Abstract

We have previously discussed the transmission and coupling losses of a Gaussian beam in a folded waveguide structure made of two hollow square waveguides placed symmetrically above a spherical mirror with a nonzero on-axis angle between. Here we investigate the additional loss due to the misalignment of the mirror (axial displacement and/or angular tilt). The theoretical results are in good agreement with experiment and provide alignment tolerances for folded waveguide design.

© 1990 Optical Society of America

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References

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  1. E. A. J. Marcatili, R. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
  2. R. L. Abrams, “Waveguide Gas Lasers,” in Laser Handbook, Vol. 3 (North-Holland, Amsterdam, 1979).
  3. L. A. Newman, R. A. Hart, “Recent R&D Advances in Sealed-off CO2 Lasers,” Laser Focus80–96 (1987).
  4. 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).
  5. R. M. Jenkins, R. W. J. Devereux, “EH1m Mode Excitation in Circular Cross Section Hollow Dielectric Waveguides,” (to be published in IEEE J. Quantum Electron. in press, (1990).
  6. J. Banerji, A. R. Davies, P. E. Jackson, R. M. Jenkins, “Transmission and Coupling Losses in a Folded Waveguide,” Appl. Opt. 28, 4637–4643 (1989).
    [CrossRef] [PubMed]
  7. P. E. Jackson, D. R. Hall, C. A. Hill, “Comparisons of Waveguide Folding Geometries in a CO2Z-Fold Laser,” Appl. Opt. 28, 935–941 (1989).
    [CrossRef] [PubMed]
  8. 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).
  9. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1970).
  10. C. A. Hill, D. R. Hall, “Waveguide Laser Resonators with a Tilted Mirror,” IEEE J. Quantum Electron. QE-22, 1078–1087 (1986).
    [CrossRef]
  11. P. J. Gorton, R. M. Jenkins, “Complex Refractive Index Measurements of Polycrystalline Alumina in the 9–11 Micron Waveband,” (to be submitted for publication in Appl. Phys. Letts).

1989

1987

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

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

1964

E. A. J. Marcatili, R. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).

Abrams, R. L.

R. L. Abrams, “Waveguide Gas Lasers,” in Laser Handbook, Vol. 3 (North-Holland, Amsterdam, 1979).

Banerji, J.

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).

Davies, A. R.

Devereux, R. W. J.

R. M. Jenkins, R. W. J. Devereux, “EH1m Mode Excitation in Circular Cross Section Hollow Dielectric Waveguides,” (to be published in IEEE J. Quantum Electron. in press, (1990).

Gorton, P. J.

P. J. Gorton, R. M. Jenkins, “Complex Refractive Index Measurements of Polycrystalline Alumina in the 9–11 Micron Waveband,” (to be submitted for publication in Appl. Phys. Letts).

Hall, D. R.

P. E. Jackson, D. R. Hall, C. A. Hill, “Comparisons of Waveguide Folding Geometries in a CO2Z-Fold Laser,” Appl. Opt. 28, 935–941 (1989).
[CrossRef] [PubMed]

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).

Hart, R. A.

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

Hill, C. A.

P. E. Jackson, D. R. Hall, C. A. Hill, “Comparisons of Waveguide Folding Geometries in a CO2Z-Fold Laser,” Appl. Opt. 28, 935–941 (1989).
[CrossRef] [PubMed]

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).

Jackson, P. E.

Jenkins, R. M.

J. Banerji, A. R. Davies, P. E. Jackson, R. M. Jenkins, “Transmission and Coupling Losses in a Folded Waveguide,” Appl. Opt. 28, 4637–4643 (1989).
[CrossRef] [PubMed]

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).

R. M. Jenkins, R. W. J. Devereux, “EH1m Mode Excitation in Circular Cross Section Hollow Dielectric Waveguides,” (to be published in IEEE J. Quantum Electron. in press, (1990).

P. J. Gorton, R. M. Jenkins, “Complex Refractive Index Measurements of Polycrystalline Alumina in the 9–11 Micron Waveband,” (to be submitted for publication in Appl. Phys. Letts).

Marcatili, E. A. J.

E. A. J. Marcatili, R. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).

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).

Schmeltzer, R. A.

E. A. J. Marcatili, R. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).

Appl. Opt.

Bell Syst. Tech. J.

E. A. J. Marcatili, R. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).

IEEE J. Quantum Electron.

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

Laser Focus

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

Proc. Soc. Photo-Opt. Instrum. Eng.

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

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1970).

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).

