cm ) = 0.01 cm 1 p = 0.08 atm T = 300 K
f n ( x , z ) = ( 2 / π ) 1 / 4 { [ 1 / 2 n n ! w 1 Q ( z ) ] } 1 / 2 [ Q * ( z ) / Q ( z ) ] n / 2 × H n ( 2 1 / 2 x / w ) exp { j [ k x 2 / 2 q ( z ) ] } ,
q ( z ) = [ 1 / R ( z ) j λ / π w 2 ( z ) ] 1 = q 1 + z z 1 , Q ( z ) = q ( z ) / q 1 .
f 0 ( x , z ) = ( 2 / π ) 1 / 4 { 1 / [ w 1 Q ( z ) ] } 1 / 2 exp { j [ ( k x 2 ) / 2 q ( z ) ] } , f 1 ( x , z ) = [ Q * ( z ) / Q ( z ) ] 1 / 2 [ ( 2 3 / 2 x ) / w ( z ) ] f 0 ( x , z ) .
f n ( x , z ) = [ Q * ( z ) n Q ( z ) ] 1 / 2 2 x w ( z ) f n 1 ( x , z ) ( n 1 n ) 1 / 2 Q * ( z ) Q ( z ) f n 2 ( x , z ) .
d d x f n ( x , z ) = [ ( 2 n 1 / 2 ) / w ( z ) ] f n 1 ( x , z ) [ 2 x / w 2 ( z ) ] f n ( x ) .
1 / q b = 1 / q a 2 / R back .
Q ( z ) = [ q ( z ) / q ( z b ) ] [ q ( z a ) / q 1 ] .
f n ( x ) = ( 2 / π ) 1 / 4 [ 1 / ( 2 n n ! w ) ] 1 / 2 H n ( 2 1 / 2 x / w ) exp ( x 2 / w 2 )
u ( x ) = n = 0 c n f n ( x ) , c n = a a u ( x ) f n ( x ) d x .
ζ max ( n ) n 1 / 2
N ( a / w ) 2 .
λ n [ ( 2 ζ max w ) / n / 2 ] ( 4 w / n 1 / 2 ) .
c 0 = ( 2 / π ) 1 / 4 ( w / a ) 1 / 2 erf ( a / w ) , c 1 = 0 , c n = [ ( n 1 ) / n ] 1 / 2 c n 2 { 2 w / [ ( n a ) 1 / 2 ] } f n 1 ( a ) ,
c 0 = ( 2 π ) 1 / 4 ( w / 4 a ) 1 / 2 erf ( a / w ) , c 1 = ( 2 / π ) 1 / 4 ( w / a ) 1 / 2 [ 1 exp ( a 2 / w 2 ) ] , c n = [ ( n 1 ) / n ] 1 / 2 c n 2 { w / [ ( n a ) 1 / 2 ] } [ f n 1 ( a ) f n 1 ( 0 ) ] .
N n 0 ( a / w ) 2
N = a a [ u ( x ) u ˆ N ( x ) ] 2 d x ,
u ˆ N ( x ) = n = 0 N c n f n ( x ) .
f n ( x ) f m ( x ) d x
Δ x 1.5 w / N 1 / 2 N points 1.5 ( a / w ) N 1 / 2 ,

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