# Expression of the angular dependence of
the quantum efficiency for a thin multi-alkali photocathode and its optical
properties

Kodai Matsuoka<EMAIL_ADDRESS>Kobayashi-Maskawa Institute,
Nagoya University, Nagoya 464-8602, Japan

###### Abstract

The dependence of the quantum efficiency on the angle and polarization of the
incident photon needs to be formulated for a precise description of the
response of photomultiplier tubes. A simplified one-step model of
photoelectron emission was derived from Spicer’s three-step model, and it
enabled the formulation of the dependence of the quantum efficiency in the
visible range for thin multi-alkali (NaKSbCs) photocathodes. The expression of
the quantum efficiency was proved by a measurement of the photocurrent for
linearly polarized light at various incident angles. Meanwhile, the
measurement revealed the complex refractive indices and thicknesses both of
the stratified photocathode and antireflection coating. It is indicated that
the angular dependence of the quantum efficiency is dictated by the optical
properties of the photocathode, which are discussed in detail on the basis of
the obtained parameters.

H02, H15

##  1 Introduction

The photocathode is a key component of photomultiplier tubes (PMTs). It
dominates the PMT performance in terms of the sensitivity to photons. As PMTs
are used in a variety of fields, photocathode properties are major subjects of
research. One of the most important properties is the efficiency of
photoelectron emission, or the quantum efficiency (QE). The QE depends not
only on the wavelength but also on the incident angle and polarization of the
light. Such dependence is of great interest especially for Cherenkov detectors
because the Cherenkov light is emitted along the definite Cherenkov angle and
is naturally 100% linearly polarized Cherenkov ; Tamm . Rigorous expression of
the dependence of the QE is demanded to precisely estimate or calibrate the
detector performance IACT ; IceCube ; TOP ; dualCAL . Furthermore, a detailed
understanding of the QE could help to find a recipe for enhancement of the QE.

The process of the photoelectron emission can be described by Spicer’s three-
step model ThreeStepModel : (1) photoexcitation of an electron in the
photocathode, (2) transport of the excited electron to the vacuum surface and
(3) escape of the electron over the surface barrier into the vacuum. Therefore
the QE is dictated by the optical properties of the PMT, the physical
properties of the photocathode material such as the band structure and the
electron scattering mechanisms during transportation, and the quality of the
photocathode such as the surface characteristics. Each of these has to be well
described for expression of the dependence of the QE. Some theoretical
expressions or practical models of photoelectron emission were proposed for
metal photoemit_Berglund ; photoemit_Dowell ; photoemit_Jensen_metal  and
semiconductor ThreeStepModel ; photoemit_Jensen_semi ; photoemit_Kane  under
assumption of infinite thickness of the photocathode, where it was not
necessary to treat the optical properties of the photocathode in detail
because the photon absorption follows a simple exponential function of the
depth (Lambert-Beer law). For thin photocathodes, which are widely used for
PMTs, however, the optical properties are of more complication, and there are
yet no expressions of photoelectron emission which have been theoretically or
experimentally verified.

The optical properties can be expressed by means of the complex refractive
index and the thickness of the photocathode. These parameters can be derived
from the standard techniques of reflectance measurement at various angles or
of ellipsometry. Such measurements were reported in Refs. index_Dolizy ;
index_Johnson ; index_Lang ; index_Moorhead ; index_Lay1 ; index_Lay2 ;
index_Shibamura  for various photocathode types but only at a few wavelengths.
Measurements over the whole visible range were done in Refs.
photoemit_multialkali ; index_Hallensleben ; index_Motta ; index_Harmer  for
bi/multi-alkali photocathodes. Among them, Refs. index_Lay1 ; index_Lay2  made
a comparison between the measured angular dependence of the photocurrent and
the predicted one of the absorptance from the measured index and thickness of
the photocathode, and found some unexplained discrepancies. Another attempt to
derive the photocathode index and thickness can be made by photocurrent
measurement at various angles. This method was adopted in Ref. index_Jones ,
but the theoretical function applied in the reference did not fit the data
precisely in terms of the angular dependence of the photocurrent. A
discrepancy on the angular dependence was also found between the measurements
of the photocurrent and the absorptance by Ref. index_Chyba . The biggest
concern in those comparisons is that some interpretations are needed to
connect the photocurrent to the optical absorptance. Hence the dependence of
the QE on the angle and polarization is yet to be investigated. In addition
available data of photocathode indices are not satisfactory for the whole
visible range.

In this paper a simplified expression of the QE for thin photocathodes is
proposed based on Spicer’s three-step model. It is verified by the measurement
of the dependence of the QE on the angle and polarization. The dependence of
the QE was examined directly by the photocurrent measurement for a thin multi-
alkali (NaKSbCs) photocathode of the transmission mode, fabricated in a
square-shaped micro-channel-plate (MCP) PMT R10754-07-M16(N) MCP-PMT  made by
HAMAMATSU PHOTONICS K.K. The photocurrent was measured at wavelengths from 320
to 680 nm and at incident angles from 0 to 80∘ both for $s$\- and
$p$-polarizations. It has to be noted that this PMT has a confidential
antireflection coating, of which optical properties are totally unknown. Hence
the optical parameters of the stratified photocathode and antireflection
coating are derived at the same time in this work. Based on the measured
parameters, in the latter part of this paper, the optical properties of the
photocathode are discussed in connection with effects on the QE.

##  2 Expression of the quantum efficiency

In this section the simplified expression of the QE is derived for thin multi-
alkali photocathodes. Based on the three-step model, the QE can be factorized
into the probabilities $A_{\rm pc}$, $P_{\rm transport}$ and $P_{\rm escape}$,
where $A_{\rm pc}$ is the absorptance of the photocathode or the fraction of
the absorbed light intensity to the incident one, and $P_{\rm transport}$ and
$P_{\rm escape}$ are the probabilities that the electron reaches the vacuum
surface and escapes into the vacuum, respectively. These three factors are
functions of the photon wavelength because $P_{\rm transport}$ and $P_{\rm
escape}$ depend on the energy of the electron, which is determined by the
photoexcitation process being subject to the energy of the absorbed photon.
$P_{\rm transport}$ and $P_{\rm escape}$ also depend on the depth $z$ where
the photon is absorbed due to a loss of the electron energy during
transportation. That demands a differential function of $A_{\rm pc}$,
represented as $A^{\prime}_{\rm pc}(z)$, which describes the absorption
distribution along $z$. Therefore the QE at a wavelength $\lambda$ is
expressed in the form

$Q\\!E(\lambda)=\int_{0}^{d}A^{\prime}_{\rm pc}(z,\lambda)P_{\rm
transport}(z,\lambda)P_{\rm escape}(z,\lambda){\rm d}z,$ (1)

where $d$ is the thickness of the photocathode. It also has to be expressed as
a function of the angle $\theta$ and the polarization $\epsilon$ ($=$ $s$ or
$p$) of the incident light. In the following, each factor of this equation is
described by elementary functions with several parameters based on theories.

###  2.1 Optical model

Figure 1: (Left) Diagram of light passing through a double thin layer of the
antireflection coating and the photocathode. Multiple reflections in the thin
layers are also drawn to some extent. (Right) Diagram of light transmitted
through the photocathode and reflected on the MCP electrode.

