We present the optical models and calculation results of thinfilm organic solar cells (OSCs) at oblique incidence of light, using the transfer matrix method. The simple expression for the optical power dissipation is derived at oblique incidence for s and ppolarized light. The spatial distribution of the electric field intensity, the optical power density, and the optical power dissipation are calculated in both s and ppolarized light with respect to the incidence angle. We identify how the light absorption efficiency for ppolarized light becomes relatively larger than that for spolarized light as the incidence angle increases.
I. INTRODUCTION
Organic solar cells (OSCs) have been intensively investigated as one of the promising green energy sources due to the feasibility of producing thin, lightweight, and flexible solar cells
[1]
. However, the power conversion efficiency should be further improved for practical use of the OSC. Optical interference effects play an important role in the efficiency of multilayer OSCs, where the thickness of each thinfilm layer should be carefully designed to maximize the optical absorption in the active region.
The optical interference effect of the multilayer solar cells has been modeled based on the finite element method (FEM)
[2
,
3]
, the rigorous coupledwave analysis (RCWA)
[4
,
5]
, and the transfer matrix method (TMM)
[6

8]
. Among them, the TMM has been widely used for thinfilm OSCs due to its relatively ease of simulation code implementation but high accuracy in calculation results. The TMMbased optical model for the OSC provides the spatial distribution of the timeaverage optical power dissipation
Q
(
z
)usually for normal incidence of sunlight
[6

8]
. Here,
Q
(
z
) is related with the amount of photon energy absorbed by the material per unit time and per unit volume and
z
is a position within the multilayer OSC.
Recently, the angular response of thinfilm OSCs with respect to the incidence angle of sunlight has been intensively investigated due to the fact that most OSCs, having the advantage of lowcost production and flexible application,are expected to be used without a sunlight tracking system. The TMMbased optical model for normal incidence has been adopted for oblique incidence to analyze the angular response of thinfilm OSCs
[9

11]
. However, the previous TMMbased optical models for oblique incidence of sunlight are still insufficient.
It was mathematically proved that
Q
(
z
) can be expressed as
Q
(
z
)= (1/2)
cε_{0}nα

