Abstract

A numerical technique has been developed for analyzing the transverse modes of waveguide lasers with external mirrors. Propagation outside the guide is computed with the Fresnel-Kirchhoff diffraction integral and within the guide by decomposing the fields into the characteristic modes of the guide structure. The transverse modes of the entire waveguide–mirror system fall into a number of distinct classes: TE0m, TM0m, EH1m, EH2m, etc. For each class of modes, the, corresponding guide modes form a complete and orthogonal set and may be used as basis vectors to describe those modes. This reduces the mode analysis of the waveguide resonator to the diagonalization of a small (5 × 5 or 10 × 10) complex matrix. Guide losses, coupling losses, and mode shapes will be discussed for a number of interesting cases, with the Fresnel number of the waveguide ranging from 0.1 to 1.0 and with various values of mirror curvature and position. It will be shown that some values of resonator parameters are particularly advantageous for achieving single mode operation.

© 1974 Optical Society of America

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References

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  1. E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).
  2. H. Kogelnik, T. Li, Appl. Opt. 54, 1550 (1966).
    [CrossRef]
  3. R. L. Abrams, IEEE J. Quantum Electron. QE-8, 838 (1972).
    [CrossRef]
  4. A. N. Chester, R. L. Abrams, Appl. Phys. Lett. 21, 576 (1972).
    [CrossRef]
  5. A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).
  6. R. L. Abrams, A. N. Chester, presented at the 1973 Spring Meeting of the Optical Society of America, paper ThD16, Denver, Colorado, March1973.
  7. J. J. Degnan, D. R. Hall, presented at the 1972 IEEE Int. Electron Devices Meeting, Washington, D.C. (Dec. 1972);IEEE J. Quantum Electron. QE-9, 901 (1973).
  8. See, for example, Eqs. (8.2.1) and (8.3.20) in M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965).
  9. R. L. Abrams, W. B. Bridges, IEEE J. Quantum Electron. QE-9, 940 (1973).
    [CrossRef]
  10. D. Pohl, Appl. Phys. Lett. 20, 266 (1972).
    [CrossRef]

1973

R. L. Abrams, W. B. Bridges, IEEE J. Quantum Electron. QE-9, 940 (1973).
[CrossRef]

1972

D. Pohl, Appl. Phys. Lett. 20, 266 (1972).
[CrossRef]

R. L. Abrams, IEEE J. Quantum Electron. QE-8, 838 (1972).
[CrossRef]

A. N. Chester, R. L. Abrams, Appl. Phys. Lett. 21, 576 (1972).
[CrossRef]

1966

H. Kogelnik, T. Li, Appl. Opt. 54, 1550 (1966).
[CrossRef]

1964

E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).

1961

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Abrams, R. L.

R. L. Abrams, W. B. Bridges, IEEE J. Quantum Electron. QE-9, 940 (1973).
[CrossRef]

R. L. Abrams, IEEE J. Quantum Electron. QE-8, 838 (1972).
[CrossRef]

A. N. Chester, R. L. Abrams, Appl. Phys. Lett. 21, 576 (1972).
[CrossRef]

R. L. Abrams, A. N. Chester, presented at the 1973 Spring Meeting of the Optical Society of America, paper ThD16, Denver, Colorado, March1973.

Born, M.

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

Bridges, W. B.

R. L. Abrams, W. B. Bridges, IEEE J. Quantum Electron. QE-9, 940 (1973).
[CrossRef]

Chester, A. N.

A. N. Chester, R. L. Abrams, Appl. Phys. Lett. 21, 576 (1972).
[CrossRef]

R. L. Abrams, A. N. Chester, presented at the 1973 Spring Meeting of the Optical Society of America, paper ThD16, Denver, Colorado, March1973.

Degnan, J. J.

J. J. Degnan, D. R. Hall, presented at the 1972 IEEE Int. Electron Devices Meeting, Washington, D.C. (Dec. 1972);IEEE J. Quantum Electron. QE-9, 901 (1973).

Fox, A. G.

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Hall, D. R.

J. J. Degnan, D. R. Hall, presented at the 1972 IEEE Int. Electron Devices Meeting, Washington, D.C. (Dec. 1972);IEEE J. Quantum Electron. QE-9, 901 (1973).

