Abstract

A multimode planar optical waveguide of homogeneous refractive index can produce real one-dimensional self-images by interference of the waveguide modes. We discuss the image formation, the linespread function, and its apodisation and optimization. Under optimum conditions, the theoretical linear resolution can reach 0.25(W λf)1/2, where W is the slab thickness and λf is the wavelength in the slab material.

© 1978 Optical Society of America

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References

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  1. O. Bryngdahl, J. Opt. Soc. Am. 63, 416 (1973).
    [CrossRef]
  2. R. Ulrich, Opt. Commun. 13, 259 (1975). This reference contains some typographical errors: In the last paranthesis of Eq. (12), the first term should read 7/9π2, and the last term should have a positive sign. These errors were kindly pointed out to us by E. Voges. Furthermore, in Eqs. (6) and (7) the term (M0 − 2) should read (M0 − 2/π), and the reflection coefficient mentioned after Eq. (9) should be exp(−2iϕm). The response function discussed is a line spread function rather than a point spread function.
    [CrossRef]
  3. R. Ulrich and G. Ankele, Appl. Phys. Lett. 27, 338 (1975).
    [CrossRef]
  4. E. Voges and R. Ulrich, Proceedings of the VIth European Microwave Conference (Microwave Exhibition and Publishers, Sevenoaks, 1976), p. 447.
  5. A. Simon and R. Ulrich, Appl. Phys. Lett. 31, 77 (1977).
    [CrossRef]
  6. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Chap: 9.5.
  7. P. Jaquinot and B. Roizen-Dossier, Progress in Optics, Vol. 3 (North-Holland, Amsterdam, 1964), p. 29.
    [CrossRef]
  8. R. Ulrich and W. Prettl, Appl. Phys. 1, 55 (1973).
    [CrossRef]
  9. A. Reisinger, Appl. Opt. 12, 1015 (1973).
    [CrossRef] [PubMed]

1977 (1)

A. Simon and R. Ulrich, Appl. Phys. Lett. 31, 77 (1977).
[CrossRef]

1975 (2)

R. Ulrich, Opt. Commun. 13, 259 (1975). This reference contains some typographical errors: In the last paranthesis of Eq. (12), the first term should read 7/9π2, and the last term should have a positive sign. These errors were kindly pointed out to us by E. Voges. Furthermore, in Eqs. (6) and (7) the term (M0 − 2) should read (M0 − 2/π), and the reflection coefficient mentioned after Eq. (9) should be exp(−2iϕm). The response function discussed is a line spread function rather than a point spread function.
[CrossRef]

R. Ulrich and G. Ankele, Appl. Phys. Lett. 27, 338 (1975).
[CrossRef]

1973 (3)

Ankele, G.

R. Ulrich and G. Ankele, Appl. Phys. Lett. 27, 338 (1975).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Chap: 9.5.

Bryngdahl, O.

Jaquinot, P.

P. Jaquinot and B. Roizen-Dossier, Progress in Optics, Vol. 3 (North-Holland, Amsterdam, 1964), p. 29.
[CrossRef]

Prettl, W.

R. Ulrich and W. Prettl, Appl. Phys. 1, 55 (1973).
[CrossRef]

Reisinger, A.

Roizen-Dossier, B.

P. Jaquinot and B. Roizen-Dossier, Progress in Optics, Vol. 3 (North-Holland, Amsterdam, 1964), p. 29.
[CrossRef]

Simon, A.

A. Simon and R. Ulrich, Appl. Phys. Lett. 31, 77 (1977).
[CrossRef]

Ulrich, R.

A. Simon and R. Ulrich, Appl. Phys. Lett. 31, 77 (1977).
[CrossRef]

R. Ulrich, Opt. Commun. 13, 259 (1975). This reference contains some typographical errors: In the last paranthesis of Eq. (12), the first term should read 7/9π2, and the last term should have a positive sign. These errors were kindly pointed out to us by E. Voges. Furthermore, in Eqs. (6) and (7) the term (M0 − 2) should read (M0 − 2/π), and the reflection coefficient mentioned after Eq. (9) should be exp(−2iϕm). The response function discussed is a line spread function rather than a point spread function.
[CrossRef]

R. Ulrich and G. Ankele, Appl. Phys. Lett. 27, 338 (1975).
[CrossRef]

R. Ulrich and W. Prettl, Appl. Phys. 1, 55 (1973).
[CrossRef]

E. Voges and R. Ulrich, Proceedings of the VIth European Microwave Conference (Microwave Exhibition and Publishers, Sevenoaks, 1976), p. 447.

Voges, E.

E. Voges and R. Ulrich, Proceedings of the VIth European Microwave Conference (Microwave Exhibition and Publishers, Sevenoaks, 1976), p. 447.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Chap: 9.5.