The absorptance $A_{\rm pc}$ and its derivation $A^{\prime}_{\rm pc}$ are
fully described by the optics. A specific configuration of the PMT shown in
Fig. 1 is considered here111In most cases light comes from the outside of the
window, and one should also consider the reflection and refraction at the
outer surface of the window.: the light passes from the quartz window (layer 1
with refractive index $n_{1}$) through a double thin layer of the
antireflection coating (layer 2 with $n_{2}$) and the photocathode (layer 3
with $n_{3}$) into the vacuum (layer 4 with $n_{4}=1$). The amplitude
reflection and transmission coefficients at the interface from layer $i$ to
$j=i+1$ ($r_{ij}$ and $t_{ij}$, respectively, and $i=1$, 2, 3) are described
by Fresnel equations:

$\begin{array}[]{lll}r_{ij}=\cfrac{n_{i}\cos\theta_{i}-n_{j}\cos\theta_{j}}{n_{i}\cos\theta_{i}+n_{j}\cos\theta_{j}},&t_{ij}=\cfrac{2n_{i}\cos\theta_{i}}{n_{i}\cos\theta_{i}+n_{j}\cos\theta_{j}}&{\rm({\it
s}\mathchar 45\relax polarization)},\\\
r_{ij}=\cfrac{n_{j}\cos\theta_{i}-n_{i}\cos\theta_{j}}{n_{j}\cos\theta_{i}+n_{i}\cos\theta_{j}},&t_{ij}=\cfrac{2n_{i}\cos\theta_{i}}{n_{j}\cos\theta_{i}+n_{i}\cos\theta_{j}}&{\rm({\it
p}\mathchar 45\relax polarization)},\end{array}$ (2)

where $\theta_{i}$ ($\theta_{j}$) is the angle between the incident
(refracted) light and the normal of the interface, which follows Snell’s law:

$n_{i}\sin\theta_{i}=n_{j}\sin\theta_{j}.$ (3)

It should be noted that Eq. (3) holds for any combination of $i$ and $j>i$.
When layer $l$ absorbs the light or has stimulated emission, $n_{l}$ and
$\theta_{l}$ are complex numbers. Taking into account the multiple reflections
on both surfaces of the photocathode, the amplitude reflection and
transmission coefficients for the photocathode layer ($\hat{r}_{3}$ and
$\hat{t}_{3}$, respectively) are described using Eq. (2) as follows:

$\begin{split}\hat{r}_{3}=r_{23}+t_{23}r_{34}t_{32}e^{i\delta_{3}}\sum_{m=0}^{\infty}(r_{32}r_{34}e^{i\delta_{3}})^{m}=\frac{r_{23}+r_{34}e^{i\delta_{3}}}{1+r_{34}r_{23}e^{i\delta_{3}}},\\\
\hat{t}_{3}=t_{23}t_{34}e^{i\delta_{3}/2}\sum_{m=0}^{\infty}(r_{34}r_{32}e^{i\delta_{3}})^{m}=\frac{t_{23}t_{34}e^{i\delta_{3}/2}}{1+r_{34}r_{23}e^{i\delta_{3}}},\quad\;\end{split}$
(4)

and similarly for the double layer:

$\hat{r}_{2+3}=\frac{r_{12}+\hat{r}_{3}e^{i\delta_{2}}}{1+\hat{r}_{3}r_{12}e^{i\delta_{2}}},\quad\hat{t}_{2+3}=\frac{t_{12}\hat{t}_{3}e^{i\delta_{2}/2}}{1+\hat{r}_{3}r_{12}e^{i\delta_{2}}},$
(5)

where

$\delta_{l}=\frac{4\pi d_{l}n_{l}}{\lambda_{0}}\cos\theta_{l}\quad(l=2,3)$ (6)

is the phase difference, $d_{l}$ is the thickness of layer $l$ and
$\lambda_{0}$ is the wavelength of the light in the vacuum. The reflectance
$R_{2+3}$ and transmittance $T_{2+3}$ of the double layer are related to the
coefficients:

$R_{2+3}=\left|\hat{r}_{2+3}\right|^{2},$ (7)
$T_{2+3}=\left\\{\begin{array}[]{ll}\cfrac{{\rm Re}(n_{4}\cos\theta_{4})}{{\rm
Re}(n_{1}\cos\theta_{1})}\left|\hat{t}_{2+3}\right|^{2}&{\rm({\it s}\mathchar
45\relax polarization)}\\\ \cfrac{{\rm Re}(n_{4}\cos\theta_{4}^{*})}{{\rm
Re}(n_{1}\cos\theta_{1}^{*})}\left|\hat{t}_{2+3}\right|^{2}&{\rm({\it
p}\mathchar 45\relax polarization)}\end{array}\right.$ (8)

where $\theta^{*}$ denotes the complex conjugate of $\theta$ (although
$\theta_{1}$ and $\theta_{4}$ are real in this case) and ${\rm Re}(x)$ denotes
the real part of $x$. Equation (7) is the base of the reflectance measurement
for estimation of the photocathode index and thickness mentioned in Sec. 1. On
the other hand the photocurrent measurement is associated with the absorptance
of the double layer $A_{2+3}$, which has the relation:

$R_{2+3}+T_{2+3}+A_{2+3}=1.$ (9)

If there is no absorption in the antireflection coating, $A_{2+3}$ is equal to
the absorptance of the photocathode $A_{3}$.

(a)  (b)

Figure 2: Reflectance $R_{2+3}$, transmittance $T_{2+3}$ and absorptance
$A_{2+3}$ of the double layer of the antireflection coating ($n_{2}=2.1$ and
$d_{2}=30$ nm) and the photocathode ($n_{3}=2.3+3.3i$ and $d_{3}=10$ nm) at a
wavelength $\lambda_{0}=360$ nm for $s$-polarization (left) and
$p$-polarization (right). The indices of quartz and vacuum are $n_{1}=1.475$
at 360 nm and $n_{4}=1$.

An example of $R_{2+3}$, $T_{2+3}$ and $A_{2+3}$ as a function of $\theta_{1}$
is shown in Fig. 2 for a given set of parameters. For $s$($p$)-polarization
$A_{2+3}$ increases (decreases) along with the incident angle from 0∘ because
at the photocathode-vacuum interface the reflection increases (decreases) and
the transmission to the vacuum decreases (increases). Accordingly $T_{2+3}$
decreases (increases) and falls down to zero at the total reflection angle for
the photocathode-vacuum interface. The incident angle $\theta_{1}$
corresponding to the total reflection angle is determined by the indices of
quartz and vacuum according to Eq. (3). It ranges from 42.4∘ at 320 nm to
43.4∘ at 660 nm, where the quartz index varies from 1.48 to 1.46. Nearly at
that angle $A_{2+3}$ peaks for $s$-polarization and drops steeply for
$p$-polarization. $A_{2+3}$ for $p$-polarization then peaks and $R_{2+3}$
drops to nearly zero. The angle of the minimum $R_{2+3}$ is 57.1∘, which is
slightly different from Brewster’s angle of
$\arctan(n_{2}/n_{1})=54.9^{\circ}$ for the first interface between the window
and the antireflection coating due to the reflections from the other
interfaces. When the angle gets close to 90∘ the reflection at the first
interface becomes dominant and thus $A_{2+3}$ declines.

For the sake of precise expression of the absorptances of both stratified
layers, the exact expression recently derived absorptance  is applied in this
work. Namely the amount of light absorbed per unit length at a depth $z$ from
the surface of each layer $l(=2,3)$ ($z=0$ on the light incident surface) is
given as

$A^{\prime}_{l}(z)=a_{l}\left|v_{l}\right|^{2}e^{-2z{\rm
Im}(\tilde{\nu}_{l})}+a_{l}\left|w_{l}\right|^{2}e^{2z{\rm
Im}(\tilde{\nu}_{l})}+b_{l}v_{l}w_{l}^{*}e^{2iz{\rm
Re}(\tilde{\nu}_{l})}+b_{l}v_{l}^{*}w_{l}e^{-2iz{\rm Re}(\tilde{\nu}_{l})},$
(10) $\begin{array}[]{ll}a_{l}=b_{l}=\cfrac{{\rm
Im}(\tilde{\nu}_{l}n_{l}\cos\theta_{l})}{{\rm
Re}(n_{1}\cos\theta_{1})}&{\rm({\it s}\mathchar 45\relax polarization)}\\\
a_{l}=\cfrac{2{\rm Im}(\tilde{\nu}_{l}){\rm Re}(n_{l}\cos\theta_{l}^{*})}{{\rm
Re}(n_{1}\cos\theta_{1}^{*})},\quad b_{l}=-\cfrac{2{\rm
Re}(\tilde{\nu}_{l}){\rm Im}(n_{l}\cos\theta_{l}^{*})}{{\rm
Re}(n_{1}\cos\theta_{1}^{*})}&{\rm({\it p}\mathchar 45\relax
polarization)}\end{array}$

where ${\rm Im}(x)$ denotes the imaginary part of $x$, $\tilde{\nu}_{l}$ is
the z-component of the wavenumber in layer $l$, and $v_{l}$ ($w_{l}$) is the
amplitude of the wave heading forward (backward) on the layer $l$ side of the
interface between $l-1$ and $l$:

