Abstract

I analyze the formation of transverse modes in a microchip solid-state laser with plane-parallel cavity mirrors, generalizing the traditional mean-field limit of laser theory. Owing to the balance between mode diffraction and focusing effects induced by the pump beam, the transverse dimension of the laser modes is shown to depend on the cavity Q factor. To provide further insight, a one-dimensional model for an end-pumped microchip laser is also presented. The model gives analytical solutions for the threshold and for the intensity profile of different transverse modes and allows us to study the effects of narrow pump beams. In particular, it is predicted that decreasing the dimensions of the pump beam will lead to longitudinal mode hopping with a corresponding discontinuity in the transverse-mode profile.

© 1994 Optical Society of America

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  1. J. J. Zayhowski and A. Mooradian, Opt. Lett. 14, 24 (1989).
    [CrossRef] [PubMed]
  2. F. Zhou and A. Ferguson, Electron. Lett. 26, 490 (1990).
    [CrossRef]
  3. P. Laporta, S. Taccheo, S. Longhi, O. Svelto, and G. Sacchi, Opt. Lett. 18, 1232 (1993).
    [CrossRef] [PubMed]
  4. H. Kogelnik and T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  5. L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Treddice, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
    [CrossRef] [PubMed]
  6. L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Marducci, P. Ru, and J. R. Treddice, Opt. Commun. 64, 167 (1987).
    [CrossRef]
  7. J. J. Zayhowski, Proc. Adv. Solid State Lasers 6, 9 (1991).
  8. W. Koechner, Appl. Opt. 9, 2548 (1970).
    [CrossRef] [PubMed]
  9. G. K. Harkness and W. J. Firth, J. Mod. Opt. 39, 2023 (1992).
    [CrossRef]
  10. J. J. Zayhowski and J. A. Keszenheimer, IEEE J. Quantum Electron. 28, 1118 (1992).
    [CrossRef]
  11. W. J. Firth, Opt. Commun. 22, 226 (1977).
    [CrossRef]
  12. A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, New York, 1991).
  13. L. W. Caperson, Phys. Rev. A 23, 248 (1981).
    [CrossRef]
  14. H. J. Carmichael, Opt. Acta 27, 147 (1980).
    [CrossRef]
  15. L. A. Lugiato and L. M. Narducci, Z. Phys. B 71, 129 (1988).
    [CrossRef]
  16. R. Bonifacio and L. A. Lugiato, Lett. Nuovo Cimento 21, 505 (1978).
    [CrossRef]
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1966).
  18. R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953).
  19. J. J. Zayhowski, IEEE J. Quantum Electron. 26, 2052 (1990).
    [CrossRef]

1993 (1)

1992 (2)

G. K. Harkness and W. J. Firth, J. Mod. Opt. 39, 2023 (1992).
[CrossRef]

J. J. Zayhowski and J. A. Keszenheimer, IEEE J. Quantum Electron. 28, 1118 (1992).
[CrossRef]

1991 (1)

J. J. Zayhowski, Proc. Adv. Solid State Lasers 6, 9 (1991).

1990 (2)

F. Zhou and A. Ferguson, Electron. Lett. 26, 490 (1990).
[CrossRef]

J. J. Zayhowski, IEEE J. Quantum Electron. 26, 2052 (1990).
[CrossRef]

1989 (1)

1988 (2)

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Treddice, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
[CrossRef] [PubMed]

L. A. Lugiato and L. M. Narducci, Z. Phys. B 71, 129 (1988).
[CrossRef]

1987 (1)

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Marducci, P. Ru, and J. R. Treddice, Opt. Commun. 64, 167 (1987).
[CrossRef]

1981 (1)

L. W. Caperson, Phys. Rev. A 23, 248 (1981).
[CrossRef]

1980 (1)

H. J. Carmichael, Opt. Acta 27, 147 (1980).
[CrossRef]

1978 (1)

R. Bonifacio and L. A. Lugiato, Lett. Nuovo Cimento 21, 505 (1978).
[CrossRef]

1977 (1)

W. J. Firth, Opt. Commun. 22, 226 (1977).
[CrossRef]

1970 (1)

1966 (1)

Bandy, D. K.

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Treddice, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
[CrossRef] [PubMed]

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Marducci, P. Ru, and J. R. Treddice, Opt. Commun. 64, 167 (1987).
[CrossRef]

Bonifacio, R.