Appl. Opt. (1)

Appl. Phys. (1)

R. Ulrich and W. Prettl, Appl. Phys. 1, 55 (1973).
[CrossRef]

Appl. Phys. Lett. (2)

A. Simon and R. Ulrich, Appl. Phys. Lett. 31, 77 (1977).
[CrossRef]

R. Ulrich and G. Ankele, Appl. Phys. Lett. 27, 338 (1975).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

R. Ulrich, Opt. Commun. 13, 259 (1975). This reference contains some typographical errors: In the last paranthesis of Eq. (12), the first term should read 7/9π2, and the last term should have a positive sign. These errors were kindly pointed out to us by E. Voges. Furthermore, in Eqs. (6) and (7) the term (M0 − 2) should read (M0 − 2/π), and the reflection coefficient mentioned after Eq. (9) should be exp(−2iϕm). The response function discussed is a line spread function rather than a point spread function.
[CrossRef]

Other (3)

E. Voges and R. Ulrich, Proceedings of the VIth European Microwave Conference (Microwave Exhibition and Publishers, Sevenoaks, 1976), p. 447.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Chap: 9.5.

P. Jaquinot and B. Roizen-Dossier, Progress in Optics, Vol. 3 (North-Holland, Amsterdam, 1964), p. 29.
[CrossRef]

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Figures (11)

FIG. 1
FIG. 1

Cross section of dielectric slab waveguide.

FIG. 2
FIG. 2

Line-spread function of a single, erect self-image, schematically. (a) the isoplanatic part; (b) the nonisoplanatic part; and (c) their sum.

FIG. 3
FIG. 3

Amplitude spectra and associated line-spread functions, (a) for sharply cutoff spectrum, (b) for Gaussian apodization.

FIG. 4
FIG. 4

Methods of limiting the number of modes contributing to the observed image. (a) aperture stop; (b) apodization by an amplitude filter; (c) apodization by lossy cladding.

FIG. 5
FIG. 5

Calculated phase spectra of various self-imaging slab waveguides. W = 50 μm; λ = 0.6328 μm; TE polarization; nc = 1.500. The curves refer to guides of different indices: (1) nf = 1.515; (2) 1.520; (3) 1.522; (4) 1.5225; (5) 1.528; (6) 1.650; (7) 2.25; (8)3.00; (9) 4.50. Curve 3′ calculated from truncated power series (22).

FIG. 6
FIG. 6

Phase spectra of the same guides as in Fig. 5, but with their lengths adjusted to yield a maximum number M of contributing modes.

FIG. 7
FIG. 7

Representation of the maximum number M of contributing modes for guides of maximally flat phase characteristic. —, first-order optimization; - - -, second-order optimization.

FIG. 8
FIG. 8

Number of contributing modes as a function of the guide index. W = 50 μm; λ = 0.6328 μm, TE polarization, nc = 1.500; h = 1. Full lines, approximate analytical results; xxoo, exact numerical results; ⊕, Eq. (28); ●, Eq. (37).

FIG. 9
FIG. 9

Index difference (nf,optnc) required for optimum imaging and resulting M as functions of guide thickness. - - -, optimization for maximum contrast; —, for maximum resolution; ○ ○, exact results; M ˆ, total number of modes.

FIG. 10
FIG. 10

Line-spread functions of the guide of Fig. 6, curve 5, in the vicinity of the first image (h = 1; L1 = 24.819 mm; M = 39).

FIG. 11
FIG. 11

Influence of various phase errors on the image of a model object function.

Equations (52)

