Improving the Point Spread Function of an Aberrated 7-Mirror

Segmented Reﬂecting Telescope using a Spatial Light Modulator

Mary Angelie Alagao, Mary Ann Go, Maricor Soriano and Giovanni Tapang

National Institute of Physics, University of the Philippines, Diliman, Quezon City, Philippines

Keywords:

Point Spread Function, Segmented Reﬂecting Telescope, Gerchberg-Saxton, Phase Retrieval.

Abstract:

We reduce the aberrations in a segmented reﬂecting telescope composed of seven identical concave mirrors

by correcting the point spread functions (PSFs) using a a spatial light modulator. We ﬁrst calculate and

compare the PSF of a segmented reﬂecting telescope and a monolithic reﬂecting telescope, both having the

same aperture diameter. We simulate the aberrations using the Zernike polynomials and add these to the PSF

of the segmented mirror. Using the Gerchberg-Saxton (GS) algorithm, we retrieve the phase information used

to correct for these aberrations. Results show an improvement in the imaging resolution of the telescope due

to the correction phase applied.

1 INTRODUCTION

The quest to see far into space has led to the con-

struction of meter-wide ground-based telescopes. In-

creasing the aperture of a telescope increases its light

gathering power, thus, allowing the detection of faint

astronomical objects. A smaller angular resolution

is also achieved with a larger aperture diameter as

shown in Equation 1, where λ is the wavelength and

D is the aperture diameter.

Θ =

(1)

There is a technological limit of 10 m in the con-

struction of large single mirror telescopes due to the

cost of building and transportation of a very large mir-

ror (G. Chanan, 2013).The solution proposed to this

problem is the segmented mirror design (G. Chanan,

2013). Small mirrors are put together to act as a sin-

gle large mirror. Each mirror has a speciﬁc shape and

a control system for precise positioning to reduce, if

not eliminate, the optical path difference between seg-

ments. While this poses structural complexity due to

the control system of each segment, this solution is

less expensive compared to the construction of one

very large mirror, making it easier to increase the

aperture diameter of a telescope.

A telescope is a diffraction-limited system. Its

imaging performance can be evaluated by calculating

its point spread function (PSF). The image produced

by an optical system is just the Fourier transform of

the aperture at the exit pupil.

Most telescopes assume a circular opening and the

image of a point due to diffraction effects is the PSF.

The transverse size of the PSF shows the image of

a point in the image plane. The diameter of the ﬁrst

dark ring of PSF, known as the Airy disk, gives a mea-

sure of the resolution and its size is the smallest size

that the telescope can detect (Goodman, 2005). The

effective PSF detected by a camera, however, may

include aberrations from imperfect alignment. The

wavefront can be corrected by getting the phase infor-

mation based on the difference between the aberrated

PSF and the ideal PSF.

In this work, we perform numerical simulations

to characterize a segmented reﬂecting telescope com-

posed of seven identical concave mirrors. We inten-

tionally add aberrations and use a spatial light mod-

ulator (SLM) to shape the wavefront in order to im-

prove the PSF.

2 METHODOLOGY

For the segmented reﬂecting telescope, we used seven

identical mirrors with a diameter of 76mm and a focal

length 300mm. The mirrors are rotated and translated

such that they focus at one point. The effective di-

ameter of the segmented mirror is 223.22 mm. We

compare our results with a monolithic reﬂecting tele-

scope. For performance comparison, we denote the

Alagao, M., Go, M., Soriano, M. and Tapang, G.

Improving the Point Spread Function of an Aberrated 7-Mirror Segmented Reﬂecting Telescope using a Spatial Light Modulator.

DOI: 10.5220/0005691900940101

In Proceedings of the 4th International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS 2016), pages 96-103

ISBN: 978-989-758-174-8

monolithic reﬂecting telescope has the same aperture

diameter as the segmented telescope. The setup of the

whole system is shown in Figure 1.

Figure 1: Optical setup for wavefront correction.

2.1 PSF Calculation

We simulate the performance of a segmented reﬂect-

ing telescope using the angular spectrum method. We

represent a plane wave located at z = 0 as U(x,y,0).

