# Optimizing the performance of an Astigmatic Defocus Sensor and its
implementation using Computer Generated Hologram

Santanu Konwar
Abhayapuri College, Abhayapuri, Bongaigaon, Assam, India
Email<EMAIL_ADDRESS>
###### Abstract

Astigmatic Defocus Sensor is a very simple and easy technique of measuring the
amount of defocus (deviation from the focal plane) present in an optical beam.
The approach use a cylindrical lens to focus the beam and the measure of the
intensity at the focal plane is used to find out the amount of defocus present
in the beam. In this paper we provide a theoretical discussion, on the
Astigmatic Defocus Sensor, based on Fourier optics. The theoretical discussion
is validated through simulation results. Two ways to optimize the performance
of the astigmatic defocus sensor were also established. Further, the process
of implementing the Astigmatic Defocus Sensor using computer generated
hologram is presented.

##  1 Introduction

An optical beam with astigmatism can be sharply focussed at two different
planes perpendicular to the propagation direction. The beam when focused on
one of the plane shows a intensity distribution of line shaped directed along
a certain direction, on the other hand the intensity distribution focussed on
the other plane is line shaped oriented in a direction rotated by $90^{o}$
degree to that of the previous one. The focal plane of the beam free from
astigmatism lies in between these two planes, where the intensity distribution
will appear as a cross due to the effect of astigmatism [2]. This unique
property of change in orientation of the intensity distribution of the beam in
presence of astigmatism can be used for the measurement of focussing error or
the amount of defocus present in an optical beam [4, 10]. The astigmatic lens
approach is a very simple technique that gives a direct measure of the defocus
present in the beam. The technique being free from any intensive computation,
provides a very fast and effective measure of the focussing error and can be
used for auto focusing an optical beam [4, 6], dynamic focus control [7],
thickness measurement [9], beam tracking [10], etc.

The defocus measurement can be carried out using a single astigmatic lens
system [4, 12] or a double astigmatic lens system [10, 1]. However, in this
paper we deal with the single astigmatic lens approach. We present an
elaborate theoretical discussion on the sensor followed by simulation
validation. The theoretical discussion is mainly based on Fourier optics[5]
and the Zernike polynomials[11]. We also deduce an expression for the
sensitivity and discuss on the cross sensitivity of of the sensor. Simulations
are carried out to test the validity of the theoretical findings. Further, two
ways of optimizing the performance of the astigmatic defocus sensor are also
established. Later, the implementation of the Astigmatic Defocus Sensor using
computer generated hologram is presented.

##  2 Theory

To begin with, let us consider that a monochromatic collimated beam with a
certain phase $\psi(r,\theta)$ is passing through a converging lens of focal
length $f$ as shown in Fig. 1.

Figure 1: Illustration of a monochromatic collimated beam with a wavefront $W$
passing through a converging lens $L$ and focussed at a point $P$ on the
detector $F$

The amplitude distribution immediately after the lens $L$ will be

$U^{\prime}_{L}(x,y)=U_{L}(x,y)circ(r)e^{-j\frac{k}{2f}(x^{2}+y^{2})}$ (1)

where, $U_{L}(x,y)$ is the amplitude distribution of the wavefront $W$
incident on the lens given by

$U_{L}(x,y)=Ae^{j\psi(r,\theta)}$ (2)

and $circ(r)$ defines a circular aperture given by

$circ(r)=\begin{cases}1,&\text{for}\;r\leq 1\\\ 0,&\text{for}\;r>1\end{cases}$

The amplitude distribution at the back focal plane $F$ of the lens can be
obtained from the Fresnel diffraction integral which can be written as [5].

$\displaystyle U(u,v)$ $\displaystyle=$ $\displaystyle\frac{e^{jkz}}{j\lambda
z}e^{j\frac{k}{2z}(u^{2}+v^{2})}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}[U(x,y)e^{j\frac{k}{2z}(x^{2}+y^{2})}]$
(3) $\displaystyle e^{-j\frac{2\pi}{\lambda z}(xu+yv)}dxdy$

Now, by taking $z=f$ equation 3 can be rewritten as

$\displaystyle U_{F}(u,v)$ $\displaystyle=$
$\displaystyle\frac{e^{jkf}}{j\lambda
f}e^{j\frac{k}{2f}(u^{2}+v^{2})}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}[U^{\prime}_{L}(x,y)e^{j\frac{k}{2f}(x^{2}+y^{2})}]$
(4) $\displaystyle e^{-j\frac{2\pi}{\lambda f}(xu+yv)}dxdy$

where $U_{F}(u,v)$ is the amplitude distribution at the back focal plane of
the lens. On removing the constant phase and further simplification, we get

$\displaystyle U_{F}(u,v)$ $\displaystyle=$
$\displaystyle\frac{e^{j\frac{k}{2f}(u^{2}+v^{2})}}{j\lambda
f}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}[U_{L}(x,y)circ(r)e^{j\frac{k}{2f}(x^{2}+y^{2})}$
$\displaystyle e^{j\frac{k}{2f}(x^{2}+y^{2})}]e^{-j\frac{2\pi}{\lambda
f}(xu+yv)}dxdy$ $\displaystyle=$
$\displaystyle\frac{e^{j\frac{k}{2f}(u^{2}+v^{2})}}{j\lambda
f}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}U_{L}(x,y)circ(r)e^{-j\frac{2\pi}{\lambda
f}(xu+yv)}dxdy$

The expression in equation LABEL:eq:refname5 can be transformed into polar
coordinates using the condition (detail derivation of the expression is
available in Annexure 1)

$\displaystyle\frac{1}{\lambda f}(ux+vy)$ $\displaystyle=$ $\displaystyle\rho
r(cos\phi cos\theta+sin\phi sin\theta)$ (6) $\displaystyle=$
$\displaystyle\rho rcos(\phi-\theta)$

