Page 1

Transfer-matrix formalism for the calculation of

optical response in multilayer systems: from

coherent to incoherent interference

M. Claudia Troparevsky,1,2* Adrian S. Sabau,2 Andrew R. Lupini,2 and Zhenyu

Zhang2,1,3

1Department of Physics and Astronomy, the University of Tennessee, Knoxville, Tennessee 37996, USA

2Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

3ICQD, University of Science and Technology of China, Hefei, Anhui, 230026, China

*mtropare@utk.edu

Abstract: We present a novel way to account for partially coherent

interference in multilayer systems via the transfer-matrix method. The

novel feature is that there is no need to use modified Fresnel coefficients or

the square of their amplitudes to work in the incoherent limit. The transition

from coherent to incoherent interference is achieved by introducing a

random phase of increasing intensity in the propagating media. This random

phase can simulate the effect of defects or impurities. This method provides

a general way of dealing with optical multilayer systems, in which coherent

and incoherent interference are treated on equal footing.

Notice: This manuscript has been authored by UT-Battelle, LLC, under Contract No.

DE-AC05-00OR22725 with the U.S. Department of Energy. The United States

Government retains and the publisher, by accepting the article for publication,

acknowledges that the United States Government retains a non-exclusive, paid-up,

irrevocable, world-wide license to publish or reproduce the published form of this

manuscript, or allow others to do so, for United States Government purposes.

©2010 Optical Society of America

OCIS codes: (030.0030) Coherence and statistical optics; (310.0310) Thin films.

References and links

1. O. Oladeji, and L. Chow, “Synthesis and processing of CdS/ZnS multilayer films for solar cell application,” Thin

Solid Films 474(1-2), 77–83 (2005).

2. D. R. Sahu, S. Y. Lin, and J. L. Huang, “Deposition of Ag-based Al-doped ZnO multilayer coatings for the

transparent conductive electrodes by electron beam evaporation,” Sol. Energy Mater. Sol. Cells 91(9), 851–855

(2007).

3. R. K. Gupta, K. Ghosh, R. Patel, and P. K. Kahol, “Properties of ZnO/W-doped In2O3/ZnO multilayer thin films

deposited at different growth temperatures,” J. Phys. D Appl. Phys. 41(21), 215309 (2008).

4. H. Cho, C. Yun, and S. Yoo, “Multilayer transparent electrode for organic light-emitting diodes: tuning its

optical characteristics,” Opt. Express 18(4), 3404–3414 (2010).

5. H. A. Atwater, and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213

(2010).

6. W. Wang, S. Wu, K. Reinhardt, Y. Lu, and S. Chen, “Broadband light absorption enhancement in thin-film

silicon solar cells,” Nano Lett. 10(6), 2012–2018 (2010).

7. X. L. Ruan, and M. Kaviany, “Photon localization and electromagnetic field enhancement in laser-irradiated,

random porous media,” Microscale Thermophys. Eng. 9(1), 63–84 (2005).

8. C. C. Katsidis, D. I. Siapkas, D. Panknin, N. Hatzopoulos, and W. Skorupa, “General transfer-matrix method for

optical multilayer systems with coherent, partially coherent,and incoherent interference,” Microelectron. Eng.

28, 439 (1995).

9. X. L. Ruan, and M. Kaviany, “Enhanced nonradiative relaxation and photoluminescence quenching in random,

doped nanocrystalline powders,” J. Appl. Phys. 97(10), 104331 (2005).

10. S. Logothetidis, and G. Stergioudis, “Studies of density and surface roughness of ultrathin amorphous carbon

films with regards to thickness with x-ray reflectometry and spectroscopic ellipsometry,” Appl. Phys. Lett.

71(17), 2463 (1997).

11. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

12. Z. Knittl, Optics of Thin Films: An Optical Multilayer Theory (Wiley, London, 1976).

#135997 - $15.00 USD

(C) 2010 OSA

Received 11 Oct 2010; revised 24 Oct 2010; accepted 25 Oct 2010; published 10 Nov 2010

22 November 2010 / Vol. 18, No. 24 / OPTICS EXPRESS 24715

Page 2

13. O. S. Heavens, Optical Properties of Thin Films (Dover, New York, 1965).

14. K. Ohta, and H. Ishida, “Matrix formalism for calculation of electric field intensity of light in stratified

multilayered films,” Appl. Opt. 29(13), 1952–1959 (1990).

15. E. Centurioni, “Generalized matrix method for calculation of internal light energy flux in mixed coherent and

incoherent multilayers,” Appl. Opt. 44(35), 7532–7539 (2005).

16. C. L. Mitsas, and D. I. Siapkas, “Generalized matrix method for analysis of coherent and incoherent reflectance

and transmittance of multilayer structures with rough surfaces, interfaces, and finite substrates,” Appl. Opt.

34(10), 1678 (1995).

17. J. S. C. Prentice, “Coherent, partially coherent and incoherent light absorption in thin-film multilayer structures,”

J. Phys. D Appl. Phys. 33(24), 3139–3145 (2000).

18. C. C. Katsidis, and D. I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent,

partially coherent, and incoherent interference,” Appl. Opt. 41(19), 3978–3987 (2002).

19. J. Cho, J. Hong, K. Char, and F. Caruso, “Nanoporous block copolymer micelle/micelle multilayer films with

dual optical properties,” J. Am. Chem. Soc. 128(30), 9935–9942 (2006).

20. S. F. Rowlands, J. Livingstone, and C. P. Lund, “Optical modelling of thin film solar cells with textured

interfaces using the effective medium approximation,” Sol. Energy 76(1-3), 301–307 (2004).

The study of multilayer films has gained increasing interest in recent years due to their many

potential uses as optical coatings and as transparent conductive electrodes in optoelectronic

devices such as flat displays, thin film transistors and solar cells [1–4]. In particular, the

unique optical properties of multilayer films play a vital role in the construction of thin-film

solar cells [5,6], where an important challenge is to increase the absorption of the near-

bandgap light, which would allow a reduction in the thickness of the cell. Consequently, the

ability to predict and tune the optical properties at the semiconductor interface can greatly

contribute to reducing the cost of thin-film solar cells. In addition, multilayer systems have

been used to study photon localization [7], model ion-implanted materials [8,9], and to

determine the thicknesses, densities and roughness of films in combination with X-ray

reflectivity measurements [10]. Therefore, developing a general method that allows the study

of the optical response of multilayer systems is of fundamental importance for designing and

tuning more efficient optoelectronic devices.

The reflectance and transmittance of a multilayer structure can be calculated by using the

transfer matrix method [11–13]. However, the form of the conventional transfer matrix

assumes coherent light propagation, which may lead to narrow oscillations in the calculated

reflectance and transmittance spectra of the system. In practice, due to interference-destroying

effects these oscillations may not be observable. Consequently, in order to have a realistic

description of the optical properties of multilayer systems these interference-destroying

effects should be introduced. There have been previous attempts to modify the transfer matrix

in order to take into account incoherent interference [14,15], as well as partial coherence [16–

18]. However, in the proposed methods the square of the amplitude of the electric field, or

Fresnel coefficients, are used to study the incoherent case. The case of partial coherence

simulates macroscopic surface or interface roughness, and is introduced by multiplying the

Fresnel coefficients by correction factors.

In this article, we present a modified transfer matrix method to calculate the optical

properties of multilayer systems that includes coherent, partially coherent, and incoherent

interference. The modeling of the partially coherent and incoherent cases is done by adding a

random phase that simulates the effects of impurities or defects in the layer. The value of the

random phase can be gradually varied in order to go from coherence to incoherence. The

capabilities of the method are illustrated by presenting calculations of the optical properties of

Si and ZnO films. The physical meaning of the random phase as well as potential methods for

modeling materials with varying impurity profiles and interface roughness are discussed. An

additional advantage of this method is that the partial coherence or incoherence can be

separately/individually introduced for each layer, without having to calculate the intensity

matrix for all the layers as it has been previously implemented.