The angular spectrum U( f x, f y,0) at this location is

just the Fourier transform of U(x,y,0) given by Equa-

tion 2 (Goodman, 2005).

U( f

, f

;0) =

∞

−∞

U(x,y,0)exp[−j2πφ]dxdy (2)

where

φ = f

x + f

y (3)

To get the ﬁeld amplitude U(x,y,0), we get the in-

verse Fourier transform as shown in Equation 4.

U(x,y,0) = F

−1

{U( f

, f

;0)} (4)

Similarly, the angular spectrum at a distance z is

given by Equation 5.

U( f

, f

;z) =

∞

−∞

U(x,y,z)exp[−j2πφ)]dxdy

(5)

where the amplitude U(x,y,z) at a distance z is also

given by

U(x,y,z) = F

−1

{U( f

, f

;z)} (6)

Comparing Equations 2 and 5, the relation be-

tween U(x,y,0) and U(x, y,z) is given by Equation

U(x,y,z) = U(x,y,0)exp( j 2π

1 − f

− f

z) (7)

The angular spectrum methods gives us the PSF at

the point (x,y,z);

2.2 Angular Spectrum of the Segmented

Mirrors

The apertures, which are actually the mirrors, are as

shown in Figure 2.

Figure 2: Top view of the conﬁguration of the seven mirror

segments where D is the diameter of the mirror segment

which is equal to 76mm and θ is the tilt angle.

To simulate the PSF of this conﬁguration, the

other mirrors are rotated along the x and y axis with

respect to Mirror 1, which is centered at the origin.

Each mirror must be tilted such that all the foci of

the mirrors coincide. The geometry for determining

the tilt angle is shown in Figure 3.

We use the sine law in 4ABC to obtain the fol-

lowing relation:

sinθ

sin(90 −

)

(8)

Using 4CBD, the left hand side in Equation 8 can

be written as

sin(180 −θ)

sin(

)

(9)

Using Equation 9 in Equation 8, we get the fol-

lowing relation:

cos(

)

sin(

)

(10)

such that the tilt angle can now be expressed as:

θ = 2tan

−1

(

) (11)

This was used to construct the rotation matrices

for each mirror. The angle is found to be 14.4

◦

.Setting

Improving the Point Spread Function of an Aberrated 7-Mirror Segmented Reﬂecting Telescope using a Spatial Light Modulator

Figure 3: Geometry for determining the tilt angle of mir-

rors.

the optical axis along z, we use the angle to rotate

about the y-axis then the x-axis depending on the po-

sition of the mirror.

(α,β) =





cosβ sinα sin β cosα sin β

0 cosα −sin α

−sin β sinα cos β cosα cos β





(12)

The rotation and translation matrices given by

Equations 12 and 14, respectively, were used to ob-

tain the x and y coordinates of the off-axis mirrors as

shown in Equation 13. The angles α and β are both

equal to θ. The variables, dx and dy, are the trans-

lations along the x and y axes, respectively. For the

diagonal mirrors, Mirror 2,4,5 and 7, dx is equal to

√

3r cos(θ) and dy is equal to r cos(θ).









= T









(13)





1 0 dx

0 0 dy

0 0 1





(14)

2.3 Aberration Simulations

Aberrations are deviations in the wavefront which re-

duce the intensity and contrast of an image. They can

be described in terms of Zernike polynomials which

are orthogonal over a unit circle.

Using the Zernike polynomials, the aberration

function can be expressed in terms of polynomials as

shown in Equation 18 (Mahajan, 1994).

W (ρ, θ) =

∞

∑

j=1

(ρ,θ) (15)

where the polynomial ordering number is given by

j, a

is the aberration coefﬁcient and

even j

(ρ,θ) =

(2n + 1)R

(ρ)cosmθ m 6= 0

(16a)

odd j

(ρ,θ) =

(2n + 1)R

(ρ)sinmθ m 6= 0

(16b)

(ρ,θ) =

(n + 1)R

(ρ) m = 0

(16c)

and

(ρ) =

(n−m)/2

∑

s=0

(−1)

(n −s)!

s!(

n+m

−s)!(

n−m

−s)!

(17)

We used the following aberrations: coma, astig-

matism, spherical aberration and defocus. We simu-

late the aberrations using Equation 18.

