Abstract

This paper discusses the transmission and coupling losses of hollow dielectric waveguide modes in a folded waveguide structure. The folded structure consists of two square-bore waveguides placed symmetrically above a spherical mirror with a nonzero on-axis angle between them. The presence of the fold renders the structure astigmatic. However, for moderate fold angles (10–20°), the effect of the fold is negligible.

© 1989 Optical Society of America

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References

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  1. P. A. Forrester, K. F. Hulme, “Laser Rangefinders,” Opt. Quantum Electron. 13, 259–293 (1981).
    [CrossRef]
  2. J. J. Degnan, “The Waveguide Laser: a Review,” Appl. Phys. 11, 1–33 (1976).
    [CrossRef]
  3. P. C. Conder, R. M. Jenkins, J. R. Redding, “Recent Advances in CO2 Laser Technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 806, 27–33 (1987).
  4. L. A. Newman, R. A. Hart, “Recent R&D Advances in Sealed-Off CO2 Lasers,” Laser Focus80–96 (1987).
  5. D. R. Hall, C. A. Hill, “RF-Excited CO2 Waveguide Lasers,” in Handbook of Molecular Lasers, P. K. Cheo, Ed. (Marcel Dekker, New York, 1987).
  6. P. E. Jackson, D. R. Hall, R. M. Jenkins, “Mode Coupling Losses at a Fold in Hollow Dielectric Waveguide,” presented at QE8, St. Andrews (22 Sept. 1987), paper 23P.
  7. J. L. Boulnois, G. P. Agrawal, “Mode Discrimination and Coupling Losses in Rectangular-Waveguide Resonators with Conventional and Phase-Conjugate Mirrors,” J. Opt. Soc. Am. 72, 853–860 (1982).
    [CrossRef]
  8. H. Krammer, “Field Configurations and Propagation Constants of Modes in Hollow Rectangular Dielectric Waveguides,” IEEE J. Quantum Electron. QE-12, 505–507 (1976).
    [CrossRef]
  9. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982).
  10. Ref. 9, pp. 254–255.
  11. S. A. Collins, “Analysis of Optical Resonators Involving Focusing Elements,” Appl. Opt. 3, 1263–1275 (1964).
    [CrossRef]
  12. K. D. Laakmann, W. H. Steier, “Waveguides: Characteristic Modes of Hollow Rectangular Dielectric Waveguides,” Appl. Opt. 15, 1334–1340 (1976).
    [CrossRef] [PubMed]
  13. C. A. Hill, D. R. Hall, “Waveguide Laser Resonators with a Tilted Mirror,” IEEE J. Quantum Electron. QE-22, 1078–1087 (1986).
    [CrossRef]
  14. D. M. Henderson, “Waveguide Lasers with Intracavity Electrooptic Modulators: Misalignment Loss,” Appl. Opt. 15, 1066–1070 (1976).
    [CrossRef] [PubMed]
  15. R. M. Jenkins, R. W. J. Devereux, “EH1m Mode Excitation in Circular Cross Section Hollow Dielectric Waveguides,” submitted to IEEE J. Quantum Electron.

1987 (2)

P. C. Conder, R. M. Jenkins, J. R. Redding, “Recent Advances in CO2 Laser Technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 806, 27–33 (1987).

L. A. Newman, R. A. Hart, “Recent R&D Advances in Sealed-Off CO2 Lasers,” Laser Focus80–96 (1987).

1986 (1)

C. A. Hill, D. R. Hall, “Waveguide Laser Resonators with a Tilted Mirror,” IEEE J. Quantum Electron. QE-22, 1078–1087 (1986).
[CrossRef]

1982 (1)

1981 (1)

P. A. Forrester, K. F. Hulme, “Laser Rangefinders,” Opt. Quantum Electron. 13, 259–293 (1981).
[CrossRef]

1976 (4)

J. J. Degnan, “The Waveguide Laser: a Review,” Appl. Phys. 11, 1–33 (1976).
[CrossRef]

D. M. Henderson, “Waveguide Lasers with Intracavity Electrooptic Modulators: Misalignment Loss,” Appl. Opt. 15, 1066–1070 (1976).
[CrossRef] [PubMed]

H. Krammer, “Field Configurations and Propagation Constants of Modes in Hollow Rectangular Dielectric Waveguides,” IEEE J. Quantum Electron. QE-12, 505–507 (1976).
[CrossRef]

K. D. Laakmann, W. H. Steier, “Waveguides: Characteristic Modes of Hollow Rectangular Dielectric Waveguides,” Appl. Opt. 15, 1334–1340 (1976).
[CrossRef] [PubMed]

1964 (1)

Agrawal, G. P.

