Abstract

There are three recognized low-loss configurations for waveguide laser resonators in which the waveguide is either closed at each end by a plane mirror (dual case I design) or one of the plane mirrors is replaced by a curved mirror at some distance from the guide exit. Some time ago, a variant of the latter design was proposed by exploiting the self-imaging properties of multimode waveguides. The resonator was predicted to produce a TEM00-like output with very low round-trip loss and excellent mode discrimination even though the curved mirror was placed much nearer to the guide exit (making the resonator more compact) than was conventional for achieving those results. In the present work, we show that the desirable features of the above design can be achieved even in a dual case I configuration by using end mirrors with suitable reflectivity profiles. Since there is no free space region between the waveguide and the mirrors, the resonator will have the additional advantages of being compact and portable. Furthermore, the absence of curved mirrors will also facilitate its realization in semiconductor integrated optics technology.

© 2009 Optical Society of America

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References

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  1. J. J. Degnan and D. R. Hall, “Finite-aperture waveguide laser resonators,” IEEE J. Quantum Electron. QE-9, 901-910 (1973).
    [CrossRef]
  2. J. Banerji, A. R. Davies, and R. M. Jenkins, “Laser resonators with self-imaging waveguides,” J. Opt. Soc. Am. B 14, 2378(1997).
    [CrossRef]
  3. L. A. Rivlin and V. S. Shul'dyaev, “Multimode waveguides for coherent light,” Radiophys. Quantum Electron. 11, 318-321 (1968).
    [CrossRef]
  4. O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Am. 63, 416-419 (1973).
    [CrossRef]
  5. R. Ulrich and G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337-339 (1975).
    [CrossRef]
  6. J. Banerji, A. R. Davies, C. A. Hill, R. M. Jenkins, and J. R. Redding, “Effects of curved mirrors in waveguide resonators,” Appl. Opt. 34, 3000-3008 (1995).
    [CrossRef] [PubMed]
  7. D. M. Henderson, “Waveguide lasers with intracavity electrooptic modulators: misalignment loss,” Appl. Opt. 15, 1066-1070 (1976).
    [CrossRef] [PubMed]
  8. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453-488 (1961).

1997 (1)

1995 (1)

1976 (1)

1975 (1)

R. Ulrich and G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337-339 (1975).
[CrossRef]

1973 (2)

J. J. Degnan and D. R. Hall, “Finite-aperture waveguide laser resonators,” IEEE J. Quantum Electron. QE-9, 901-910 (1973).
[CrossRef]

O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Am. 63, 416-419 (1973).
[CrossRef]

1968 (1)

L. A. Rivlin and V. S. Shul'dyaev, “Multimode waveguides for coherent light,” Radiophys. Quantum Electron. 11, 318-321 (1968).
[CrossRef]

1961 (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453-488 (1961).

Ankele, G.

R. Ulrich and G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337-339 (1975).
[CrossRef]

Banerji, J.

Bryngdahl, O.

Davies, A. R.

Degnan, J. J.

J. J. Degnan and D. R. Hall, “Finite-aperture waveguide laser resonators,” IEEE J. Quantum Electron. QE-9, 901-910 (1973).
[CrossRef]

Fox, A. G.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453-488 (1961).

Hall, D. R.

J. J. Degnan and D. R. Hall, “Finite-aperture waveguide laser resonators,” IEEE J. Quantum Electron. QE-9, 901-910 (1973).
[CrossRef]

Henderson, D. M.

Hill, C. A.

Jenkins, R. M.

Li, T.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453-488 (1961).

Redding, J. R.

Rivlin, L. A.

L. A. Rivlin and V. S. Shul'dyaev, “Multimode waveguides for coherent light,” Radiophys. Quantum Electron. 11, 318-321 (1968).
[CrossRef]

Shul'dyaev, V. S.

L. A. Rivlin and V. S. Shul'dyaev, “Multimode waveguides for coherent light,” Radiophys. Quantum Electron. 11, 318-321 (1968).
[CrossRef]

Ulrich, R.

R. Ulrich and G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337-339 (1975).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

R. Ulrich and G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337-339 (1975).
[CrossRef]

Bell Syst. Tech. J. (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453-488 (1961).

IEEE J. Quantum Electron. (1)

J. J. Degnan and D. R. Hall, “Finite-aperture waveguide laser resonators,” IEEE J. Quantum Electron. QE-9, 901-910 (1973).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

Radiophys. Quantum Electron. (1)

L. A. Rivlin and V. S. Shul'dyaev, “Multimode waveguides for coherent light,” Radiophys. Quantum Electron. 11, 318-321 (1968).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic of a self-imaging waveguide resonator as reported in an earlier publication [2]. The resonator was formed of a waveguide of square cross section 2 a × 2 a and of length 2 a 2 / λ = L / 2 , closed at one end ( z = 0 ) by a curved mirror at a distance a 2 / 2 λ = L / 8 from the guide exit, and bounded at the other end ( z = L / 2 ), by a plane mirror.

