Abstract

A theoretical method is described from which the thickness of an arbitrary layer in a dielectric multilayer stack is determined if all other thicknesses and all refractive indices are given, such that a prescribed effective index for a given mode of the stack is obtained. The same theory can determine the optimal cover thickness of such a stack for prism coupling. Experimental verification shows effective indices that are in agreement with the designed values.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. de Jong, “Multilayered Slab Waveguide Design Using a Hybrid Field Vector,” Appl. Opt. 28, 3567–3576 (1989).
    [CrossRef] [PubMed]
  2. J. Chillwell, I. Hodgkinson, “Thin-Films Field-Transfer Matrix Theory of Planar Multilayer Waveguides and Reflection from Prism-Loaded Waveguides,” J. Opt. Soc. Am. A 1, 742–753 (1984).
    [CrossRef]
  3. H.-G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977).
  4. P. K. Tien, R. Ulrich, “Theory of Prism-Film Coupler and Thin-Film Light Guides,” J. Opt. Soc. Am. 60, 1325–1336 (1970).
    [CrossRef]
  5. T. Tamir, Ed., Integrated Optics (Springer-Verlag, Berlin, 1979).

1989 (1)

1984 (1)

1970 (1)

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Stack configuration.

Fig. 2
Fig. 2

Layer thickness dj as a function of β for a structure with J = 2, nt = ns = 1.460, n1 = 2.25, n2 = 1.66, λv = 632.8 nm. Parameters indicate mode numbers: TE-modes: solid lines; TM-modes: dashed lines. (a) Periodic d1 for β < n1. (b) Periodic d2 for β < n2, aperiodic d2 for β > n2.

Fig. 3
Fig. 3

Layer thickness d1 as a function of coupling length L for λv = 632.8 nm and nt = 1.7796: a, for the TE0-mode in the structure of Fig. 4 (β = 1.6400); b, for the TE0-mode in a single-film structure (β = 1.6146) with ns = 1.457, n1 = 1.000, n2 = 1.705, d2 = 380 nm; and c, same as b but calculated with first-order perturbation theory.

Fig. 4
Fig. 4

Designed stack.

Tables (2)

Tables Icon

Table I Definition of the Components of the Hybrid Field Vector F

Tables Icon

Table II Effective Refractive Indices for the Stack of Fig. 4

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

F ( 0 ) = ( j = 1 J M j ) F ( x J ) = M F ( x J ) ,
M j = [ cos ( Φ j ) 1 q j sin ( Φ j ) - q j sin ( Φ j ) cos ( Φ j ) ]             ( j = 1 , 2 , j )
q j = 1 n j 2 ρ n j 2 - β 2 ,
Φ j = k v α j d j ,
α j = n j 2 - β 2 .
( m 11 + m 12 Q s ) Q t + m 21 + m 22 Q s = 0.
M j = [ I + U j tan ( Φ j ) ] cos ( Φ j ) ,
U j = [ 0 1 q j - q j 0 ]             ( β n j ) .
S j = M j + 1 M j + 2 M J = i = j + 1 J M i             ( 0 < j < J ) ,
T j = M 1 M 2 M j - 1 = i = 1 j - 1 M i             ( 1 < j < J ) .
M = T j [ I + U j tan ( Φ j ) ] cos ( Φ j ) S j             ( β n j ) .
m p q = [ a p q + b p q tan ( Φ j ) ] cos ( Φ j )             ( p , q = 1 or 2 ) .
tan ( Φ j ) = - ( a 11 + a 12 Q s ) Q t + a 21 + a 22 Q s ( b 11 + b 12 Q s ) Q t + b 21 + b 22 Q s ,
d j = ( ½ + p ) π k v α j             ( p = 0 , 1 , 2 , ) .
τ j = - ( a 11 + a 12 Q s ) Q t + a 21 + a 22 Q s ( b 11 + b 12 Q s ) Q t + b 21 + b 22 Q s .
d j = Φ j k v α j = arctan ( τ j ) + p π k v α j             ( p = 0 , 1 , 2 , ) ,
d j , 0 = { arctan ( τ j ) k v α j if τ j 0 , arctan ( τ j ) + π k v α j if τ j < 0 ,
d j , p = d j , 0 + p λ v 2 α j             ( p = 1 , 2 , ) .
Λ j = λ v α j ,
tan ( Φ j ) = τ j = i tanh ( k v α j d j i )
d j = arctanh ( τ j / i ) k v α j / i = 1 2 k v α j / i ln ( 1 + τ j / i 1 - τ j / i ) .
L = 1 k v Im ( Δ β ) .
[ 2 w L cos ( θ t ) ] opt = 1.36.
Re ( Φ 1 ) = 1 2 arg [ 1 + i τ 1 1 - i τ 1 ] , Im ( Φ 1 ) = - 1 2 ln | 1 + i τ 1 1 - i τ 1 | .
Re ( Φ 1 ) Im ( α 1 ) - Im ( Φ 1 ) Re ( α 1 ) = 0.

Metrics