$\tilde{\nu}_{l}=\frac{2\pi n_{l}}{\lambda_{0}}\cos\theta_{l},$ (11)
$\begin{split}v_{l}=v_{l-1}e^{i\delta_{l-1}/2}t_{l-1,l}+w_{l}r_{l,l-1},\;\;\\\
w_{l}=w_{l+1}t_{l+1,l}e^{i\delta_{l}/2}+v_{l}r_{l,l+1}e^{i\delta_{l}}.\end{split}$
(12)

In the case of four layers considered,

$\begin{array}[]{lll}&v_{1}=1,&w_{1}=r_{12},\\\
&v_{2}=\cfrac{\left(1-r_{32}r_{34}e^{i\delta_{3}}\right)t_{12}}{1-r_{21}r_{23}e^{i\delta_{2}}-r_{32}r_{34}e^{i\delta_{3}}-r_{21}r_{34}e^{i(\delta_{2}+\delta_{3})}},&w_{2}=\cfrac{r_{23}+r_{34}e^{i\delta_{3}}}{1-r_{32}r_{34}e^{i\delta_{3}}}e^{i\delta_{2}}v_{2},\\\
&v_{3}=\cfrac{t_{23}e^{i\delta_{2}/2}}{1-r_{32}r_{34}e^{i\delta_{3}}}v_{2},&w_{3}=r_{34}e^{i\delta_{3}}v_{3},\\\
&v_{4}=t_{34},&w_{4}=0.\\\ \end{array}$ (13)

The absorptance of layer $l$ is given by integrating Eq. (10) over the
thickness:

$A_{l}=\int_{0}^{d_{l}}A^{\prime}_{l}(z){\rm d}z.$ (14)

It is also necessary to consider the transmitted light into the vacuum, which
is then reflected on the MCP electrode back to the photocathode as shown in
Fig. 1 (right). Hereafter the illumination from the window/vacuum side is
called front/back-illumination. The back-illumination needs to be taken into
account only below the total reflection angle for the photocathode-vacuum
interface. The gap spacing between the photocathode and the MCP is 2 mm, and
the back-illumination hits the photocathode at a different position from the
front-illumination depending on the incident angle. The reflectance $R_{\rm
el}$ of the electrode with refractive index $n_{5}$ is calculated using Eq.
(2),

$R_{\rm el}=|r_{45}|^{2}.$ (15)

The MCP has micro pores of 10 $\mathrm{\SIUnitSymbolMicro}$m diameter and the
open area ratio ($\mathcal{R}_{\rm OA}$) is about 0.6. Hence the fraction of
the reflected light accounts for $(1-\mathcal{R}_{\rm OA})R_{\rm el}$. The
absorptance of each layer for the back-illumination $A_{\bar{l}}$ and its
derivation $A^{\prime}_{\bar{l}}(z)$ are given in the same manner as described
above by reversing the order of the layers (namely $\bar{l}=\bar{1}$,
$\bar{2}$, $\bar{3}$, $\bar{4}$ for the vacuum, photocathode, antireflection
coating and window, respectively; and the indices $n_{\bar{l}}$ and
thicknesses $d_{\bar{l}}$ should be defined correspondingly) and using
$\theta_{4}$ as the incident angle. The sum of the front- and back-
illuminations is the total absorption of the photocathode,

$A_{\rm pc}=A_{3}+T_{2+3}(1-\mathcal{R}_{\rm OA})R_{\rm el}A_{\bar{2}}.$ (16)

Subsequent reflections of the back-illumination in the vacuum layer are
omitted in this work because they should have little contribution to $A_{\rm
pc}$.

###  2.2 Dispersion model

In order to describe dispersion functions of refractive indices, the following
two models and an empirical equation are applied in this work.

The one for the photocathode and the antireflection coating is the Lorentz
dispersion model Lorentz , which represents dielectric response of inter-band
transitions by the classical damped harmonic oscillator. As Ref.
photoemit_multialkali  infers that the photon absorption in NaKSbCs is
probably of direct transition, the Lorentz model is expected to describe the
dispersion well. In this model, the complex dielectric function of the
frequency $\omega$ can be expressed as

$\varepsilon(\omega)=\varepsilon_{\infty}+\frac{(\varepsilon_{s}-\varepsilon_{\infty})\omega_{t}^{2}}{\omega_{t}^{2}-\omega^{2}-i\varGamma_{0}\omega},$
(17)

where $\varepsilon_{\infty}$ is the high frequency limit of $\varepsilon$,
$\varepsilon_{s}=\varepsilon_{\infty}+\omega_{p}^{2}/\omega_{t}^{2}$ is the
static dielectric function at $\omega=0$, $\omega_{p}$ is the plasma
frequency, $\omega_{t}$ is the resonant frequency of the oscillator
corresponding to the absorption peak, and $\varGamma_{0}$ is the damping
factor corresponding to the full width at half maximum of the peak. The real
and imaginary parts of $\varepsilon$ are described as follows:

${\rm
Re}(\varepsilon)=1+\frac{(\varepsilon_{s}-1)\omega_{t}^{2}\left(\omega_{t}^{2}-\omega^{2}\right)}{\left(\omega_{t}^{2}-\omega^{2}\right)^{2}+\varGamma_{0}^{2}\omega^{2}},\quad{\rm
Im}(\varepsilon)=\frac{(\varepsilon_{s}-1)\omega_{t}^{2}\varGamma_{0}\omega}{\left(\omega_{t}^{2}-\omega^{2}\right)^{2}+\varGamma_{0}^{2}\omega^{2}}.$
(18)

Here $\varepsilon_{\infty}$ is fixed to 1 as in general material. The complex
refractive index ($n+ik$) as a function of $\lambda(=2\pi c/\omega$, where $c$
is the speed of light) is derived as follows:

$n(\lambda)=\sqrt{\frac{{\rm Re}(\varepsilon)+|\varepsilon|}{2}},\quad
k(\lambda)=\sqrt{\frac{-{\rm Re}(\varepsilon)+|\varepsilon|}{2}}.$ (19)

The other dispersion model for the electrode is described by the Drude free
electron theory Drude1 ; Drude2  as follows:

$\varepsilon(\omega)=\varepsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega^{2}+i\varGamma_{d}\omega},$
(20) ${\rm
Re}(\varepsilon)=1-\frac{\omega_{p}^{2}}{\omega^{2}+\varGamma_{d}^{2}},\quad{\rm
Im}(\varepsilon)=\frac{\omega_{p}^{2}\varGamma_{d}/\omega}{\omega^{2}+\varGamma_{d}^{2}},$
(21)

where $1/\varGamma_{d}$ is the relaxation time or DC conductivity and
$\varepsilon_{\infty}$ is fixed to 1. The parameters of NiFe, namely
$\omega_{p}=14.79$ eV and $\varGamma_{d}=4.78$ eV Horiba_TN09 , are applied in
this work.

