Abstract

A method for designing and recording a holographic optical element that is used as a waveguide focusing grating coupler is presented. It is based on recording the holographic coupler with two predistorted wave fronts, derived from interim holograms, whose readout and recording geometries are different. The corrected holographic coupler has almost aberration-free performance even for couplers with very small f/numbers or with large wavelength shift between recording and readout.

© 1991 Optical Society of America

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References

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  1. G. N. Lawrence, P. J. Cronkite, “Physical optics analysis of the focusing grating coupler,” Appl. Opt. 27, 672–678 (1988).
    [CrossRef] [PubMed]
  2. J. Seligson, “Modeling of a focusing coupler using vector scattering theory,” Appl. Opt. 27, 684–692 (1988).
    [CrossRef] [PubMed]
  3. S. Ura, T. Suhara, H. Nishihara, J. Koyana, “An integrated-optic disk pickup device,” IEEE J. Lightwave Technol. LT-4, 913–918 (1986).
    [CrossRef]
  4. D. Heitmann, R. V. Pole, “Two-dimensional focusing holographic grating coupler,” Appl. Phys. Lett. 37, 585–587 (1980).
    [CrossRef]
  5. T. Tamir, Integrated Optics, Vol. 7 of Applied Physics Series, (Springer-Verlag, Berlin, 1975).
  6. G. N. Lawrence, K. E. Moore, P. J. Cronkite, “Rotationally symmetric construction optics for a waveguide focusing grating,” Appl. Opt. 29, 2315–2319 (1990).
    [CrossRef] [PubMed]
  7. J. Kedmi, A. A. Friesem, “Optimized holographic optical elements,” J. Opt. Soc. Am. A 3, 2011–2018 (1986).
    [CrossRef]
  8. Y. Amitai, A. A. Friesem, “Design of holographic optical elements by using recursive techniques,” J. Opt. Soc. Am. 5, 702–712 (1988).
    [CrossRef]
  9. Y. Amitai, A. A. Friesem, V. Weiss, “Designing Holographic Lenses with Different Recording and Readout Wavelengths,” J. Opt. Soc. Am. A 7, 80–86 (1990).
    [CrossRef]
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 5.
  11. J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85–88 (1964).
    [CrossRef]
  12. T. Tamir, S. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
    [CrossRef]

1990 (2)

1988 (3)

1986 (2)

S. Ura, T. Suhara, H. Nishihara, J. Koyana, “An integrated-optic disk pickup device,” IEEE J. Lightwave Technol. LT-4, 913–918 (1986).
[CrossRef]

J. Kedmi, A. A. Friesem, “Optimized holographic optical elements,” J. Opt. Soc. Am. A 3, 2011–2018 (1986).
[CrossRef]

1980 (1)

D. Heitmann, R. V. Pole, “Two-dimensional focusing holographic grating coupler,” Appl. Phys. Lett. 37, 585–587 (1980).
[CrossRef]

1977 (1)

T. Tamir, S. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

1964 (1)

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85–88 (1964).
[CrossRef]

Amitai, Y.

Y. Amitai, A. A. Friesem, V. Weiss, “Designing Holographic Lenses with Different Recording and Readout Wavelengths,” J. Opt. Soc. Am. A 7, 80–86 (1990).
[CrossRef]

Y. Amitai, A. A. Friesem, “Design of holographic optical elements by using recursive techniques,” J. Opt. Soc. Am. 5, 702–712 (1988).
[CrossRef]

Cronkite, P. J.

Friesem, A. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 5.

Heitmann, D.

D. Heitmann, R. V. Pole, “Two-dimensional focusing holographic grating coupler,” Appl. Phys. Lett. 37, 585–587 (1980).
[CrossRef]

Kedmi, J.

Koyana, J.

S. Ura, T. Suhara, H. Nishihara, J. Koyana, “An integrated-optic disk pickup device,” IEEE J. Lightwave Technol. LT-4, 913–918 (1986).
[CrossRef]

Lawrence, G. N.

Moore, K. E.

Nishihara, H.

S. Ura, T. Suhara, H. Nishihara, J. Koyana, “An integrated-optic disk pickup device,” IEEE J. Lightwave Technol. LT-4, 913–918 (1986).
[CrossRef]

Peng, S.

T. Tamir, S. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

Pole, R. V.

D. Heitmann, R. V. Pole, “Two-dimensional focusing holographic grating coupler,” Appl. Phys. Lett. 37, 585–587 (1980).
[CrossRef]

Rayces, J. L.

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85–88 (1964).
[CrossRef]

Seligson, J.

Suhara, T.

S. Ura, T. Suhara, H. Nishihara, J. Koyana, “An integrated-optic disk pickup device,” IEEE J. Lightwave Technol. LT-4, 913–918 (1986).
[CrossRef]

Tamir, T.

T. Tamir, S. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

T. Tamir, Integrated Optics, Vol. 7 of Applied Physics Series, (Springer-Verlag, Berlin, 1975).

Ura, S.

S. Ura, T. Suhara, H. Nishihara, J. Koyana, “An integrated-optic disk pickup device,” IEEE J. Lightwave Technol. LT-4, 913–918 (1986).
[CrossRef]

Weiss, V.