R. Bonifacio and L. A. Lugiato, Lett. Nuovo Cimento 21, 505 (1978).
[CrossRef]

Caperson, L. W.

L. W. Caperson, Phys. Rev. A 23, 248 (1981).
[CrossRef]

Carmichael, H. J.

H. J. Carmichael, Opt. Acta 27, 147 (1980).
[CrossRef]

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953).

Ferguson, A.

F. Zhou and A. Ferguson, Electron. Lett. 26, 490 (1990).
[CrossRef]

Firth, W. J.

G. K. Harkness and W. J. Firth, J. Mod. Opt. 39, 2023 (1992).
[CrossRef]

W. J. Firth, Opt. Commun. 22, 226 (1977).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1966).

Harkness, G. K.

G. K. Harkness and W. J. Firth, J. Mod. Opt. 39, 2023 (1992).
[CrossRef]

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953).

Keszenheimer, J. A.

J. J. Zayhowski and J. A. Keszenheimer, IEEE J. Quantum Electron. 28, 1118 (1992).
[CrossRef]

Koechner, W.

Kogelnik, H.

Laporta, P.

Li, T.

Longhi, S.

Lugiato, L. A.

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Treddice, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
[CrossRef] [PubMed]

L. A. Lugiato and L. M. Narducci, Z. Phys. B 71, 129 (1988).
[CrossRef]

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Marducci, P. Ru, and J. R. Treddice, Opt. Commun. 64, 167 (1987).
[CrossRef]

R. Bonifacio and L. A. Lugiato, Lett. Nuovo Cimento 21, 505 (1978).
[CrossRef]

Marducci, L. M.

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Marducci, P. Ru, and J. R. Treddice, Opt. Commun. 64, 167 (1987).
[CrossRef]

Moloney, J. V.

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, New York, 1991).

Mooradian, A.

Narducci, L. M.

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Treddice, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
[CrossRef] [PubMed]

L. A. Lugiato and L. M. Narducci, Z. Phys. B 71, 129 (1988).
[CrossRef]

Newell, A. C.

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, New York, 1991).

Prati, F.

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Treddice, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
[CrossRef] [PubMed]

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Marducci, P. Ru, and J. R. Treddice, Opt. Commun. 64, 167 (1987).
[CrossRef]

Ru, P.

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Treddice, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
[CrossRef] [PubMed]

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Marducci, P. Ru, and J. R. Treddice, Opt. Commun. 64, 167 (1987).
[CrossRef]

Sacchi, G.

Svelto, O.

Taccheo, S.

Treddice, J. R.

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Treddice, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
[CrossRef] [PubMed]

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Marducci, P. Ru, and J. R. Treddice, Opt. Commun. 64, 167 (1987).
[CrossRef]

Zayhowski, J. J.

J. J. Zayhowski and J. A. Keszenheimer, IEEE J. Quantum Electron. 28, 1118 (1992).
[CrossRef]

J. J. Zayhowski, Proc. Adv. Solid State Lasers 6, 9 (1991).

J. J. Zayhowski, IEEE J. Quantum Electron. 26, 2052 (1990).
[CrossRef]

J. J. Zayhowski and A. Mooradian, Opt. Lett. 14, 24 (1989).
[CrossRef] [PubMed]

Zhou, F.

F. Zhou and A. Ferguson, Electron. Lett. 26, 490 (1990).
[CrossRef]

Appl. Opt. (2)

Electron. Lett. (1)

F. Zhou and A. Ferguson, Electron. Lett. 26, 490 (1990).
[CrossRef]

IEEE J. Quantum Electron. (2)

J. J. Zayhowski and J. A. Keszenheimer, IEEE J. Quantum Electron. 28, 1118 (1992).
[CrossRef]

J. J. Zayhowski, IEEE J. Quantum Electron. 26, 2052 (1990).
[CrossRef]

J. Mod. Opt. (1)

G. K. Harkness and W. J. Firth, J. Mod. Opt. 39, 2023 (1992).
[CrossRef]

Lett. Nuovo Cimento (1)

R. Bonifacio and L. A. Lugiato, Lett. Nuovo Cimento 21, 505 (1978).
[CrossRef]

Opt. Acta (1)

