Supplementary Materials for “Vertical waveguides integrated with silicon photodetectors:
towards high efficiency and low cross-talk image sensors,” by Turgut Tut et al
Supplementary Figure S1. a). Sine condition: each ray converging to focus of aplanatic system
intersects its conjugate ray on a sphere of radius f, where f is focal length of lens. b). Surface
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area element dA of reference sphere. c). Ray tracing model of light pipe device. d). Ray tracing
model of light pipe free device.
To model the fabricated devices fully rigorously, one would need to use numerical
electromagnetics simulations, for example the finite difference time domain (FDTD) method.
While the results obtained in this manner would be rigorous, they would not necessarily provide
physical insight. We instead therefore use ray-tracing, as while it is not as exact as full-field
simulations, it facilitates physical interpretation of the differences between the devices.
Our method is as follows. We assume the sine condition, which states that each ray converging
to the focus of an aplanatic optical system intersects its conjugate ray on a sphere of radius f,
where f is the focal length of the lens.1 This is illustrated as Supplementary Figure S1a. This
sphere is also referred to as the Gaussian reference sphere.1 The conjugate ray is the incident ray
propagating parallel to the optical axis.1 In Supplementary Figure S1a, θNA denotes largest
incident angle available from lens with numerical aperture NA, i.e. 𝑁𝐴 = 𝑛1 𝑠𝑖𝑛𝜃. The goal of
our ray tracing analysis is to predict the ratio between the photodetector signal occurring in the
light pipe device to that occurring in the light pipe free device. We assume that this can be done
by integrating the power collected by the photodetector for rays with incident angles ranging
from zero to θNA. The way in which this is performed can be understood by considering the
reference sphere (Supplementary Figure S1b). The power incident on the photodetector for a
̂ 𝑑𝐴, where 𝑺 is the
surface element 𝑑𝐴 of the reference sphere is assumed equal to −𝑺 ∙ 𝒏
̂ is the normal to the
Poynting vector of the illumination striking the reference sphere, and 𝒏
surface element 𝑑𝐴. With the magnitude of the Poynting vector normalized to unity for
̂ 𝑑𝐴 = 2𝜋 𝑓2 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 𝑑𝜃 . As we describe below, this must be
simplicity, 𝑺 = 𝒛̂, and we have 𝑺 ∙ 𝒏
modified to account for the transmission coefficient at each interface. In addition, for the light pipe free
device, some of the rays at larger angles are incident on the metal portion of the bottom surface of the
device, and therefore not collected by the photodetector. These rays are therefore not included in the
power calculation. It should be noted that this method differs from the vector diffraction theory of
Richards and Wolf ,2,3 in which the fields at the focus of a lens are found by summation of plane waves
converging at different angles. We instead integrate intensities, rather than amplitudes, and polarization
is therefore not accounted for. This simplifies the analysis. It should be noted that, even with its
additional complexity, one would also need to make simplifying assumptions if one were to use vector
diffraction theory, as its basic formulation is for focusing in a homogeneous medium.
We now consider the application of this ray-tracing model to the light pipe and light pipe-free
devices. The former is illustrated as Supplementary Figure S1c. Incident rays from air (n1 =1)
are focused onto top surface of SiO2, and transmitted with intensity transmission coefficient Ta1.
These rays then encounter the light pipe, where they are transmitted with coefficient Ta2. The
light pipe comprises an SiNx cylinder (n3 = 1.8, determined by ellipsometry) with diameter 5 µm
and height 7 m. This is surrounded by SiO2 (n2 = 1.46), which is 8.7 m thick. Snell’s Law is
applied at each interface to determine the refraction angle. θmax1 denotes the incident angle of the
ray that strikes the Si at the edge of the metal aperture. A metal layer is formed at the bottom of
2
the device, with a circular opening whose diameter matches that of the light pipe. Rays incident
within this opening are transmitted into the underlying silicon with intensity transmission
coefficient Ta3. Ta1, Ta2 and Ta3 are dependent on angle, and determined from the Fresnel
reflection coefficients in a manner described below. The path of the ray incident at θNA is shown.