The transfer matrix method is used to solve Maxwell’s equations in a multilayer system

subject to a uniform incident field E. The field in the medium is divided into two components,

#135997 - $15.00 USD

(C) 2010 OSA

Received 11 Oct 2010; revised 24 Oct 2010; accepted 25 Oct 2010; published 10 Nov 2010

22 November 2010 / Vol. 18, No. 24 / OPTICS EXPRESS 24716

Page 3

the forward (transmitted) component E+ and the backward (reflected) component E- (see Fig.

1).

Fig. 1. Multilayer system composed of N layers with complex refractive indices, ni, and N + 1

interfaces. The sign + and - on the electric field amplitudes, Ei, indicate left- and right-going

waves, respectively. The prime is used for waves at the right hand side of an interface.

The amplitudes of the field at the left- and right-hand side of an interface are related by:

1

1,

1

m

m

mm

mm

E

E

E

E

I

(1)

where,

1,

1,

1,

1,

1

1

1

mm

mm

mm

mm

r

r

t

I

(2)

and tm-1,m and rm-1,m are the transmission and reflection Fresnel coefficients respectively.

The field amplitudes at the left- and right-hand side of the mth layer are related by:

'

'

,

mm

m

mm

E

E

E

E

P

(3)

where,

0

.

0

m

m

i

m

i

e

e

P

(4)

δ is the phase shift due to the wave passing through the film m, and is defined by

2cos,

mmmm

n d

(5)

where σ is the wave number, nm is the complex refractive index of the mth layer, dm is the

thickness of the mth layer, and m is the complex propagation angle following Snell’s law

(n0 sin0 = n1 sin1 = … = nN+1 sinN+1). The above matrix transformations can be applied

for the N layers and N + 1 interfaces resulting in:

11 12

0

11

01 1 12

I PI P I

2 23(1)

2122

011

.

N

N

NN N

NN

T

T

T

T

E

E

E

E

E

E

P I

(6)

T0,(N+1), defined by

0,(1) 01 1 12

I PI P I

2 23(1),

NNN N

TP I

(7)

is the system transfer matrix.

The reflection and transmission coefficients of the multilayer system can be calculated

from the elements of the transfer matrix as follows:

#135997 - $15.00 USD

(C) 2010 OSA

Received 11 Oct 2010; revised 24 Oct 2010; accepted 25 Oct 2010; published 10 Nov 2010

22 November 2010 / Vol. 18, No. 24 / OPTICS EXPRESS 24717

Page 4

0

21

11

0

,

E

E

T

T

r

(8)

'

1

11

0

1.

T

N

E

t

E

(9)

Since the transmission and reflection coefficients are related to the elements of the

transfer matrix, the matrix can be written as follows:

1,0

r

0,(1)

0,1 0,1 1,0

0,11,0

0,1

1

1

.

N

N

NNNNN

N

r

rttr

t

T

(10)

In previous work [18,15], the incoherence was treated by replacing the reflection and

transmission vectors by the squares of their amplitudes. In this way the conventional

(coherent) transfer matrix was replaced by an intensity matrix, as:

2

1,0

0,( 1)

2

222

0,1

0,10,1 1,0

0,1 1,0

1

1

.

N

N

N

NNNNN

r

t

rttrr

T

(11)

The way in which the partial coherence and incoherence are introduced in our method is

as follows. A random phase is added to the phase shift in the selected layer. Therefore, Eq. (5)

is re-written as follows:

2 cos,

mmmm

n d Rand

(12)

where takes values between 0 and , and Rand is a randomly generated number between

1 and 1. The randomly generated numbers are uniformly distributed. The final transmittance

is obtained by averaging the calculated transmittances with different sequences of random

numbers.

