Abstract

Out-of-phase properties of optical-waveguide Bragg-grating lenses caused by the linearly approximated grating lines and by mismatching of the propagation vectors of guided waves were theoretically investigated with a coupled-wave analysis. Two types of lens with straight and parabolic grating lines were fabricated by photolithography. Agreement of the experimental results with the theory enabled us to design a lens with the required properties.

© 1984 Optical Society of America

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References

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  1. G. Hatakoshi, H. Inoue, K. Naito, S. Umegaki, S. Tanaka, Opt. Acta 26, 961 (1979).
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  3. Z.-Q. Lin, S.-T. Zhou, W. S. C. Chang, S. Forouhar, J.-M. Delavaux, IEEE Trans. Microwave Theory Tech. MTT-29, 881 (1981).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  7. J. Van Roey, P. E. Lagasse, J. Opt. Soc. Am. 72, 337 (1982).
    [CrossRef]
  8. G. Hatakoshi, S. Tanaka, J. Opt. Soc. Am. 71, 121 (1981).
    [CrossRef]
  9. W. R. Klein, Proc. IEEE 54, 803 (1966).
    [CrossRef]

1982

1981

1979

G. Hatakoshi, H. Inoue, K. Naito, S. Umegaki, S. Tanaka, Opt. Acta 26, 961 (1979).
[CrossRef]

1977

L. Solymar, Appl. Phys. Lett. 31, 820 (1977).
[CrossRef]

1966

W. R. Klein, Proc. IEEE 54, 803 (1966).
[CrossRef]

Chang, W. S. C.

Z.-Q. Lin, S.-T. Zhou, W. S. C. Chang, S. Forouhar, J.-M. Delavaux, IEEE Trans. Microwave Theory Tech. MTT-29, 881 (1981).

Delavaux, J.-M.

Z.-Q. Lin, S.-T. Zhou, W. S. C. Chang, S. Forouhar, J.-M. Delavaux, IEEE Trans. Microwave Theory Tech. MTT-29, 881 (1981).

Forouhar, S.

Z.-Q. Lin, S.-T. Zhou, W. S. C. Chang, S. Forouhar, J.-M. Delavaux, IEEE Trans. Microwave Theory Tech. MTT-29, 881 (1981).

Hatakoshi, G.

Inoue, H.

G. Hatakoshi, H. Inoue, K. Naito, S. Umegaki, S. Tanaka, Opt. Acta 26, 961 (1979).
[CrossRef]

Klein, W. R.

W. R. Klein, Proc. IEEE 54, 803 (1966).
[CrossRef]

Lagasse, P. E.

Lin, Z.-Q.

Z.-Q. Lin, S.-T. Zhou, W. S. C. Chang, S. Forouhar, J.-M. Delavaux, IEEE Trans. Microwave Theory Tech. MTT-29, 881 (1981).

Naito, K.

G. Hatakoshi, H. Inoue, K. Naito, S. Umegaki, S. Tanaka, Opt. Acta 26, 961 (1979).
[CrossRef]

Solymar, L.

Syms, R. R. A.

Tanaka, S.

Umegaki, S.

G. Hatakoshi, H. Inoue, K. Naito, S. Umegaki, S. Tanaka, Opt. Acta 26, 961 (1979).
[CrossRef]

Van Roey, J.

Zhou, S.-T.

Z.-Q. Lin, S.-T. Zhou, W. S. C. Chang, S. Forouhar, J.-M. Delavaux, IEEE Trans. Microwave Theory Tech. MTT-29, 881 (1981).

Appl. Opt.

Appl. Phys. Lett.

L. Solymar, Appl. Phys. Lett. 31, 820 (1977).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

Z.-Q. Lin, S.-T. Zhou, W. S. C. Chang, S. Forouhar, J.-M. Delavaux, IEEE Trans. Microwave Theory Tech. MTT-29, 881 (1981).

J. Opt. Soc. Am.

Opt. Acta

G. Hatakoshi, H. Inoue, K. Naito, S. Umegaki, S. Tanaka, Opt. Acta 26, 961 (1979).
[CrossRef]

Proc. IEEE

W. R. Klein, Proc. IEEE 54, 803 (1966).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Focusing of a wave by a lens with linearly approximated grating lines.

Fig. 2
Fig. 2

Variation of ( π W / 2 ) R 2 and ( π W / 2 ) S 1 2 as functions of ξ. Parameters used are K1 = 15, θ = 0°, W = 0.075 and (a) η0 = 0.07 and (b) 0.15. Solid lines are for a parabolic line-grating lens and dashed lines for a straight line lens.