Kogelnik, H.

H. Kogelnik, T. Li, Appl. Opt. 54, 1550 (1966).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, Appl. Opt. 54, 1550 (1966).
[CrossRef]

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Marcatili, E. A. J.

E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).

Pohl, D.

D. Pohl, Appl. Phys. Lett. 20, 266 (1972).
[CrossRef]

Schmeltzer, R. A.

E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).

Wolf, E.

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

Appl. Opt.

H. Kogelnik, T. Li, Appl. Opt. 54, 1550 (1966).
[CrossRef]

Appl. Phys. Lett.

A. N. Chester, R. L. Abrams, Appl. Phys. Lett. 21, 576 (1972).
[CrossRef]

D. Pohl, Appl. Phys. Lett. 20, 266 (1972).
[CrossRef]

Bell Syst. Tech. J.

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).

IEEE J. Quantum Electron.

R. L. Abrams, W. B. Bridges, IEEE J. Quantum Electron. QE-9, 940 (1973).
[CrossRef]

R. L. Abrams, IEEE J. Quantum Electron. QE-8, 838 (1972).
[CrossRef]

Other

R. L. Abrams, A. N. Chester, presented at the 1973 Spring Meeting of the Optical Society of America, paper ThD16, Denver, Colorado, March1973.

J. J. Degnan, D. R. Hall, presented at the 1972 IEEE Int. Electron Devices Meeting, Washington, D.C. (Dec. 1972);IEEE J. Quantum Electron. QE-9, 901 (1973).

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

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Figures (8)

Fig. 1
Fig. 1

Hollow waveguide resonator.

Fig. 2
Fig. 2

Lens–guide arrangement equivalent to the resonator shown in Fig. 1.

Fig. 3
Fig. 3

Analytic expressions and characteristic electric field patterns for waveguide modes.

Fig. 4
Fig. 4

Half-symmetric resonator geometry.

Fig. 5
Fig. 5

Equivalent lens–guide system for round trip through the half-symmetric geometry.

Fig. 6
Fig. 6

Far-field radiation pattern for mode No. 1 of Table I.

Fig. 7
Fig. 7

Round-trip resonator loss vs Fresnel number of the half-guide for the lowest loss mode of each resonator mode class. The dashed line is the guiding loss of the EH1m resonator mode. The guiding loss of the EH11 waveguide mode is shown for comparison. For these curves, R = 2B, Z = B, υ = √3, and a/λ = 75.

Fig. 8
Fig. 8

Round-trip resonator loss vs Fresnel number of the half-guide for the two lowest loss modes of the EH1m class. The circles and triangles are the two lowest loss EH1m modes for Z = R = 2B. Other parameters used in the calculation are a/λ = 75 and υ = √3. The dashed line (from Fig. 7) is included for comparison.

Tables (4)

Tables Icon

Table I Round-Trip Resonator Loss for the First Four Modes of the EH1m Classa

Tables Icon

Table II Relative Energy in Each Waveguide Mode for Mode No. 1 from Table I, Resolved Into EH1m Guide Modes at the Exit of the Waveguide

Tables Icon

Table III Round-Trip Resonator Loss for the Modes of the EH1m Classa

Tables Icon

Table IV Relative Energy in Each Waveguide Mode for Mode No. 1 from Table III, Resolved into EH1m Guide Modes at the Exit of the Waveguide

Equations (40)