PSF

aberrated

= F {F

−1

{PSF

segmented

}exp(i

2π

Φ)}}

(18)

where Φ is given by the following Zernike poly-

nomials (M. Born, 1999)

sphericalaberration

= 5λρ

(19)

coma

= 0.3λρ

cosθ (20)

astigmatism

= 10λρ

cos2θ (21)

de f ocus

= 8

(3)(2ρ

−1) (22)

2.4 Wavefront Correction

To correct the wavefront, we needed to calculate the

correcting phase that will serve as the input to the

SLM. The capability of the SLM for wavefront shap-

ing was demonstrated in the photopolymerization of

microgear patterns by encoding in the SLM the topo-

logical charge and phase level of an optical vortexp

(G. Bautista and Daria, 2009). Another work uses the

SLM to reshape light in three dimensions (P.L.A. Hi-

lario and Tapang, 2014). This makes it possible to

correct the aberrations inherent to the telescope. The

phase information from intensity image was retrieved

using the Gerchberg-Saxton (GS) algorithm. From

the intensity image of the PSF, the amplitudes of the

PHOTOPTICS 2016 - 4th International Conference on Photonics, Optics and Laser Technology

input data can be obtained by taking the square root

of the measured intensities (Goodman, 2005).

U(x,y) =

I(x, y)exp(iφ(x,y)) (23)

Using Equation 23, we ﬁrst take a guess phase

φ(x,y). We multiply this phase to the amplitude of

the source image and take the Fourier transform. We

take the phase of the resulting complex ﬁeld and mul-

tiply it with the amplitude of the target image. We

get the inverse Fourier transform and check if the tar-

get image has already been reconstructed. This proce-

dure is repeated until a tolerable reconstruction error

is obtained (R.W. Gerchberg, 1972). A diagram of the

algorithm is shown in Figure 4.

Figure 4: Schematic of the GS algorithm.

For our simulations, we used aberrated PSF of

the segmented mirror as our source and the unaber-

rated PSF of the segmented mirrors as the target. The

phase information retrieved using the GS algorithm

serves as the correction phase. To check if the cor-

rection successful, we denote the correction phase as

correction

and use Equation 24.

PSF

corrected

= F {|F

−1

{PSF

aberrated

}|exp(iΦ

correction

)}

(24)

The quality of the reconstruction is evaluated us-

ing the Linfoot’s criteria of merit given by Equation

27. When all these values are 1, it means that the

signal was perfectly recovered. If C <1, the recon-

structed proﬁle is narrower than the target proﬁle.

When Q <1, it means that the reconstructed and target

proﬁles are erroneously close to each other (G.Tapang

and Saloma, 2002).

F = 1 −

< (I

target

−I

reconstructed

)

< (I

target

)

(25)

Q =

< (I

reconstructed

)

< (I

target

)

(26)

C =

< |I

reconstructed

||I

target

| >

< (I

reconstructed

)

(27)

3 RESULTS AND DISCUSSION

Figure 5 shows the transverse PSFs of a single mirror

segment, the segmented mirrors and the monolithic

mirror. Adding the ﬁelds due to the 7 mirrors, we get

a PSF that is much smaller than the individual mirror

segment. This is expected since the effective aperture

diameter of the segmented mirrors is now larger than

a single mirror.

Taking the cross-section of the PSFs, we see that

the width of the Airy disk of the segmented mirror

is broader than that of the monolithic mirror despite

having the same aperture diameter. This is due to the

segmentation of the reﬂecting telescope.

Figures 6 and 7 show the results of the aberration

correction. Using the GS algorithm, the added aberra-

tions were corrected. The quality of the correction is

evaluated by computing the Linfoot’s criteria of merit

values of both the aberrated PSF and the corrected

PSF as shown in Table 1.

The SLM in Figure 1 is a transmitting twisted

nematic spatial light modulator such as the Holoeye

LC2012. Modulation in phase occurs when an elec-

tric ﬁeld is applied causing the molecules to align

themselves along its direction. The correcting phase

obtained using the GS algorithm will be sent to the

SLM via a video signal from the computer. In the ac-

tual experiment, the PSF should be scaled such that

4f =

λ f M

where λ is the wavelength, f is the focal

length, M is equal to the pixel dimension and a is the

aperture diameter. We computed 4f to be equal to

640µm.