Sellmeier equation Sellmeier  is used for the refractive index of quartz:

$n^{2}-1=\frac{B_{1}\lambda^{2}}{\lambda^{2}-C_{1}}+\frac{B_{2}\lambda^{2}}{\lambda^{2}-C_{2}}+\frac{B_{3}\lambda^{2}}{\lambda^{2}-C_{3}},$
(22)

where $B_{i}$ and $C_{i}$ ($i=$ 1, 2, 3) are the dispersion constants listed
below for $\lambda$ in units of $\mathrm{\SIUnitSymbolMicro}$m at
20$\mathrm{\SIUnitSymbolCelsius}$ Suprasil :

$\begin{array}[]{l}B_{1}=0.473115591,\quad C_{1}=\phantom{0}0.0129957170,\\\
B_{2}=0.631038719,\quad C_{2}=\phantom{0}0.00412809220,\\\
B_{3}=0.906404498,\quad C_{3}=98.7685322.\\\ \end{array}$ (23)

###  2.3 Model of photoelectron emission

The other factors in Eq. (1) are considered in the case of thin semiconductors
and visible light in the following. The loss of the electron during
transportation depends on the inelastic and elastic scattering lengths. The
electron-phonon scattering can be regarded as the elastic one because the
energy loss by the scattering is quite small photoemit_alkali . Scattering
with electrons in the valence band can take place only if the electron in the
conduction band has a sufficient energy to produce an electron pair over the
band gap. The threshold energy was found to be more than twice the band gap
above the bottom of the conduction band photoelectronic_Sommer . It was
measured for (Cs)Na3KSb photocathodes to be 3.0 eV scatter_ee , or 4.0 eV
above the edge of the valence band, which corresponds to 310 nm wavelength of
the photon absorbed. Scattering with impurities or defects could take place,
but its probability should be small if the photocathode thickness is
sufficiently less than the scattering lengths for the impurities and defects.
Therefore, the loss of the electron by the scatterings is expected to be
small, and hence

$P_{\rm transport}\approx 1.$ (24)

When the electron reaches the vacuum surface, it can escape to the vacuum if
it has a momentum of which the normal component with respect to the vacuum
surface is greater than a critical value photoemit_Berglund . Namely the
electron trajectory is required to fall within the escape cone of the apex
angle $2\theta_{c}$ defined by
$\cos\theta_{c}=\sqrt{\varPhi/E_{e}}>\sqrt{\varPhi/h\nu}$, where $\varPhi$ is
the work function, $E_{e}(>\varPhi)$ is the energy of the electron at the
vacuum surface and $h\nu(>E_{e})$ is the energy of the absorbed photon. This
requirement could be a significant reduction factor for the QE when the
electron went straight from the initial position to the vacuum surface without
any scatterings. The acceptance of the electron escape in this case would be
$(1-\cos\theta_{c})/2<0.18$ for $\varPhi=1.55$ eV ThreeStepModel  and
$\lambda=320$ nm. It means that the QE at 320 nm or longer wavelengths would
be limited to 0.18 at most. However the elastic electron-phonon scattering can
enhance the escape probability because it gives the electron many chances to
reach the vacuum surface at different angles as in the case of random walk
photoemit_review . Once the electron falls within the escape cone, the escape
probability can be regarded as unity photoemit_Jensen_semi . As far as only
electrons above the vacuum level are counted, $P_{\rm escape}$ therefore could
be unity. The proportion of the electrons above the vacuum level is described
by the photoexcitation probability $P_{\rm excite}(\lambda)$ which is equal to
the ratio of the probability of transitions above the vacuum level to the one
of all possible transitions by absorbing a photon of the wavelength $\lambda$.
Thus

$P_{\rm escape}(\lambda)=P_{\rm excite}(\lambda).$ (25)

When one takes account of the energy loss by the scatterings, $P_{\rm excite}$
can be interpreted as the ratio of the probability of transitions above the
vacuum level plus the amount of energy loss to the one of all possible
transitions.

Consequently the three-step model could be reduced to only one step for thin
photocathodes: photoexcitation of an electron above a certain energy level
needed for the escape. This presumption shall be examined in Sec. 3. $P_{\rm
excite}$ depends on the work function and normally increases monotonically as
the photon energy increases. To exactly describe the $P_{\rm excite}$
function, one also has to consider the electron density of states which
governs the transition probability as studied in Refs. band_calc_Ettema ;
band_calc_Murtaza ; band_XPS . However it is beyond the scope of this work,
and $P_{\rm excite}$ is taken as a parameter to be determined independently at
each wavelength.

For thick photocathodes where the approximations above do not hold, $P_{\rm
transport}$ and $P_{\rm escape}$ also depend on $z$. It was mentioned that
$P_{\rm transport}$ and $P_{\rm escape}$ can be lumped together and

$P_{\rm transport}(z,\lambda)P_{\rm
escape}(z,\lambda)=\left\\{\begin{array}[]{ll}G(\lambda)e^{-(d-z)/L_{e}}&(\mathrm{front\mathchar
45\relax illumination})\\\ G(\lambda)e^{-z/L_{e}}&(\mathrm{back\mathchar
45\relax illumination})\end{array}\right.$ (26)

is a good approximation ThreeStepModel , where $G(\lambda)$ is an undefined
function of $\lambda$ and $L_{e}$ is the escape length of the excited electron
in the photocathode. This approximation shall also be investigated in Sec. 3.

###  2.4 QE function

In summary the QE for a thin photocathode is expressed by a function

$Q\\!E(\lambda,\theta,\epsilon)=P_{\rm excite}(\lambda)A_{\rm
pc}(\lambda,\theta,\epsilon),$ (27) $A_{\rm
pc}(\lambda,\theta,\epsilon)=\int_{0}^{d_{3}}A^{\prime}_{3}(z,\lambda,\theta,\epsilon){\rm
d}z+T_{2+3}(\lambda,\theta,\epsilon)(1-\mathcal{R}_{\rm OA})R_{\rm
el}(\lambda,\theta,\epsilon)\int_{0}^{d_{\bar{2}}}A^{\prime}_{\bar{2}}(z,\lambda,\theta,\epsilon){\rm
d}z.$ (28)

The parameters to be determined are $P_{\rm excite}$ for each $\lambda$, and
respective $d$, $\varepsilon_{s}$, $\omega_{t}$ and $\varGamma_{0}$ of the
photocathode and the antireflection coating. $\mathcal{R}_{\rm OA}$ is
supposed to be 0.6. As a consequence of the one-step model of photoelectron
emission, the dependence of the QE on the angle $\theta$ and the polarization
$\epsilon$ is dictated only by the optical properties.

If the photocathode thickness is comparable to the escape length, the
dependence of $P_{\rm transport}P_{\rm escape}$ on $z$ expressed in Eq. (26)
has to be taken into account in the integral:

$\begin{split}Q\\!E(\lambda,\theta,\epsilon)=G(\lambda)e^{\beta}\left[a|v|^{2}\int_{0}^{d}e^{(-2{\rm
Im}(\tilde{\nu})+\alpha)z}{\rm d}z+a|w|^{2}\int_{0}^{d}e^{(2{\rm
Im}(\tilde{\nu})+\alpha)z}{\rm d}z\right.\quad\quad\quad\\\ \left.+2b{\rm
Re}(vw^{*})\int_{0}^{d}\cos(2{\rm Re}(\tilde{\nu})z)e^{\alpha z}{\rm
d}z-2b{\rm Im}(vw^{*})\int_{0}^{d}\sin(2{\rm Re}(\tilde{\nu})z)e^{\alpha
z}{\rm d}z\right]\\\ =G(\lambda)e^{\beta}\left[a|v|^{2}\frac{e^{(-2{\rm
Im}(\tilde{\nu})+\alpha)d}-1}{-2{\rm
Im}(\tilde{\nu})+\alpha}+a|w|^{2}\frac{e^{(2{\rm
Im}(\tilde{\nu})+\alpha)d}-1}{2{\rm
Im}(\tilde{\nu})+\alpha}\right.\quad\quad\quad\quad\;\;\>\\\ +2b{\rm
Re}(vw^{*})\frac{e^{\alpha d}\\{\alpha\cos(2{\rm Re}(\tilde{\nu})d)+2{\rm
Re}(\tilde{\nu})\sin(2{\rm Re}(\tilde{\nu})d)\\}-\alpha}{4({\rm
Re}(\tilde{\nu}))^{2}+\alpha^{2}}\quad\\\ \left.-2b{\rm
Im}(vw^{*})\frac{e^{\alpha d}\\{\alpha\sin(2{\rm Re}(\tilde{\nu})d)-2{\rm
Re}(\tilde{\nu})\cos(2{\rm Re}(\tilde{\nu})d)\\}+2{\rm
Re}(\tilde{\nu})}{4({\rm Re}(\tilde{\nu}))^{2}+\alpha^{2}}\right],\end{split}$
(29)
$\begin{array}[]{lll}\alpha=\cfrac{1}{L_{e}},&\beta=-\cfrac{d}{L_{e}}&{\rm(front\mathchar
45\relax illumination)}\\\
\alpha=-\cfrac{1}{L_{e}},&\beta=0&{\rm(back\mathchar 45\relax
illumination)}\end{array}$

Here the second term of Eq. (28) for the back-illumination is not explicitly
written and the subscript $l$ is omitted. As $L_{e}$ gets larger, $G$ becomes
closer to $P_{\rm excite}$, and Eq. (29) with $L_{e}=\infty$ is identical to
Eq. (27).

