Abstract

An approximate calculation method for light propagation in random multilayer films is presented. It is applied to a particular structure consisting of alternating lower- and higher-refractive-index materials with one type of layer having random thickness. An analytical expression for the localization length is derived. It is found to be in excellent agreement over a broad wavelength range, with numerical calculations performed by use of the transfer matrix formalism without any simplifying assumptions. Furthermore, this approximation accounts very well for anomalous reflectance effects that have been reported in experimental studies of amorphous silicon–silicon nitride multilayer films with random-thickness layers. Within the approximation presented, one can identify separate terms that are responsible for localization and for anomalous reflectance. This separation is helpful in clarifying the origins of both effects.

© 1994 Optical Society of America

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References

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  1. S. Nitta, T. Itoh, S. Nonomura, H. Ohta, K. Morigaki, “Optical properties of amorphous semiconductor multilayer films with random-thickness well layers,” Philos. Mag. B 60, 119–125 (1989).
    [CrossRef]
  2. A. Kondilis, P. Tzanetakis, “Reflectance of multilayer amorphous semiconductor films with random thickness layers,” Philos. Mag. Lett. 62, 299–304 (1990).
    [CrossRef]
  3. A. Kondilis, P. Tzanetakis, “Numerical calculations on optical localization in multilayer structures with random-thickness layers,” Phys. Rev. B 46, 15426–15431 (1992).
    [CrossRef]
  4. S. Nitta, S. Takeuchi, K. Ogawa, T. Furukawa, T. Itoh, S. Nonomura, “Anomalous reflectance in random amorphous multilayers a-Si:H/Si1−xNxand classical localization of light?” J. Non-Cryst. Solids 137&138, 1095–1098 (1991).
    [CrossRef]
  5. Z. Knittl, Optics of Thin Films (Wiley, New York, 1976), Chap. 2, pp. 47–52.
  6. J. E. Sipe, P. Sheng, B. S. White, M. H. Cohen, “Brewster anomalies: a polarization-induced delocalization effect,” Phys. Rev. Lett. 60, 108–111 (1988).
    [CrossRef] [PubMed]
  7. P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Wave localization and multiple scattering in randomly-layered media,” in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1990), pp. 563–619.
    [CrossRef]
  8. P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Minimum wave-localization length in a one-dimensional random medium,” Phys. Rev. B 34, 4757–4761 (1986).
    [CrossRef]

1992

A. Kondilis, P. Tzanetakis, “Numerical calculations on optical localization in multilayer structures with random-thickness layers,” Phys. Rev. B 46, 15426–15431 (1992).
[CrossRef]

1991

S. Nitta, S. Takeuchi, K. Ogawa, T. Furukawa, T. Itoh, S. Nonomura, “Anomalous reflectance in random amorphous multilayers a-Si:H/Si1−xNxand classical localization of light?” J. Non-Cryst. Solids 137&138, 1095–1098 (1991).
[CrossRef]

1990

A. Kondilis, P. Tzanetakis, “Reflectance of multilayer amorphous semiconductor films with random thickness layers,” Philos. Mag. Lett. 62, 299–304 (1990).
[CrossRef]

1989

S. Nitta, T. Itoh, S. Nonomura, H. Ohta, K. Morigaki, “Optical properties of amorphous semiconductor multilayer films with random-thickness well layers,” Philos. Mag. B 60, 119–125 (1989).
[CrossRef]

1988

J. E. Sipe, P. Sheng, B. S. White, M. H. Cohen, “Brewster anomalies: a polarization-induced delocalization effect,” Phys. Rev. Lett. 60, 108–111 (1988).
[CrossRef] [PubMed]

1986

P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Minimum wave-localization length in a one-dimensional random medium,” Phys. Rev. B 34, 4757–4761 (1986).
[CrossRef]

Cohen, M. H.

J. E. Sipe, P. Sheng, B. S. White, M. H. Cohen, “Brewster anomalies: a polarization-induced delocalization effect,” Phys. Rev. Lett. 60, 108–111 (1988).
[CrossRef] [PubMed]

Furukawa, T.