R. M. Jenkins, R. W. J. Devereux, “EH1m Mode Excitation in Circular Cross Section Hollow Dielectric Waveguides,” (to be published in IEEE J. Quantum Electron. in press, (1990).

R. L. Abrams, “Waveguide Gas Lasers,” in Laser Handbook, Vol. 3 (North-Holland, Amsterdam, 1979).

P. J. Gorton, R. M. Jenkins, “Complex Refractive Index Measurements of Polycrystalline Alumina in the 9–11 Micron Waveband,” (to be submitted for publication in Appl. Phys. Letts).

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Figures (13)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Experimental setup.

Fig. 3
Fig. 3

Self- and cross-coupling efficiencies as functions of the angular tilt.

Fig. 4
Fig. 4

Self- and cross-coupling efficiencies as functions of the axial displacement.

Fig. 5
Fig. 5

Transmission as a function of angular tilt for a set of perpendicular and parallel loss factors of the guide material.

Fig. 6
Fig. 6

Maximum values of mode numbers as a function of angular tilt for a set of values of Δ1.

Fig. 7
Fig. 7

Maximum values of mode numbers as a function of axial displacement for a set of values of Δ1.

Fig. 8
Fig. 8

Output powers in various waveguide modes as functions of angular tilt.

Fig. 9
Fig. 9

Output powers in various waveguide modes as functions of axial displacement.

Fig. 10
Fig. 10

Transmission as a function of angular tilt with inbuilt axial displacements.

Fig. 11
Fig. 11

Transmission as a function of axial displacement with inbuilt angular tilt.

Fig. 12
Fig. 12

Transmission as a function of angular tilt. Comparison of theory to experiment.

Fig. 13
Fig. 13

Transmission as a function of axial displacement. Comparison of theory to experiment.

Equations (46)

