Abstract

We present an approach to the analysis of a holographically produced grating coupler, in which we separate the coupling problem into independent slab waveguide and diffraction grating problems. Considering the guiding structure as a slab waveguide, we obtain the parameters of the propagating modes of the coupler through conventional analysis. The diffraction efficiency of the grating orders are then calculated by using the thin grating decomposition method (TGD). Utilizing the propagating mode angles of the waveguide and the calculated diffraction efficiencies of the grating, we are able to calculate the radiation loss coefficient and therefore the coupling efficiency of the holographic waveguide coupler.

© 1977 Optical Society of America

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References

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  1. K. Ogawa, W. S. C. Chang, Appl. Opt. 12, 2167 (1973).
    [CrossRef] [PubMed]
  2. S. T. Peng, T. Tamir, H. L. Bertoni, IEEE Trans. Microwave Theory Tech. 23, 123 (1975).
    [CrossRef]
  3. R. Alferness, Appl. Phys. 7, 29 (1975).
    [CrossRef]
  4. H. Kogelnik, T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).
  5. R. Alferness, J. Opt. Soc. Am. 66, 533 (1976).
    [CrossRef]

1976 (1)

R. Alferness, J. Opt. Soc. Am. 66, 533 (1976).
[CrossRef]

1975 (2)

S. T. Peng, T. Tamir, H. L. Bertoni, IEEE Trans. Microwave Theory Tech. 23, 123 (1975).
[CrossRef]

R. Alferness, Appl. Phys. 7, 29 (1975).
[CrossRef]

1973 (1)

1970 (1)

H. Kogelnik, T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).

Alferness, R.

R. Alferness, J. Opt. Soc. Am. 66, 533 (1976).
[CrossRef]

R. Alferness, Appl. Phys. 7, 29 (1975).
[CrossRef]

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, IEEE Trans. Microwave Theory Tech. 23, 123 (1975).
[CrossRef]

Chang, W. S. C.

Kogelnik, H.

H. Kogelnik, T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).

Ogawa, K.

Peng, S. T.

S. T. Peng, T. Tamir, H. L. Bertoni, IEEE Trans. Microwave Theory Tech. 23, 123 (1975).
[CrossRef]

Sosnowski, T. P.

H. Kogelnik, T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).

Tamir, T.

S. T. Peng, T. Tamir, H. L. Bertoni, IEEE Trans. Microwave Theory Tech. 23, 123 (1975).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. (1)

R. Alferness, Appl. Phys. 7, 29 (1975).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).

IEEE Trans. Microwave Theory Tech. (1)

S. T. Peng, T. Tamir, H. L. Bertoni, IEEE Trans. Microwave Theory Tech. 23, 123 (1975).
[CrossRef]

J. Opt. Soc. Am. (1)

R. Alferness, J. Opt. Soc. Am. 66, 533 (1976).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of grating waveguide configuration with guided mode (zeroth order) Poynting vector [P(0)] and radiation mode (first and second order) Poynting vectors [P(−1),P(−2)].

Fig. 2
Fig. 2

One unit of the sectionalization of the grating structure, illustrating the decoupled energy Ao(ηd1)1/2 and Ao(ηd2)1/2 with the remaining coupled energy. Loss occurs only over one half of the section where the guided mode satisfies the Bragg condition.

Fig. 3
Fig. 3

The Bragg diagram for second order diffraction in a slanted grating (represented by grating vector K) which is constructed by interference of plane waves A and B.

Fig. 4
Fig. 4

Radiation loss coefficient (μm−1) vs n1 for the thickness D = 2.0 μm of a case (1) slanted grating.

Fig. 5
Fig. 5

Radiation loss coefficient (μm−1) vs n1 for the thickness D = 2.0 μm of a case (1) unslanted grating.

Fig. 6
Fig. 6

Radiation loss coefficient (μm−1) vs n1 for the thickness D = 2.0 μm of a case (2) slanted grating.

Fig. 7
Fig. 7

Radiation loss coefficient (μm−1) vs n1 for the thickness D = 2.0 μm for a case (2) unslanted grating.

Fig. 8
Fig. 8

TE0 radiation loss coefficient (μm−1) vs n1 with n2 = 0 for the thickness D = 2.0 μm of a case (3) slanted grating.

Fig. 9
Fig. 9

TE radiation loss coefficient (μm−1) vs n1 with n2 = −n1/3 for the thickness D = 2.0 μm of a case (4) slanted grating.

Equations (43)

