Abstract

Diagrams of complex reflectance or circle diagrams are used to design optical coatings at normal incidence. The technique can be extended to oblique incidence, including the design of coatings which take advantage of total internal reflectance. Use of these diagrams is helpful in visualizing and compensating for polarization effects in such coatings.

© 1989 Optical Society of America

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References

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  1. J. H. Apfel, “Graphics in Optical Coating Design,” Appl. Opt. 11, 1303–1312 (1972).
    [CrossRef] [PubMed]
  2. P. H. Berning, “Theory and Calculations of Optical Thin Films,” Phys. Thin Films 1, 69–121 (1963).
  3. Y. O. Dovgii, B. V. Mykytyuk, “Circle Diagram Method for Practical Calculations for Multilayer Thin-Film Systems,” Opt. Spektrosk. 58, 432–436 (1985) [Opt. Spectrosc. U.S.S.R. 58, 258–260 (1985)].
  4. R. M. A. Azzam, M. E. R. Khan, “Complex Reflection Coefficients for the Parallel and Perpendicular Polarizations of a Film–Substrate System,” Appl. Opt. 22, 253–264 (1983).
    [CrossRef] [PubMed]
  5. H. A. Macleod, Thin-Film Optical Filters (Macmillan, New York, 1986), pp. 59, 33.
  6. H. A. Macleod, Thin-Film Optical Filters (Adam Hilger, Ltd., London, 1969), p. 14.
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 66.
  8. J. H. Apfel, “Graphical Method to Design Internal Reflection Phase Retarders,” Appl. Opt. 23, 1178–1183 (1984).
    [CrossRef] [PubMed]
  9. Z. Knittl, Optics of Thin Films (Wiley, New York, 1976), p. 351.
  10. The fact that ρM = −ρE at critical and grazing incidence is a consequence of defining the incident and reflected unit vectors so as to make ρM = ρE at normal incidence. The unit vectors are sometimes defined so that ρM = −ρE at normal incidence; in this case, we would have ρM = ρE at critical and grazing incidence. This is important to understand because, in the later convention, the incident and reflected unit vectors point in the same direction for both polarizations at grazing incidence while in the former convention (the one used in this paper), the incident and reflected unit vectors for the TM polarization point in opposite directions. Thus, ρM = −ρE at grazing incidence is a result of reversing the positive direction for TM but not for TE; physically, there is a 180° phase shift at grazing incidence for both polarizations.
  11. E. S. Shakaryan et al., “Frustrated Total Internal Reflection Refractometry,” Opt. Mekh. Prom. 44, 47–53 (1977) [Sov. J. Opt. Technol. 44, 748–754 (1977)].
  12. P. Leurgans, A. F. Turner, “Frustrated Total Reflection Interference Filters,” J. Opt. Soc. Am. 37, 983A (1947).
  13. B. H. Billings, M. A. Pittman, “A Frustrated Total Reflection Filter for the Infra-Red,” J. Ot. Soc. Am. 39, 978–983 (1949).
    [CrossRef]
  14. B. H. Billings, “A Birefringent Frustrated Total Reflection Filter,” J. Opt. Soc. Am. 40, 471–476 (1950).
    [CrossRef]
  15. P. G. Kard, “On Elimination of the Doublet Structure of the Transmission Band in a Total-Reflection Light-Filter,” Opt. Spektrosk. 6, 339 (1959) [Opt. Spectrosc. U.S.S.R. 6, 244–246 (1959)].
  16. H. Kitajima, K. Fujita, H. Cizmic, “Multiple Resonance FTR Filters for Nonpolarizing Bandpass Filters,” Appl. Opt. 23, 3487–3492 (1984).
    [CrossRef] [PubMed]
  17. P. W. Baumeister, “Optical Tunneling and Its Application to Optical Filters,” Appl. Opt. 6, 897–905 (1967).
    [CrossRef] [PubMed]

1985 (1)

Y. O. Dovgii, B. V. Mykytyuk, “Circle Diagram Method for Practical Calculations for Multilayer Thin-Film Systems,” Opt. Spektrosk. 58, 432–436 (1985) [Opt. Spectrosc. U.S.S.R. 58, 258–260 (1985)].

1984 (2)

1983 (1)

1977 (1)

E. S. Shakaryan et al., “Frustrated Total Internal Reflection Refractometry,” Opt. Mekh. Prom. 44, 47–53 (1977) [Sov. J. Opt. Technol. 44, 748–754 (1977)].

1972 (1)

1967 (1)

1963 (1)

P. H. Berning, “Theory and Calculations of Optical Thin Films,” Phys. Thin Films 1, 69–121 (1963).

1959 (1)

P. G. Kard, “On Elimination of the Doublet Structure of the Transmission Band in a Total-Reflection Light-Filter,” Opt. Spektrosk. 6, 339 (1959) [Opt. Spectrosc. U.S.S.R. 6, 244–246 (1959)].