E
(
z
)
^{2}
for normal incidence of sunlight
[6]
. Here,
c
is the speed of light in free space,
ε
_{0}
is the permittivity in free space,
n
is the real part of the complex refractive index,
α
is the absorption coefficient, and
E
(
z
) is the electric field amplitude at the position
z
. In the case of oblique incidence,
Q
(
z
) was also assumed to be proportional to 
E
(
z
)
^{2}
for both s and ppolarized sunlight
[10
,
11]
. However, we have not yet found any text to provide a clear derivation for the relation between
Q
(
z
) and 
E
(
z
)
^{2}
at oblique incidence, especially for ppolarized light. In addition, the effect of the incidence angle and light polarization on the distribution of
Q
(
z
) has not yet been investigated quantitatively.
In this paper, we present comprehensive optical modeling and calculation results of thinfilm OSCs at oblique incidence of sunlight. By applying the TMM at oblique incidence, we derive a simple expression for
Q
(
z
) for both s and ppolarized light. We calculate the spatial distribution of the electric field intensity, the timeaverage Poynting vector(optical power density), and the optical power dissipation with respect to the incidence angle. We investigate how the optical interference effect of thinfilm OSCs is affected by the incidence angle and light polarization.
II. OPTICAL MODEL
Fig. 1
shows a stack of M layers, where a plane wave with an incident angle
θ
and a wavelength
λ
is incident from the semiinfinite transparent ambient on the left (j=0). The multilayer stack is assumed to be isotropic and homogeneous with plane and parallel interfaces. The orientation of the electromagnetic field oscillations is composed of spolarization and ppolarization that are perpendicular to or parallel to the incident plane, respectively. Each thinfilm layer j (j=1, 2, …, M) has a thickness of
d_{j}
and a complex refractive index of
where
n_{j}
and
Kj
are the refractive index and extinction coefficient, respectively.
In
Fig. 1
, the wave vector in the transparent ambient of the 0th layer is given by
where
n
_{0}
is the refractive index of the 0th layer and
k
_{0}
=2
π/λ
. In the jth layer, the wave vector is written as
Schematic diagram of M multilayers with forward and backwardpropagating electric field amplitude (E). A plane wave with an incident angle of θ and a wavelength of λ is incident from the semiinfinite transparent ambient on the left (j=0). The orientation of the electromagnetic field oscillations is composed of spolarization and ppolarization that are perpendicular to or parallel to the incident plane respectively.
where the relation
k_{x,j}=k_{x,0}
is ascribed to the boundary condition that the transverse wave vector should be continuous at the interface. The terms
and
correspond to the effective refractive index and the extinction coefficient for the z direction. From the dispersion relation
we have
Consequently, we obtain
 2.1. Internal Reflection and Transmission Coefficients in Layer j
In
Fig. 1
, we assign a positive (negative) direction to the light propagating from left (right) to right (left), designating the + () superscripts. The electric field amplitudes on the interface between the adjacent layer j and k=j+1 are described by an interface matrix I
^{jk}
:
where
and
are the forwardpropagating electric field amplitudes at the right boundary of the layer j and the left boundary of the layer k, respectively. The terms
r_{jk}
and
t_{jk}
are the complex Fresnel reflection and transmission coefficients on the interface of the adjacent layer. For spolarized light, they are defined as
[12]
and for ppolarized light as
The propagation of the electric field amplitude between the left and right boundaries of the jth layer is described by a layer matrix
L
^{j}
:
where we assume the time dependence of e
^{iwt}
.
In
Fig. 1
, the electric field amplitude passing through all the M multilayers from the ambient on the left (layer 0) to the right (layer M+1) can be written as
where the amplitude scattering matrix S
^{0/(M+1)}
is defined as
If there is no incidence of light from the ambient on the right (layer M+1) to the left (layer 0),
and the frontreflection and fronttransmission coefficients from the layer 0 to M+1 can be obtained by
In the same manner, we can define the complex reflection and transmission coefficients for the jth layer in terms of the matrix elements
As shown in
Fig. 1
, t0j is the internal transmission coefficient from the ambient on the left (layer 0) to layer j. The terms r
_{j0}
and r
_{j(M+1) }
are the internal reflection coefficients through the partial front multilayers from layer j to layer 0 and through the partial rear multilayers from layer j to layer M+1, respectively.
 2.2. Optical Power Dissipation for Spolarized Light
The y component of the electric field in the jth layer is expressed as
where
is an internal transfer coefficient, relating the incident plane wave to the internal electric field propagating in the positive (negative) z direction at the interface between the layer j1 and j
[13]
. The values of
and
are given by
where t
_{0j}
, r
_{j(M+1)}
, and rj0 are defined in Eqs. (11). The corresponding magnetic fields are
The timeaverage Poynting vector in the z direction is expressed as
where we can define
Finally, the optical power dissipation in the z direction is given by
where
and
α j,θ=2_{qj}k_{0}
in Eq. (18) are the real part of the complex refractive index and the absorption coefficient at the incident angle
θ
, respectively. We can obtain Eq. (19) by substituting Eq. (12) into Eq. (18). Similar to normal incidence, the optical power dissipation at the incident angle
θ
is also proportional to the electric field intensity for spolarized light. It is noticeable that the values of
p_{j}
and
α _{j,θ}
are modified for oblique incidence compared with the case of normal incidence.
 2.3. Optical Power Dissipation for Ppolarized Light
The x and z component of the magnetic field in the jth layer can be written as
The corresponding magnetic field is given by
The timeaverage Poynting vector in the z direction is given by
Finally, the optical power dissipation in the z direction is written as
Rearrangement of Eq. (24) together with Eq. (3) leads to
By substituting the expressions for E
_{x,j}
and E
_{z,j }
in Eqs.(20) and (21) into Eq. (25), we obtain
It is remarkable that the optical power dissipation for ppolarized light at oblique incidence is also proportional to the electric field intensity, which is composed of the x and z components of the electric field amplitude for ppolarized light. Thus, information on the electric field intensity with respect to light polarization and incident angle is sufficient to calculate the optical power dissipation in the OSC.
 2.4. Total Light Absorptance at the Active Region
Because light absorbed in the active region of the OSC can contribute to the electric power generation, we have to maximize the light absorption efficiency at the active region to enhance power conversion efficiency of the OSC. The light absorptance of the layer j (d
_{jl}
≤
z
≤
d
_{j}
) can be expressed as
[8]
where
is the optical power of sunlight light with s(p) polarization, which is incident from the ambient on the left.
III. CALCULATION RESULTS
Fig. 2
shows the multilayer structure of the OSC along with its corresponding complex refractive index, which is
Multilayer structure of the OSC together with its corresponding complex refractive index. The thickness of the optical spacer layer is varied as x = 50 100 and 150 nm.
taken from Ref. 14. The wavelength of light is assumed to be 550 nm, where the P3HT:PCBM active layer has a peak absorption. We consider the incident angles of 0°, 30°, 45°, and 60° in this calculation. The thickness of the optical spacer layer is varied as x = 50, 100, and 150 nm to adjust the optical interference effect and maximize the electric field intensity distribution within the active region.
Fig. 3
shows the y component of the normalized electric field intensities at the spacer thickness of 50, 100, and 150 nm. The light green area in the middle of
Fig. 3
indicates the P3HT:PCBM active region, where the electric field intensity distribution is important for determining the efficiency of the OSC. The value of the electric field intensity is normalized to spolarized light incident from the transparent glass on the left ambient when the incident angle is (a) 0°, (b) 30° and (c) 60°. From the ITO toward the Al layers, the oscillating electric field intensities show the general interference pattern
[6