##  3 Experimental verification

This section describes the measurement of the dependence of the QE on the
angle and polarization. The QE function derived in the previous section shall
be verified experimentally.

###  3.1 Measurement setup

The QE was measured as the photocurrent on the PMT photocathode under
illumination of linearly polarized light. The data were taken at 81 incident
angles (2∘ intervals from $-80^{\circ}$ to 80∘) and at 18 wavelengths (20-nm
intervals from 320 to 660 nm) for both polarizations in the setup shown in
Fig. 3.

Figure 3: Schematic top view of the setup of the QE measurement.

The incident angle was selected by rotating the PMT on a rotary stage while
fixing the optical system. The normal incident angle was defined as 0∘. The
light source was a xenon lamp, and the wavelength was selected by a
monochromator. Two types of sharp cut filters were placed in a switchable
holder and one of them or a blank was selected according to the wavelength in
order to cut out stray light from the monochromator. Following a round
continuously variable neutral density (ND) filter and two slits, a Glan laser
prism was placed. It polarized the light with an extinction ratio less than
$5\times 10^{-6}$. The photocathode was illuminated through a semi-cylinder
whose radius of the curvature was 15 mm and through the 1.5 mm thick PMT
window. The semi-cylinder and the window were both made of synthetic quartz
and were coupled together via index-matching optical oil made of fused silica.
Therefore the light passed from the air to the photocathode without refraction
at any orientations of the PMT rotation as the PMT was rotated about the axis
of the semi-cylinder. At 0∘ of the incident angle, the light spotted at the
center of the photocathode and the size of the light spot was about 1 mm in
diameter. Since the photocathode was located off from the rotation center, the
light spot shifted and ovalized on the photocathode at oblique angles. At $\pm
80^{\circ}$, the profile of the light spot was nearly on the verge of the
photocathode, of which the size was $23\times 23$ mm2. The photoinduced
current on the photocathode $I_{\rm PMT}$ was measured as the mean of ten
readings by a picoammeter without any amplification. The light intensity was
adjusted by the ND filter so that $I_{\rm PMT}$ did not exceed 10 nA to avoid
saturation due to the space charge effect in the photocathode. The counter-
current of the photoelectrons was got out to the MCP electrode by applying
+200 V on it. A silicon photodiode was used to measure the light intensity. It
was mounted on a moving stage together with the PMT rotary stage and took
turns measuring the light. The photocurrent of the photodiode $I_{\rm PD}$ was
measured by the same picoammeter (60 times each in this instance) as a relay
switch selected either of the photodevices. Therefore $I_{\rm PMT}\cdot
Q\\!E_{\rm PD}/I_{\rm PD}$ amounts to the PMT QE, where the absolute QE
spectrum of the photodiode $Q\\!E_{\rm PD}(\lambda)$ was calibrated in advance
with a precision of 0.5-1.7% depending on the wavelength. In this measurement
the PMT QE was defined as the efficiency of detecting the incident light onto
the quartz semi-cylinder, and about 4% loss by the reflection at the air-
quartz interface was counted as the inefficiency.

A misalignment of the centers of the semi-cylinder and the rotary stage could
systematically deviate the incident angle on the photocathode due to the
refraction at the surface of the semi-cylinder. The measurement for both
rotation directions (from 0∘ to $\pm 80^{\circ}$) was intended for canceling
out the error of the misalignment as well as one of the offset angle. In the
analysis the QEs at the same absolute angle were averaged, and the difference
between the QEs, which was typically $\Delta Q\\!E/Q\\!E\approx 1$%, was
considered as the systematic error. The misalignment of the centers along the
light axis however cannot be compensated by averaging the QEs, and it was
counted in the error on the incident angle. The uncertainty of the alignment
was 0.25 mm and the error on the incident angle was estimated to be 0-0.7∘
depending on the angle. The other errors on the angle were 0.3∘ due to the
spread of the light spot and 0.2∘ for the positioning accuracy of the rotary
stage.

The degree of polarization could be another major source of the systematic
error on the QE. It was determined mostly by the alignment of the polarization
axis of the Glan laser prism relative to the plane of incidence for the
photocathode. The uncertainty of the alignment was estimated to be
$0.25^{\circ}$, and the systematic error on the QE was typically 0.1%,
depending on the difference in the QEs between $s$\- and $p$-polarizations at
the same incident angle. The statistical error in this measurement was
typically 0.2%, which was quoted from the standard deviation of the ten
readings of the picoammeter. The accuracy of the picoammeter, 0.2% of the
reading, was also added in the error. The errors on $Q\\!E_{\rm PD}$ and
$I_{\rm PD}$, in total 0.5-2.5% depending on the wavelength, were treated as
the systematic error on the normalization of the PMT QE because they do not
distort the angular dependence of the QE.

Compared to the standard technique of reflectance measurement, this
photocurrent measurement has some advantages. Another photodevice to measure
the reflected light is unnecessary, and therefore the difficulties can be
avoided in aligning that photodevice in accordance with the angle of the
incident light, in particular around 0∘ where the incident and reflected light
overlaps. In addition the measurement uncertainties related to that
photodevice can be omitted. Regarding the dependence of the QE on the angle
and polarization, it should be measured directly by the photocurrent as it is
in general not trivial to interpret the absorptance deduced by the reflectance
measurement as the QE.

###  3.2 Results

Eight PMTs (called hereafter PMT1-8) of the same type of photocathode were
measured. Though all the photocathodes were produced in the same way, some
differences in the QE spectra measured at 0∘ are recognizable in Fig. 5. While
PMT1-7 have similar spectra, PMT8 differs from the others: the former spectra
peak around 360 nm and the latter peaks around 420 nm. The QE at the peak
varies PMT-by-PMT from 0.249 to 0.306. Within the individual PMTs the QE is
uniform over the photocathode. It was confirmed in advance by measuring the QE
at $18\times 18$ points over the photocathode including the fringe at 0∘
without the semi-cylinder. A typical example is shown in Fig. 5 for PMT3.

Figure 4: Measured QE spectra at 0∘. The error bars of the data are omitted in
this figure.

Figure 5: QE distribution of PMT3 at 0∘ and 360 nm. The color represents the
QE.

The QEs measured as a function of the incident angle are shown in Fig. 6. The
broad feature of the curve is similar for all the PMTs except PMT8. The
prominent rise (valley) of the QE for $s$($p$)-polarization around 43∘
mentioned in Sec. 2.1 cannot be seen for PMT8.

The function of Eq. (27) was fitted to the QE data of each PMT. The fitting
was done simultaneously for both polarizations, all the 18 wavelengths
(320-660 nm) and all the 41 angles (0-80∘) using chi-squared minimization.
Comparisons between the data and the fitted function can be seen for PMT5 as
an example in Fig. 7. The error bars on the data points in this figure include
all the statistical and systematical errors. It is remarkable that the single
function fits all the data of the PMT accurately. For the other PMTs the
agreement between the data and the fitted function is similar or better except
for PMT8. The function of Eq. (29), which has the additional parameter
$L_{e}$, fits better to PMT8. It was also fitted for the other PMTs, and
consistent results with Eq. (27) were obtained.

The best fit parameters of Eq. (27) for PMT1-7 and those of Eq. (29) for PMT8
are shown in Table 8 and Fig. 8. No significant differences in the thickness
and the optical parameters are identified for the antireflection coating. The
errors on $\omega_{t}$ and $\varGamma_{0}$ of the antireflection coating are
sizable. That is because the measurement was done in the wavelength range of
320-660 nm (or 1.9-3.9 eV) and had little sensitivity to $\omega_{t}$ and
$\varGamma_{0}$. Nevertheless the refractive index of the antireflection
coating in that wavelength range should be reliable since it is derived mainly
from $\varepsilon_{s}$. The photocathode has a variation of the thickness.
Especially the one of PMT8 is thicker than the others. It distinguishes the
photocathode response as being different from the others as seen in Figs. 5
and 6; a thicker photocathode is more sensitive at longer wavelengths, and has
less QE variation along with the angle. It is notable that the photocathode of
PMT8 is thicker than $L_{e}$, which requires incorporating the scattering loss
into the QE function as discussed in Sec. 2.3. On the other hand, for the thin
photocathodes of PMT1-7, the approximation by the one-step model holds well.
The optical parameters of the photocathodes are consistent with each other
while $P_{\rm excite}$ differs PMT-by-PMT. $P_{\rm excite}$ and $G$ seem to be
a monotonically increasing function of the photon energy.