Fig. 3
Fig. 3

Focusing of a wave in a Bragg lens. The deviation of the effective propagation vector of the incident wave from the Bragg condition causes the shift of the focusing point of the diffracted wave from O to O′.

Fig. 4
Fig. 4

Directional selectivity of a Bragg lens. Parameters used are K1 = 15, θ = 0°, δβ = 0, W = 0.1, η0 = 0.2 and (a) δθ = 0′, (b) 1′, and (c) 3′. In (c) almost no conversion occurs.

Fig. 5
Fig. 5

Focusing properties of (a) type 1 and (b) type 2 lenses.

Fig. 6
Fig. 6

Directional selectivities of type 1 and 2 lenses. The normalized diffraction efficiency E as a function of the deviation δθ of the incident plane wave from the Bragg angle is shown.

Fig. 7
Fig. 7

Wavelength selectivities of type 1 and 2 lenses. The abscissa represents the relative change of the effective propagation constant β.

Fig. 8
Fig. 8

Relation between the coupling coefficient K and the depth of corrugation Δd for TE0 and TE1 modes. The effective refractive index N or the thickness of the waveguide d is used as a parameter.

Fig. 9
Fig. 9

Grating lines of the type 1 lens at η = 0.08 taken with a scanning electron microscope.

Tables (1)

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Table I Specifications of Bragg Lenses

Equations (20)

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R ψ - i K 1 η 1 / 2 1 ( sin ψ ) 1 / 2 1 tan ψ S 1 exp [ i δ Ω ( ψ , η ) ] = 0 , S 1 η + i K 1 η 1 / 2 1 ( sin ψ ) 1 / 2 1 tan ψ R exp [ - i δ Ω ( ψ , η ) ] = 0 , }
δ Ω ( ψ , η ) = Ω ( ψ , η ) - Ω G ( ψ , η ) .
ψ = π - ( ϕ - θ ) , η = r f sin ( ϕ - θ ) . }
Ω ( x , y ) = β ( x 2 + y 2 + x ) ,
Ω G ( x , y ) = Ω ( - a , y 0 ) = β ( a 2 + y 0 2 - a ) .
y - y 0 = α ( x + a ) ,
[ δ Ω ( x , y ) β ] 3 - ( 3 x 2 + y 2 + x + 2 a ) [ δ Ω ( x , y ) β ] 2 + [ 2 ( x + 2 a ) x 2 + y 2 + ( x + a ) 2 + 2 ( x 2 + y 2 ) ] δ Ω ( x , y ) β - ( x + a ) 2 ( x 2 + y 2 + x ) = 0.
δ Ω ( x , y ) β ( x + a ) 2 ( x 2 + y 2 + x ) 2 ( x + 2 a ) x 2 + y 2 + ( x + a ) 2 + 2 ( x 2 + y 2 ) .
R 0 ( η ) = ( 2 π W ) 1 / 2 exp [ - 2 ( η - η 0 ) 2 W 2 ]
β = k 0 N s = ( 2 π / λ 0 ) N s ,
Ω G ( r , ϕ ) = β p · r + β c r ,
Ω ( r , ϕ ) = ( β p + δ β p ) · r + β c + δ β c r - δ r 0 ,
δ Ω ( r , ϕ ) = r { β δ θ sin ( ϕ - θ ) + δ β [ 1 + cos ( ϕ - θ ) ] } - β δ r 0 cos ( ϕ - ϕ 0 ) .
δ Ω ( r , ϕ ) r = β δ θ sin ( ϕ - θ ) + δ β [ 1 + cos ( ϕ - θ ) ] ,
δ Ω ( r , ϕ ) ϕ = ( r β δ θ sin θ - r δ β cos θ + δ r 0 β cos ϕ 0 ) sin ϕ + ( r β δ θ cos θ + r δ β sin θ - δ r 0 β sin ϕ 0 ) cos ϕ .
δ r 0 cos ϕ 0 = - r ( δ θ sin θ - δ β β cos θ ) , δ r 0 sin ϕ 0 = r ( δ θ cos θ + δ β β sin θ ) . }
δ Ω ( r , ϕ ) = ( r - r 0 ) [ β δ θ sin ( ϕ - θ ) + δ β cos ( ϕ - θ ) ] + r δ β .
E = P S P R + P S × 100 ,
P R = - R 2 ( S P ) x d y d z , P S = - S 2 ( S C ) x d y d z , }
P R = cos θ - R 2 d y , P S = - - S 1 2 cos ϕ ( r / f ) d y . }

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