Abstract

A simple geometrical method to measure the aberration of astigmatism present in the wavefront that emerges from a telescope is presented. The method is based on the analysis of the external contour of the image of a slightly defocused star. An expression elliptical edge is obtained, which links the rms value of Z22 to the geometric parameters of the ellipse. This expression is tested as a function of introduced defocus and astigmatism aberration in telescopes of the San Pedro Martir Observatory. It is shown that the method gives comparable results to wavefront tests, being capable of measuring astigmatism values of 60  nm and larger without the need for auxiliary optics.

© 2007 Optical Society of America

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References

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  1. E. Hecht and A. Zajac, "More on geometrical optics," in Optics, 1st ed. (Addison-Wesley, 1974), Chap. 6.
  2. A. E. Conrady, "Decentred lens-systems," Mon. Not. R. Astron. Soc. 79, 384-390 (1919).
  3. I. Ghozeil, "Hartmann and Other Screen Tests," in Optical Shop Testing, 2nd ed., D.Malacara, ed. (Wiley, 1992), Chap. 10.
  4. A. Cornejo-Rodriguez, "Ronchi test," in Optical Shop Testing, 2nd ed., D.Malacara, ed. (Wiley, 1992), Chap. 9.
  5. M. R. Teague, "Irradiance moments: their propagation and use for unique retrieval of phase," J. Opt. Soc. Am. 72, 1199-1209 (1982).
    [CrossRef]
  6. F. Roddier, C. Roddier, and N. Roddier, "Curvature sensing: a new wavefront sensing method," Proc. SPIE 976, 203-209 (1988).
  7. B. A. McLeod, "Collimation of fast wide-field telescopes," Publ. Astron. Soc. Pac. 108, 217-219 (1996).
    [CrossRef]
  8. E. O'Neill, "The geometrical theory of aberrations," in Introduction to Statistical Optics, 2nd ed. (Dover Publications, 1992), Chap. 4.
  9. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-211 (1976).
    [CrossRef]
  10. E. Luna, A. Cordero, J. Valdez, L. Gutierrez, and L. Salas, "Telescope alignment by out-of-focus stellar image analysis," Publ. Astron. Soc. Pac. 111, 104-110 (1999).
    [CrossRef]
  11. L. Salas, L. Gutiérrez, M. H. Pedrayes, J. Valdez, C. Carrasco, M. Carrillo, B. Orozco, B. García, E. Luna, E. Ruiz, S. Cuevas, A. Iriarte, A. Cordero, O. Harris, F. Quiroz, E. Sohn, and L. A. Martínez, "Active primary mirror support for the 2.1 m telescope at the San Pedro Mártir Observatory," Appl. Opt. 36, 3708-3716 (1997).
    [CrossRef] [PubMed]

1999 (1)

E. Luna, A. Cordero, J. Valdez, L. Gutierrez, and L. Salas, "Telescope alignment by out-of-focus stellar image analysis," Publ. Astron. Soc. Pac. 111, 104-110 (1999).
[CrossRef]

1997 (1)

1996 (1)

B. A. McLeod, "Collimation of fast wide-field telescopes," Publ. Astron. Soc. Pac. 108, 217-219 (1996).
[CrossRef]

1988 (1)

F. Roddier, C. Roddier, and N. Roddier, "Curvature sensing: a new wavefront sensing method," Proc. SPIE 976, 203-209 (1988).

1982 (1)

1976 (1)

1919 (1)

A. E. Conrady, "Decentred lens-systems," Mon. Not. R. Astron. Soc. 79, 384-390 (1919).

Carrasco, C.

Carrillo, M.

Conrady, A. E.

A. E. Conrady, "Decentred lens-systems," Mon. Not. R. Astron. Soc. 79, 384-390 (1919).

Cordero, A.

Cornejo-Rodriguez, A.

A. Cornejo-Rodriguez, "Ronchi test," in Optical Shop Testing, 2nd ed., D.Malacara, ed. (Wiley, 1992), Chap. 9.

Cuevas, S.

García, B.

Ghozeil, I.

I. Ghozeil, "Hartmann and Other Screen Tests," in Optical Shop Testing, 2nd ed., D.Malacara, ed. (Wiley, 1992), Chap. 10.

Gutierrez, L.

E. Luna, A. Cordero, J. Valdez, L. Gutierrez, and L. Salas, "Telescope alignment by out-of-focus stellar image analysis," Publ. Astron. Soc. Pac. 111, 104-110 (1999).
[CrossRef]

Gutiérrez, L.

Harris, O.

Hecht, E.

E. Hecht and A. Zajac, "More on geometrical optics," in Optics, 1st ed. (Addison-Wesley, 1974), Chap. 6.

Iriarte, A.

Luna, E.

Martínez, L. A.

McLeod, B. A.

B. A. McLeod, "Collimation of fast wide-field telescopes," Publ. Astron. Soc. Pac. 108, 217-219 (1996).
[CrossRef]

Noll, R. J.

O'Neill, E.

E. O'Neill, "The geometrical theory of aberrations," in Introduction to Statistical Optics, 2nd ed. (Dover Publications, 1992), Chap. 4.