It can be seen that rays with incident angles from zero to θNA (≈64o) are collected. It is assumed
that rays transmitted into the silicon are absorbed and converted to photocurrent. The
photocurrent signal is therefore given by:
𝛳
𝐼𝑙𝑖𝑔ℎ𝑡 𝑝𝑖𝑝𝑒 = 𝐶 ∫0 𝑁𝐴 𝑇𝑎1 (𝜃) 𝑇𝑎2 (𝜃) 𝑇𝑎3 (𝜃) 𝑠𝑖𝑛𝛳𝑐𝑜𝑠𝛳𝑑𝛳
(S1)
where 𝐶 is a constant that accounts for the laser power and photodetector responsivity.
In Supplementary Figure S1d, we consider the case of the light pipe free device. The
geometrical parameters are identical, except that the light pipe is omitted. Rays with incident
angle between zero and θmax2 (≈23.8o ) are collected, i.e. transmitted into Si. Tb1 and Tb2 denote
intensity transmission coefficients at the air-SiO2 and SiO2-Si interfaces. The photocurrent signal
in this case is given by:
𝛳
𝐼𝑛𝑜 𝑙𝑖𝑔ℎ𝑡 𝑝𝑖𝑝𝑒 = 𝐶 ∫0 𝑚𝑎𝑥2 𝑇𝑏1 (𝜃) 𝑇𝑏2 (𝜃)𝑠𝑖𝑛𝛳𝑐𝑜𝑠𝛳𝑑𝛳
(S2)
We now discuss the method by which the transmission coefficients are modeled. We begin with
the Fresnel reflection coefficients. These specify the amplitude reflectivity for plane waves
incident from a medium with index 𝑛𝑖 upon a medium with index 𝑛𝑡 at angle 𝜃𝑖 , for polarization
in the plane of incidence (∥) and perpendicular to it (⊥). These are given by
𝑟⊥ =
𝑟∥ =
3
𝑛i𝑐𝑜𝑠𝛳i−𝑛t𝑐𝑜𝑠𝛳t
𝑛i𝑐𝑜𝑠𝛳i+𝑛t𝑐𝑜𝑠𝛳t
𝑛t𝑐𝑜𝑠𝛳i−𝑛i𝑐𝑜𝑠𝛳t
𝑛i𝑐𝑜𝑠𝛳t+𝑛t𝑐𝑜𝑠𝛳i
(S3)
(S4)
The transmission coefficients are then found using:
𝑇⊥ = 1 − 𝑟⊥ 𝑟⊥∗
(S5)
𝑇∥ = 1 − 𝑟∥ 𝑟∥∗
(S6)
The transmission coefficients are therefore polarization-dependent. To simplify matters, in
determining Ta1, Ta2, Ta3, Tb1, Tb2, we average these, i.e.
𝑇𝑎𝑣𝑔 =
1
2
(𝑇∥ + 𝑇⊥ )
(S7)
As discussed in the main body of the paper, the ratio of the photocurrents (𝐼𝑙𝑖𝑔ℎ𝑡 𝑝𝑖𝑝𝑒 /
𝐼𝑛𝑜 𝑙𝑖𝑔ℎ𝑡 𝑝𝑖𝑝𝑒 ) found in this manner, for the case of the input beam focused at the SiO2 surface, is
~3.6 times. We also perform calculations for the case of device moved vertically upward by 6
m, so that the focus is within the device. The ray tracing analysis predicts that the photocurrent
for the light pipe device is ~1.2 times larger when the focus is at the SiO2 surface than when the
device is translated 6 m. Similarly, the analysis predicts that the photocurrent for the light pipefree device is ~2.3 times larger when the device is translated 6 m than when the focus is at the
SiO2 surface.
Supplementary Materials References
1
L. Novotny and B. Hecht, Principles of Nano-Optics Cambridge University Press, Cambridge,
(2006).
2
E. Wolf, Proc Roy. Soc. A 253, pp. 349-357 (1959).
3
B. Richards and E. Wolf, Proc. Roy. Soc. A 253, pp. 358-379 (1959).
4