Thus, in polar coordinates, equation LABEL:eq:refname5 can be written as

$U_{F}(\rho,\phi)=\frac{e^{j\frac{k}{2f}\rho^{2}}}{j\lambda
f}\int^{2\pi}_{0}\int^{\infty}_{0}U_{L}(r,\theta)e^{-j2\pi\rho
rcos(\phi-\theta)}rdrd\theta$ (7)

where, $\rho^{2}=u^{2}+v^{2}$ and

$U_{L}(r,\theta)=Ae^{j\psi(r,\theta)}$ (8)

The term $circ(r)$ is omitted keeping in view that the co-ordinates taken now
are polar coordinates for a circular window. The intensity distribution at the
back focal plane of the lens can thus be written as

$\displaystyle I_{F}(\rho,\phi)$ $\displaystyle=$
$\displaystyle\left[\frac{e^{j\frac{k}{2f}\rho^{2}}}{j\lambda
f}\int^{2\pi}_{0}\int^{\infty}_{0}U_{L}(r,\theta)e^{-j2\pi\rho
rcos(\phi-\theta)}rdrd\theta\right]$ (9) $\displaystyle\times$
$\displaystyle\left[\frac{e^{-j\frac{k}{2f}\rho^{2}}}{-j\lambda
f}\int^{2\pi}_{0}\int^{\infty}_{0}U_{L}(r,\theta)e^{-j2\pi\rho
rcos(\phi-\theta)}rdrd\theta\right]$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda^{2}f^{2}}\left|\int^{2\pi}_{0}\int^{\infty}_{0}U_{L}(r,\theta)e^{-j2\pi\rho
rcos(\phi-\theta)}rdrd\theta\right|^{2}$

The phase profile of the incident beam $\psi(r,\theta)$ can be represented in
terms of Zernike modes as [11].

$\psi(r,\theta)=\sum^{\infty}_{k=0}a_{k}Z_{k}(r,\theta)$ (10)

where $Z_{k}(r,\theta)$ are Zernike modes that can represent the optical
aberrations modes correctly. The Zernike modes can be represented as [11]

$\begin{array}[]{l}{\left.\begin{array}[]{l}Z_{even\,k}=\sqrt{n+1}R_{n}^{m}(r)\sqrt{2}\cos(m\theta)\\\
Z_{odd\,k}=\sqrt{n+1}R_{n}^{m}(r)\sqrt{2}\sin(m\theta)\end{array}\right\\}\,\mbox{for
m$\neq$0}}\\\ Z_{k}=\sqrt{n+1}R_{n}^{m}(r)\,\mbox{\; \; for m=0}\end{array}$
(11)

and

$R_{n}^{m}(r)=\sum_{q=0}^{\frac{(n-m)}{2}}\frac{(-1)^{q}(n-q)!}{q!\left[\frac{n+m}{2}-q\right]!\left[\frac{n-m}{2}-q\right]!}r^{n-2q}$
(12)

Few low order Zernike polynomials are shown in table 1 below.

Table 1: Zernike polynomials representing few low order aberrations index ($j$) | $Z_{j}(r,\theta)$ | common aberration
---|---|---
$k=4$ | $\sqrt{3}(2r^{2}-1$ | Defocus
$k=5$ | $\sqrt{6}r^{2}sin{2\theta}$ | Astigmatism $\pm 45^{o}$
$k=6$ | $\sqrt{6}r^{2}cos{2\theta}$ | Astigmatism $0^{o}$
$k=7$ | $\sqrt{8}(3r^{2}-2r)sin\theta$ | y coma
$k=8$ | $\sqrt{8}(3r^{2}-2r)cos\theta$ | x coma
$k=9$ | $\sqrt{8}r^{3}sin{3\theta}$ | Trefoil y
$k=10$ | $\sqrt{8}r^{3}cos{3\theta}$ | Trefoil x
$k=11$ | $\sqrt{5}(6r^{4}-6r^{2}+1)$ | Spherical aberration

###  2.1 Astigmatic Defocus Sensor

If a beam with a plane wavefront passes through an astigmatic lens rotated at
$45^{o}$, i.e. the lens $L$ of Fig 1 is replaced by an astigmatic lens, the
amplitude distribution immediately after the lens will be

$U_{L}(r,\theta)=Ae^{jaZ_{5}(r,\theta)}$ (13)

where $Z_{5}(r,\theta)$ is the Zernike mode that represent astigmatism with
$45^{o}$ tilt and $a_{5}$ is taken as $a$ for simplicity. If the Zernike modes
are defined over an unit circular aperture, the intensity at the detector will
be

$I(\rho,\phi)=\frac{A}{\lambda^{2}f^{2}}\left|\int^{2\pi}_{0}\int^{1}_{0}e^{jaZ_{5}(r,\theta)}e^{-j2\pi\rho
rcos(\phi-\theta)}rdrd\theta\right|^{2}$ (14)

For a finite size detector of radius $\rho_{p}$ defined in Fourier
coordinate[5], the intensity recorded by the detector can be obtained by
taking the integration of the whole detector area

$I_{T}(\rho,\phi)=\int^{2\pi}_{0}\int^{\rho_{p}}_{0}I(\rho,\phi)\rho d\rho
d\phi$ (15)

Figure 2: Illustration of the intensity distribution at the detector plane
with $a=1$ radian and (i) $b=0$ radian, (ii) $b=1$ radian and (iii) $b=-1$
radian

If we divide the detector aperture into four equal parts as shown in figure 2,
the intensities at each sub-aperture will be

$\displaystyle I_{1}(\rho,\phi)$ $\displaystyle=$
$\displaystyle\int^{\frac{\pi}{2}}_{0}\int^{\rho_{p}}_{0}I(\rho,\phi)\rho
d\rho d\phi$ (16) $\displaystyle I_{2}(\rho,\phi)$ $\displaystyle=$
$\displaystyle\int^{\pi}_{\frac{\pi}{2}}\int^{\rho_{p}}_{0}I(\rho,\phi)\rho
d\rho d\phi$ (17) $\displaystyle I_{3}(\rho,\phi)$ $\displaystyle=$
$\displaystyle\int^{\frac{3\pi}{2}}_{\pi}\int^{\rho_{p}}_{0}I(\rho,\phi)\rho
d\rho d\phi$ (18) $\displaystyle I_{4}(\rho,\phi)$ $\displaystyle=$
$\displaystyle\int^{2\pi}_{\frac{3\pi}{2}}\int^{\rho_{p}}_{0}I(\rho,\phi)\rho
d\rho d\phi$ (19)