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A ( z ) = m a m m ( z ) .
( z ) = m a m τ m m ( z ) .
τ m ( L ) = τ m exp ( i β m L ) .
ϕ m ( L ) ( β m - β 0 ) L = - π ( L / L 1 ) ( m 2 + 2 m + Δ m ) .
ϕ m , i d ( L h ) = ( β m , i d - β 0 , i d ) L h = - h π ( m 2 + 2 m ) .
L 1 4 W 2 / λ f .
L = h L 1 + l .
ψ n = - h π Δ m - π ( l / L 1 ) ( m 2 + 2 m + Δ m ) .
τ m = s m t m ,
s m = exp [ - i h π ( m 2 + 2 m ) ] ,
t m = τ m exp ( i ψ m ) .
( z , z 0 ) = 2 m ( τ m / W eq , m ) m ( z ) m ( z 0 ) .
( z ) = A ( z 0 ) ( z , z 0 ) d z 0 .
L ( z , z 0 ) = - ( z - z 0 ) - + ( z + z 0 ) ,
- ( z - z 0 ) = C 0 + m ( τ m / W eq , m ) cos [ κ m ( z - z 0 ) ] ,
+ ( z + z 0 ) = C 0 + m ( τ m / W eq , m ) × cos [ κ m ( z + z 0 ) + π - 2 ϕ m ] .
κ m ( m + 1 ) π / W a = ( m + 1 ) κ 0 .
R ( z ) C 0 + W a - 1 m τ m cos ( m + 1 ) κ 0 z .
( z , z 0 ) = S - * T - - S + * T + ,
T ( z ) = ( 2 W a ) - 1 sin [ ( 2 M + 1 ) κ 0 z / 2 ] / sin ( κ 0 z / 2 ) .
r 0.89 W a / M .
T ( z ) = ( π 1 / 2 M c / 2 W a ) exp [ - ( π M c z / 2 W a ) 2 ] .
μ m = ( m + 1 ) / M ˆ .
ψ ( μ ) π M ˆ 2 [ - μ 2 l L 1 - μ 4 ( h + l L 1 ) ( M ˆ M ˆ 0 ) 2 ( A 2 4 - 2 3 π M ˆ 0 ) - μ 6 ( h + l L 1 ) ( M ˆ M ˆ 0 ) 4 ( A 4 8 - A 2 3 π M ˆ 0 - 3 10 π M 0 + 7 9 π 2 M ˆ 0 2 ) ] .
L 1 = 4 W a 2 / λ f
I : A 3 ( 4 π / 3 ) ( λ f / W ) , II : A 3 ( 4 π / 3 ) ( λ f / W ) , III : A ( 4 π / 3 ) ( λ f / W ) .
I :             M 1.75 h - 1 / 4 ( A W / h f ) 3 / 4 - 1 ,
III :             M 1.41 h - 1 / 4 ( W / λ f ) 1 / 2 - 1.
n f , opt [ n c 2 + ( 4 n c 2 λ / 3 π W ) 2 / 3 ] 1 / 2 .
II :             M 1.65 h - 1 / 6 ( n c W / λ ) 5 / 9 - 1.
ψ ( μ ) / ψ max = q 2 ( μ / μ M ) 2 + q 4 ( μ / μ M ) 4 P 24 ( μ / μ M ) ,
I :             l opt + 0.326 h - 1 / 2 L 1 ) a w . λ f ) - 1 / 2 ,
M 2.72 h - 1 / 4 ( A W / λ f ) 3 / 4 - 1.
III :             l opt - h 1 / 2 W ,
M 2.20 h - 1 / 4 ( W / λ f ) 1 / 2 - 1.
III :             r 0.403 h 1 / 4 ( W λ f ) 1 / 2 .
A = [ C h - 1 / 3 ( λ / 2 n f W ) 4 / 3 + ( 8 / 3 π ) ( λ / 2 n f W ) A 1 / 3 ] 0.3 .
II opt :             l opt - 0.931 h 1 / 2 W ,
M 3.47 h - 1 / 4 ( W / λ f ) 1 / 2 - 1 ,
r 0.255 h 1 / 4 ( W λ f ) 1 / 2 .
{ m ( z ) = cos ϕ m exp [ ( β m 2 - β c 2 ) 1 / 2 z ] , if z 0 cos ( κ m z - ϕ m ) , if 0 z W ( - 1 ) m cos ϕ m exp [ ( β m 2 - β c 2 ) 1 / 2 ( W - z ) ] , if W z . κ m = ( β f 2 - β m 2 ) 1 / 2
tan ϕ m = ( n f / n c ) 2 σ κ m - 1 ( β m 2 - β c 2 ) 1 / 2 .
κ m W - 2 ϕ m = m π ,
a m = 2 W eq , m - + ( n f n ( z ) ) 2 σ A ( z ) m ( z ) d z ,
W eq , m = 2 - + ( n f n ( z ) ) 2 σ m 2 ( z ) d z .
W a = W + ( n c / n f ) 2 σ ( λ f / π A ) .
β ¯ ( μ ) = β f ( 1 - μ 2 A 2 ) 1 / 2 .
β ¯ ( μ ¯ ) = β ( μ ) .
μ ¯ = μ - ( 2 / M ˆ π ) arctan [ ( n c / n f ) 2 σ μ ¯ ( 1 - μ ¯ 2 ) - 1 / 2 ] .
μ ¯ = μ - ( 2 / M ˆ π ) arcsin μ ¯ .
μ ¯ ( μ ) = ( M ˆ M ˆ 0 ) μ - 1 3 π M ˆ 0 ( M ˆ M ˆ 0 ) 3 μ 3 + ( 1 3 π 2 M ˆ 0 2 - 3 20 π M ˆ 0 ) ( M ˆ M ˆ 0 ) 5 μ 5 + .
β ( μ ) β f = 1 - A 2 2 ( M ˆ M ˆ 0 ) 2 μ 2 + ( A 2 3 π M ˆ 0 - A 4 8 ) ( M ˆ M ˆ 0 ) 4 μ 4 - ( 7 A 2 18 π 2 M ˆ 0 2 - 3 A 2 20 π M ˆ 0 - A 4 6 π M ˆ 0 + A 6 16 ) ( M ˆ M ˆ 0 ) 6 μ 6 + .