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x = x cos θ - - z sin θ - + δ sin θ , y = y , z = z cos θ - + x sin θ - - d ^ , x = x cos θ + + z sin θ + - δ sin θ , y = y , z = z cos θ + - x sin θ + - d ^ ,
θ ± = θ ± ψ ,             d ^ = d + δ cos θ .
E p q ( 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 E p q ( x , y ) E p , q ( x , y ) d x d y = δ p p δ q q ,
E ( P ) = F ( x 0 ) G ( y 0 ) .
E ( R ) = - 1 4 λ 2 - a a - a a - M x M s - M y M y F ( x 0 ) G ( 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 2 q 2 = - a a - a a E ( R ) E p 2 q 2 ( x 0 , y 0 ) d x 0 d y 0 .
r 01 = d ^ - x 1 sin θ - + ( x 0 - δ sin θ ) 2 + y 0 2 2 d ^ - ( x 0 - δ sin θ ) x 1 cos θ - + y 0 y 1 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 + δ sin θ ) 2 + y 0 2 2 d ^ - ( x 0 + δ sin θ ) x 1 cos θ + + y 0 y 1 d ^ + [ 1 - β + ( 1 ) ] 2 d ^ x 1 2 cos 2 θ + + [ 1 - β + ( 2 ) ] 2 d ^ y 1 2 ,
β ± ( j ) = { d ^ / ( R cos θ ± ) , if j = 1 ; d ^ / ( R sec θ ± ) , if j = 2.
E ( R ) = ρ 1 a exp ( i k d ^ ρ 2 ) I ( u 2 ) J ( v 2 ) ,
ρ 1 = - N ^ ( cos θ - + cos θ + ) ( 1 + 1 i ) ( 1 + 2 i ) Γ 1 Γ 2 8 ,
N ^ = a 2 λ d ^ ,
Γ 1 = [ 1 - β - ( 1 ) ] 2 cos 2 θ - + [ 1 - β + ( 1 ) ] 2 cos 2 θ + = 1 Γ 1 ,
Γ 2 = [ 1 - β - ( 2 ) ] 2 + [ 1 - β + ( 2 ) ] 2 = 2 Γ 2 ,
ρ 2 = 2 + ( δ d ^ sin θ ) 2 - ( δ d ^ sin 2 θ + cos θ ) 2 sin 2 ψ Γ 1 ,
I ( u 2 ) = a - 1 1 F ( a u 1 ) exp [ i π N ^ P ( u 1 , u 2 ) / 2 ] d u 1 = p 2 I p 2 E p 2 ( a u 2 ) ,
J ( v 2 ) = a - 1 1 G ( a v 1 ) exp [ i π N ^ Q ( v 1 , v 2 ) / 2 ] d v 1 = q 2 J q 2 E q 2 ( a v 2 ) .
P ( u 1 , u 2 ) = a 1 u 1 2 + a 2 u 2 2 + a 3 u 1 u 2 + a 4 u 1 + a 5 u 2 ,
Q ( v 1 , v 2 ) = b 1 v 1 2 + b 2 v 2 2 + b 3 v 1 v 2 ,
a 1 = 2 - cos 2 θ - Γ 1 , a 2 = 2 - cos 2 θ + Γ 1 , a 3 = - 2 Γ 1 cos θ - cos θ + , a 4 = - 4 δ a sin θ + 2 γ Γ 1 cos θ - , a 5 = 4 δ a sin θ + 2 γ Γ 1 cos θ + , γ = 2 a ( δ + d cos θ ) sin ψ , b 1 = b 2 = 2 - 1 Γ 2 , b 3 = - 2 Γ 2 .
χ p 2 q 2 = ρ 1 exp ( i k d ^ ρ 2 ) I p 2 J q 2 ,
I p 2 = - 1 1 I ( u 2 ) E p 2 ( a u 2 ) d u 2 ,
J q 2 = - 1 1 J ( v 2 ) E q 2 ( a v 2 ) d v 2 .
p 2 I p 2 2 = - 1 1 I ( u 2 ) 2 d u 2 ;
q 2 J q 2 2 = - 1 1 J ( v 2 ) 2 d v 2 .
P out ( 2 ) = ρ 1 2 p 2 q 2 I p 2 J q 2 2 exp ( - 2 α p 2 q 2 L ) ,
α p q = α p ( 1 ) + α q ( 2 ) with α n ( j ) = ( λ / 4 a ) 2 n 2 L ( j ) / a .
L ( j ) = { Re { ( - 1 ) - 1 / 2 } , if j = 1 , Re { ( - 1 ) - 1 / 2 } , if j = 2 ,
C p = { ( 2 π ) 1 / 4 ( ω a ) - 1 / 2 - a a exp [ - ( x / ω ) 2 ] cos ( p π x 2 a ) d x , for odd p ; 0 , for even p .
P in ( 1 ) = - a a - a a E in ( 1 ) ( r ) 2 d x 0 d y 0 = p , q A p q 2 ,
E in ( 1 ) ( r ) = exp [ - ( r / ω ) 2 ] ( π ω 2 / 2 ) 1 / 2 ,
r 2 = x 0 2 + y 0 2 .
p C p 2 = erf ( a 2 / ω ) .
erf ( z ) = 2 π 0 z exp ( - t 2 ) d t .
β p = π λ ( λ 4 a ) 2 p 2 ,
F ( x 0 ) = 1 a p ʹ C p E p ( x 0 ) exp { i L [ k / 2 - β p + i α p ( 1 ) ] } ,
G ( y 0 ) = 1 a p ʹ C p E p ( y 0 ) exp { i L [ k / 2 - β p + i α p ( 2 ) ] } .
I ( u 2 ) = exp [ i π N ^ ( a 2 u 2 2 + a 5 u 2 ) / 2 ] × p ʹ C p exp { i L [ k / 2 - β p + i α p ( 1 ) ] } × { M [ π p / 2 - π N ^ ( a 3 u 2 + a 4 ) / 2 , π N ^ a 1 / 2 ] + M [ π p / 2 + π N ^ ( a 3 u 2 + a 4 ) / 2 , π N ^ a 1 / 2 ] } ,
J ( v 2 ) = exp ( i π N ^ b 2 v 2 2 / 2 ) p ʹ C p exp { i L [ k / 2 - β p + i α p ( 2 ) ] } × [ M ( π p / 2 - π N ^ b 3 v 2 / 2 , π N ^ b 1 / 2 ) + M ( π p / 2 + π N ^ b 3 v 2 / 2 , π N ^ b 1 / 2 ) ] ,
M ( A , C ) = 0 1 cos ( A v ) exp ( i C v 2 ) d v = 1 2 π 2 C exp ( - i A 2 / 4 C ) × [ f { 2 C π + A 2 π C } + f { 2 C π - A 2 π C } ] ,
f ( x ) = 0 x exp ( i π t 2 / 2 ) d t .
T = output power input power ,
p 2 = 1 p 2 ( max ) I p 2 2 Δ 1 - 1 1 I ( u 2 ) 2 d u 2 ,
q 2 = 1 q 2 ( max ) J q 2 2 Δ 2 - 1 1 J ( v 2 ) 2 d v 2 ,

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