Abstract

The recently proposed 1-N-way resonator based on beam splitting and beam combining effects in rectangular cross-sectional multimode waveguides offers a valuable way in which N low-power laser elements can be combined in a coherent fashion. We develop a theory of such resonators in the presence of perturbations in the N-element array. We demonstrate that despite the presence of perturbations there is only one possible mode of the resonator. The theory is used to provide an understanding of the effects of a number of possible perturbations that could arise as a result of manufacturing processes and operational effects. The results give scaling laws for the design tolerances on array element mirror tilt, array element optical path length control, and the effects of array element malfunction and their need for gain balance.

© 2001 Optical Society of America

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References

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  1. F. Talbot, “Facts relating to optical science no. IV,” Philos. Mag. 9, 401–407 (1836).
  2. D. Mehuys, W. Streifer, R. G. Waarts, D. F. Welch, “Modal analysis of linear Talbot-cavity semiconductor lasers,” Opt. Lett. 16, 823–825 (1991).
    [CrossRef] [PubMed]
  3. R. M. Jenkins, J. Banerji, A. R. Davies, J. M. Heaton, “1-N-way phased array resonator,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 228–229.
  4. R. M. Jenkins, J. M. Heaton, “Optical device,” international patent applicationPCT/GB91/02129 (1992); UK patent application9,027,657.7 (priority date 20December1990).
  5. R. M. Jenkins, R. W. J. Devereux, J. M. Heaton, “Waveguide beam splitters and recombiners based on multimode propagation phenomena,” Opt. Lett. 17, 991–993 (1992).
    [CrossRef] [PubMed]
  6. J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
    [CrossRef]
  7. R. M. Jenkins, R. W. J. Devereux, J. M. Heaton, “A novel waveguide Mach–Zehnder interferometer based on multimode interference phenomena,” Opt. Commun. 110, 410–424 (1994).
    [CrossRef]
  8. J. Banerji, A. R. Davies, R. M. Jenkins, “Comparison of Talbot and 1-to-N way phase locked resonators,” Appl. Opt. 36, 1604–1609 (1997).
    [CrossRef] [PubMed]
  9. E. K. Gorton, R. M. Jenkins, “Theory of 1-N-way phase-locked resonators,” Appl. Opt. 40, 916–920 (2001).
    [CrossRef]
  10. K. D. Laakmann, W. H. Steier, “Waveguides: characteristic modes of hollow rectangular dielectric waveguides,” Appl. Opt. 15, 1334–1340 (1976).
    [CrossRef] [PubMed]
  11. S. Wolfram, Mathematica: a System for Doing Mathematics by Computer, 2nd ed. (Addison–Wesley, Reading, Mass., 1992).

2001 (1)

1997 (1)

1994 (1)

R. M. Jenkins, R. W. J. Devereux, J. M. Heaton, “A novel waveguide Mach–Zehnder interferometer based on multimode interference phenomena,” Opt. Commun. 110, 410–424 (1994).
[CrossRef]

1992 (2)

R. M. Jenkins, R. W. J. Devereux, J. M. Heaton, “Waveguide beam splitters and recombiners based on multimode propagation phenomena,” Opt. Lett. 17, 991–993 (1992).
[CrossRef] [PubMed]

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

1991 (1)

1976 (1)

1836 (1)

F. Talbot, “Facts relating to optical science no. IV,” Philos. Mag. 9, 401–407 (1836).

Banerji, J.

J. Banerji, A. R. Davies, R. M. Jenkins, “Comparison of Talbot and 1-to-N way phase locked resonators,” Appl. Opt. 36, 1604–1609 (1997).
[CrossRef] [PubMed]

R. M. Jenkins, J. Banerji, A. R. Davies, J. M. Heaton, “1-N-way phased array resonator,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 228–229.

Birbeck, J. C. H.

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

Davies, A. R.

J. Banerji, A. R. Davies, R. M. Jenkins, “Comparison of Talbot and 1-to-N way phase locked resonators,” Appl. Opt. 36, 1604–1609 (1997).
[CrossRef] [PubMed]

R. M. Jenkins, J. Banerji, A. R. Davies, J. M. Heaton, “1-N-way phased array resonator,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 228–229.

Devereux, R. W. J.

R. M. Jenkins, R. W. J. Devereux, J. M. Heaton, “A novel waveguide Mach–Zehnder interferometer based on multimode interference phenomena,” Opt. Commun. 110, 410–424 (1994).
[CrossRef]

R. M. Jenkins, R. W. J. Devereux, J. M. Heaton, “Waveguide beam splitters and recombiners based on multimode propagation phenomena,” Opt. Lett. 17, 991–993 (1992).
[CrossRef] [PubMed]

Gorton, E. K.

Heaton, J. M.

R. M. Jenkins, R. W. J. Devereux, J. M. Heaton, “A novel waveguide Mach–Zehnder interferometer based on multimode interference phenomena,” Opt. Commun. 110, 410–424 (1994).
[CrossRef]

R. M. Jenkins, R. W. J. Devereux, J. M. Heaton, “Waveguide beam splitters and recombiners based on multimode propagation phenomena,” Opt. Lett. 17, 991–993 (1992).
[CrossRef] [PubMed]

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

R. M. Jenkins, J. M. Heaton, “Optical device,” international patent applicationPCT/GB91/02129 (1992); UK patent application9,027,657.7 (priority date 20December1990).