Boulnois, J. L.

Collins, S. A.

Conder, P. C.

P. C. Conder, R. M. Jenkins, J. R. Redding, “Recent Advances in CO2 Laser Technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 806, 27–33 (1987).

Degnan, J. J.

J. J. Degnan, “The Waveguide Laser: a Review,” Appl. Phys. 11, 1–33 (1976).
[CrossRef]

Devereux, R. W. J.

R. M. Jenkins, R. W. J. Devereux, “EH1m Mode Excitation in Circular Cross Section Hollow Dielectric Waveguides,” submitted to IEEE J. Quantum Electron.

Forrester, P. A.

P. A. Forrester, K. F. Hulme, “Laser Rangefinders,” Opt. Quantum Electron. 13, 259–293 (1981).
[CrossRef]

Hall, D. R.

C. A. Hill, D. R. Hall, “Waveguide Laser Resonators with a Tilted Mirror,” IEEE J. Quantum Electron. QE-22, 1078–1087 (1986).
[CrossRef]

D. R. Hall, C. A. Hill, “RF-Excited CO2 Waveguide Lasers,” in Handbook of Molecular Lasers, P. K. Cheo, Ed. (Marcel Dekker, New York, 1987).

P. E. Jackson, D. R. Hall, R. M. Jenkins, “Mode Coupling Losses at a Fold in Hollow Dielectric Waveguide,” presented at QE8, St. Andrews (22 Sept. 1987), paper 23P.

Hart, R. A.

L. A. Newman, R. A. Hart, “Recent R&D Advances in Sealed-Off CO2 Lasers,” Laser Focus80–96 (1987).

Henderson, D. M.

Hill, C. A.

C. A. Hill, D. R. Hall, “Waveguide Laser Resonators with a Tilted Mirror,” IEEE J. Quantum Electron. QE-22, 1078–1087 (1986).
[CrossRef]

D. R. Hall, C. A. Hill, “RF-Excited CO2 Waveguide Lasers,” in Handbook of Molecular Lasers, P. K. Cheo, Ed. (Marcel Dekker, New York, 1987).

Hulme, K. F.

P. A. Forrester, K. F. Hulme, “Laser Rangefinders,” Opt. Quantum Electron. 13, 259–293 (1981).
[CrossRef]

Jackson, P. E.

P. E. Jackson, D. R. Hall, R. M. Jenkins, “Mode Coupling Losses at a Fold in Hollow Dielectric Waveguide,” presented at QE8, St. Andrews (22 Sept. 1987), paper 23P.

Jenkins, R. M.

P. C. Conder, R. M. Jenkins, J. R. Redding, “Recent Advances in CO2 Laser Technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 806, 27–33 (1987).

P. E. Jackson, D. R. Hall, R. M. Jenkins, “Mode Coupling Losses at a Fold in Hollow Dielectric Waveguide,” presented at QE8, St. Andrews (22 Sept. 1987), paper 23P.

R. M. Jenkins, R. W. J. Devereux, “EH1m Mode Excitation in Circular Cross Section Hollow Dielectric Waveguides,” submitted to IEEE J. Quantum Electron.

Krammer, H.

H. Krammer, “Field Configurations and Propagation Constants of Modes in Hollow Rectangular Dielectric Waveguides,” IEEE J. Quantum Electron. QE-12, 505–507 (1976).
[CrossRef]

Laakmann, K. D.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982).

Newman, L. A.

L. A. Newman, R. A. Hart, “Recent R&D Advances in Sealed-Off CO2 Lasers,” Laser Focus80–96 (1987).

Redding, J. R.

P. C. Conder, R. M. Jenkins, J. R. Redding, “Recent Advances in CO2 Laser Technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 806, 27–33 (1987).

Steier, W. H.

Appl. Opt. (3)

Appl. Phys. (1)

J. J. Degnan, “The Waveguide Laser: a Review,” Appl. Phys. 11, 1–33 (1976).
[CrossRef]

IEEE J. Quantum Electron. (2)

H. Krammer, “Field Configurations and Propagation Constants of Modes in Hollow Rectangular Dielectric Waveguides,” IEEE J. Quantum Electron. QE-12, 505–507 (1976).
[CrossRef]

C. A. Hill, D. R. Hall, “Waveguide Laser Resonators with a Tilted Mirror,” IEEE J. Quantum Electron. QE-22, 1078–1087 (1986).
[CrossRef]

J. Opt. Soc. Am. (1)

Laser Focus (1)

L. A. Newman, R. A. Hart, “Recent R&D Advances in Sealed-Off CO2 Lasers,” Laser Focus80–96 (1987).

Opt. Quantum Electron. (1)

P. A. Forrester, K. F. Hulme, “Laser Rangefinders,” Opt. Quantum Electron. 13, 259–293 (1981).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

P. C. Conder, R. M. Jenkins, J. R. Redding, “Recent Advances in CO2 Laser Technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 806, 27–33 (1987).

