Abstract

It is well known that the spectral transmittance and reflectance of a thin film can be influenced by even small inhomogeneities or variations in its complex refractive-index profile. Formulas are derived that describe the theoretical variation of the spectral characteristics for small changes in the refractive index and the extinction coefficient of a homogeneous thin film. These formulas, accurate to the first order in the change in the complex refractive index, are compared with exact calculations for a number of different types of inhomogeneities. It is shown that specific qualitative features in the refractive-index profile of a nearly homogeneous thin film frequently can be determined from an examination of the change in the spectral transmittance and reflectance at normal incidence.

© 1997 Optical Society of America

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References

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  1. J. A. Dobrowolski, P. G. Verly, eds., Inhomogeneous and quasi-inhomogeneous coatings, Proc. SPIE2046 (1993).
  2. R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. 5, Sec. 2.
  3. R. Jacobsson, “Inhomogeneous and co-evaporated homogeneous films for optical applications,” in Physics of Thin Films, G. Hass, M. H. Francombe, R. W. Hoffman, eds. (Academic, New York, 1975), Vol. 8, pp. 51–98.
  4. P. S. Epstein, “Reflection of waves in an inhomogeneous absorbing medium,” Proc. Natl. Acad. Sci. 17, 627–637 (1930).
    [CrossRef]
  5. S. F. Monaco, “Reflectance of an inhomogeneous thin film,” J. Opt. Soc. Am. 51, 280–282 (1961).
    [CrossRef]
  6. Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).
  7. G. Koppelmann, K. Krebs, “Die optischen Eigenschaften dielektrischer Schichten mit kleinen Homogenitätsstörungen,” Z. Phys. 164, 539–556 (1961).
    [CrossRef]
  8. H. Schröder, “Bemerkung zur Theorie des Lichtdurchgangs durch inhomogene durchsichtige Schichten,” Ann. Phys. 5th Ser. 39, 55–58 (1941).
    [CrossRef]
  9. J. P. Borgogno, B. Lazarides, E. Pelletier, “Automatic determination of optical constants of inhomogeneous thin films,” Appl. Opt. 21, 4020–4029 (1982).
    [CrossRef] [PubMed]
  10. I. V. Zuev, A. V. Tikhonravov, “The uniqueness of the determination of the parameters of a stratified medium from the energy reflectance,” Computat. Math. Math. Phys. 33, 387–395 (1993).
  11. Sh. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvett, 1992).

1993

I. V. Zuev, A. V. Tikhonravov, “The uniqueness of the determination of the parameters of a stratified medium from the energy reflectance,” Computat. Math. Math. Phys. 33, 387–395 (1993).

1982

1961

S. F. Monaco, “Reflectance of an inhomogeneous thin film,” J. Opt. Soc. Am. 51, 280–282 (1961).
[CrossRef]

G. Koppelmann, K. Krebs, “Die optischen Eigenschaften dielektrischer Schichten mit kleinen Homogenitätsstörungen,” Z. Phys. 164, 539–556 (1961).
[CrossRef]

1941

H. Schröder, “Bemerkung zur Theorie des Lichtdurchgangs durch inhomogene durchsichtige Schichten,” Ann. Phys. 5th Ser. 39, 55–58 (1941).
[CrossRef]

1930

P. S. Epstein, “Reflection of waves in an inhomogeneous absorbing medium,” Proc. Natl. Acad. Sci. 17, 627–637 (1930).
[CrossRef]

Borgogno, J. P.

Epstein, P. S.

P. S. Epstein, “Reflection of waves in an inhomogeneous absorbing medium,” Proc. Natl. Acad. Sci. 17, 627–637 (1930).
[CrossRef]

Furman, Sh.

Sh. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvett, 1992).

Jacobsson, R.

R. Jacobsson, “Inhomogeneous and co-evaporated homogeneous films for optical applications,” in Physics of Thin Films, G. Hass, M. H. Francombe, R. W. Hoffman, eds. (Academic, New York, 1975), Vol. 8, pp. 51–98.

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. 5, Sec. 2.

Knittl, Z.

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

Koppelmann, G.

G. Koppelmann, K. Krebs, “Die optischen Eigenschaften dielektrischer Schichten mit kleinen Homogenitätsstörungen,” Z. Phys. 164, 539–556 (1961).
[CrossRef]

Krebs, K.

G. Koppelmann, K. Krebs, “Die optischen Eigenschaften dielektrischer Schichten mit kleinen Homogenitätsstörungen,” Z. Phys. 164, 539–556 (1961).
[CrossRef]

Lazarides, B.

