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Measurement of the Atmospheric Primary Aberrations by 4Apertures HARTMANN
Ramin Shomali,1 Sadollah Nasiri,1,2 Ahmad Darudi,1,3 and Davoud Afshari1
2Institute for Advanced Studies in Basic Science (IASBS), Zanjan, Iran 3Lund Observatory, Lund, Sweden

1Physics Department, Zanjan University, Zanjan, Iran.


Hartmann Test Using a Screen With Four Holes





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Hartmann test with four holes screen
The theoretical concept of the primary aberrations measurement by four apertures Hartmann test introduced by "Malacara" * and "SalasPeimbert et. al. " **



* Malacara D, Malacara Z Opt. Eng. 31. 155155(1992). ** SalasPeimbert et. al. A ppl. Opt. 44, 422838(2005).

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Theory
Let us assume at Hartmann screen, each aperture is located on the corners of a square with side d . The aberrations of the distorted wavefront at four apertures screen are defined by:

2 2 W ( x , y ) Bx Cy D x 2 y 2 E x 2 y 2 Fxy G x 2 y 2 d y H x 2 y 2 d x ,





















B is tilt coefficient about the y axis, C is tilt coefficient about the x axis, D is the defocusing coefficient, E is the astigmatisms coefficient with the axis at 0 or 90, F is the astigmatisms coefficient with the axis at 45, G is the Coma coefficient about the y axis, H is the Coma coefficient about the x axis.


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The x and y components of the transverse aberrations are given by:
W x , y ta x 2 B 2 Dx 2 Ex Fy 2Gxy H 3 x 2 y 2 d , x Ftel





Fi Where t els the four apertures screen distance from CCD

W x , y ta y 2 c 2 Dy 2 Ey Fx G x 2 3 y 2 d 2 Hxy , y Ftel









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We can determine seven aberration coefficients with eight transverse aberrations equations for all apertures as:
tilts:
B ta
y

ta

y

ta
l

y

ta

y

4 Fte

,

C

ta

x

ta

x

ta
l

x

ta

x

4 Fte

,

Defocus:



D



ta

x

ta

x



ta

x

ta

x

t

a

y

ta

y

t

a

y

ta

y



8 Ftel d

.

Astigmatism:
E



ta

x

ta

x



ta

x

ta

x



ta

y

ta

y



ta

y

ta

y



8 Ftel d

.

F



ta

x

ta

x



ta x ta

x



ta

y

ta

y



ta

y

ta

y



4 Ftel d

.



Coma:



H

t

a

y

ta

y



ta

y

ta

y



2 Ftel d 2

.

G



ta

x

ta

x



ta

x

ta

x



2 Ftel d 2

.

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By this definition for the wavefront aberrations the centroid of these four spots (the average coordinates of the four spots) is not shifted by defocusing, astigmatism and coma terms and it is shifted only by the two tilts, The configuration of the system of four spots depends on the coefficients D, E, F, G, H

The global position depends on the coefficients B & C.
"If there are no local tilts, the centroid of the four spots is located exactly in the position of ideal four spots centroid, so if we locate ideal four spots centroid in four spots centroid position and reconstruct ideal four spot by measured distance for each ideal spots, we could ignore tilt terms."




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Simulation
To test the ability of four apertures Hartmann for the measurement of atmospheric primary aberrations, We perform a numerical simulation. Here we explain:

1How We Simulate of four spots images at the telescope focal plane 2The result of Primary aberrations measurement by the four apertures Hartmann
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Simulation of four spots images at the telescope focal plane





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generated phase screen by Harding et. al. code's

phase screen includes 8 modes of Zernike aberrations



*Harding C M, Johnston R. A, Lane R G 1999 "Fast Simulation of a Kolmogorov Phase Screen" Appl. Opt. 38, 216170.

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Defocus

Defocus modes of Zernike aberration. Black curve show the calculated coefficients for phase screens which have 8 modes of Zernike aberrations and red curve show the calculated coefficient for original phase screen.
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Astigmatisms



).

Astigmatisms modes of Zernike aberration. Black curve show the calculated coefficients for phase screens which have 8 modes of Zernike aberrations and red curve show the calculated coefficient for original phase screen. (a) astigmatism 45 (b) astigmatism 090
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Coma



Coma modes of Zernike aberration. Black curve show the calculated coefficients for phase screens which have 8 modes of Zernike aberrations and red curve show the calculated coefficient for original phase screen. (a) coma y (b) coma x .
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(a) Defocus (b) Astigmatism 45 (c) Astigmatism 090 (d) Coma y (e) Coma x





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(a) Ideal (b) Defocus (c) Astigmatism 45 (d) Astigmatism 090 (e) Coma y (f) Coma x (g) Trefoil 30 deg. (h) Trefoil 0 deg.





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The ideal four spots images

The ideal four spots images can be obtained by two methods:
1Pointing the telescope on faint star and capture an image, the

exposure time of this image must be long enough to average out turbulence effects but short enough to avoid any degradation that is due to telescope tracking errors.
2Averaging on the centroid of large number of short exposure time

images which captured for Fried parameters measurement to eliminate the seeing effects and finding the position of the centeroid of 4spots images.
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Conclusion

we discuss the ability of Hartmann test with four apertures to measure the atmospheric primary aberrations both by the simulation procedure. We show by numerical simulations that this method is able to measure the defocus and astigmatism aberrations correctly.

However, the results obtained for the coma aberration are not desirable.





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