Now, let us consider that the incident beam consist of certain amount (say $b$
radian) of defocus ($Z_{4}$) as shown in figure 2(ii - iii). As seen in the
figure, the presence of defocus incorporated directionality to the intensity
profile at the detector plane. Further the intensity profile changes the
orientation for opposite polarity of defocus. The intensity at the detector
plane now becomes

$I(\rho,\phi)=\left|\int^{2\pi}_{0}\int^{1}_{0}e^{jaZ_{5}(r,\theta)+jbZ_{4}(r,\theta)}e^{-j2\pi\rho
rcos(\phi-\theta)}rdrd\theta\right|^{2}$ (20)

The constant term $\frac{A}{\lambda^{2}f^{2}}$ is taken as 1 to keep the
expression simple. The amount of defocus present in the incident beam can be
obtained from the record of the intensities of the different sub-apertures.
The sensor output can thus be obtained from the expression [12]

$S=\frac{(I_{1}+I_{3})-(I_{2}+I_{4})}{(I_{1}+I_{3})+(I_{2}+I_{4})}$ (21)

It is quite clear from figure 2(ii - iii) that there is symmetry between the
diagonally opposite sub-apertures and the total intensities of the diagonally
opposite sub-apertures are same. Thus, $I_{1}=I_{3}$ and $I_{2}=I_{4}$ in both
the cases of $b=1$ and $b=-1$. Therefore the expression of sensor output can
be simplified as

$S=\frac{(I_{1}-I_{2})}{(I_{1}+I_{2})}$ (22)

###  2.2 Sensitivity of Astigmatic Defocus Sensor

The sensitivity of a sensor is a measure of the effectiveness of the sensor
while measuring small aberration. It can be calculated by differentiating the
sensor output by the amplitude of the input aberration mode ($Z_{4}$ in this
case) i.e. $b$, at the amplitude of the bias aberration mode ($Z_{5}$ in this
case) $a=0$ [12]. Therefore proceeding accordingly we first calculate
$\frac{\partial S}{\partial b}$ as

$\displaystyle\frac{\partial S}{\partial b}$ $\displaystyle=$
$\displaystyle\frac{(I_{1}+I_{2})\frac{\partial}{\partial
b}(I_{1}-I_{2})-(I_{1}-I_{2})\frac{\partial}{\partial
b}(I_{1}+I_{2})}{(I_{1}+I_{2})^{2}}$ (23) $\displaystyle=$
$\displaystyle\frac{2I_{2}\frac{\partial}{\partial
b}I_{1}-2I_{1}\frac{\partial}{\partial b}I_{2}}{(I_{1}+I_{2})^{2}}$

And,

$\displaystyle\frac{\partial}{\partial b}\left\\{I(u,v)\right\\}$
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial
b}\left|\int^{2\pi}_{0}\int^{1}_{0}e^{jaZ_{5}(r,\theta)+jbZ_{4}(r,\theta)}e^{-j2\pi\rho
rcos(\phi-\theta)}rdrd\theta\right|^{2}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
b}\left[\int^{2\pi}_{0}\int^{1}_{0}e^{j[aZ_{5}(r,\theta)+bZ_{4}(r,\theta)-2\pi\rho
rcos(\phi-\theta)}rdrd\theta\right.$
$\displaystyle\times\left.\int^{2\pi}_{0}\int^{1}_{0}e^{-j[aZ_{5}(r,\theta)+bZ_{4}(r,\theta)-2\pi\rho
rcos(\phi-\theta)}rdrd\theta\right]$ $\displaystyle=$
$\displaystyle-j\int^{2\pi}_{0}\int^{1}_{0}e^{j[aZ_{5}(r,\theta)+bZ_{4}(r,\theta)-2\pi\rho
rcos(\phi-\theta)}rdrd\theta$
$\displaystyle\times\int^{2\pi}_{0}\int^{1}_{0}Z_{4}(r,\theta)e^{-j[aZ_{5}(r,\theta)+bZ_{4}(r,\theta)-2\pi\rho
rcos(\phi-\theta)}rdrd\theta$
$\displaystyle+\>j\int^{2\pi}_{0}\int^{1}_{0}e^{-j[aZ_{5}(r,\theta)+bZ_{4}(r,\theta)-2\pi\rho
rcos(\phi-\theta)}rdrd\theta$
$\displaystyle\times\int^{2\pi}_{0}\int^{1}_{0}Z_{4}(r,\theta)e^{j[aZ_{5}(r,\theta)+bZ_{4}(r,\theta)-2\pi\rho
rcos(\phi-\theta)}rdrd\theta$

which gives

$\displaystyle\left.\frac{\partial}{\partial
b}\left\\{I(u,v)\right\\}\right|_{b=0}$ (25) $\displaystyle=$
$\displaystyle-j\left[\int^{2\pi}_{0}\int^{1}_{0}e^{j[aZ_{5}(r,\theta)-2\pi\rho
rcos(\phi-\theta)}rdrd\theta\right.$
$\displaystyle\times\int^{2\pi}_{0}\int^{1}_{0}Z_{4}(r,\theta)e^{-j[aZ_{5}(r,\theta)-2\pi\rho
rcos(\phi-\theta)}rdrd\theta$
$\displaystyle-\>\int^{2\pi}_{0}\int^{1}_{0}e^{-j[aZ_{5}(r,\theta)-2\pi\rho
rcos(\phi-\theta)}rdrd\theta$
$\displaystyle\times\left.\int^{2\pi}_{0}\int^{1}_{0}Z_{4}(r,\theta)e^{j[aZ_{5}(r,\theta)-2\pi\rho
rcos(\phi-\theta)}rdrd\theta\right]$

Since at $b=0$, $I_{1}=I_{2}$ and $\frac{\partial}{\partial
b}(I_{1})=-\frac{\partial}{\partial b}(I_{2})$, therefore we have