Other (5)

D. R. Hall, C. A. Hill, “RF-Excited CO2 Waveguide Lasers,” in Handbook of Molecular Lasers, P. K. Cheo, Ed. (Marcel Dekker, New York, 1987).

P. E. Jackson, D. R. Hall, R. M. Jenkins, “Mode Coupling Losses at a Fold in Hollow Dielectric Waveguide,” presented at QE8, St. Andrews (22 Sept. 1987), paper 23P.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982).

Ref. 9, pp. 254–255.

R. M. Jenkins, R. W. J. Devereux, “EH1m Mode Excitation in Circular Cross Section Hollow Dielectric Waveguides,” submitted to IEEE J. Quantum Electron.

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Figures (7)

Fig. 1
Fig. 1

Schematic of the waveguide and mirror geometry along with the coordinate systems used.

Fig. 2
Fig. 2

Transmission and coupling efficiencies as functions of the mirror curvature for 2θ = 5°.

Fig. 3
Fig. 3

Optimum coupling efficiency C1(C2) from EH11 to EH11 and the corresponding mirror curvature R1(R2) as functions of the fold angle for t = 1.0 mm and 2a = 1.5(1.0) mm.

Fig. 4
Fig. 4

Coupling efficiency C(R) from EH11 to EH11 as a function of the mirror curvature R (both quantities normalized with respect to their optimum values) for 2θ = 1°, 2.5°, 10° and 15°.

Fig. 5
Fig. 5

Optimum transmission Topt of a Gaussian beam and the corresponding value of Ropt as functions of the fold angle.

Fig. 6
Fig. 6

Optimum coupling efficiency Copt from EH11 to EH11 and the corresponding value of βopt as functions of the Fresnel number N. Broken lines represent results for a folded structure. The corresponding results for a straight structure are represented by + for Copt and by * forβopt.

Fig. 7
Fig. 7

Coupling efficiencies as functions of the inverse curvature of the mirror for a folded structure with 2θ = 60° and for a straight structure with the corresponding value of the Fresnel number N.

Equations (42)