Fig. 2
Fig. 2

New design for self-imaging waveguide resonators in which the end-mirrors are plane with step-reflectance profiles. We assume that the reflectivity of the mirror M 1 at the z = 0 plane is r 1 on a square block ( 2 b × 2 b ) centered at the origin and r 2 elsewhere with r 1 1 and r 2 0 . The reflectivity of the mirror M 2 at the z = L / 2 plane, on the other hand, is taken to be r 2 on a square block ( 2 c × 2 c ) centered at the origin and r 1 elsewhere.

Fig. 3
Fig. 3

Round-trip fractional loss of the first six resonator modes (all of which are symmetric solutions) in the x z plane as a function of b / a when c = 0 . The thick line corresponds to the lowest-loss mode, and the thin lines represent the next five modes (in ascending order of fractional loss). All calculations reported in this paper have been carried out in Mathematica 6.0.

Fig. 4
Fig. 4

Profiles for the two lowest-loss resonator modes in the x z plane at (a)  z = 0 and at (b)  z = L / 2 when b / a = 0.44 and c = 0 . The dotted line corresponds to the lowest-loss mode, whereas the solid line corresponds to the next mode (in ascending order of fractional loss).

Fig. 5
Fig. 5

Round-trip fractional loss of the first six resonator modes in the x z plane, as a function of c / a when b / a = 0.44 . The thick line corresponds to the lowest-loss mode, and the thin lines represent the next five modes (in ascending order of fractional loss).

Fig. 6
Fig. 6

Round-trip fractional loss of the first six resonator modes in the x z plane, as a function of the resonator length when b / a = c / a = 0.44 . The thick line corresponds to the lowest-loss mode, and the thin lines represent the next five modes (in ascending order of fractional loss).

Fig. 7
Fig. 7

Transverse intensity profiles in the x z plane at each end of the resonator and inside the waveguide for the two lowest-loss resonator modes when b / a = c / a = 0.44 and the resonator length is set at L / 2 = 2 a 2 / λ . The left column represents the profiles for the lowest-loss mode in the + z direction, and the right column represents the profiles for the next mode (in ascending order of fractional loss). In each column, the bottom figure displays the intensity profile at z = 0 , the middle figure is the contour plot of the intensity profile inside the waveguide, and the top figure is the intensity profile at z = L / 2 .

Fig. 8
Fig. 8

Mirror reflectivity is set at 0.9 for the high-reflectivity segments and at 0.1 for the low-reflectivity segments. (a) Round-trip fractional loss of the first six resonator modes in the x z plane as a function of the resonator length. All other details are as in Fig. 6. (b) Intensity profiles for the lowest-loss mode at z = 0 (bottom figure) and at z = L / 2 (top figure). (c) Same as in (b) for the next mode (in ascending order of fractional loss).

Equations (16)

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E m ( x , a ) = 1 a { cos ( π m x / 2 a ) , if     m   is odd sin ( π m x / 2 a ) , if     m   is even .
a a E m ( x , a ) E n ( x , a ) d x = δ m n .
k m = 2 π λ ( 1 1 2 ( m λ 4 a ) 2 )
F ( x , 0 ) = m a m E m ( x , a ) .
F ( x , z ) = m a m E m ( x , a ) exp ( i β m z ) .
F s ( x , L ) = F s ( x , 0 ) exp ( i π / 4 ) , F a ( x , 2 L ) = F a ( x , 0 ) , F g ( x , 8 L ) = F g ( x , 0 ) .
F a ( x , L ) = 1 2 [ F a ( x a , 0 ) + F a ( x + a , 0 ) ] .
F s ( x , L / 2 ) = F s ( x a / 2 , 0 ) + F s ( x + a / 2 , 0 ) .
F 0 ( x , 0 ) = m a m E m ( x , a ) .
F 1 ( x , a ) = m , n , p a m exp ( i β m L / 2 ) X m n ( 2 ) exp ( i β n L / 2 ) X n p ( 1 ) E p ( x , a ) ,
m , n X n q ( 1 ) exp ( i β n L / 2 ) X m n ( 2 ) exp ( i β m L / 2 ) a m = σ a q .
X m n ( 1 ) = [ r 2 ( a b d x + b a d x ) + r 1 b b d x ] E m ( x , a ) E n ( x , a ) = r 2 δ m n + ( r 1 r 2 ) I m n ( b ) ,
I m n ( b ) = b b E m ( x , a ) E n ( x , a ) d x ,
X m n ( 2 ) = [ r 1 ( a c d x + c a d x ) + r 2 c c d x ] E m ( x , a ) E n ( x , a ) = r 1 δ m n ( r 1 r 2 ) I m n ( c ) .
I m n ( b ) = { γ m n ( b / a ) γ m + n ( b / a ) , if     m   and     n   are both even γ m n ( b / a ) + γ m + n ( b / a ) , if     m   and   n   are both odd 0 , otherwise
γ j ( x ) = { sin ( π j x / 2 ) π j / 2 , for     j 0 x , otherwise .

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