$\displaystyle\left.\frac{\partial S}{\partial b}\right|_{b=0}$
$\displaystyle=$ $\displaystyle\frac{2I_{2}\left.\frac{\partial}{\partial
b}I_{1}\right|_{b=0}-2I_{1}\left.\frac{\partial}{\partial
b}I_{2}\right|_{b=0}}{\left.(I_{1}+I_{2})^{2}\right|_{b=0}}$ (26)
$\displaystyle=$ $\displaystyle\frac{2I_{1}\left.\frac{\partial}{\partial
b}I_{1}\right|_{b=0}+2I_{1}\left.\frac{\partial}{\partial
b}I_{1}\right|_{b=0}}{\left.(2I_{1})^{2}\right|_{b=0}}$ $\displaystyle=$
$\displaystyle\frac{\left.\frac{\partial}{\partial
b}I_{1}\right|_{b=0}}{\left.I_{1}\right|_{b=0}}$

Thus, equation 26 gives the sensitivity of the sensor. Now using equation 16,
20, 25 and 26, the sensitivity of the sensor can be calculated for different
RMS amplitude of $Z_{5}$ mode ($a$) and also for different detector radius in
Fourier plane coordinates ($\rho_{p}$) [5].

Figure 3: Plot of sensitivity verses (i) RMS amplitude of astigmatism $45^{o}$
(i.e. $a$) in radian and detector radius in Fourier coordinate (i.e.
$\rho_{p}$) and (ii) RMS amplitude of astigmatism $45^{o}$ (i.e. $a$) in
radian.

Figure 3(i) shows the plot of the sensitivity of the sensor for different RMS
amplitude of $Z_{5}$ mode ($a$) and detector radius in Fourier plane
coordinates ($\rho_{p}$). In the figure $a$ is taken along the horizontal axis
and $\rho_{p}$ is taken along the vertical axis, while the sensitivity is
represented in different colors ranging from -1.5 to +1.5. The physical
detector radius can be calculated from the Fourier plane coordinates
($\rho_{p}$) by using the expression $R=\rho_{p}\lambda f$, where $\lambda$ is
the wavelength of the light beam and $f$ is the focal length of the focusing
lens used. It is observed that for RMS amplitudes of $Z_{5}$ ranging from
$a=0.8$ – $1.4$ radian and $\rho_{p}>0.8$ we get a comparatively good
sensitivity (i.e. $>1$). It is found that a sensitivity of $\approx 1.2$ is
achieved at $\rho_{p}=2.1$. Figure 3(ii) shows a plot of the sensitivity of
the sensor verses RMS amplitudes of $Z_{5}$ ($a$) taking $\rho_{p}=2.1$. It is
observed that the maximum sensitivity of $1.2238$ is obtained at $a=1.1$
radian.

###  2.3 Cross-sensitivity of Astigmatic Defocus Sensor

In order to test the effectiveness of the sensor, let us check whether the
sensor shows any sensitivity towards other Zernike modes (aberration modes),
i.e. other than the defocus mode. We termed it as cross-sensitivity. We tested
the cross-sensitivity considering one of the six low order aberration modes to
be present in the beam. The intensity distribution of the beam, comprising
astigmatism $0^{o}$ ($Z_{6}$), y coma ($Z_{7}$), x coma ($Z_{8}$), y trefoil
($Z_{9}$), x trefoil ($Z_{10}$) and spherical aberration ($Z_{11}$) at the
detector plane when focused by an astigmatic lens is shown in Fig. 4
(i)$\rightarrow$ (vi).

The evaluation of the cross-sensitivity of each of the modes is discussed
below -

  * •

From Fig 4(i) it is seen that in case of astigmatism $0^{o}$ (Zernike mode
$Z_{6}$) the intensities of sub-aperture 1, 2, 3, and 4 are equal, i.e.
$I_{1}=I_{2}=I_{3}=I_{4}$ . Thus, from equation 21 the sensor output, $S=0$
and hence the sensitivity $\left.\frac{\partial S}{\partial
b}\right|_{b=0}=0$.

  * •

From Fig 4(ii) it is seen that in case of y coma (Zernike mode $Z_{7}$) the
intensity distribution of sub-aperture 1 $\&$ 2 and 3 $\&$ 4 are symmetric and
thus $I_{1}=I_{2}$ and $I_{3}=I_{4}$. Therefore, from equation 21 the sensor
output, $S=0$ and hence the sensitivity $\left.\frac{\partial S}{\partial
b}\right|_{b=0}=0$.

  * •

From Fig 4(iii) it is seen that in case of x coma (Zernike mode $Z_{8}$) the
intensity distribution of sub-aperture 1 $\&$ 4 and 2 $\&$ 3 are symmetric and
thus $I_{1}=I_{4}$ and $I_{2}=I_{3}$. Therefore, from equation 21 the sensor
output, $S=0$ and hence the sensitivity $\left.\frac{\partial S}{\partial
b}\right|_{b=0}=0$.

  * •

From Fig 4(iv) it is seen that, similar to that of y coma, in case of y
trefoil (Zernike mode $Z_{9}$) the intensity distribution of sub-aperture 1
$\&$ 2 and 3 $\&$ 4 are symmetric and thus $I_{1}=I_{2}$ and $I_{3}=I_{4}$.
Therefore, from equation 21 the sensor output, $S=0$ and hence the sensitivity
$\left.\frac{\partial S}{\partial b}\right|_{b=0}=0$.

  * •

From Fig 4(v) it is seen that, similar to that of x coma, in case of x trefoil
(Zernike mode $Z_{10}$) the intensity distribution of sub-aperture 1 $\&$ 4
and 2 $\&$ 3 are symmetric and thus $I_{1}=I_{4}$ and $I_{2}=I_{3}$.
Therefore, from equation 21 the sensor output, $S=0$ and hence the sensitivity
$\left.\frac{\partial S}{\partial b}\right|_{b=0}=0$.