The reduced $\chi^{2}$ of the fit is also listed in Table 8. Since the number
of degrees of freedom is as large as 1450 or 1449 with $L_{e}$, a reduced
$\chi^{2}$ which deviates from 1 by about 0.1 or more is statistically
improbable. The small reduced $\chi^{2}$ for PMT1-7 could be due to
overestimation of the systematic errors. Nevertheless the good agreement
between the data and the fitted function indicates validity of the models. The
Lorentz dispersion model is appropriate for this type of photocathodes. The
one-step model of the photoelectron emission is applicable for photocathodes
much thinner than the escape length. For thicker photocathodes the
approximation of $P_{\rm transport}P_{\rm escape}$ in Eq. (26) holds well. A
little deviation of the fitted function from the data was found around
$42^{\circ}$ only in the wavelengths not longer than 340 nm for
$p$-polarization as shown in Fig. 7. It was found in every PMT systematically.
It could be an indication of the effect of the electron-electron scattering,
which we have neglected for 310 nm or longer wavelengths as discussed in Sec.
2.3. The large reduced $\chi^{2}$ for PMT8 is also attributed mainly to that
deviation. It is reasonable that a larger deviation by the scattering loss is
seen in a thicker photocathode.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 6: Measured QE as a function of the incident angle for $s$-polarized
(left) and $p$-polarized (right) light. The data only at 360 nm (top), 420 nm
(middle) and 520 nm (bottom) wavelengths are shown in this figure. The error
bars of the data are omitted. Figure 7: Fitted function (blue lines for
$s$-polarization and red ones for $p$-polarization) overlaid on the data (dots
with error bars) for PMT5. Only those at 340, 380, 420, 460, 500, 540, 580 and
620 nm wavelengths are shown in this figure. Table 1: Thickness and the
optical parameters of the photocathode and the antireflection coating obtained
by the fitting with Eq. (27) for PMT1-7 and Eq. (29) for PMT8. The reduced
$\chi^{2}$ of the fit is also shown in the last column. The last row shows the
average of the eight PMTs.

| Photocathode |
---|---|---
PMT |  $d$ (nm) | $\varepsilon_{s}$ |  $\omega_{t}$ (eV) |  $\varGamma_{0}$ (eV) |  $L_{e}$ (nm)
1 | 11.2$\pm$2.6 | 8.5$\pm$1.8 | 3.17$\pm$0.11 | 1.22$\pm$0.19 | –
2 | 6.4$\pm$2.7 | 9.7$\pm$4.0 | 3.30$\pm$0.14 | 1.23$\pm$0.20 | –
3 | 5.5$\pm$2.3 | 10.9$\pm$4.2 | 3.22$\pm$0.13 | 1.10$\pm$0.19 | –
4 | 8.2$\pm$1.5 | 8.3$\pm$1.4 | 3.23$\pm$0.07 | 1.08$\pm$0.12 | –
5 | 9.9$\pm$2.9 | 8.2$\pm$2.1 | 3.24$\pm$0.10 | 0.98$\pm$0.14 | –
6 | 6.6$\pm$3.9 | 8.7$\pm$4.8 | 3.25$\pm$0.10 | 0.98$\pm$0.15 | –
7 | 8.5$\pm$0.9 | 7.8$\pm$0.8 | 3.25$\pm$0.05 | 0.99$\pm$0.10 | –
8 | 34.1$\pm$3.1 | 6.8$\pm$0.6 | 3.32$\pm$0.05 | 1.04$\pm$0.06 | 27.1$\pm$2.5
Avg. | NA | 7.4$\pm$0.4 | 3.26$\pm$0.03 | 1.05$\pm$0.04 | NA
| Antireflection coating |
PMT |  $d$ (nm) | $\varepsilon_{s}$ |  $\omega_{t}$ (eV) |  $\varGamma_{0}$ (eV) |  $\chi^{2}$/ndf0
1 | 36.8$\pm$5.7 | 4.3$\pm$0.6 | 17.2$\pm$28.4 | 3.4$\pm$13.6 | 0.91
2 | 38.6$\pm$8.0 | 4.0$\pm$0.5 | 19.7$\pm$24.3 | 5.9$\pm$11.7 | 0.71
3 | 40.6$\pm$6.6 | 4.1$\pm$0.5 | 16.4$\pm$12.4 | 4.4$\pm$07.4 | 0.67
4 | 36.9$\pm$5.5 | 4.2$\pm$0.4 | 26.6$\pm$16.9 | 10.6$\pm$13.5 | 0.85
5 | 37.6$\pm$5.9 | 4.5$\pm$0.6 | 24.7$\pm$25.3 | 7.3$\pm$15.8 | 0.91
6 | 30.5$\pm$8.4 | 4.5$\pm$0.8 | 12.5$\pm$10.3 | 3.6$\pm$05.4 | 0.56
7 | 31.7$\pm$4.3 | 4.7$\pm$0.5 | 29.5$\pm$29.1 | 14.5$\pm$12.5 | 0.91
8 | 48.3$\pm$6.9 | 3.6$\pm$0.3 | 11.7$\pm$13.5 | 1.7$\pm$01.4 | 1.22
Avg. | 36.9$\pm$2.1 | 4.1$\pm$0.2 | 16.7$\pm$05.7 | 2.3$\pm$01.3 | –

Figure 8: $P_{\rm excite}$ or $G$ for PMT8 obtained by the fitting.

###  3.3 Effects of the scattering loss

(a)  (b)

(c)  (d)

Figure 9: Expected angular dependence of the QE at 360 nm (left) and 520 nm
(right) with different escape lengths $L_{e}$ for PMT7 (upper) and PMT8
(lower). The QE curves both for $s$\- and $p$-polarizations are plotted in the
same upper panels. The dots with error bars represent the data. The
differences of the data from each curve are shown only for $p$-polarization in
the lower panels.

In order to compare the angular dependence of the QE with different $L_{e}$’s,
Eq. (29) is plotted for PMT7 and 8 in Fig. 9, where the optical parameters are
fixed to those obtained by the fitting but $G$ is adjusted to normalize the QE
to the one of the data at $\theta=0^{\circ}$. $G$ to be applied as a function
of $L_{e}$ is shown in Fig. 10 (left). The effect of $L_{e}$ on the angular
dependence of the QE is found to be little for $s$-polarization, which
indicates that the optical parameters are strongly constrained by the angular
dependence for $s$-polarization independently of $L_{e}$. The one for
$p$-polarization is rather visible, but the angular dependence does not start
to differ clearly from the data down to $L_{e}\approx 10$ nm. That
corroborates the consequence of the one-step model that the angular dependence
is dictated only by the optical properties for thin photocathodes. It also
indicates that $L_{e}$ for PMT1-7 is larger than the photocathode thickness.
Actually $L_{e}$’s obtained by the fitting with Eq. (29) for PMT1-7 are
19.9-32.3 nm though the errors are sizable. Moreover a distinct lower limit on
$L_{e}$ can be obtained from Fig. 10 (left) as $G<1$ by definition: for
example, $L_{e}>4.6$ nm (PMT7) or 21.4 nm (PMT8) at 360 nm wavelength. It is
consistent with the measured $L_{e}$ reported in Ref. escapelength_Dolizy .

According to Eq. (29), the QE largely depends on $L_{e}$ when $L_{e}$ is
roughly equal to or less than the photocathode thickness as shown in Fig. 10
(right), where $\theta=0^{\circ}$ and the parameters other than $L_{e}$ are
fixed. With $L_{e}=10$ nm, for example, the QE at 0∘ and 360 nm wavelength is
reduced to 78% of the one with $L_{e}=$ 27.1 nm for PMT7. Since the effect of
the scattering loss is implicitly incorporated into $P_{\rm excite}$ in the
one-step model, one of the causes for the PMT-by-PMT difference of $P_{\rm
excite}$ therefore could be a difference of $L_{e}$ especially due to the
scattering with impurities or defects.