The physical meaning of this random phase is to simulate impurities or defects in the

layer, which would introduce a dephasing or loss of coherence such as the one we are

introducing by adding the term

Rand

. If the concentration of defects is large then the loss

of coherence would be complete and the system would be in the completely incoherent case

as it has been dealt with before in Refs. 15 and 18. In our method, the total incoherence is

represented by introducing a random phase with = , while in the partial coherence cases

0< < .

The calculation of the optical response in the incoherent limit is illustrated in Fig. 2. The

figure shows the calculated transmittance vs. wavelength for a crystalline Si film of 150 nm

thickness in air. The curves corresponding to the coherent case, = 0, and the incoherent

limit, = , are displayed. Also, as a reference, the curve for the incoherent limit

calculated from the intensity matrix method is shown. It can be observed that the curve

corresponding to the coherent limit presents oscillations while these oscillations have

vanished in the incoherent limit. In addition, the agreement between this method and the

intensity matrix method can be clearly observed from the superposition of the two curves

representing the incoherent limit.

#135997 - $15.00 USD

(C) 2010 OSA

Received 11 Oct 2010; revised 24 Oct 2010; accepted 25 Oct 2010; published 10 Nov 2010

22 November 2010 / Vol. 18, No. 24 / OPTICS EXPRESS 24718

Page 5

Fig. 2. Transmittance vs. wavelength for a crystalline Si film of 150 nm thickness in the

coherent and incoherent limits. The solid red line corresponds to the coherent limit; the solid

green line corresponds to the incoherent limit calculated using the squares of the amplitudes of

the transmission coefficients; and the dashed blue line corresponds to the incoherent limit

calculated using the added random phase with β = π. A filter of 10 moving averages was used

to smooth the blue curve.

In our calculations of the incoherent limit using the random phase method, a random

phase with = was used. The calculations of the transmittance vs. the wavelength were

repeated 30 times, with 30 sequences of randomly generated numbers, and the results of the

30 runs were averaged to obtain the incoherent curve. The calculated curve (β = π) displayed

in Fig. 2 was further smoothed by using a filter of 10 moving averages. The application of

this filter did not distort the original results and it was only applied to reduce the noise. This

noise is due to the limited number of runs we averaged to obtain the incoherent curve. It can

be clearly observed from Fig. 2 that with only 30 runs the transmittance curve calculated

using the random phase method has converged to the curve calculated using the intensity

matrix. This figure clearly displays that by introducing a random phase one can successfully

simulate the coherence-destroying effects that lead to complete incoherence.

The partial coherence cases can be simulated by introducing a random phase of smaller

magnitude. In these cases a certain loss of coherence is introduced; the partial coherence

cases represent an intermediate situation between the coherent limit and the complete

incoherence. Figure 3 shows the transmittance for a 150 nm film of crystalline Si as a

function of the wavelength. The coherent limit and the complete incoherent limit are shown

as a reference. It can be observed from this plot two intermediate cases of partial coherence,

one with β = π/3 and one with β = π/4. It can be seen that as the magnitude of β decreases the

curves approach that of the complete coherent limit. Therefore, by changing β from π to 0 one

can calculate the transmittance from the complete incoherent case to the coherent limit.

#135997 - $15.00 USD

(C) 2010 OSA

Received 11 Oct 2010; revised 24 Oct 2010; accepted 25 Oct 2010; published 10 Nov 2010

22 November 2010 / Vol. 18, No. 24 / OPTICS EXPRESS 24719

Page 6

Fig. 3. Transmittance vs. wavelength for a crystalline Si film of 150 nm thickness in the

coherent and partially coherent regimes. The dash-dotted red curve corresponds to the coherent

limit. The solid blue and green curves, which correspond to the partially coherent regime, were

calculated using the random phase method with β = π/4 and β = π/3 respectively. As a

reference the incoherent limit is also plotted (dashed green curve). The curves corresponding to

the partial coherence limit were smoothed using a filter of 5 moving averages.