H. J. Carmichael, Opt. Acta 27, 147 (1980).
[CrossRef]

Opt. Commun. (2)

W. J. Firth, Opt. Commun. 22, 226 (1977).
[CrossRef]

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Marducci, P. Ru, and J. R. Treddice, Opt. Commun. 64, 167 (1987).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (2)

L. W. Caperson, Phys. Rev. A 23, 248 (1981).
[CrossRef]

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Treddice, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
[CrossRef] [PubMed]

Proc. Adv. Solid State Lasers (1)

J. J. Zayhowski, Proc. Adv. Solid State Lasers 6, 9 (1991).

Z. Phys. B (1)

L. A. Lugiato and L. M. Narducci, Z. Phys. B 71, 129 (1988).
[CrossRef]

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1966).

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953).

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, New York, 1991).

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Figures (5)

Fig. 1
Fig. 1

(a) Mode width, (b) pump parameter, and (c) laser detuning as functions of the pump width for various low-order transverse modes. Curve 1, TEM0 mode; curve 2, TEM1 mode; curve 3, TEM2 mode. The other parameters are T = 0.01, β = 1, and δC = 0.

Fig. 2
Fig. 2

Intensity profiles of low-order transverse modes for various pump widths: (a) γ = 0.01, (b) γ = 0.1, (c) γ = 1. Curve 1, TEM0 mode; curve 2, TEM1 mode; curve 3, TEM2 mode. The other parameters are T = 0.01, β = 1, and δC = 0. The dashed curves represent the transverse pump profiles.

Fig. 3
Fig. 3

(a) Mode width, (b) pump parameter, and (c) laser detuning of the fundamental transverse mode as functions of the cavity detuning for various pump widths. Curve 1, γ = 0.1; curve 2, γ = 0.5; curve 3, γ = 1; curve 4, γ = 2. The other parameters are T = 0.01 and β = 1.

Fig. 4
Fig. 4

(a) Mode width, (b) pump parameter, and (c) laser detuning of the fundamental transverse mode as functions of the pump width, with cavity detuning as a parameter. Curve 1, δC = −1; curve 2, δC = 0; curve 3, δC = 1; curve 4, δC = 2. The other parameters are T = 0.01 and β = 1/2π.

Fig. 5
Fig. 5

Mode width as function of pump width for the lowest threshold mode. The discontinuity in the curve corresponds to the longitudinal mode hopping between the central (δC = 0) and the lateral (δC = 1) modes [see also Fig. 4(b)].

Equations (78)