(a)  (b)

Figure 10: (Left) $G$ which makes the QE function normalized to the data at 0∘
depending on the escape length. (Right) Variation of the QE at 0∘ as a
function of the escape length when $G$ is fixed to the one for $L_{e}$ $=27.1$
nm. $G$ for PMT7 is determined to have the same QE at $L_{e}=27.1$ nm as the
one of the data at 0∘. Those for PMT7 and 8 at 360 and 520 nm wavelengths are
shown in these plots.

##  4 Optical properties of the photocathode

As denoted in Eq. (27) the optical properties of the photocathode as well as
those of the antireflection coating are of great importance in the QE and its
angular dependence. In the following they are discussed using the optical
parameters obtained in the previous section.

###  4.1 Refractive index and absorptance of the photocathode

Figure 11 shows the refractive index ($n+ik$) of PMT5 photocathode derived
from Eqs. (18) and (19) using the best fit parameters in Table 8. The peak of
$n$ and $k$ is around 420 nm and 360 nm, respectively. The spectrum of $k$ is
similar to the one of the QE shown in Fig. 5 as $k$ relates to the absorption.
For comparison Fig. 11 also includes previously published data for multi-
alkali photocathodes composed of the same elements (NaKSbCs) and windows of
two different materials: Jones index_Jones  and Chyba index_Chyba  for
borosilicate windows; Ghosh photoemit_multialkali , Hallensleben
index_Hallensleben  and Harmer index_Harmer  for fused silica windows. The
index measured in this work differs at the shorter wavelengths from Ghosh,
Hallensleben and Harmer. Especially the discrepancy in $k$ with Harmer is
quite large, but $k\sim 0$ below 350 nm or insensitivity to near-ultraviolet
light is rather strange for the multi-alkali photocathode. The dispersion
curves for the other PMTs are plotted in Fig. 12. They are consistent within
the errors since there are no significant differences in the optical
parameters among the PMTs.

(a)  (b)

Figure 11: Real (left) and imaginary (right) parts of the refractive index of
the photocathode for PMT5. The solid and dotted green lines represent the best
fit dispersions and the 1$\sigma$ allowed bands, respectively. The previously
published data for the same type of photocathodes are also plotted for
comparison.

(a)  (b)

Figure 12: Real (left) and imaginary (right) parts of the refractive index of
the photocathode. Only the best fit dispersions are plotted in this figure.

The total absorptions of the photocathodes at 0∘ calculated using Eq. (16) are
shown in Fig. 13 (left). They are nearly the same at wavelengths below 400 nm.
The difference above 400 nm comes mainly from the difference of the
photocathode thickness. Given the same thickness of 10.0 nm for all the
photocathodes, the absorption spectrum is similar to each other as shown in
Fig. 13 (right). Compared to the QE spectrum in Fig. 5, therefore, the
variation of the QE can be attributed not to the optical property of the
photocathode other than the thickness but to $P_{\rm excite}$. It can be
clearly seen in Fig. 14, where the QE at 0∘ is plotted relative to the total
absorption or $P_{\rm excite}$ at 360 nm for the eight PMTs. The QE has a
correlation not with the total absorption but with $P_{\rm excite}$, which is
subject to the work function and the escape length.

(a)  (b)

Figure 13: Total absorption of the photocathode at 0∘ including the back-
illumination. (Left) Calculated with the best fit parameters in Table 8.
(Right) Calculated with the same parameters other than the photocathode
thickness, which is fixed to 10.0 nm for all the photocathodes.

(a)  (b)

Figure 14: Correlation of the QE at 0∘ with the total absorption (left) and
with $P_{\rm excite}$ (right) at 360 nm. $P_{\rm excite}$ for PMT8 is derived
from the QE divided by the total absorption.

###  4.2 Effect of the back-illumination

Figure 15: Intensity fraction of 360 nm light transmitted through PMT7
photocathode (dotted line), one of the back-illumination after reflection on
the electrode (dashed line) and one of the back-illumination absorbed by the
photocathode (solid line) with respect to the initial light. The blue (red)
lines represent those for $s$($p$)-polarization.

Figure 16: QE at 360 nm as a function of the incident angle calculated with
and without the back-illumination (solid and dashed lines, respectively) for
PMT7. The blue (red) lines represent those for $s$($p$)-polarization.

The QE was measured as a sum of the front- and back-illuminations. As denoted
in Eq. (16) the back-illumination is reduced by the factors $T_{\rm 2+3}$,
$1-\mathcal{R}_{\rm OA}$ and $R_{\rm el}$, and it should have less
contribution to the QE. Figure 16 shows $T_{\rm 2+3}$, $T_{\rm
2+3}(1-\mathcal{R}_{\rm OA})R_{\rm el}$ and $T_{\rm 2+3}(1-\mathcal{R}_{\rm
OA})R_{\rm el}A_{\bar{2}}$ at 360 nm as a function of the incident angle
$\theta_{1}$. $T_{\rm 2+3}$ is 0.232 at 0∘ and increases as the angle for
$p$-polarization, while it decreases for $s$-polarization. Hence the
contribution of the back-illumination is larger for $p$-polarization. The
following discussion takes the case of the normal incidence at 360 nm. The
factor $(1-\mathcal{R}_{\rm OA})R_{\rm el}$ reduces the fraction of the back-
illumination down to 0.052. In addition the reflection at the vacuum-
photocathode interface for the back-illumination is higher than the one at the
quartz-photocathode interface for the front-illumination due to a larger
mismatch of the refractive indices. For example, the reflectance of the
photocathode $|\hat{r}|^{2}$ for the back- and front-illumination is 0.405 and
0.224, respectively, where the antireflection coating is omitted for
simplicity and the parameters of PMT7 photocathode are used. It further
reduces the fraction of the absorbed back-illumination down to 0.013,
resulting in the QE of 0.006 regarding only the back-illumination. The effect
of the back-illumination is feeble at any angle as shown in Fig. 16.

Even if a more reflective material is put behind the photocathode to fully
reflect the transmitted light, the enhancement of the QE cannot be
significant. The maximum possible contribution of the back-illumination at 0∘,
namely under assumption of $(1-\mathcal{R}_{\rm OA})R_{\rm el}=1$, is shown in
Fig. 17. The factor of enhancement by the full back-illumination is only 1.09
at 360 nm for this photocathode with the antireflection coating. It is
inversely related to the QE for the front-illumination since the transmittance
of the photocathode decreases as the absorptance increases. Therefore the
fraction of the back-illumination becomes larger away from the wavelength of
the peak QE, but the net QE increase by the full back-illumination is nearly
constant at about 0.01-0.03 regardless of the wavelength. In the same manner
the ratio of the back-illumination becomes larger without the antireflection
coating. The increase of the QE by the antireflection coating is, however,
larger than the one by the back-illumination at the peak wavelength, and
removing the antireflection coating to increase the back-illumination
adversely decreases the QE.

Another attempt is to put the second photocathode of the reflection mode which
detects transmitted light through the first photocathode JUNO_MCP-PMT . For
the same reason mentioned above, however, one can expect a little improvement
of the QE, while the dark noise rate increases in proportion to the
photocathode area.

Since the contribution of the back-illumination is little for the PMTs in this
work, only the front-illumination is considered in the following discussions.

(a)  (b)

Figure 17: (Left) Ratio of the absorbed back- to front-illumination at 0∘ when
$(1-\mathcal{R}_{\rm OA})R_{\rm el}$ is assumed to be unity. It is calculated
for PMT7 photocathode with and without the antireflection coating (AR).
(Right) Respective contributions of the front- and back-illuminations to the
QE at 0∘ for PMT7 photocathode with the antireflection coating. Both the
actual and the maximum possible back-illumination are plotted.