Orozco, B.

Pedrayes, M. H.

Quiroz, F.

Roddier, C.

F. Roddier, C. Roddier, and N. Roddier, "Curvature sensing: a new wavefront sensing method," Proc. SPIE 976, 203-209 (1988).

Roddier, F.

F. Roddier, C. Roddier, and N. Roddier, "Curvature sensing: a new wavefront sensing method," Proc. SPIE 976, 203-209 (1988).

Roddier, N.

F. Roddier, C. Roddier, and N. Roddier, "Curvature sensing: a new wavefront sensing method," Proc. SPIE 976, 203-209 (1988).

Ruiz, E.

Salas, L.

Sohn, E.

Teague, M. R.

Valdez, J.

Zajac, A.

E. Hecht and A. Zajac, "More on geometrical optics," in Optics, 1st ed. (Addison-Wesley, 1974), Chap. 6.

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Mon. Not. R. Astron. Soc. (1)

A. E. Conrady, "Decentred lens-systems," Mon. Not. R. Astron. Soc. 79, 384-390 (1919).

Publ. Astron. Soc. Pac. (2)

B. A. McLeod, "Collimation of fast wide-field telescopes," Publ. Astron. Soc. Pac. 108, 217-219 (1996).
[CrossRef]

E. Luna, A. Cordero, J. Valdez, L. Gutierrez, and L. Salas, "Telescope alignment by out-of-focus stellar image analysis," Publ. Astron. Soc. Pac. 111, 104-110 (1999).
[CrossRef]

Other (5)

E. Hecht and A. Zajac, "More on geometrical optics," in Optics, 1st ed. (Addison-Wesley, 1974), Chap. 6.

E. O'Neill, "The geometrical theory of aberrations," in Introduction to Statistical Optics, 2nd ed. (Dover Publications, 1992), Chap. 4.

I. Ghozeil, "Hartmann and Other Screen Tests," in Optical Shop Testing, 2nd ed., D.Malacara, ed. (Wiley, 1992), Chap. 10.

A. Cornejo-Rodriguez, "Ronchi test," in Optical Shop Testing, 2nd ed., D.Malacara, ed. (Wiley, 1992), Chap. 9.

F. Roddier, C. Roddier, and N. Roddier, "Curvature sensing: a new wavefront sensing method," Proc. SPIE 976, 203-209 (1988).

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Figures (5)

Fig. 1
Fig. 1

(Color online) Ellipse eccentricity versus major axis (A) for each assumed value of astigmatism = 100 , 200 , , 1000   nm .

Fig. 2
Fig. 2

Several out-of-focus images produced by defocusing the secondary mirror of the 1.5 m telescope. Astigmatism is present, and it is constant through the sequence. Note that the geometry goes from circular near focus, to a maximum elongation, and approaches circular again as the size of the image increases.

Fig. 3
Fig. 3

(Color online) Experimental data obtained with the 1.5   m telescope are shown as points. The theoretical curve of Eq. (10) (Fig. 1) is shown for a central value of 1.2 λ ± λ / 10 .

Fig. 4
Fig. 4

Defocused stellar images obtained for a series of astigmatism values, of different magnitudes and directions, introduced by the actively controlled M1 cell at the 2.1   m telescope. Each concentric circle is separated by 1λ. At the position of each chosen amount of astigmatism, we show the corresponding image observed.

Fig. 5
Fig. 5

Introduced and recovered astigmatism by analyzing the defocused images obtained when the primary mirror is intentionally deformed. Error bars shown are 189   nm , and the slope of the fit is 0.91.

Equations (15)

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u 2 + v 2 + w 2 = ( R Δ ) 2 .
u 2 + v 2 + w 2 + 2 R Δ R 2 = 0 .
cos   α = u R Δ / u 1 / 2 | grad   S | = x u Q Q ¯ ,
cos   β = v R Δ / v 1 / 2 | grad   S | = y v Q Q ¯ ,
cos   γ = w 1 / 2 | grad   S | = z w Q Q ¯ .
x u u R Δ / u = y v v R Δ / v = z w w .
x u z R + R Δ u , y v z R + R Δ v .
Δ ( r , θ ) = Z 20 ( 2 3 r 2 1 ) + Z 22 6 r 2   cos ( 2 θ ) ,
Δ ( u , v ) = ( Z 20 2 3 a 2 + Z 22 6 a 2 ) u 2 + ( Z 20 2 3 a 2 Z 22 6 a 2 ) v 2 Z 20 ,
x 2 [ ( z + 2 R 2 a 2 [ Z 20 2 3 + Z 22 6 ] ) a R ] 2     + y 2 [ ( z + 2 R 2 a 2 [ Z 20 2 3 Z 22 6 ] ) a R ] 2 = r 2 .
A = { z + 2 R 2 a 2 [ Z 20 2 3 + Z 22 6 ] } a R ,
B = { z + 2 R 2 a 2 [ Z 20 2 3 Z 22 6 ] } a R .
Z 22 = ( A B ) a 4 6 R .
e 2 = A 2 B 2 A 2 ,
Z 22 min = A min a 4 6 R e min 1 + 1 e min .

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