  * •

From Fig 4(vi) it is seen that in case of spherical aberration (Zernike mode
$Z_{11}$) the intensity distribution of sub-aperture 1 $\&$ 3 and 2 $\&$ 4 are
symmetric and thus $I_{1}=I_{3}$ and $I_{2}=I_{4}$. The situation here is
similar to that of defocus (Zernike mode $Z_{4}$). Therefore, there will a non
zero cross-sensitivity in case of spherical aberration. The cross-sensitivity
can be calculated by replacing Zernike mode $Z_{4}$ by $Z_{11}$ in equation 20
and 25 and by using equation 16 and 26.

Figure 4: Illustration of the intensity distribution at the detector plane
with $a=1$ radian and $1$ radian RMS amplitude of (i) astigmatism $0^{o}$
($Z_{6}$), (ii) y coma ($Z_{7}$), (iii) x coma ($Z_{8}$), (iv) y trefoil
($Z_{9}$), (v) x trefoil ($Z_{10}$) and (vi) spherical aberration ($Z_{11}$)

The cross-sensitivities due to the Zernike modes $Z_{6}$, $Z_{7}$, $Z_{8}$,
$Z_{9}$, $Z_{10}$ and $Z_{11}$ are shown in Table 2.

Table 2: cross-sensitivity obtained in a astigmatic defocus sensor due to different Zernike modes Zernike modes | $Z_{6}$ | $Z_{7}$ | $Z_{8}$ | $Z_{9}$ | $Z_{10}$ | $Z_{11}$
---|---|---|---|---|---|---
Cross-sensitivity | 0 | 0 | 0 | 0 | 0 | -0.497

It is observed from the Table 2 that the sensitivities of all other modes
($Z_{6}$, $Z_{7}$, $Z_{8}$, $Z_{9}$ and $Z_{10}$ ) are zero, while in case of
spherical aberration ($Z_{11}$) there is a maximum cross-sensitivity of
-0.497, which is less than about $58\%$ from that of the sensitivity of the
defocus aberration.

##  3 Simulation

Simulations are carried out in support of the above theoretical study. A plane
wavefront of dimension $256\times 256$ pixels is simulated. A fixed amount,
say 0.001 radian RMS amplitude of Defocus aberration (i.e. Zernike mode
$Z_{4}$) is added to the simulated plane wavefront. The beam with this
wavefront now passes through an astigmatic lens which adds certain amount of
astigmatism to the wavefront. Thus a certain amount $a$ of astigmatism
$45^{o}$ (refer to table 1) is further added to the wavefront. Now apart from
adding astigmatism to the wavefront the astigmatic lens focuses the beam and
the detector is placed at the focal plane of the lens. This can be achieved
numerically through Fourier Transformation of the wavefront at the plane of
the lens. The Fourier plane resembles the focal plane of the lens [5]. For
better visibility a magnification factor of 20 along horizontal axis and
another 20 along vertical axis, i.e. a total of 40, is incorporated in the
Fourier plane.

###  3.1 Sensitivity of Astigmatic Defocus Sensor

The detector plane is divided as shown in Fig 2 and equation 21 is used to
obtain the sensor output. Now, since the sensitivity of a sensor is a measure
of the effectiveness of the sensor while measuring small aberration and we
have already incorporated a very small aberration (i.e. 0.001 radian RMS
amplitude of $Z_{4}$) into the wavefront, the sensor output divided by 0.001
(i.e. $S/0.001$) gives the sensitivity of the sensor. The sensitivity of the
sensor is then calculated for different value of $a$ (i.e. RMS amplitude of
astigmatism $45^{o}$) and plotted with sensitivity along vertical axis and $a$
along horizontal axis.

The plot of sensitivity verses RMS amplitude of astigmatism $45^{o}$ (i.e.
$a$) in radian is shown in Fig 5(i) (in case of a detector with circular
aperture) and in Fig 6(i) (in case of a detector with square aperture). Figure
5(ii) shows the plot of sensitivity verses RMS amplitude of astigmatism
$45^{o}$ (i.e. $a$) in radian and detector radius of a circular aperture in
number of pixels. The sensitivity is represented in different colors ranging
from -1.5 to +1.5. The plot is obtained by varying the value of $a$ as well as
the number of pixels defining the radius of the detector aperture.

It is observed that for RMS amplitudes of $Z_{5}$ ranging from $a=0.8$ – $1.4$
radian and radius $>30$ pixels we get a comparatively good sensitivity (i.e.
$>1$). It is found that a sensitivity of $\approx 1.3$ is achieved at a
detector radius of $85$ pixels. Therefore in the plot shown in Fig. 5(i), i.e.
the plot of the sensitivity of the sensor verses RMS amplitudes of $Z_{5}$
($a$), the radius of the detector aperture is taken as $85$ pixels. It is
observed that the maximum sensitivity of $1.269128$ is obtained at $a=1.1$
radian.

Figure 5: Simulation plot of sensitivity verses (i) RMS amplitude of
astigmatism $45^{o}$ (i.e. $a$) in radian and (ii) RMS amplitude of
astigmatism $45^{o}$ (i.e. $a$) in radian and detector radius in number of
pixels.

Similarly, Fig. 6(ii) shows the plot of sensitivity verses RMS amplitude of
astigmatism $45^{o}$ (i.e. $a$) in radian and number of pixels defining the
length of a sub-aperture of a detector with square aperture. The sensitivity
is represented in different colors ranging from -1.5 to +1.5. The plot is
obtained by varying the value of $a$ as well as the number of pixels defining
the length of a sub-aperture of the detector.

Figure 6: Simulation plot of sensitivity verses (i) RMS amplitude of
astigmatism $45^{o}$ (i.e. $a$) in radian and (ii) RMS amplitude of
astigmatism $45^{o}$ (i.e. $a$) in radian and length of a sub-aperture of a
square detector in number of pixels.

In this case it is observed that for RMS amplitudes of $Z_{5}$ ranging from
$a=0.6$ – $1.7$ radian and length of a sub-aperture $>50$ pixels we get a
comparatively good sensitivity (i.e. $>1$). It is found that a sensitivity of
$\approx 1.3$ is achieved at a sub-aperture length of $60$ pixels. Figure 6(i)
shows a plot of the sensitivity of the sensor verses RMS amplitudes of $Z_{5}$
($a$) taking the length of a sub-aperture of the detector $=60$ pixels. It is
observed that the maximum sensitivity of $1.271803$ is obtained at $a=1.1$
radian.