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k a | ν | u n m ,
γ k [ 1 1 2 ( u n m λ 2 π a ) 2 ( 1 i ν n λ π a ) ] ,
ν n = { 1 ( ν 2 1 ) 1 / 2 for T E 0 m modes ( n = 0 ) ν 2 ( ν 2 1 ) 1 / 2 for T M 0 m modes ( n = 0 ) 1 2 ( ν 2 + 1 ) ( ν 2 1 ) 1 / 2 for E H n m modes ( n 0 ) .
β n m = Re ( γ ) = 2 π λ { 1 1 2 ( u n m λ 2 π a ) 2 [ 1 + Im ( ν n λ π a ) ] } , α n m = Im ( γ ) = ( u n m 2 π ) 2 λ 2 a 3 Re ( ν n ) .
U ( r , θ , Z ) = u l ( r , Z ) exp ( i l θ ) .
u l ( r , Z ) = ( i ) l + 1 k Z 0 r d r u l ( r , 0 ) J 1 ( k r r Z ) exp [ i k 2 Z ( r 2 + r 2 ) ] ,
u l ( r , Z ) = ( i ) l + 1 k Z 0 r d r u l ( r , 0 ) J 1 ( k r r Z ) exp [ i k 2 Z ( r 2 + r 2 ) ] ,
E θ = J 1 ( u 0 m r / a ) , E r = 0.
E x = J 1 ( u 0 m r / a ) sin θ , E y = J 1 ( u 0 m r / a ) cos θ .
u m ( r ) = A J 1 ( u 0 m r / a ) ,
A = 1 / [ ( π ) 1 / 2 J n ( u n m ) ] ,
J n 1 ( u n m ) = 0.
For T E 0 m or T M 0 m modes , l = 1 and u m ( r ) = 1 ( π ) 1 / 2 J 2 ( u 0 m ) J 1 ( u 0 m r / a )
For E H n m modes , l = n 1 and u m ( r ) = 1 ( π ) 1 / 2 J n ( u n m ) J n 1 ( u n m r / a ) .
υ ( 1 ) = υ j ( 1 ) ; j = 1 , M ,
υ ( 2 ) = υ j ( 2 ) ; j = 1 , N ,
Δ r = a / ( N 1 ) ,
r j = Δ r ( j 1 ) ,
υ j ( 2 ) = k = 1 M υ k ( 1 ) u k ( r j ) , j = 1 , N .
A ( 1,2 ) j k = u k ( r j )
υ ( 5 ) ( r ) = ( i ) l + 1 k Z 1 2 g 0 r d r υ ( 2 ) ( r ) J l ( k r r 2 g Z ) × exp [ i k 2 Z ( r 2 + r 2 ) ( 1 1 2 g ) ] ,
υ j ( 5 ) = k = 1 N P ( 2,5 ) j k υ k ( 2 ) , k = 1 , N ,
P ( 2,5 ) j k = ( i ) l + 1 π g ( k 1 ) Δ 2 ( a 2 λ Z ) × J i [ π g Δ 2 ( k 1 ) ( j 1 ) ( a 2 λ Z ) ] × exp { i π Δ 2 [ ( j 1 ) 2 + ( k 1 ) 2 ] [ 1 1 2 g ] ( a 2 λ Z ) } ,
υ ( 5 ) ( r ) = ( 1 ) l + 1 exp [ i k r 2 / Z υ ( 2 ) ( r ) ] .
P ( 2,5 ) j k = ( 1 ) l + 1 δ j k exp i [ 2 π ( a 2 λ Z ) Δ 2 ( j 1 ) 2 ] .
υ j ( 6 ) = k = 1 N A ( 5,6 ) j k υ k ( 5 ) ,
A ( 5,6 ) j k = 2 π Δ 2 ( k 1 ) u j ( r k ) .
υ j ( 7 ) = k = 1 M P ( 6,7 ) j k υ k ( 6 ) , j = 1 , M ,
P ( 6,7 ) j k = δ j k exp { i u n m 2 2 π ( a 2 / λ L ) [ 1 i ν n π ( a / λ ) ] } .
C i j = P ( 6,7 ) i k A ( 5,6 ) k l P ( 2,5 ) l m A ( 1,2 ) m j ,
k = 1 M C j k υ k ( 1 ) = Λ υ k ( 1 ) j = 1 , M .
resonator loss = 1 | Λ | 2 ,
P ( 7 ) = j = 1 M | υ j ( 7 ) | 2 = power exiting from guide ,
P ( 6 ) = j = 1 M | υ j ( 7 ) / P ( 6,7 ) j j | 2 = power entering guide ,
L GUIDE = 1 P ( 7 ) / P ( 6 )
ρ = Z + B 2 / Z ,
B = π w 0 2 / λ
Re ( ν n ) = 2 , Im ( ν n ) = 0.
a / λ = 75 ,
0.1 < a 2 / λ L < 1.0.

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