Another interesting feature of this method is the capability of gradually varying the

incoherence degree independently on individual layers. The loss of coherence is introduced in

each layer by introducing a random phase to each phase shift as shown in Eq. (12). In this

case δ1 and δ2, the phase shifts of layer 1 and layer 2 respectively, are written as:

1 1 1

nd

111

2cos,

Rand

(13)

222222

2cos.

n d Rand

(14)

The capability of introducing different incoherence degrees in each layer is displayed in

Fig. 4, where the transmittance of a two layer system is presented. This figure shows the

transmittance of a 150 nm ZnO film on a 150 nm Si film vs. wavelength. It shows the

complete incoherent limit, where

2

= π, the coherent limit where

two intermediate cases. For the first case,

is treated as coherent, and in the second case

last case that the transmittance is rapidly approaching the coherent limit.

1 =

1 =

2

= 0, and

1 = 0 and

1 = 0 and

2

= π, which means that the ZnO layer

= π/2. It can be seen from this

2

#135997 - $15.00 USD

(C) 2010 OSA

Received 11 Oct 2010; revised 24 Oct 2010; accepted 25 Oct 2010; published 10 Nov 2010

22 November 2010 / Vol. 18, No. 24 / OPTICS EXPRESS 24720

Page 7

Fig. 4. Transmittance vs. wavelength for a two layer system consisting of a ZnO film of 150

nm thickness and a crystalline Si film of 150 nm thickness. The black curve represents the

complete incoherent limit; the (dash-dotted) green curve represents a coherent ZnO layer on an

incoherent Si layer; the (dashed) blue curve represents a coherent ZnO layer on a partially

coherent Si layer; and the solid red curve represents the complete coherent limit. The curves

corresponding to the partial coherence and incoherent limit were smoothed using a filter of 5

moving averages (for wavenumbers > 0.6 μm).

This method presents a unique way to study the optical response of multilayer systems

since the transition from coherence to incoherence can be easily achieved. For instance, this

method provides the potential to simulate layered films with impurities or scattering center

defects. Moreover, as the degree of incoherence can be modified in each layer independently,

it can be very useful to study systems with varying profiles of impurity concentrations by

varying the value of β in each film. Also, this method could present an alternative route to

study multilayered nanoporous films with different degrees of nanoporosity [19].

In addition, the introduction of partial coherence is expected to be very useful for

simulating the effects of interface roughness on the scale of the wavelength. For instance, the

interface morphology can be a key variable for optimizing the performance of thin film solar

cells since the reflectivity of the semiconductor interface can critically affect the performance

of the cell [20]. The surface roughness in a multilayer system can, if large enough, cause

phase differences between the reflected and transmitted beams. This effect can also be

simulated by the inclusion of a random phase of varying intensity.

In summary, we have developed a method for calculating the optical response of

multilayer systems, which can deal with coherent, partially coherent, and incoherent

interference on equal footing. This method is based on the transfer matrix method employed

in its usual way via Fresnel coefficients in a 2x2 matrix configuration. The novel feature is

that there is no need to use modified Fresnel coefficients or the square of their amplitudes to

work in the incoherent limit. The transition from coherent, to partially coherent, to incoherent

interference is achieved by introducing a random phase of increasing intensity in the

propagating media. This random phase can account for the effect of defects or impurities in

the layer. The capabilities of the method were presented by calculating the optical properties

of Si and ZnO films from the coherent to the incoherent limit.

Acknowledgments

This work was supported in part by NSF (Grant No. DMR-0906025), DOE (the Division of

Material Sciences and Engineering, Office of Basic Sciences, and BES-CMSN), and DOE

(Office of Energy Efficiency and Renewable Energy, Industrial Technologies Program) under

contract DE-AC05-00OR22725.

#135997 - $15.00 USD

(C) 2010 OSA

Received 11 Oct 2010; revised 24 Oct 2010; accepted 25 Oct 2010; published 10 Nov 2010

22 November 2010 / Vol. 18, No. 24 / OPTICS EXPRESS 24721