Figure 7 shows the simulation plot of sensitivity verses detector dimension in
number of pixels when RMS amplitude of the bias aberration, i.e. a is 1.1
radian. Here, Fig. 7(i) is the plot of sensitivities obtained for a circular
aperture of different radius while Fig. 7(ii) is the plot of sensitivities
obtained for a square aperture of different sub-aperture lengths. It is seen
that for a bias aberration of 1.1 radian a sensitivity of approximately 1.2
and higher can be achieved using a detector of circular aperture of radius
equal to or greater than 40 pixels while the same can be achieved using a
detector of square aperture of sub-aperture length equal to or greater than 30
pixels. It is also observed that, with increase in the dimension of the
detector there is a very slight (almost negligible) and gradual decrease in
the sensitivity after obtaining a peak value at a radius of 85 pixels for a
circular aperture and a sub-aperture length of 60 pixels for a square
aperture.

Figure 7: Simulation plot of sensitivity verses (i) detector radius of a
circular aperture and (ii) detector length of a square sub-aperture in number
of pixels when RMS amplitude of the bias aberration, i.e. a = 1.1 radian.

###  3.2 Cross-sensitivity of Astigmatic Defocus Sensor

The cross-sensitivity of the Astigmatic Defocus Sensor is checked through
simulation by introducing different aberration modes into the incident
wavefront. The simulation for cross-sensitivity is carried out considering
$a=1.1$ radian and the radius of circular aperture equal to 60 pixels. Table 2
shows the cross-sensitivity (C.S.) observed in a astigmatic defocus sensor due
to the presence of different aberration modes (Zernike modes) in the test
wavefront. It is observed that apart from $Z_{11}$ mode all other modes shows
a very low cross-sensitivity ($<0.007$). The $Z_{11}$ mode due to its
symmetric resemblance with the $Z_{4}$ (i.e. defocus) mode shows a higher
cross-sensitivity. The table so obtained through simulation is very much in
agreement to that of the theoretical prediction.

Table 3: Cross-sensitivity, obtained through simulation, in a Astigmatic Defocus Sensor due to the presence of different Zernike modes in the test beam $Z_{k}$ | $Z_{6}$ | $Z_{7}$ | $Z_{8}$ | $Z_{9}$ | $Z_{10}$ | $Z_{11}$
---|---|---|---|---|---|---
C.S. | 0.0061 | 0.0063 | 0.0063 | 0.0061 | 0.0061 | -0.3499

Figure 8 shows the response of the Astigmatic Defocus Sensor towards the
presence of different Zernike modes. These are the plots of the sensor output
verses RMS amplitude of the Zernike modes $Z_{4}$ to $Z_{11}$ taking bias (a)
as 1.1 radian and radius as 60 pixels, i.e. for good sensitivity. From the
figure we can see that the astigmatic defocus sensor shows a good response to
the defocus mode ($Z_{4}$) as the curve coincides with the ideal curve to some
extent. However, the sensor shows no response to Zernike modes $Z_{6}$,
$Z_{7}$, $Z_{8}$, $Z_{9}$ and $Z_{10}$ as the curve is a near straight line at
sensor output ’0’. Further, it shows a very low but distinct response to
Zernike modes $Z_{11}$ but in an opposite sense to that of the defocus mode
($Z_{4}$). This is in agreement with the above table as well as the
theoretical prediction.

Figure 8: Simulation plot of sensor output verses RMS amplitude (i.e. $b$ in
radian) of Zernike modes $Z_{k}$ for $k=4\rightarrow 11$.

###  3.3 Linear response of Astigmatic Defocus Sensor

Figure 9: Simulation plot of sensor output verses RMS amplitude (i.e. $b$ in
radian) of defocus aberration mode (Zernike mode $Z_{4}$) present in the input
beam.

Liner response is a measure of how linear is the change in sensor output to a
linear change in the input to the sensor. Figure 9 shows the simulation plot
of the response curve of the sensor and the ideal response. These are
basically the plots of the sensor output verses RMS amplitude (i.e. $b$ in
radian) of defocus aberration mode (Zernike mode $Z_{4}$) present in the input
beam.

The sensor can be considered as showing a linear response, if its response
curve coincides with the ideal response curve. It is observed that the
response curve of the sensor coincides with the ideal response curve for a
limited range on the either side of the zero RMS amplitude of $Z_{4}$. The
response curve of the sensor shown in Fig 9 coincides with the ideal curve
from RMS amplitude of -0.2 radian to +0.2 radian approximately, i.e. linear
for a range of 0.2 radian of input aberration mode.

Figure 10 shows the response curves of the sensor with a circular aperture of
different radius and with different RMS amplitude of the bias aberration mode
(i.e. Zernike mode $Z_{5}$). Figure 10(i $\rightarrow$ iv) are plots of the
response curve with RMS amplitude of bias $a=1,2,3$ and $4$ radians
respectively. The response curve with aperture of different radius ($r$) are
represented by different colours, i.e. black are the ideal curves, green are
curves for $r=50$ pixels, red are curves for $r=100$ pixels, yellow are curves
for $r=150$ pixels and violet are curves for $r=200$ pixels.

Similarly, Fig. 11 shows the response curves of the sensor with a square
aperture of different sub-aperture lengths and with different RMS amplitude of
the bias aberration mode (i.e. Zernike mode $Z_{5}$). Figure 11(i
$\rightarrow$ iv) are plots of the response curve with RMS amplitude of bias
$a=1$, $2$, $3$ and $4$ radians respectively. The response curve with sub-
aperture of different lengths ($l$) are represented by different colours, i.e.
black, green, red, yellow and violet are the ideal curves, curves for $l=50$
pixels, curves for $l=100$ pixels, curves for $l=150$ pixels and curves for
$l=200$ pixels respectively.