1950 (1)

1949 (1)

B. H. Billings, M. A. Pittman, “A Frustrated Total Reflection Filter for the Infra-Red,” J. Ot. Soc. Am. 39, 978–983 (1949).
[CrossRef]

1947 (1)

P. Leurgans, A. F. Turner, “Frustrated Total Reflection Interference Filters,” J. Opt. Soc. Am. 37, 983A (1947).

Apfel, J. H.

Azzam, R. M. A.

Baumeister, P. W.

Berning, P. H.

P. H. Berning, “Theory and Calculations of Optical Thin Films,” Phys. Thin Films 1, 69–121 (1963).

Billings, B. H.

B. H. Billings, “A Birefringent Frustrated Total Reflection Filter,” J. Opt. Soc. Am. 40, 471–476 (1950).
[CrossRef]

B. H. Billings, M. A. Pittman, “A Frustrated Total Reflection Filter for the Infra-Red,” J. Ot. Soc. Am. 39, 978–983 (1949).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 66.

Cizmic, H.

Dovgii, Y. O.

Y. O. Dovgii, B. V. Mykytyuk, “Circle Diagram Method for Practical Calculations for Multilayer Thin-Film Systems,” Opt. Spektrosk. 58, 432–436 (1985) [Opt. Spectrosc. U.S.S.R. 58, 258–260 (1985)].

Fujita, K.

Kard, P. G.

P. G. Kard, “On Elimination of the Doublet Structure of the Transmission Band in a Total-Reflection Light-Filter,” Opt. Spektrosk. 6, 339 (1959) [Opt. Spectrosc. U.S.S.R. 6, 244–246 (1959)].

Khan, M. E. R.

Kitajima, H.

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976), p. 351.

Leurgans, P.

P. Leurgans, A. F. Turner, “Frustrated Total Reflection Interference Filters,” J. Opt. Soc. Am. 37, 983A (1947).

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Adam Hilger, Ltd., London, 1969), p. 14.

H. A. Macleod, Thin-Film Optical Filters (Macmillan, New York, 1986), pp. 59, 33.

Mykytyuk, B. V.

Y. O. Dovgii, B. V. Mykytyuk, “Circle Diagram Method for Practical Calculations for Multilayer Thin-Film Systems,” Opt. Spektrosk. 58, 432–436 (1985) [Opt. Spectrosc. U.S.S.R. 58, 258–260 (1985)].

Pittman, M. A.

B. H. Billings, M. A. Pittman, “A Frustrated Total Reflection Filter for the Infra-Red,” J. Ot. Soc. Am. 39, 978–983 (1949).
[CrossRef]

Shakaryan, E. S.

E. S. Shakaryan et al., “Frustrated Total Internal Reflection Refractometry,” Opt. Mekh. Prom. 44, 47–53 (1977) [Sov. J. Opt. Technol. 44, 748–754 (1977)].

Turner, A. F.

P. Leurgans, A. F. Turner, “Frustrated Total Reflection Interference Filters,” J. Opt. Soc. Am. 37, 983A (1947).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 66.

Appl. Opt. (5)

J. Opt. Soc. Am. (2)

B. H. Billings, “A Birefringent Frustrated Total Reflection Filter,” J. Opt. Soc. Am. 40, 471–476 (1950).
[CrossRef]

P. Leurgans, A. F. Turner, “Frustrated Total Reflection Interference Filters,” J. Opt. Soc. Am. 37, 983A (1947).

J. Ot. Soc. Am. (1)

B. H. Billings, M. A. Pittman, “A Frustrated Total Reflection Filter for the Infra-Red,” J. Ot. Soc. Am. 39, 978–983 (1949).
[CrossRef]

Opt. Mekh. Prom. (1)

E. S. Shakaryan et al., “Frustrated Total Internal Reflection Refractometry,” Opt. Mekh. Prom. 44, 47–53 (1977) [Sov. J. Opt. Technol. 44, 748–754 (1977)].

Opt. Spektrosk. (2)

Y. O. Dovgii, B. V. Mykytyuk, “Circle Diagram Method for Practical Calculations for Multilayer Thin-Film Systems,” Opt. Spektrosk. 58, 432–436 (1985) [Opt. Spectrosc. U.S.S.R. 58, 258–260 (1985)].

P. G. Kard, “On Elimination of the Doublet Structure of the Transmission Band in a Total-Reflection Light-Filter,” Opt. Spektrosk. 6, 339 (1959) [Opt. Spectrosc. U.S.S.R. 6, 244–246 (1959)].