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E ( x , t ) = μ ( T 1 T 2 ) 1 / 2 { F + ( x , t ) exp [ i ( ω 0 t - k 0 z ) ] + F - ( x , t ) exp [ i ( ω 0 t + k 0 z ) ] + c . c . } ,
P ( x , t ) = i N μ ( T 2 / T 1 ) 1 / 2 [ P ( x , t ) exp ( i ω 0 t ) - c . c ] ,
Δ N ( x , t ) = N Δ ( x , t ) ,
1 2 i k 0 t 2 F + - F + z - 1 c F + t = - α 0 2 P exp ( i k 0 z ) ,
1 2 i k 0 t 2 F - + F - z - 1 c F - t = - α 0 2 P exp ( - i k 0 z ) ,
P t = - 1 T 2 { ( 1 + i δ AC ) P - Δ [ F + exp ( - i k 0 z ) + F - exp ( i k 0 z ) ] } ,
Δ t = - Δ - Δ e T 1 - 1 T 1 [ P * F + exp ( - i k 0 z ) + P * F - exp ( i k 0 z ) + c . c ] .
P exp ( i k 0 z ) = ( 1 - i δ AC ) Δ e exp ( i ϑ + ) × 1 2 π - π π d η p + ( η ) ,
P exp ( - i k 0 z ) = ( 1 - i δ AC ) Δ e exp ( i ϑ - ) × 1 2 π - π π d η p - ( η ) ,
p + ( η ) = F + + F - cos ( 2 η ) 1 + δ AC 2 + 2 [ F + 2 + F - 2 + 2 F + F - cos ( 2 η ) ] , p - ( η ) = F - + F + cos ( 2 η ) 1 + δ AC 2 + 2 [ F + 2 + F - 2 + 2 F + F - cos ( 2 η ) ] ,
1 c F + t + F + z = - i 2 k 0 t 2 F + + α 0 2 ( 1 - i δ AC ) Δ e exp ( i ϑ + ) × 1 2 π - π π d η p + ( η ) ,
1 c F - t - F - z = - i 2 k 0 t 2 F - + α 0 2 ( 1 - i δ AC ) Δ e exp ( i ϑ - ) × 1 2 π - π π d η p - ( η ) .
F - ( x , y , 0 , t ) = F + ( x , y , 0 , t ) , F - ( x , y , l , t ) = R 1 / 2 F + ( x , y , l , t ) exp ( - 2 i k 0 l ) ,
φ ( x , y , z , t ) = { F + ( x , y , z , t ) exp ( - i k 0 l - Γ z / 2 ) 0 < z < l F - ( x , y , 2 l - z , t ) exp [ i k 0 l + ( 2 l - z ) Γ / 2 ) ] l < z < 2 l ,
φ z + 1 c φ t = - Γ 2 φ - i 2 k 0 t 2 φ + 1 2 α 0 ( 1 - i δ AC ) Θ ,
φ ( x , y , 2 l , t ) = φ ( x , y , 0 , t ) exp ( 2 i k 0 l ) .
Θ ( x , y , z , t ) = { Δ e ( x , y , z , t ) exp ( i τ ) 1 2 π - π π d η φ + Φ exp ( - Γ z ) cos ( 2 η ) 1 + δ AC 2 + 2 [ φ 2 exp ( Γ z ) + Φ 2 exp ( - Γ z ) + 2 φ Φ cos ( 2 η ) ] 0 < z < l Δ e ( x , y , 2 l - z , t ) exp ( i τ ) 1 2 π - π π d η φ + ( 1 / R ) Φ exp ( - Γ z ) cos ( 2 η ) 1 + δ AC 2 + 2 [ R φ 2 exp ( Γ z ) + ( 1 / R ) Φ 2 exp ( - Γ z ) + 2 φ Φ cos ( 2 η ) ] l < z < 2 l ,
T 0 ,             α 0 l 0 ,             α 0 l T = C arbitrary .
φ z = - T 4 l φ - i 2 k 0 l 2 φ + α 0 2 ( 1 - i δ AC ) Θ .
φ ˜ ( k x , k y , z ) = - + d x d y φ ( x , y , z ) exp [ i ( k x x + k y y ) ] .
φ ( x , y , z ) = 1 ( 2 π ) 2 - + d k x d k y φ ˜ ( k x , k y , z ) × exp [ - i ( k x x + k y y ) ] .
k 0 l = n π + O ( T ) ,
φ z = O ( T ) ;
k x 2 + k y 2 = k 0 2 l T .
Θ = φ Δ e 1 π - π π d η cos 2 η 1 + δ AC 2 + 8 φ 2 cos 2 η ,
2 i ( k 0 l - n π ) φ = - T 2 φ - i l k 0 t 2 φ + α 0 2 ( 1 - i δ AC ) φ × ( 0 2 l d z Δ e ) × ( 1 π - π π d η cos 2 η 1 + δ AC 2 + 8 φ 2 cos 2 η ) .