Monaco, S. F.

Pelletier, E.

Schröder, H.

H. Schröder, “Bemerkung zur Theorie des Lichtdurchgangs durch inhomogene durchsichtige Schichten,” Ann. Phys. 5th Ser. 39, 55–58 (1941).
[CrossRef]

Tikhonravov, A. V.

I. V. Zuev, A. V. Tikhonravov, “The uniqueness of the determination of the parameters of a stratified medium from the energy reflectance,” Computat. Math. Math. Phys. 33, 387–395 (1993).

Sh. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvett, 1992).

Zuev, I. V.

I. V. Zuev, A. V. Tikhonravov, “The uniqueness of the determination of the parameters of a stratified medium from the energy reflectance,” Computat. Math. Math. Phys. 33, 387–395 (1993).

Ann. Phys. 5th Ser.

H. Schröder, “Bemerkung zur Theorie des Lichtdurchgangs durch inhomogene durchsichtige Schichten,” Ann. Phys. 5th Ser. 39, 55–58 (1941).
[CrossRef]

Appl. Opt.

Computat. Math. Math. Phys.

I. V. Zuev, A. V. Tikhonravov, “The uniqueness of the determination of the parameters of a stratified medium from the energy reflectance,” Computat. Math. Math. Phys. 33, 387–395 (1993).

J. Opt. Soc. Am.

Proc. Natl. Acad. Sci.

P. S. Epstein, “Reflection of waves in an inhomogeneous absorbing medium,” Proc. Natl. Acad. Sci. 17, 627–637 (1930).
[CrossRef]

Z. Phys.

G. Koppelmann, K. Krebs, “Die optischen Eigenschaften dielektrischer Schichten mit kleinen Homogenitätsstörungen,” Z. Phys. 164, 539–556 (1961).
[CrossRef]

Other

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

J. A. Dobrowolski, P. G. Verly, eds., Inhomogeneous and quasi-inhomogeneous coatings, Proc. SPIE2046 (1993).

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. 5, Sec. 2.

R. Jacobsson, “Inhomogeneous and co-evaporated homogeneous films for optical applications,” in Physics of Thin Films, G. Hass, M. H. Francombe, R. W. Hoffman, eds. (Academic, New York, 1975), Vol. 8, pp. 51–98.

Sh. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvett, 1992).

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Figures (7)

Fig. 1
Fig. 1

Schematic of an inhomogeneous layer.

Fig. 2
Fig. 2

Measured transmittance of an electron-beam-deposited ZrO2 thin film. Note the modulation of the transmittance minima, indicating that there is an inhomogeneity present in the film.

Fig. 3
Fig. 3

Comparison of the results of exact calculations (solid curves) and approximate formulas (dashed curves) of the variation in transmittance for nonabsorbing layers. View A represents high-index inhomogeneity at the substrate boundary; view B represents linear inhomogeneity along the film thickness; and view C represents low-index inhomogeneity at the ambient medium boundary.

Fig. 4
Fig. 4

Comparison of the results of exact calculations (solid curves) and approximate formulas (dashed curves) of a homogeneous film with a constant extinction coefficient, χ = 0.001. Variation in view A is reflectance and in view B is transmittance.

Fig. 5
Fig. 5

Effects of different types of inhomogeneities on the spectral transmittance: column 1, schematic refractive-index profile representing the inhomogeneity (for all examples, z a = 700 nm, z 1 = z a - z 2 = 30 nm); column 2, exact calculations of spectral transmittance are solid curves, envelopes for inhomogeneous layers are dashed curves, envelopes for homogeneous layers are dotted lines; column 3, equations for inhomogeneous layer envelopes. The rows are view A, linear inhomogeneity; views B and C, positive and negative inhomogeneity, respectively, at the substrate boundary; views D and E, positive and negative inhomogeneity, respectively, at the ambient medium boundary; view F, negative inhomogeneities at both the substrate and ambient medium boundaries.

Fig. 6
Fig. 6

Schematic refractive-index profile and the calculated spectral transmittance curve and envelopes for a film in which z a = 700 nm, z 1 = 60 nm, and z a - z 2 = 30 nm. The dotted lines represent the envelopes of a homogeneous layer.

Fig. 7
Fig. 7

Effect of absorption; comparison of spectral transmittance based on exact calculation (solid curve) and on Eq. (41)(dashed curve) for an extinction coefficient χ = 0.001.