S. Nitta, S. Takeuchi, K. Ogawa, T. Furukawa, T. Itoh, S. Nonomura, “Anomalous reflectance in random amorphous multilayers a-Si:H/Si1−xNxand classical localization of light?” J. Non-Cryst. Solids 137&138, 1095–1098 (1991).
[CrossRef]

Itoh, T.

S. Nitta, S. Takeuchi, K. Ogawa, T. Furukawa, T. Itoh, S. Nonomura, “Anomalous reflectance in random amorphous multilayers a-Si:H/Si1−xNxand classical localization of light?” J. Non-Cryst. Solids 137&138, 1095–1098 (1991).
[CrossRef]

S. Nitta, T. Itoh, S. Nonomura, H. Ohta, K. Morigaki, “Optical properties of amorphous semiconductor multilayer films with random-thickness well layers,” Philos. Mag. B 60, 119–125 (1989).
[CrossRef]

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976), Chap. 2, pp. 47–52.

Kondilis, A.

A. Kondilis, P. Tzanetakis, “Numerical calculations on optical localization in multilayer structures with random-thickness layers,” Phys. Rev. B 46, 15426–15431 (1992).
[CrossRef]

A. Kondilis, P. Tzanetakis, “Reflectance of multilayer amorphous semiconductor films with random thickness layers,” Philos. Mag. Lett. 62, 299–304 (1990).
[CrossRef]

Morigaki, K.

S. Nitta, T. Itoh, S. Nonomura, H. Ohta, K. Morigaki, “Optical properties of amorphous semiconductor multilayer films with random-thickness well layers,” Philos. Mag. B 60, 119–125 (1989).
[CrossRef]

Nitta, S.

S. Nitta, S. Takeuchi, K. Ogawa, T. Furukawa, T. Itoh, S. Nonomura, “Anomalous reflectance in random amorphous multilayers a-Si:H/Si1−xNxand classical localization of light?” J. Non-Cryst. Solids 137&138, 1095–1098 (1991).
[CrossRef]

S. Nitta, T. Itoh, S. Nonomura, H. Ohta, K. Morigaki, “Optical properties of amorphous semiconductor multilayer films with random-thickness well layers,” Philos. Mag. B 60, 119–125 (1989).
[CrossRef]

Nonomura, S.

S. Nitta, S. Takeuchi, K. Ogawa, T. Furukawa, T. Itoh, S. Nonomura, “Anomalous reflectance in random amorphous multilayers a-Si:H/Si1−xNxand classical localization of light?” J. Non-Cryst. Solids 137&138, 1095–1098 (1991).
[CrossRef]

S. Nitta, T. Itoh, S. Nonomura, H. Ohta, K. Morigaki, “Optical properties of amorphous semiconductor multilayer films with random-thickness well layers,” Philos. Mag. B 60, 119–125 (1989).
[CrossRef]

Ogawa, K.

S. Nitta, S. Takeuchi, K. Ogawa, T. Furukawa, T. Itoh, S. Nonomura, “Anomalous reflectance in random amorphous multilayers a-Si:H/Si1−xNxand classical localization of light?” J. Non-Cryst. Solids 137&138, 1095–1098 (1991).
[CrossRef]

Ohta, H.

S. Nitta, T. Itoh, S. Nonomura, H. Ohta, K. Morigaki, “Optical properties of amorphous semiconductor multilayer films with random-thickness well layers,” Philos. Mag. B 60, 119–125 (1989).
[CrossRef]

Papanicolaou, G.

P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Minimum wave-localization length in a one-dimensional random medium,” Phys. Rev. B 34, 4757–4761 (1986).
[CrossRef]

P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Wave localization and multiple scattering in randomly-layered media,” in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1990), pp. 563–619.
[CrossRef]

Sheng, P.

J. E. Sipe, P. Sheng, B. S. White, M. H. Cohen, “Brewster anomalies: a polarization-induced delocalization effect,” Phys. Rev. Lett. 60, 108–111 (1988).
[CrossRef] [PubMed]

P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Minimum wave-localization length in a one-dimensional random medium,” Phys. Rev. B 34, 4757–4761 (1986).
[CrossRef]

P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Wave localization and multiple scattering in randomly-layered media,” in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1990), pp. 563–619.
[CrossRef]

Sipe, J. E.

