Abstract

A 1-N-way resonator based on beam splitting and beam combining effects in rectangular cross-sectional multimode waveguides was recently proposed. Such a resonator structure offers a valuable way in which N low-power laser elements may be combined in a coherent fashion. We examine the case of passive 1-N-way resonators. We develop a theory of these 1-N-way structures to show that there is only one possible mode of these resonators. The theory is used to give a scaling law for the design tolerances of the beam splitting and beam combining region of the resonator.

© 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 application PCT/GB91/02129 (1992); UK patent application 9,027,657.7 (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-N-way integrated optical beamsplitters 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-N-way phase locked resonators,” Appl. Opt. 36, 1604–1609 (1997).
    [CrossRef] [PubMed]
  9. O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Am. 63, 416–419 (1973).
    [CrossRef]
  10. R. Ulrich, G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337–339 (1975).
    [CrossRef]
  11. K. D. Laakmann, W. H. Steier, “Waveguides: characteristic modes of hollow rectangular dielectric waveguides,” Appl. Opt. 15, 1334–1340 (1976).
    [CrossRef] [PubMed]
  12. S. Wolfram, Mathematica—A System for Doing Mathematics by Computer, 2nd ed. (Addison-Wesley, Reading, Mass., 1992).

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-N-way integrated optical beamsplitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

1991 (1)

1976 (1)

1975 (1)

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

1973 (1)

1836 (1)

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

Ankele, G.

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

Banerji, J.

J. Banerji, A. R. Davies, R. M. Jenkins, “Comparison of Talbot and 1-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-N-way integrated optical beamsplitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

Bryngdahl, O.

Davies, A. R.

J. Banerji, A. R. Davies, R. M. Jenkins, “Comparison of Talbot and 1-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]

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-N-way integrated optical beamsplitters 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 application PCT/GB91/02129 (1992); UK patent application 9,027,657.7 (priority date 20December1990).

Hilton, K. P.

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

Jenkins, R. M.

J. Banerji, A. R. Davies, R. M. Jenkins, “Comparison of Talbot and 1-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]

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

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 application PCT/GB91/02129 (1992); UK patent application 9,027,657.7 (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-N-way integrated optical beamsplitters 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).

Ulrich, R.

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

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-N-way integrated optical beamsplitters 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. (2)

Appl. Phys. Lett. (2)

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

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

J. Opt. Soc. Am. (1)

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 application PCT/GB91/02129 (1992); UK patent application 9,027,657.7 (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 (2)

Fig. 1
Fig. 1

Schematic of a 1-N-way resonator.

Fig. 2
Fig. 2

Mode loss as a function of rectangular guide length for some values of array number.

Equations (35)

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E1y, a=1acosπy2a,  |y|a,
Eqy, b=1bcosqπy2bq odd1bsinqπy2bq even,  |y|b.
Cmq=y=ym-ay=ym+a E1ym, aEqy, bdy.
Fy, z=0=amq=1q= CmqEqy, b,
βq=2πλ1-12λq2b2.
cq=y=-ay=a Eqy, bE1y, ady.
Fy, z=d=amq=1 cqCmq exp-iβqdE1y, a,
Fy, z=d=amr2 exp-2ilβaq=1 cqCmq×exp-iβqdE1y, a.
Fy, z=0=amr2 exp-2ilβaq=1 cqCmq×exp-iβqdr=1 crEry, bexp-iβrd.
Fy, z=0=amr1r2 exp-2il+lβa×q=1 cqCmq exp-iβqdr=1 crCnr×exp-iβrdexp-2πiϕnE1y-yn, a.
anm=amr1r2 exp-2il+lβaBnm exp-2πiϕn,
Bnm=q=1 cqCmq exp-iβqdr=1 crCnr exp-iβrd.
σan=m=1N anm
gan=m=1n Bnmam exp-2πiϕn,
g=σ exp2il+lβa/r1r2.
gã=B · ã,
B · ã-gã=0
B-g · ã=0,
Bnm2=BnnBmm.
D=DNj=1N Bjj exp-2πiϕj.
DN=αNDN-1+αN-1DN-1,
DN=αN-1DN-1.
DN=j=1N-1 αj.
DN=αNDN-1+j=1N-1 αj.
DN=1+k=1N 1αkj=1N αj.
D=-1NgσN-gσN-1j=1N Bjj exp-2πiϕj,
σ=1gj=1N Bjj exp-2πiϕj, σ=0.
am=anBmmBnm.
Bnm=N-1 expiπϕnm,
ϕnm=N+1-2n2+N+1-2m2/4N-1/2.
σ=1gj-1N Bjj exp-2πiϕj=1gexp-iπN.
1-|σ|2=1-R1R2,
δdplotdN=2N.
δddN=f2N
δd=f2ρ2λ.

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