Equations (58)

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rλ=nanino1/2-nsnoni1/2cos φ+inansnoni1/2-noni1/2sin φnanino1/2+nsnoni1/2cos ϕ+inansnoni1/2+noni1/2sin φ.
φ=2πλ0zanzdz,
n=1za0zanzdz.
δ=no-nino+ni,
no/ni1/21+δ,ni/no1/21-δ.
rλ=na-ns+δna+nscos φ+inansn-nsin φna+ns+δna-nscos φ+inansn+nsin φ.
λm=2nzdm, m=1, 2,.
r=ras+rio1+rasrio,
Tλ=nsnatλ2,
Rλ=rλ2,
tλ=2nana+nscos φ+inansn+nsin φ,
rλ=na-nscos φ+inansn-nsin φna+nscos φ+inansn+nsin φ
φ=2πλnza.
η˜z=ηz-iχz,
δRλ=-4πλnnsTλ0zaIm×rλcos2πnzλ+insnsin2πnzλ2η˜zdz,
δTλ=-δRλ+4πλnns Tλ0zacos2πnzλ+insnsin2πnzλ2Imη˜zdz.
rλ=ns-nacos φ+inansn-nsin φns+nacos φ+inansn+nsin φ
δTλ=-δRλ=4πλnnsTλ×0zaImrλcos2πnzλ+insnsin2πnzλ2ηzdz.
δRaλ=4πλnnsTλ0zaRerλcos2πnzλ+insnsin2πnzλ2χzdz,
Taλ=-δRaλ-4πλnnsTλ0zacos2πnzλ+insnsin2πnzλ2χzdz.
λe=4nzam, m=1, 2,,
Imrλe=0
Rerλe=ns-nans+na
Rerλe=nans-n2nans+n2
Iλ=0zaImrλ121-ns2n2+12×1+ns2n2cos4πnzλ+Rerλnsnsin4πnzλ×ηzdz.
Iλe=λens4πn2Rerλeη0-cos4πnzaλeηza.
δTλmax=Tλmaxnns-nans+naη0-ηza,
δTλmin=Tλminnnans-n2nans+n2η0+ηza.
cos2πnzλ+insnsin2πnzλ21+i2πnszλ21+i4πnszλ.
δTk=-δRk4πλnnsTkImrk0z1ηzdz+4πλnsRerk0z1zηzdz.
δTλmax=16π2Tλmaxnλmax2ns-nans+na0zlzηzdz,
δTλmin=16π2Tλminnλmin2nans-n2nans+n20zlzηzdz.
δTλ=-δRλ=4πλnnaTk Imrλ0zacos2πnza-zλ+inansin2πnza-zλ2ηzdz.
cos2πnza-zλ+inansin2πnza-zλ21+i4πnaza-zλ.
δTλ=-δRλ4πλnnaTλImrλz2zaηzdz+4πλnaRerλz2zaza-zηzdz.
δTλmax=16π2Tλmaxnλmax2na-nsna+ns×z2zaza-zηzdz,
δTλmin=16π2Tλminnλmin2nans-n2nans+n2×z2zaza-zηzdz.
δRaλ2πλnns-nsnTλRerλ0zaχzdz,
δTaλ-δRaλ-2πλnns+nsnTλ0Zaχzdz.
δA(λmaxδRaλ=n2+ns2n2-ns21maxrλ=9.5.
δTaλ-δAλ=-2πλnns+nsnTλ0zaχzdz.
ñz, λ=nz, λ-iχz, λ,
nz, λ=qznλ,
ñz, λ=qznλ-iχλ.
dudz=ikv, dvdz=ikn2u,
u0, k=1, v0, k=ns,
rk=nauza, k-vza, knauza, k+vza, k,
Rk=rk2=rkr*k.
dhdz=ikg, dgdz=ikn2h+2iknuη˜z,
h0, k=g0, k=0.
δrk=12tk1-rkhza, k-12natk[1+rk[gza, k.
δRk=2Rer*kδrk.
dφdz=-ikn2ψ,dψdz=-ikφ,
φza,k=r*ktk1-rk,ψzak=-1nar*ktk1+rk.
δRk=Rehza, kφza, k+gza, kψza, k.
ddzhφ+gψ=2iknuψη˜.
hza, kφza, k+gza, kψza, k=2ikn0zauz, kψz, kη˜(z)dz.
δRk=-2kn Im0Zauz, kψz, kη˜zdz.

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