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x = x cos θ z sin θ , y = y , z = z cos θ + x sin θ d , x = x cos θ + z sin θ , y = y , z = z cos θ x sin θ d .
pq ( x , y ) = 1 a E p ( x ) E q ( y ) ,
E p ( x ) = { cos ( π p x / 2 a ) , if p is odd ; sin ( π p x / 2 a ) , if p is even .
a a a a pq ( x , y ) p q ( x , y ) dxdy = δ p p δ q q .
E ( P ) = p 1 q 1 ( x 0 , y 0 ) .
d E ( Q ) = i 2 λ p 1 q 1 ( x 0 , y 0 ) exp ( i k r 01 ) r 01 ( 1 + cos γ 1 ) d x 0 d y 0 ,
d E ( R ) = i 2 λ d E ( Q ) exp ( i k r 02 ) r 02 ( cos γ 2 + cos γ 3 ) d x 1 d y 1 ,
E ( R ) = 1 4 λ 2 a a a a M x M x M y M y p 1 q 1 ( x 0 , y 0 ) × exp [ i k ( r 01 + r 02 ) ] r 01 r 02 × ( 1 + cos γ 1 ) ( cos γ 2 + cos γ 3 ) d x 0 d y 0 d x 1 d y 1 .
χ p 1 q 1 p 2 q 2 = a a a a E ( R ) p 2 q 2 ( x 0 , y 0 ) d x 0 d y 0 .
ρ 2 2 R ρ
r 01 = d x 1 sin θ + x 0 2 + y 0 2 2 d x 1 x 0 cos θ + y 1 y 0 d + ( 1 β 1 ) 2 d x 1 2 cos 2 θ + ( 1 β 2 ) 2 d y 1 2 ,
r 02 = d + x 1 sin θ + x 0 2 + y 0 2 2 d x 1 x 0 cos θ + y 1 y 0 d + ( 1 β 1 ) 2 d x 1 2 cos 2 θ + ( 1 β 2 ) 2 d y 1 2 ,
β j = d R μ j and μ j = { cos θ , if j = 1 ; sec θ , if j = 2 .
χ p 1 q 1 p 2 q 2 = N exp ( 2 ikd ) F 1 ( p 1 , p 2 ) F 2 ( q 1 , q 2 ) ,
F j ( p , p ) = d x 1 1 d x 1 1 d x exp { i π [ η 2 ( x 2 + x 2 ) / 4 + 2 ( 1 β j ) x 2 η ( x + x ) x ] } × E p ( a x ) E p ( a x ) ,
η = 2 N = 2 a 2 λ d .
u = x + x 2 and p ± = p ± p 2 ,
F j ( p , p ) = { 1 + j i η | 1 β j | 0 1 exp ( 2 π i N β j 1 β j u 2 ) G ( p + , p , u ) d u if β j 1 ; G ( p + , p , 0 ) η 2 , if β j = 1 .
G ( p , p , u ) = cos ( π p u ) I ( p , u ) ± cos ( π p u ) I ( p , u ) ,
I ( p , u ) = exp ( i π p 2 / 2 η 2 ) [ f ( η η u + p / η ) + f ( η η u + p / η ] ,
X 2 2 R cos θ + Y 2 2 R sec θ = Z ,
E in ( 1 ) = exp [ ( r / ω ) 2 ] ( π w 2 / 2 ) 1 / 2 ,
C n = { ( 2 π ) 1 / 4 ( ω a ) 1 / 2 a a exp [ ( x / ω ) 2 ] cos ( n π x 2 a ) d x , for odd n ; 0 , for even n .
E out ( 2 ) p 1 , q 1 odd p 2 , q 2 odd A p 1 q 1 exp ( i γ p 1 q 1 L ) χ p 1 q 1 p 2 q 2 exp ( i γ p 2 q 2 L ) p 2 q 2 ,
γ p q = β p q + i α p q ,
β p q = k β p β q with β n = π λ ( λ 4 a ) 2 n 2 ,
α p q = α p ( 1 ) + α q ( 2 ) with α n ( j ) = 1 4 a ( λ / 4 a ) 2 n 2 L ( j ) .
L ( j ) = { Re [ ( 1 ) 1 / 2 ] , if j = 1 ; Re [ ( 1 ) 1 / 2 ] , if j = 2 .
T = a a a a | E out ( 2 ) | 2 dxdy a a a a | E in ( 1 ) | 2 dxdy .
T = p 2 , q 2 odd | p 1 , q 1 odd A p 1 q 1 exp ( i γ p 1 q 1 L ) χ p 1 q 1 p 2 q 2 | 2 exp ( 2 α p 2 q 2 L ) p 1 , q 1 odd | A p 1 q 1 | 2 .
d = ( a + t ) cot θ
β | χ p 1 q 1 p 2 q 2 | 2 = 0 .
μ 1 + μ 2 2 β ( β μ 1 ) ( β μ 2 ) + 2 Im { H 1 ( p 1 , p 2 ) Ĥ 1 ( p 1 , p 2 ) ] | H 1 ( p 1 , p 2 ) | 2 + 2 Im [ H 2 ( q 1 , q 2 ) Ĥ 2 ( q 1 , q 2 ) ] | H 2 ( q 1 , q 2 ) | 2 = 0 ,
H j ( p , p ) = 0 1 exp ( 2 π i N β j 1 β j u 2 ) G ( p + , p , u ) d u ,
Ĥ j ( p , p ) = 2 π N μ j ( β j 1 ) 2 0 1 exp ( 2 π i N β j 1 β j u 2 ) u 2 G * ( p + , p , u ) d u .
μ 1 + μ 2 2 β ( β μ 1 ) ( β μ 2 ) + 2 p odd Im [ S 1 ( p ) Ŝ 1 ( p ) ] exp [ 2 α p ( 1 ) L ] p odd | S 1 ( p ) | 2 exp [ 2 α p ( 1 ) L ] + 2 p odd Im [ S 2 ( p ) Ŝ 2 ( p ) ] exp [ 2 α p ( 2 ) L ] p odd | S 2 ( p ) | 2 exp [ 2 α p ( 2 ) L ] = 0 ,
S j ( p ) = q odd C q H j ( q , p ) exp { [ α q ( j ) + i β q ] L } ,
Ŝ j ( p ) = q odd C q Ĥ j ( q , p ) exp { [ α q ( j ) i β q ] L } .
R opt = ( a + t ) cot θ + σ 2 a 4 tan θ λ 2 ( a + t ) .
| 1 β | | μ j 1 | .
| 1 β | 1 cos θ .
tan θ 1 + t / a .

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