Phys. Thin Films (1)

P. H. Berning, “Theory and Calculations of Optical Thin Films,” Phys. Thin Films 1, 69–121 (1963).

Other (5)

H. A. Macleod, Thin-Film Optical Filters (Macmillan, New York, 1986), pp. 59, 33.

H. A. Macleod, Thin-Film Optical Filters (Adam Hilger, Ltd., London, 1969), p. 14.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 66.

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976), p. 351.

The fact that ρM = −ρE at critical and grazing incidence is a consequence of defining the incident and reflected unit vectors so as to make ρM = ρE at normal incidence. The unit vectors are sometimes defined so that ρM = −ρE at normal incidence; in this case, we would have ρM = ρE at critical and grazing incidence. This is important to understand because, in the later convention, the incident and reflected unit vectors point in the same direction for both polarizations at grazing incidence while in the former convention (the one used in this paper), the incident and reflected unit vectors for the TM polarization point in opposite directions. Thus, ρM = −ρE at grazing incidence is a result of reversing the positive direction for TM but not for TE; physically, there is a 180° phase shift at grazing incidence for both polarizations.

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Figures (16)

Fig. 1
Fig. 1

Normal incidence circle diagram for an alumina layer (index 1.54) with silicon (index 3.43) as the incident medium. The trajectory of r along the amplitude circles is clockwise as the film thickness increases.

Fig. 2
Fig. 2

Transverse electric (TE) and transverse magnetic (TM) interface reflectances (ρE and ρM, respectively) on the complex plane for silicon incident on alumina: ×, interface reflectance at 5° increments of incident angle.

Fig. 3
Fig. 3

(a) TE and (b) TM circle diagrams for an alumina layer with silicon as the incident medium at 45° angle of incidence. The trajectory of r along the amplitude circles is toward ρ as the film thickness increases. Isothickness curves are shown for d/λ divisible by 0.01.

Fig. 4
Fig. 4

Combined polarizations circle diagram for an alumina layer with silicon as the incident medium at 35.4° angle of incidence. The trajectory of r along the amplitude circles is toward ρE for TE polarized light and toward ρM for TM polarized light. Isothickness curves are shown for d/λ divisible by 0.01.

Fig. 5
Fig. 5

Tunnel bandpass transmitting wavelengths around λo. The present design assumes θo = 35.4°.

Fig. 6
Fig. 6

Circle diagram for a silicon layer with silicon as the incident medium for any polarization and angle of incidence. The trajectory of r along the amplitude circles is clockwise as the film thickness increases.

Fig. 7
Fig. 7

Stack diagram for a three-layer TE bandpass filter inside silicon at 35.4° angle of incidence. rj represents the stack reflectance after j layers have been applied. The reflectance traced out by layer 2 makes a complete revolution from r1 to r1, then continues to r2.

Fig. 8
Fig. 8

TM stack diagram for the TE bandpass of Fig. 7. The reflectance traced out by layer 2 makes a complete revolution from r1 to r1, then continues to r2.

Fig. 9
Fig. 9

Spectral performance of the three-layer TE bandpass of Fig. 7.

Fig. 10
Fig. 10

(a) TE and (b) TM stack diagrams for a five-layer bandpass filter inside silicon for both polarizations at 35.4° angle of incidence.

Fig. 11
Fig. 11

Spectral performance of the five-layer bandpass of Fig. 10.

Fig. 12
Fig. 12

Spectral performance of two periods of the five-layer bandpass of Fig. 10, with compensator layers adjusted to make the center wavelengths of the two polarizations coincide.

Fig. 13
Fig. 13

(a) TE and (b) TM stack diagrams for a five-layer dichroic beam splitter inside silicon for both polarizations at 35.4° angle of incidence.

Fig. 14
Fig. 14

Spectral performance of the five-layer dichroic beam splitter of Fig. 13.

Fig. 15
Fig. 15

(a) TE and (b) TM stack diagrams for a five-layer dichroic beam splitter inside silicon for both polarizations at 45° angle of incidence.

Fig. 16
Fig. 16

Spectral performance of the dichroic beam splitter of Fig. 15, after adding a second period, substituting 3.28 as the index of thin film silicon, and optimizing. The angle of incidence is 45°.

Equations (8)

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r = ρ + r sub τ 2 1 + ρ r sub τ 2 ,
τ = exp [ i 2 π n ( cos θ ) d / λ ] ,
n sin θ = n o sin θ o ,
ρ = ( η o η ) / ( η o + η ) ,
η = n · cos θ for the transverse electric ( TE ) polarization , or η = n / cos θ for the transverse magnetic ( TM ) polarization .
r = ρ ( 1 ρ r s ) + ( r s ρ ) τ 2 1 ρ r s + ρ ( r s ρ ) τ 2 .
θ g = cos 1 [ ( n o 2 n 2 ) / ( n o 2 + n 2 ) ] 1 / 2 .
ϕ M θ o = ϕ E θ o ,

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