The use of identical mirrors reduces the cost and

complexity in constructing telescopes with very large

apertures. It eliminates the need to precisely shape

the mirrors which requires each mirror segment to be

polished well up to the edge in order to seamlessly

form one large mirror. Thus, increasing the quantity

production of the mirrors. Mounting the mirror will

also be easier since the structural support will not de-

pend on the individual shape of the mirror segments.

Furthermore, placing an adaptive optics device, such

as the SLM, before the camera allows the incoming

wavefront to be shaped to compensate for any mis-

Improving the Point Spread Function of an Aberrated 7-Mirror Segmented Reﬂecting Telescope using a Spatial Light Modulator

Single Mirror Segment 7 Mirrors

Monolithic Cross-section of the PSFs

Figure 5: PSFs of a single mirror segment, segmented mirror and the monolithic mirror.

Table 1: Linfoot’s criteria of merit for aberrated and the corrected PSF using the angular spectrum method.

Aberration Fidelity Structural Content Correlation

Spherical Aberration Aberrated 0.9670 0.7005 0.8337

Corrected 1.0000 1.0000 0.9988

Coma Aberrated 0.9956 0.9990 0.9973

Corrected 1.0000 1.0000 1.0000

Astigmatism Aberrated 0.0179 0.0137 0.0158

Corrected 0.9843 1.0000 0.9922

Defocus Aberrated 0.0045 0.0022 0.0033

Corrected 0.9843 1.0000 0.9922

alignment and aberrations inherent in the telescope

system.

4 CONCLUSION AND

RECOMMENDATION

We have demonstrated a technique to correct aberra-

tions in a basic segmented reﬂecting telescope by us-

ing the GS algorithm to obtain the correction phase.

We have numerically evaluated the technique and

have shown that it was able to reduce the effects of

aberrations. Using this technique, we remove the

need for shaping of the individual mirror segments

precisely. The intensity of the side lobes in the PSF

due to errors in alignment can be compensated using

a single SLM. The wavefront can be shaped such that

the imaging resolution can be improved to make it at

par with the performance of one large mirror.

PHOTOPTICS 2016 - 4th International Conference on Photonics, Optics and Laser Technology

100

Coma Astigmatism

Figure 6: Aberrated (1st row), corrected (2nd row) and the difference between the target and the corrected PSF of the seg-

mented reﬂecting telescope.

Improving the Point Spread Function of an Aberrated 7-Mirror Segmented Reﬂecting Telescope using a Spatial Light Modulator

101

Spherical Aberration Defocus

Figure 7: Aberrated (1st row), corrected (2nd row) and the difference between the target and the corrected PSF of the seg-

mented reﬂecting telescope.

PHOTOPTICS 2016 - 4th International Conference on Photonics, Optics and Laser Technology

102

REFERENCES

G. Bautista, M.J. Romero, G. T. and Daria, V. (2009). Paral-

lel two-photon photopolymerization of microgear pat-

terns. Optics Communications, 282(18):3746–3750.

G. Chanan, J. Nelson, T. M. (2013). Semented mirror tele-

scopes. In Planets, Stars and Stellar Systems, volume

Volume 1: Telescope and Instrumentation, pages 99–

136. Springer.

Goodman, J. (2005). Introduction to Fourier Optics.

Roberts and Compan Publishers, 3rd edition.

G.Tapang and Saloma, C. (2002). Behavior of the

point-spread function in photon-limited confocal mi-

croscopy. Applied Optics, 41(8):1534–1540.

M. Born, R. W. (1999). Principle of Optics. Cambridge

University Press.

Mahajan, V. (1994). Zernike circle polynomials and optical

aberrations of systems with circular pupils. Supple-

ment to Applied Optics, pages 8121–8124.

P.L.A. Hilario, M. V. and Tapang, G. (2014). Independent

light ﬁelds generated using a phase only spatial light

modulator. Optics Letters, 39(7):2036–2039.

R.W. Gerchberg, W. S. (1972). A practical algorithm for

the determination of phase from image and diffraction

plane pictures. Optik, 35(2):237–246.

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