8]
. For normal incidence (
θ
= 0°), the spatial distribution of 
E_{y}

^{2}
in the P3HT:PCBM active region is the largest at the spacer thickness of 100 nm, where the OSC expects to have the best light absorption efficiency. As the incident angle increases to 30° and further up to 60°, the overall oscillating interference pattern of 
E_{y}

^{2}
moves toward the ITO layer. This behavior is ascribed to the fact that the real part of the complex refractive index,
becomes smaller at the larger incident angle.
The y component of the normalized electric field intensities at the spacer thicknesses of 50 100 and 150 nm. The light green area indicates the P3HT:PCBM active region. The value of the electric field intensity is normalized to spolarized light incident from the transparent glass at the incident angles of (a) 0° (b) 30° and (c) 60°.
Fig. 4
shows the timeaverage Poynting vectors
S_{z}
(
z
) for s and ppolarized light at the incident angles of 0°, 30°, 45°, and 60°. The thickness of the optical spacer layer is fixed as x = 100 nm. The calculated timeaverage Poynting
Calculated timeaverage Poynting vectors for s and ppolarized light at the incident angles of 0° 30° 45° and 60°. The light green area indicates the P3HT:PCBM active region. The thickness of the optical spacer layer is fixed as x= 100 nm. The calculated timeaverage Poynting vectors are normalized in reference to the incident optical power from the transparent glass.
Comparison of the calculated normalized electric field intensities between s and ppolarized light at the incident angles of (a) 30° (b) 45° and (c) 60°. The light green area indicates the P3HT:PCBM active region. For reference the electric field intensity distribution at normal incidence is shown in the black dotted line. The optical spacer layer is fixed as x = 100 nm.
vectors are normalized in reference to the incident optical power from the transparent glass. As the incident angle increases, the value of Sz(z) at the glassITO interface (z = 0 nm) decreases because the frontreflection coefficient of the OSC multilayer, r
_{0(M+1)}
in Eq. (10a), becomes larger at the higher incident angle. The value of
Sz
(
z
) with normal incidence (θ = 0°) is identical and degenerate for s and ppolarized light. However, the value of
Sz
(
z
)\with oblique incidence of ppolarized light is larger than that with oblique incidence of spolarized light. This higher transmittance of ppolarized light over spolarized light at oblique incidence is similar to the transmission behavior at the boundary between two dielectric media, where the Brewster’s angle(100% transmission) is existent for ppolarized light
[12]
.
Fig. 5
shows the comparison of the calculated normalized electric field intensities between s and ppolarized light at the incident angles of (a) 30°, (b) 45°, and (c) 60°. For reference, the normalized electric field intensity distribution at normal incidence is shown in the black dotted line. The optical spacer layer thickness is x = 100 nm. The electric field intensity for ppolarized light 
E^{p}

^{2}
= 
E_{x}

^{2}
+ 
E_{z}

^{2}
is discontinuous at the interfaces because it has the electric field intensity component of 
E_{x}

^{2}
normal to the interface. On the other hand, the electric field intensity for spolarized light 
E^{s}

^{2}
= 
E_{y}

^{2}
is continuous at the interfaces because it has only a transverse electric field intensity component.
Fig. 6
shows the calculated optical power dissipations in the z direction for s and ppolarized light as a function of
Calculated optical power dissipations in the z direction for s and ppolarized light as a function of the incident angle. The light green area indicates the P3HT:PCBM active region. The optical spacer layer is fixed as x = 100 nm.
the incident angle. According to the definition of light absorptance in Eq. (27), ppolarized light shows the relatively higher light absorptance in the P3HT/PCBM active region than spolarized light does This is more pronounced as the incidence angle increases from 30° to 60°.
IV. CONCLUSION
We presented comprehensive optical modeling and calculation results of thinfilm OSC at oblique incidence of light. Based on the TMM, the simple expression for the optical power dissipation was derived at oblique incidence for s and ppolarized light. Depending on light polarization, we calculated the spatial distribution of the electric field intensity, the timeaverage Poynting vector, and the optical power dissipation at various incidence angles. As the incidence angle increases, the optical power dissipation in the active region becomes smaller for both s and ppolarized light. In the P3HT/PCBM active region, ppolarized light shows relatively higher light absorptance than spolarized light does. This is more significant as the incidence angle increases up to 60°. We expect that our optical modeling results help to design more efficient OSC with light absorption efficiency improved,considering the optical interference effect of the OSC at oblique incidence.
Acknowledgements
This research was supported by the Kyung Hee University Research Fund in 2011 (KHU20110921)
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