###  4.3 Effect of the antireflection coating

Figure 18 shows the refractive index ($n+ik$) of the antireflection coating
derived from Eqs. (18) and (19) using the best fit parameters in Table 8. The
dispersion in this range of wavelength is quite small; $n$ is about 2.1 and
$k$ is nearly 0. The thickness of the antireflection coating ranges from 30.5
to 48.3 nm. These values of the index and the thickness are adequate for
destructive interference of the reflected light waves to reduce the
reflection. The effect of the antireflection coating on the QE is manifest as
shown in Fig. 19. The factor of enhancement of the QE is 1.16 at 360 nm and
0∘. More significant enhancement can be seen at large incident angles for
$s$-polarization. On the other hand the QE at 460 nm or longer is reduced by
the antireflection coating due to constructive interference of the reflected
light waves to increase the reflection. Although the antireflection coating
has a nonzero value of $k$ and absorbs the light, it is not the main cause of
the reduction. The absorptance of the antireflection coating is, for example
in the case of PMT7, 0.074 at 360 nm and almost constant at 0.016 above 500
nm.

(a)  (b)

Figure 18: Real (left) and imaginary (right) parts of the refractive index of
the antireflection coating. The one of quartz is also shown for comparison.

(a)  (b)

Figure 19: Comparisons of the QE for PMT7 calculated with and without the
antireflection coating (AR). Only the front-illumination is taken into
account. (Left) QE at 0∘ as a function of the wavelength. (Right) QE at 360 nm
as a function of the incident angle. The blue (red) lines represent the QE for
$s$($p$)-polarization.

Adjustment of the thickness of the antireflection coating could help to
enhance the QE at a desired range of the wavelength. Given the refractive
indices of the window, antireflection coating and photocathode, the best
matching thickness of the antireflection coating can be estimated. Figure 21
shows the dependence of the absorptance of the photocathode on the thickness
of the antireflection coating, where the other parameters than the thickness
are the same as PMT7. The absorptance is maximized at the thickness of 28.9
nm, 51.4 nm and 87.0 nm for the wavelength of 360 nm, 420 nm and 520 nm,
respectively. By adjusting the thickness to these values, the absorptance
increases by 0.3%, 7.6% and 20.3%, respectively, from the one at the actual
thickness of 31.7 nm.

Figure 20: Absorptance of PMT7 photocathode at 0∘ as a function of the
thickness of the antireflection coating. There are three curves for 360, 420
and 520 nm wavelengths. Only the front-illumination is taken into account.

Figure 21: Absorptance of PMT7 photocathode at 0∘ as a function of the
photocathode thickness. There are three curves for 360, 420 and 520 nm
wavelengths. Only the front-illumination is taken into account.

###  4.4 Photon absorption in depth

Another parameter which could be tuned to enhance the QE is the thickness of
the photocathode. In terms of the absorptance a thicker photocathode is better
especially for longer wavelengths. It is also better in terms of the QE under
the condition of the one-step model, where the QE is simply proportional to
the absorptance. Nevertheless for a thick photocathode comparable to the
electron escape length, one has to take into account the scattering loss. The
optimal thickness can be foreseen in Fig. 21, where the absorptance is plotted
as a function of the photocathode thickness by using Eq. (14). The absorptance
at 10 nm thickness is close to its maximum for 360 nm wavelength, and a
thicker photocathode could result in a less QE as only the scattering loss
increases. Therefore the thickness around 10 nm is optimal for the QE at 360
nm. To enhance the QE at longer wavelengths, for example up to 520 nm, the
optimal thickness should be around 70 nm or less.

(a)  (b)

Figure 22: Absorptance per unit length at 360 nm (left) and 520 nm (right)
wavelengths as a function of the depth from the photocathode surface. There
are four curves for different photocathode thicknesses, 10, 20, 40 and 80 nm.
The refractive indices of the photocathode and antireflection coating for PMT7
are used in these plots. Only the front-illumination is taken into account.

The absorptance per unit length expressed by Eq. (10) is shown in Fig. 22. It
is different from the often-used Lambert-Beer law, $e^{-\alpha z}$, described
by an absorption coefficient $\alpha$. That is due to the light waves
reflected at the interfaces and their interference. Only when the photocathode
is enough thick, the absorptance can be approximated by $e^{-\alpha z}$ as the
light wave heading forward is fully absorbed before reflected. An example can
be found in Fig. 22 for 80 nm thickness and 360 nm wavelength. The assumption
of an infinite thickness, which is usually taken, is therefore not appropriate
for thin photocathodes.

###  4.5 Prospect for enhancement of the QE

A straightforward way to enhance the QE is to put the photocathode aslant so
that the light strikes at 50-80∘, where the QE is higher than the one at 0∘ as
shown in Fig. 6. Because it is discussed elsewhere QE_angle1 ; QE_angle2 ,
other possibilities are mentioned here in terms of enhancement of the
intrinsic QE for the same type of photocathodes. As discussed above there is a
little room for improvement of the peak QE by adjusting the thicknesses of the
antireflection coating and the photocathode. The strong correlation of the QE
with $P_{\rm excite}$ found in Fig. 14 indicates that one of the dominant
factors dictating the QE could be the work function. Because the low work
function of the NaKSbCs photocathode is derived from a Cs-Sb dipole monolayer
S20_Holtom , matters to the QE could be, for example, the arrangement of
surface atoms workfunction_Smoluchowski , the monolayer coverage
workfunction_Gyftopoulos , or the surface roughness photocathode_roughness .
Less significant in the QE should be the bulk of the photocathode, which
dictates the optical properties, though impurities and defects in the bulk
could be another dominant factor of the QE reduction.

##  5 Conclusion

A precise description of the PMT response requires formulation of the
dependence of the QE on the angle and polarization of the incident photon,
which was yet to be studied. Hence I proposed the one-step model of
photoelectron emission, or photoexcitation of an electron to an energy above a
certain threshold for the emission, which is adequate for thin multi-alkali
photocathodes and for visible light. In this model the dependence of the QE on
the angle and polarization is fully involved in the absorptance, and thus it
can be rigorously described by the optics theory. The theoretical function of
the QE derived in this work accurately fit the measured dependence of the
photocurrent on the angle and polarization at all the wavelengths from 320 to
680 nm simultaneously. That demonstrated the picture of the photoelectron
emission rendered by the one-step model.

The measurement furnished the new data of the refractive index of the NaKSbCs
photocathode over the whole visible range. It revealed the optical properties
of the photocathode and the following new insights were obtained. It was
demonstrated for the first time that the dispersion of the refractive index of
NaKSbCs can be described by the Lorentz model. The gain in QE by reabsorbing
the transmitted light through the photocathode is feeble because of the low
transmittance of the efficient photocathode and the large reflection at the
vacuum-photocathode interface. The QE differences among the photocathodes of
the same type are attributed dominantly to the thickness of the photocathode
and $P_{\rm excite}$. It implies that a key of the QE enhancement could be the
condition of the photocathode surface on the vacuum or the degree of
crystalline perfection of the bulk.

In terms of the methodology for deducing the refractive index and thickness of
the photocathode, the measurement of the photocurrent has proven to be useful
in this work. It was also demonstrated for the first time that this method can
be applied for the stratified antireflection coating and photocathode and that
the refractive indices and thicknesses of both layers can be deduced at the
same time.

To conclude, the expression of the QE derived in this work enables the
accurate measurement of the optical properties for thin multi-alkali
photocathodes in the visible range and hence the precise description of the
PMT response. It should be valid for the other types of thin photocathodes as
long as the applicable range of the wavelength is reevaluated based on the
threshold energy for the electron-electron scattering during the
transportation, and if the Lorentz dispersion model is appropriate for those
photocathodes.

## Acknowledgment

This work was supported by JSPS KAKENHI Grant-in-Aid for Challenging
Exploratory Research (Grant Number JP26610068). The MCP-PMTs measured in this
work were produced for the Belle II TOP counter with support from JSPS KAKENHI
Grant-in-Aid for Scientific Research (S) (Grant Number JP26220706). I would
like to thank my colleagues at Nagoya University for discussion, in particular
T. Iijima and K. Inami. I have also benefited from discussion with H.
Watanabe, Y. Hasegawa, H. Nishizawa and H. Yamaguchi (HAMAMATSU PHOTONICS
K.K.).

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