Appl. Opt. (3)

Appl. Phys. (1)

T. Tamir, S. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

Appl. Phys. Lett. (1)

D. Heitmann, R. V. Pole, “Two-dimensional focusing holographic grating coupler,” Appl. Phys. Lett. 37, 585–587 (1980).
[CrossRef]

IEEE J. Lightwave Technol. (1)

S. Ura, T. Suhara, H. Nishihara, J. Koyana, “An integrated-optic disk pickup device,” IEEE J. Lightwave Technol. LT-4, 913–918 (1986).
[CrossRef]

J. Opt. Soc. Am. (1)

Y. Amitai, A. A. Friesem, “Design of holographic optical elements by using recursive techniques,” J. Opt. Soc. Am. 5, 702–712 (1988).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85–88 (1964).
[CrossRef]

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 5.

T. Tamir, Integrated Optics, Vol. 7 of Applied Physics Series, (Springer-Verlag, Berlin, 1975).

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Figures (6)

Fig. 1
Fig. 1

Side-view geometry of a FGC.

Fig. 2
Fig. 2

Holographic recording of a FGC with two spherical waves.

Fig. 3
Fig. 3

Transferring the wave fronts with an intermediate hologram: (a) recording the first-step holograms (p = o, r); (b) reconstructing the first-step hologram and recording the intermediate hologram with the output from the first-step holgram and a plane wave ϕrint as a reference wave; (c) reconstructing the intermediate hologran with ϕcint = − ϕrint and recording the final hologram.

Fig. 4
Fig. 4

Calculated spot sizes, normalized to the diffraction limit, as a function of κ normalized to 1/d.

Fig. 5
Fig. 5

Calculated spot sizes for noncorrected FGC (dashed line); corrected FGC with κ = 1 (dash–dot line); corrected FGC with κ = 0.2 (dotted line); and the diffraction limit (solid line) as a function of wavelength shift.

Fig. 6
Fig. 6

Diffraction limit (dashed line) and the calculated spot sizes for different FGC's with κ = 0.2 (solid lines) and with κ = 1 (dotted line) as a function of the f/number; n represents the number of interim holograms during the recording process.

Equations (25)

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( k i ) = ( k c ± K G ) ,
K G = k o k r .
( k i ) = [ k c ± ( k o k r ) ] .
z ̂ × k q = z ̂ × k q , q = i , o , r ,
1 λ c ( υ ̂ i ) = 1 λ o [ ν μ υ ̂ c ± ( υ ̂ o υ ̂ r ) ] ,
δ = ( υ ̂ d υ ̂ i ) = [ υ ̂ d ν υ ̂ c μ ( υ ̂ o υ ̂ r ) ] .
( υ ̂ r ) = x ̂ sin β r ,
( υ ̂ o ) = r ̂ r ( R o 2 + r 2 ) 1 / 2 = r ̂ r ( R o 2 + r 2 ) 1 / 2 = r ̂ r [ R o ( 1 + r 2 R o 2 ) 1 / 2 ] = r ̂ m = 1 , 3 , α m r m R o m ,
( υ ̂ d ) = r ̂ m = 1 , 3 α m r m R i m ,
( υ ̂ c ) = x ̂ .
R o = μ R i , sin β r = ν μ .
μ ν
δ ( r ) = r ̂ m = 3 , 5 α m [ 1 ( R i ) m μ 1 ( R o ) m ] r m ,
υ ̂ p υ ̂ i p = υ ̂ c p + υ ̂ o p υ ̂ r p , p = o , r ,
( υ ̂ r p ) = x ̂ sin β r p , ( υ ̂ q p ) = r ̂ m = 1 , 3 α m r m ( R q p ) m ,
δ ( r ) = [ υ ̂ d ν υ ̂ c + μ ( υ ̂ c o + υ ̂ o o υ ̂ r o υ ̂ c r υ ̂ o r + υ ̂ r r ) ] = x ̂ [ μ ( sin β r o sin β r r ) ν ] r ̂ m = 1 , 3 α m r m { 1 ( R i ) m μ [ 1 ( R o o ) m + 1 ( R c o ) m 1 ( R o r ) m 1 ( R c r ) m ] } .
sin β r o sin β r r = ν μ ,
μ ( 1 R o o + 1 R c o 1 R o r 1 R c r ) = 1 R i ,
S m = 1 ( R i ) m μ [ 1 ( R o o ) m + 1 ( R c o ) m 1 ( R o r ) m 1 ( R c r ) m ] , m = 3 , 5
S 7 = S 5 = S 3 = 0 ,
U f ( x f , y f ) = + + P ( x , y ) exp [ j 2 π λ c f ( x x f + y y f ) ] × exp [ j Δ ( x , y ) ] d x d y ,
P ( x , y ) = { 1 for ( x , y ) inside the FGC , 0 for ( x , y ) outside the FGC ,
Δ ( x , y ) = 2 π λ c 0 r δ ( ρ ) · d ρ = 2 π λ c 0 r m = 9 , 11 α m m + 1 S m r m + 1 ,
U i ( x , y ) = U i 0 exp ( κ x ) ,
U f ( x f , y f ) = + + P ( x , y ) U i 0 exp ( κ x ) exp [ j 2 π λ c f ( x x f + y y f ) ] × exp [ j Δ ( x , y ) ] d x d y .

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