R. M. Jenkins, J. Banerji, A. R. Davies, J. M. Heaton, “1-N-way phased array resonator,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 228–229.

Hilton, K. P.

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

Jenkins, R. M.

E. K. Gorton, R. M. Jenkins, “Theory of 1-N-way phase-locked resonators,” Appl. Opt. 40, 916–920 (2001).
[CrossRef]

J. Banerji, A. R. Davies, R. M. Jenkins, “Comparison of Talbot and 1-to-N way phase locked resonators,” Appl. Opt. 36, 1604–1609 (1997).
[CrossRef] [PubMed]

R. M. Jenkins, R. W. J. Devereux, J. M. Heaton, “A novel waveguide Mach–Zehnder interferometer based on multimode interference phenomena,” Opt. Commun. 110, 410–424 (1994).
[CrossRef]

R. M. Jenkins, R. W. J. Devereux, J. M. Heaton, “Waveguide beam splitters and recombiners based on multimode propagation phenomena,” Opt. Lett. 17, 991–993 (1992).
[CrossRef] [PubMed]

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

R. M. Jenkins, J. Banerji, A. R. Davies, J. M. Heaton, “1-N-way phased array resonator,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 228–229.

R. M. Jenkins, J. M. Heaton, “Optical device,” international patent applicationPCT/GB91/02129 (1992); UK patent application9,027,657.7 (priority date 20December1990).

Laakmann, K. D.

Mehuys, D.

Parker, J. T.

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

Steier, W. H.

Streifer, W.

Talbot, F.

F. Talbot, “Facts relating to optical science no. IV,” Philos. Mag. 9, 401–407 (1836).

Waarts, R. G.

Welch, D. F.

Wight, D. R.

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

Wolfram, S.

S. Wolfram, Mathematica: a System for Doing Mathematics by Computer, 2nd ed. (Addison–Wesley, Reading, Mass., 1992).

Appl. Opt. (3)

Appl. Phys. Lett. (1)

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

Opt. Commun. (1)

R. M. Jenkins, R. W. J. Devereux, J. M. Heaton, “A novel waveguide Mach–Zehnder interferometer based on multimode interference phenomena,” Opt. Commun. 110, 410–424 (1994).
[CrossRef]

Opt. Lett. (2)

Philos. Mag. (1)

F. Talbot, “Facts relating to optical science no. IV,” Philos. Mag. 9, 401–407 (1836).

Other (3)

R. M. Jenkins, J. Banerji, A. R. Davies, J. M. Heaton, “1-N-way phased array resonator,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 228–229.

R. M. Jenkins, J. M. Heaton, “Optical device,” international patent applicationPCT/GB91/02129 (1992); UK patent application9,027,657.7 (priority date 20December1990).

S. Wolfram, Mathematica: a System for Doing Mathematics by Computer, 2nd ed. (Addison–Wesley, Reading, Mass., 1992).

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Figures (5)

Fig. 1
Fig. 1

Schematic of a 1-N-way resonator.

Fig. 2
Fig. 2

Resonator mode loss as a function of tilt for some array number values.

Fig. 3
Fig. 3

Comparison of scaling laws for small-gain reductions caused by tilt.

Fig. 4
Fig. 4

Resonator mode gain probability density plot for normally distributed random phase errors in the array elements: s p = 0.1π, N = 16.

Fig. 5
Fig. 5

Probability density plots for the resonator mode gain in the presence of array element gain perturbations: N = 16.

Equations (29)

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ϕn=1/2N+N+1/4+nn-N-1/N
E1y, a=1a cosπy2a,  |y|a,
Eqy, b=1b cosqπy2bif q is odd1b sinqπy2bif q is even  |y|b.
Cmq=y=ym-ay=ym+a E1ym, aEqy, bdy.
βq=2πλ1-12λq2b2.
cq=y=-ay=a Eqy, bE1y, ady.
βa=2πλ 1-12λ2a2.
ϕn=-2πiϕn+δn.
anm=amr1r2 exp-2il+lβaBnm expϕn,
Bnm=q=1 cqCmq exp-iβqdr=1 crCnr exp-iβrd.
σan=m=1N anm,
gσan=m=1n Bnmam expiϕn
gσã=B˜˜ · ã,
D=DNj=1N Bjj expϕj,
D=-1NgσN-gσN-1j=1N Bjj expϕj.
σ=1gj=1N Bjj expϕj,  σ=0.
am=anBmmBnm.
σ=1gj=1N Bjj exp-2πiϕj,
ϕntilt=2 2πλyn-y1χ=4πρn-1λ χ.
ϕn=-2πiϕn-4πi ρn-1λ χ.
Bnm=N-1 expiπϕnm,
σ=expi4πρχλ-π2NgNn=1N exp-4πi ρnχλ=exp-iπ2NgN1-exp-4πi ρNχλ1-exp-4πi ρχλ.
|σ|2=R1R2N2sin22πρNχ/λsin22πρχ/λ,
σ=1gj=1N2 Bjjexpϕj+expϕN+1-j=exp-iπ2NgNj=1N2expiδj+expiδN+1-j,
|σ|2=R1R2N2j=1N cos δj2+j=1N sin δj2.
|σ|2=R1R2N2N+2 j=1N-1k=j+1N cosδj-δk.
|σ|2=R1R2K+1-KN.
|σ|2=R1R2N2j=1N exp δj2,
|σ|perturbed2|σ|2=1-f2 exp2γ,

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