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B m = ( 2 W m tan θ m ) - 1 .
η d ( q ) = cos θ e ( q ) A ( q ) 2 cos θ m A i 2 ,
A = [ 1 - η d ( 1 ) - η d ( 2 ) ] 1 / 2 A o .
A = [ 1 - η d ( 1 ) - η d ( 2 ) ] 1 / 2
A exp { - [ η d ( 1 ) + η d ( 2 ) ] l 2 }
= exp { - [ η d ( 1 ) + η d ( 2 ) 2 ] ( 2 2 W m tan θ m ) }
= exp ( - α a z ) ,
α a = η d ( 1 ) + η d ( 2 ) 4 W m tan θ m .
1 = 0 E r 2 ( z ) d x ,
E r ( z ) = E o exp ( - α a z ) .
E r ( z ) = ( 2 α a ) 1 / 2 exp ( - α a z ) .
η c = [ 0 L E i ( z ) E r * ( z ) d z ] 2 0 E i E i * d z 0 E r E r * d z .
η c = η d ( 2 ) η d ( 1 ) + η d ( 2 ) ( 2 α a L ) [ 1 - exp ( - α a L ) ] 2 .
n ( z ) = n g ( z ) + n 1 cos [ 2 π f ( z - z x ) ] + n 2 cos [ 4 π f ( z - z x ) ] ,
T A ( z , x ) = exp [ j 2 π n ( z ) Δ x / λ cos θ m ] = exp ( j 2 π n g Δ x / λ cos θ m ) q = - ( j ) q J q ( b 1 ) × exp [ j 2 π q f ( z - z n ) ] p = - ( j ) p J p ( b 2 ) × exp [ j 4 π p f ( z - z n ) ] exp ( j 2 π n g Δ x cos θ m / λ ) ,
A ( M ) = H M [ 1 0 0 ] ,
H = [ 1 j δ 1 C 1 exp ( j Φ 1 ) j δ 2 C 2 j δ 1 C 0 exp ( j Φ 1 ) j δ 1 C 2 j δ 2 C 0 j δ 1 C 1 exp ( j Φ 1 ) 1 ] ,
Φ 1 = ( C 1 - C 0 - λ f z tan ϕ ) 2 π Δ x n g λ , ϕ = θ A + θ B 2 , f z = 1 λ ( sin θ A - sin θ B ) , δ i = π n i Δ x λ ( i - 1 , 2 ) ,
det ( H - σ i I ) = 0.
ψ i 3 + p ψ i 2 + q ψ i + r = 0
p = Φ 1 , q = - δ 2 2 C 0 C 2 - δ 1 2 C 0 C 1 - δ 1 2 C 1 C 2 , r = 2 δ 1 2 δ 2 C 0 C 1 C 2 - δ 2 2 Φ 1 C 0 C 2 .
ψ 0 = U + V - P / 3 ,
ψ 1 = - ( U + V ) 2 + ( U - V ) 2 ( - 3 ) 1 / 2 - P 3 ,
ψ 2 = - ( U + V ) 2 - ( U - V ) 2 ( - 3 ) 1 / 2 - P 3 ,
U = [ - b 2 + ( b 2 4 + a 3 27 ) 1 / 2 ] 1 / 3 ,
V = [ - b 2 - ( b 2 4 + a 3 27 ) 1 / 2 ] 1 / 3 ,
a = 1 3 ( 3 q - p 2 ) ,
b = 1 27 ( 2 p 3 - 9 p q + 27 r ) .
E ( M ) = [ H ] M E ( 0 ) = [ exp ( j ψ 0 ) , 0 0 0 exp ( j ψ 1 ) 0 0 0 exp ( j ψ 2 ) ] M E ( 0 )
= [ exp ( j M ψ 0 ) 0 0 0 exp ( j M ψ 1 ) 0 0 0 exp ( j M ψ 2 ) ] E ( 0 ) = [ exp ( j ψ 0 ) 0 0 0 exp ( j ψ 1 ) 0 0 0 exp ( j ψ 2 ) ] E ( 0 ) .
( H - σ i I ) E i = 0
[ E 0 E 1 E 2 ] = [ 1 μ 0 ν 0 1 μ 1 ν 1 1 μ 2 ν 2 ] [ A 0 A 1 A 2 ] ,
μ i = - ( γ 3 ψ i - γ 1 γ 2 ) γ 1 γ 3 - γ 2 ( ψ i + 2 ξ 1 ) ( C 1 C 0 ) 1 / 2 ,
ν i = - C 2 C 0 + ( ψ i + 2 ξ 1 ) ( γ 3 ψ i - γ 1 γ 2 ) γ 1 [ γ 1 γ 3 - γ 2 ( ψ i + 2 ξ 1 ) ] ( C 1 C 0 ) ,
γ 1 = M δ 1 ( C 0 C 1 ) 1 / 2 = π n 1 D λ ( C 0 C 1 ) 1 / 2 , γ 2 = M δ 2 ( C 0 C 2 ) 1 / 2 = π n 2 D λ ( C 0 C 1 ) 1 / 2 , γ 3 = M δ 1 ( C 1 C 2 ) 1 / 2 = π n 1 D λ ( C 1 C 2 ) 1 / 2 , ξ 1 = M Φ 1 2 = ( C 1 - C 0 - λ f z tan ϕ ) π D n g λ .
p = M p = 2 ξ 1 , q = M 2 q = - γ 1 2 - γ 2 2 - γ 3 2 , r = M 3 r = 2 γ 1 γ 2 γ 3 - 2 γ 2 2 ξ 1 .
E ( M ) = [ E 0 E 1 E 2 ] = [ exp ( j ψ 0 ) 0 0 0 exp ( j ψ 1 ) 0 0 0 exp ( j ψ 2 ) ] [ 1 1 1 ] ,
A 0 = 1 R 0 [ ( ν 1 E 0 - ν 0 E 1 μ 0 ν 1 - μ 1 ν 0 ) - ( ν 2 E 0 - ν 0 E 2 μ 0 ν 2 - μ 2 ν 0 ) ] ,
A 1 = 1 R 1 [ ( ν 1 - ν 2 ) E 0 + ( ν 2 - ν 0 ) E 1 + ( ν 0 - ν 1 ) E 2 ] ,
A 2 = 1 R 2 [ ( μ 1 - μ 2 ) E 0 + ( μ 2 - μ 0 ) E 1 + ( μ 0 - μ 1 ) E 2 ] ,
R 0 = [ ( ν 1 - ν 0 μ 0 ν 1 - μ 1 ν 0 ) - ( ν 2 - ν 0 μ 0 ν 2 - μ 2 ν 0 ) ] ,
R 1 = μ 0 ( ν 1 - ν 2 ) + μ 1 ( ν 2 - ν 0 ) + μ 2 ( ν 0 - ν 1 ) ,
R 2 = ν 0 ( μ 1 - μ 2 ) + ν 1 ( μ 2 - μ 0 ) + ν 2 ( μ 0 - μ 1 ) .

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