J. E. Sipe, P. Sheng, B. S. White, M. H. Cohen, “Brewster anomalies: a polarization-induced delocalization effect,” Phys. Rev. Lett. 60, 108–111 (1988).
[CrossRef] [PubMed]

Takeuchi, S.

S. Nitta, S. Takeuchi, K. Ogawa, T. Furukawa, T. Itoh, S. Nonomura, “Anomalous reflectance in random amorphous multilayers a-Si:H/Si1−xNxand classical localization of light?” J. Non-Cryst. Solids 137&138, 1095–1098 (1991).
[CrossRef]

Tzanetakis, P.

A. Kondilis, P. Tzanetakis, “Numerical calculations on optical localization in multilayer structures with random-thickness layers,” Phys. Rev. B 46, 15426–15431 (1992).
[CrossRef]

A. Kondilis, P. Tzanetakis, “Reflectance of multilayer amorphous semiconductor films with random thickness layers,” Philos. Mag. Lett. 62, 299–304 (1990).
[CrossRef]

White, B.

P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Minimum wave-localization length in a one-dimensional random medium,” Phys. Rev. B 34, 4757–4761 (1986).
[CrossRef]

P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Wave localization and multiple scattering in randomly-layered media,” in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1990), pp. 563–619.
[CrossRef]

White, B. S.

J. E. Sipe, P. Sheng, B. S. White, M. H. Cohen, “Brewster anomalies: a polarization-induced delocalization effect,” Phys. Rev. Lett. 60, 108–111 (1988).
[CrossRef] [PubMed]

Zhang, Z. Q.

P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Minimum wave-localization length in a one-dimensional random medium,” Phys. Rev. B 34, 4757–4761 (1986).
[CrossRef]

P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Wave localization and multiple scattering in randomly-layered media,” in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1990), pp. 563–619.
[CrossRef]

J. Non-Cryst. Solids

S. Nitta, S. Takeuchi, K. Ogawa, T. Furukawa, T. Itoh, S. Nonomura, “Anomalous reflectance in random amorphous multilayers a-Si:H/Si1−xNxand classical localization of light?” J. Non-Cryst. Solids 137&138, 1095–1098 (1991).
[CrossRef]

Philos. Mag. B

S. Nitta, T. Itoh, S. Nonomura, H. Ohta, K. Morigaki, “Optical properties of amorphous semiconductor multilayer films with random-thickness well layers,” Philos. Mag. B 60, 119–125 (1989).
[CrossRef]

Philos. Mag. Lett.

A. Kondilis, P. Tzanetakis, “Reflectance of multilayer amorphous semiconductor films with random thickness layers,” Philos. Mag. Lett. 62, 299–304 (1990).
[CrossRef]

Phys. Rev. B

A. Kondilis, P. Tzanetakis, “Numerical calculations on optical localization in multilayer structures with random-thickness layers,” Phys. Rev. B 46, 15426–15431 (1992).
[CrossRef]

P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Minimum wave-localization length in a one-dimensional random medium,” Phys. Rev. B 34, 4757–4761 (1986).
[CrossRef]

Phys. Rev. Lett.

J. E. Sipe, P. Sheng, B. S. White, M. H. Cohen, “Brewster anomalies: a polarization-induced delocalization effect,” Phys. Rev. Lett. 60, 108–111 (1988).
[CrossRef] [PubMed]

Other

P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Wave localization and multiple scattering in randomly-layered media,” in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1990), pp. 563–619.
[CrossRef]

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976), Chap. 2, pp. 47–52.

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Figures (5)

Fig. 1
Fig. 1

Probability density functions of ln(E0) for ML’s of 100, 500, and 1000 cells. The smooth curves are Gaussian distribution functions fitted to the data points (jagged lines). A set of 10,000 random sequences is used in each case. E0 is computed by the TM method. The optical path in the barrier layer is equal to that in the average-thickness well layer. The random thickness of the well layers obeys a Gaussian distribution. The standard deviation σw of the well layer thickness, expressed in units of average cell thickness, is taken equal to 10/60. cn = 2 and φ = 0.2π. Cn is the ratio of the refractive index of barrier layers to that of the well layers. φ is the phase shift inside any barrier layer.