Figure 10: Simulation plot of sensor output verses RMS amplitude (i.e. $b$ in
radian) of defocus mode (Zernike mode $Z_{4}$) for RMS amplitude of the bias
mode (Zernike mode $Z_{5}$) (i) $a=1$ radian, (ii) $a=2$ radians, (iii) $a=3$
radians and (iv) $a=4$ radians with a circular detector of radius (r) of 50,
100, 150 and 200 pixels Figure 11: Simulation plot of sensor output verses RMS
amplitude (i.e. $b$ in radian) of defocus mode (Zernike mode $Z_{4}$) for RMS
amplitude of the bias mode (Zernike mode $Z_{5}$) (i) $a=1$ radian, (ii) $a=2$
radians, (iii) $a=3$ radians and (iv) $a=4$ radians with a square detector
sub-aperture length (l) of 50, 100, 150 and 200 pixels Figure 12: Bar diagram
of the linear range of the response curves shown in Fig. 10 and Fig. 11 in
case of (i) a circular aperture and (ii) a square aperture.

Figure 12(i) shows the bar diagram of the linear range of the response curves
shown in Fig. 10 i.e. for a circular aperture, at an error of $\pm 0.01$
radian. The different set of bars with different colours are for bias values
$a=1$, $2$, $3$ and $4$ radians respectively while bars of different colour
represents the linear range for circular aperture of different radius. The
colour codes are same as that of Fig. 10. Similarly, Fig. 12(ii) shows the bar
diagram of the linear range of the response curves shown in Fig. 11 i.e. for a
square aperture, at an error of $\pm 0.01$ radian. The different set of bars
with different colours are for bias values $a=1$, $2$, $3$ and $4$ radians
respectively while bars of different colour represents the linear range for
square aperture of different sub-aperture lengths. The colour codes are same
as that of Fig. 11.

From Fig. 10, 11 and 12 it is found that a maximum linear range of 1.4 is
achieved for a detector radius of 100 pixels in case of circular aperture
while a maximum linear range of 2.1 is achieved for a sub-aperture length of
100 pixels in case of square aperture. Further, it is also seen that the
linear range irrespective of the shape of the detector aperture increase
gradually with increase in the amplitude of the bias aberration, reaches a
peak and then falls back (clearly shown in Fig. 13). Figure 13 shows the plot
of linear range verses RMS amplitude of the bias aberration (b) for an error
of $\pm 0.01$ radians for both circular (green curve) and square (red curve)
aperture. It is observed that a maximum linear range (peak of the curves) is
obtained at a bias amplitude of 3 radian in case of a circular aperture and at
a bias amplitude of 4 radian in case of a square aperture.

Figure 13: Plot of linear range verses RMS amplitude of the bias aberration
(b) for an error of $\pm 0.01$ radians.

##  4 Performance Optimization of the Astigmatic Defocus Sensor

As seen in the above discussions, the performance of the Astigmatic Defocus
Sensor can be optimized both in terms of its sensitivity as well as its linear
response. Let us consider both the cases one by one.

###  4.1 Optimization for higher Sensitivity

It is observed that in order to achieve a good sensitivity (i.e. $>1$), the
RMS amplitude of $Z_{5}$ is to be kept in between $a=0.8$ – $1.4$ radian and
radius $r>60$ pixels in case of a circular aperture, while the same can be
achieved for an RMS amplitude of $Z_{5}$ in between $a=0.6$ – $1.7$ radian and
length of a side $l>50$ pixels in case of a square aperture. It is also
observed that, irrespective of the shape of the detector aperture, there is a
very small change in the sensitivities with increase in the dimension of the
detector. However, a maximum sensitivity is observed at a radius of 85 pixels
for a circular aperture and a length of 60 pixels for a square aperture with
amplitude of the bias aberration $b=1.1$ radian in both the case. Further a
sensor with square shaped aperture shows a slightly higher sensitivity
(sensitivity = 1.271803) than that of a sensor with circular shaped aperture
(sensitivity = 1.269128)

Thus a sensor with comparatively good sensitivity can be designed by keeping
the detector dimension larger that 50 pixels and the RMS amplitude of the bias
aberration (i.e. astigmatism) at around 1.1 radian irrespective of the shape
of the detector aperture.

###  4.2 Optimization for higher linearity

It is observed that a maximum linear range is achieved for a detector
dimension of 200 pixels irrespective of the shape of the detector aperture.
Further, a maximum linear range of 1.4 is achieved for an amplitude of 3
radian of the bias aberration in case of circular aperture while a maximum
linear range of 2.1 is achieved for an amplitude of 4 radian of the bias
aberration in case of square aperture. Thus a sensor with a square shaped
detector aperture gives a better linear response compared to that of a sensor
with a circular shaped detector aperture.

Thus a sensor with comparatively good linear response can be designed by
keeping the detector dimension at around 200 pixels and the RMS amplitude of
the bias aberration (i.e. astigmatism) at around 3 radian in case of a
circular shaped detector aperture and at around 4 radian in case of a square
shaped detector aperture. The linear response can be further improved by
choosing a square shaped detector aperture rather than a circular one.

##  5 Implementation of Astigmatic Defocus Sensor using Computer Generated
Hologram

The Astigmatic Defocus Sensor can be easily implemented using computer
generated hologram (CGH). A CGH is the record of a numerically computed
interference pattern based on the known information of the phase profile of
the object beam (the beam assumed to be reflected from an object) and the
reference beam (normally a beam with a plane wavefront)[8]. When a beam with a
phase profile same as that of the reference beam pass through the CGH and
focused by a lens, the beam is diffracted giving rise to diffraction orders,
such that a zero order at the center and other higher orders ($\pm 1,\pm 3,\pm
5...$) on the either side of the zero order. Among these diffraction orders,
the $+1$ order carries the phase profile of the object beam. A tilt function
is used to locate the diffraction orders at spatially different positions[3].
This property of the CGH can be used to manipulate the phase profile of a
light beam.