x = ( k 0 l ) 1 / 2 x ,             y = ( k 0 l ) 1 / 2 y ,             z = 1 2 l z ,
t 2 φ + ( δ L - δ C - i T 2 ) φ + 2 α 0 l ( i + β δ L ) ( 1 + β 2 δ L 2 + 8 φ 2 ) 1 / 2 [ ( 1 + β 2 δ L 2 + 8 φ 2 ) 1 / 2 + ( 1 + β 2 δ L 2 ) 1 / 2 ] φ 0 1 d z Δ e = 0.
δ L = 2 l ( ω 0 - Ω ) / c , δ C = 2 l ( ω C - Ω ) / c , β = T 2 c / 2 l ,
- + d x d y φ * t 2 φ = - - + d x d y t φ 2 ,
- + d x d y φ 2 = 4 C - + d x d y 0 1 d z φ 2 Δ e ( 1 + β 2 δ L 2 + 8 φ 2 ) 1 / 2 [ ( 1 + β 2 δ L 2 + 8 φ 2 ) 1 / 2 + ( 1 + β 2 δ L 2 ) 1 / 2 ] ,
( 1 + β T 2 ) δ L - δ C = - + d x d y t φ 2 - + d x d y φ 2 .
ϑ d ( T 2 l k 0 ) 1 / 2 .
ϑ d λ 0 / w ,
w ( 4 π λ 0 l T ) 1 / 2 ,
w Q λ 0 .
Δ e ( x , z ) = exp ( - 2 l α p z ) sech 2 ( γ x ) ,
d 2 φ d x 2 + f ( x ) φ = 0.
f ( x ) = A + B sech 2 ( γ x ) ,
A = δ L - δ C - i T 2 ,             B = p α 0 l i + β δ L 1 + β 2 δ L 2 , p = 1 - exp ( - α p l ) α p l .
( 1 - ξ 2 ) φ - 2 ξ φ + A + B ( 1 - ξ 2 ) ( 1 - ξ 2 ) γ 2 φ = 0
φ ( 1 ) = 0 ,             φ ( - 1 ) = 0.
φ ( ξ ) = ( 1 - ξ 2 ) ν H ( ξ ) ,
( 1 - ξ 2 ) H - 2 ( 1 + 2 ν ) ξ H + λ H = 0 ,
A + B γ 2 - 2 ν = λ ,
B γ 2 - 2 ν ( 2 ν + 1 ) = λ ,
λ N = N ( N + 1 ) + 4 ν N ,
φ N ( x ) = 1 [ cosh ( γ x ) ] 2 ν H N [ tanh ( γ x ) ] .
A = - 4 γ 2 ν 2 , A + B γ 2 - 2 ν = N ( N + 1 ) + 4 ν N .
z = p α 0 l 1 + β 2 δ L 2 - T 2 ,
a 0 z 5 + a 1 z 4 + a 2 z 3 + a 3 z 2 + a 4 z + a 5 = 0.
φ ˜ z = [ - T 4 l + i 2 k 0 ( k x 2 + k y 2 ) ] φ ˜ + α 0 g ˜ ,
g = 1 2 Θ ( 1 - i δ AC ) .
φ ˜ ( k x , k y , 2 l ) = φ ˜ ( k x , k y , 0 ) exp ( 2 i k 0 l ) .
φ ˜ ( k x , k y , z ) = n = - c n ( k x , k y ) exp [ - i ( n π l - k 0 ) z ] .
1 2 l 0 2 l d z exp [ i π l ( n - m ) z ] = δ n ,
c n = α 0 l g ˜ n T 4 - i ( n π - k 0 l ) - i l 2 k 0 ( k x 2 + k y 2 ) ,
g ˜ n = 1 2 l 0 2 l d z g ˜ exp [ i ( n π l - k 0 ) z ] .
T 0 ,             α 0 l = O ( T ) ,             g ˜ n = O ( 1 ) ,
c n = O ( 1 ) if k 0 l - n π = O ( T ) c n = O ( T ) otherwise ,
φ ( x , y , z ) = exp [ - i ( n π l - k 0 ) z ] ( 2 π ) 2 - d k x d k y c n ( k x , k y ) × exp [ - i ( k x x + k y y ) ] ,
k 0 l - n π = O ( T )
H ( ξ ) = n - 0 a n ξ n .
a n + 2 = - λ - λ n ( n + 1 ) ( n + 2 ) a n ,
λ n = n ( n + 1 ) + 4 ν n .
H 0 = 1 , H 1 = ξ , H 2 = 1 - ( 3 + 4 ν ) ξ 2 , H 3 = ξ - 5 + 4 ν 3 ξ 3 .
ϑ = γ 2 N ( N + 1 ) ,
ρ = γ ( 2 N + 1 ) ,
a 0 = β ,
a 1 = ( 1 + β T / 2 ) ,
a 2 = β δ c ρ 2 ,
a 3 = - ρ 2 ( ϑ - β δ C T / 2 ) ,
a 4 = - T ρ 4 4 ( 1 + β T / 4 ) ,
a 5 = - ( 1 + β T / 2 ) ( ρ 2 T / 4 ) 2 .
δ L = ( ρ 2 T / 4 z ) + δ C + ϑ 1 + β ( z + T / 2 ) ,
p α 0 l = ( 1 + β 2 δ L 2 ) ( z + T / 2 ) ,
ν = T 8 z ( 1 + 2 N ) + i z 2 γ 2 ( 1 + 2 N ) .
z > 0.

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