Fig. 2
Fig. 2

Ensemble average of ln(E0) plotted versus the number of cells N. Curve A corresponds to one random sequence; curves B and C correspond to ensembles of 10 and 100 sequences, respectively. All other parameters are identical to those of Fig. 1.

Fig. 3
Fig. 3

Localization length l in units of average cell thickness, plotted as a function of the phase shift inside a barrier layer. Squares correspond to exact numerical calculations and solid curves to the results of relation (11). The optical path in the barrier layers is equal to that in the average-thickness well layer. The corresponding σw’s are shown with each curve. cn = 2.

Fig. 4
Fig. 4

Reflectance spectra and vector-approximation function (VAf) versus wavelength of light for a random-well-thickness ML consisting of 400 well layers alternating with 401 barrier ones. Graph I shows reflectance in the exact numerical calculation. The reflectance spectrum in the VA is shown in graph II. In graph III the VAf is plotted versus wavelength. All barrier layers have the same thickness (4.0 nm). Well layer thicknesses have been randomly chosen from a Gaussian distribution centered at 2.0 nm and having 1.0 nm standard deviation. The wavelength dependence of the complex refractive indices for the two types of layer is shown in Fig. 5. We assume a nonabsorbing substrate with refractive index ns = 1.5.

Fig. 5
Fig. 5

Optical properties of the well, a-Si:H, and barrier, a-SiN:H, layers used for the calculations presented in Fig. 4. The upper graph shows the real parts, and the lower one shows the imaginary parts, of the refractive indices of the two materials as functions of wavelength in the region of interest. These functions closely follow the actual ones of the a-Si:H/a-Si1−xNx:H films studied experimentally.1

Equations (19)

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P N ( ln E 0 ) = 1 2 π σ exp [ ( ln E 0 ln E 0 2 σ ) 2 ] ,
σ r = σ ln E 0 = 1 ( ln E 0 ) 1 / 2 .
l 1 = ln E 0 ( N ) N ,
r υ = j = 1 N + 1 r j exp ( 2 i k = 1 j ϕ k 1 ) , ϕ 0 = 0 .
r = 0 1 ( r j ) + 0 3 ( r j r k r m ) + .
E 0 ( N ) = ( n s n b ) 1 / 2 | 1 + r N | ( 1 | r N | 2 ) 1 / 2 ,
ln E 0 ( N ) = 1 2 ln n s n b + Re ( r N ) + 1 2 [ | r N | 2 Re ( r N 2 ) ] .
l = | 1 + X | 2 2 r 0 2 sin 2 φ ( 1 | X | 2 ) ,
X = exp ( 2 i φ ) exp ( 2 i φ w , j ) .
r 0 = 1 c n 1 + c n .
l = 1 + exp ( 4 σ φ w 2 ) 2 exp ( 2 σ φ w 2 ) cos 2 φ c ( 2 r 0 2 sin 2 φ ) [ 1 exp ( 4 σ φ w 2 ) ] , φ c = φ + φ w , j .
l = 1 2 r 0 2 φ c 2 φ 2 1 σ φ w 2 .
l = 1 / [ 2 r 0 2 sin 2 ( φ ) ] .
l min = 1 / 2 r 0 2 .
r = r 1 + r exp ( 2 i φ 1 ) 1 + r 1 r exp ( 2 i φ 1 ) ,
R = | r | 2 .
R = { ( 1 n b ) 2 ( 1 + n b ) 2 + 8 n b ( 1 n b ) ( n b n s ) ( 1 n b ) 3 ( n b + n s ) × Re [ exp ( 2 i k L ) ] } + 8 n b ( n b 1 ) ( 1 + n b ) 3 r υ 0 ,
r υ 0 = Re [ r υ exp ( 2 i k 0 n b d b ) ] .
k = k 0 L 0 L n ( ξ ) d ξ .

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