The basic principle of implementing the Astigmatic Defocus Sensor using CHG is
shown in Fig 14. Since the sensor is designed using hologram, it can also be
called a Holographic Astigmatic Defocus Sensor. Here the astigmatic lens is
replaced by a CGH and a converging lens immediately after it as shown. If the
CGH is designed with Zernike mode $Z_{5}$ as the object beam phase profile and
a plane wavefront as reference beam phase profile, we get an astigmatic
intensity profile at the $+1$ diffracted order. A detector (bordered green in
Fig 14) is used to capture the $+1$ diffracted order. If +1 radian RMS
amplitude of defocus aberration is present in the beam incident on the CGH,
intensity distribution of the +1 order at the detector plane will appear as
shown in Fig 15(ii). Similarly, if -1 RMS amplitude of defocus aberration is
present in the beam incident on the CGH, intensity distribution of the +1
order at the detector plane will appear as shown in Fig 15(iii) respectively.
Figure 15(i) shows a representative CGH placed at the hologram plane. The
process of obtaining the amount of defocus present in the incident beam is
similar to that of the conventional Astigmatic Defocus Sensor discussed above.

Figure 14: Illustration of the basic principle of a Holographic Astigmatic
Defocus Sensor. Figure 15: Illustration of (i) a Computer Generated Hologram
and intensity distribution of the +1 order for a beam with (ii) 1 radian and
(iii) -1 radian of Defocus aberration. Figure 16: Response curves of a
Holographic Astigmatic Defocus Sensor for bias $a=1.1$ radian (red), $a=4.0$
radian (green) and the ideal response (black).

Figure 16 shows the simulated response curve of the Holographic Astigmatic
Defocus Sensor. The response curves are obtained for a detector of square
aperture of sub aperture length $l=100$ pixels. The black curve represents the
ideal response, the red curve is the response curve of the sensor for bias
$a=1.1$ radian (i.e. for maximum sensitivity) and the green curve is the
response curve of the sensor for bias $a=4.0$ radian (i.e. for maximum
linearity). It is clear form the figure that the response curves are in
agreement with the response of the conventional sensor, i.e. the red curves of
Fig. 11(i) and 11(iv).

##  6 Conclusion

In this paper we put forward a theoretical explanation of the principle of an
Astigmatic Defocus Sensor using the concept of Fourier optics. The Zernike
polynomials are used to represent the aberration modes. An expression for
sensitivity of the sensor is also established. The effect of the presence of
other aberration modes, i.e. other than defocus mode, is explored by obtaining
the cross-sensitivity due to the other modes. A significant affect on the
sensor output is seen due to the presence of the spherical aberration in the
beam, while the presence of other modes is insignificant. A simulation study
on the sensitivity, cross-sensitivity and linear response of the sensor is
also carried out. The plots obtained from the theoretical expressions are in
agreement with the simulation plots.

Further, the criteria for optimizing the sensor in terms of sensitivity and
linear response is also discussed. It is found that higher sensitivities can
be achieved by keeping the amplitude of the bias aberration at 1.1 radian
while higher linear response can be achieved by keeping the amplitude of the
bias aberration at 3 radian for a circular aperture and 4 radian for a square
aperture. Further, a sensor designed for higher sensitivity will give low
linear response and vice versa. It is also observed that there is a very
little or no effect of the shape of the detector on the sensitivities,
however, a comparatively higher linear response can be achieved by using a
detector of square aperture than that of a circular aperture.

Later, we discussed how the sensor can be implemented using computer generated
holograms. The response of such Holographic Astigmatic Defocus Sensor is in
agreement with the conventional one.

## References

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Annexure 1

Now let us see how we can obtain the equation

$\displaystyle\frac{1}{\lambda f}(ux+vy)$ $\displaystyle=$ $\displaystyle\rho
r(cos\phi cos\theta+sin\phi sin\theta)$ (27) $\displaystyle=$
$\displaystyle\rho rcos(\phi-\theta)$

Figure 17: Cartesian and polar coordinate representation of (i) $xy$ or
$r\theta$ plane and (ii) $uv$ or $\rho\phi$ plane and (iii) representation of
$dr$ and $d\theta$ in the $r\theta$ plane

The conversion of the Cartesian coordinates $x$, $y$, $u$ and $v$ into its
polar form is shown in Fig. 17. The Cartesian coordinates of the plane of the
lens ($x,y$) can be represented in terms of polar coordinates ($r,\theta$) as

$\displaystyle x=rcos\theta$ $\displaystyle y=rsin\theta$ (28)

The Cartesian coordinates ($u,v$) of the back focal plane of the lens can be
represented in terms of polar coordinates ($\rho^{\prime},\phi$) as

$\displaystyle u=\rho^{\prime}cos\phi$ $\displaystyle v=\rho^{\prime}sin\phi$
(29)

The back focal plane of the lens can be numerically obtained by Fourier
transforming the entity of the plane of the lens (i.e. $xy$ plane). In that
case ($u,v$) coordinates will lie on the Fourier plane and
$\rho^{\prime}=\rho\lambda f$, where $\rho$ is the Fourier plane
coordinate[5]. Thus the coordinates ($u,v$) can be represented in terms of
polar coordinates ($\rho,\phi$) of the Fourier plane as

$\displaystyle u=\rho\lambda fcos\phi$ $\displaystyle v=\rho\lambda fsin\phi$
(30)

which gives

$\displaystyle\frac{u}{\lambda f}=\rho cos\phi$ $\displaystyle\frac{v}{\lambda
f}=\rho sin\phi$ (31)

Now, from equation 28 and 31 we get

$\displaystyle\left(\frac{u}{\lambda f}\right)x+\left(\frac{v}{\lambda
f}\right)y$ $\displaystyle=$ $\displaystyle\left(\rho
cos\phi\right).\left(rcos\theta\right)+\left(\rho
sin\phi\right).\left(rsin\theta\right)$ $\displaystyle\frac{1}{\lambda
f}(ux+vy)$ $\displaystyle=$ $\displaystyle\rho r(cos\phi cos\theta+sin\phi
sin\theta)$ (32) $\displaystyle=$ $\displaystyle\rho rcos(\phi-\theta)$