Abstract

We present an algorithm that uses a square grid in a Ronchi test. We assume that the point coordinates of this pattern (termed a bironchigram) are affected by Gaussian errors. To calculate the optical path difference, we apply only one nonlinear least-squares fit to the dot coordinates. The relevant equations are deduced, and experimental results are shown.

© 2001 Optical Society of America

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References

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  1. V. Ronchi, “Le frange di combinazioni nello studio delle superficie e dei sistemi ottici,” Riv. Ottica Mecc. Precis. 2, 9 (1923).
  2. V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437–451 (1964).
    [CrossRef]
  3. K. Stultz, H. P. Stahl, “A discussion of techniques that separate orthogonal data produced by Ronchi cross grating patterns,” in Current Developments in Optical Design and Optical Engineering, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2263, 226–232 (1994).
  4. W. S. Meyers, H. P. Stahl, “Contouring of a free oil surface,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y’Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 84–94 (1992).
  5. W. S. Meyers, H. P. Stahl, “Sensitivity of two-channel Ronchi test to grating misalignment,” in Fabrication and Testing of Optics and Large Optics, V. J. Doherty, ed., Proc. SPIE1994, 90–101 (1993).
  6. A. Cordero, E. Luna, S. Zárate, O. Harris, “Evaluación de la calidad de la imagen del telescopio de 2.1 m,” (Instituto de Astronómia, UNAM, Mexico City, 1995).
  7. A. Cordero-Dávila, E. Luna-Aguilar, S. Vázquez-Montiel, S. Zárate-Vázquez, M. E. Percino-Zacarías, “Ronchi test with a square grid,” Appl. Opt. 37, 672–675 (1998).
    [CrossRef]
  8. A. Cordero-Dávila, A. Cornejo Rodríguez, O. Cardona-Nuñez, “Polynomial fitting of interferograms with Gaussian errors on the fringe coordinates. I. Computer simulations,” Appl. Opt. 33, 7339–7342 (1994).
  9. A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, R. Díaz-Uribe, “Polynomial fitting of interferograms with Gaussian errors on the fringe coordinates. II. Analytical study,” Appl. Opt. 33, 7343–7349 (1994).
    [CrossRef]
  10. A. Cordero-Dávila, O. Cardona-Nuñez, A. Cornejo-Rodríguez, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. III. Nonlinear solution,” Appl. Opt. 37, 7983–7987 (1998).
    [CrossRef]
  11. W. H. Jefferys, “On the method of least squares,” Astron. J. 85, 177–181 (1980).
    [CrossRef]
  12. W. H. Jefferys, “On the method of least squares. II,” Astron. J. 86, 149–155 (1981).
    [CrossRef]
  13. D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 13.
  14. L. Salas, “Variable separation in curvature sensing: fast method for solving the irradiance transport equation in the context of optical telescopes,” Appl. Opt. 35, 1593–1596 (1996).
    [CrossRef] [PubMed]
  15. L. Salas, L. Gutierrez, M. H. Pedrayes, J. Valdez, C. Carrasco, M. Carrillo, B. Orozco, B. Garcia, E. Luna, E. Ruiz, S. Cuevas, A. Iriarte, A. Cordero, O. Harris, F. Quiroz, E. Sohn, L. A. Martinez, “Active primary mirror support for the 2.1-m telescope at the San Pedro Martir Observatory,” Appl. Opt. 36, 3708–3716 (1997).
    [CrossRef] [PubMed]

1998 (2)

1997 (1)

1996 (1)

1994 (2)

1981 (1)

W. H. Jefferys, “On the method of least squares. II,” Astron. J. 86, 149–155 (1981).
[CrossRef]

1980 (1)

W. H. Jefferys, “On the method of least squares,” Astron. J. 85, 177–181 (1980).
[CrossRef]

1964 (1)

1923 (1)

V. Ronchi, “Le frange di combinazioni nello studio delle superficie e dei sistemi ottici,” Riv. Ottica Mecc. Precis. 2, 9 (1923).

Cardona-Nuñez, O.

Carrasco, C.

Carrillo, M.

Cordero, A.

Cordero-Dávila, A.

Cornejo Rodríguez, A.

Cornejo-Rodríguez, A.

Cuevas, S.

DeVore, S. L.

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 13.

Díaz-Uribe, R.

Garcia, B.

Gutierrez, L.

Harris, O.

Iriarte, A.

Jefferys, W. H.

W. H. Jefferys, “On the method of least squares. II,” Astron. J. 86, 149–155 (1981).
[CrossRef]

W. H. Jefferys, “On the method of least squares,” Astron. J. 85, 177–181 (1980).
[CrossRef]

Luna, E.

Luna-Aguilar, E.

Malacara, D.

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 13.

Martinez, L. A.

Meyers, W. S.

W. S. Meyers, H. P. Stahl, “Sensitivity of two-channel Ronchi test to grating misalignment,” in Fabrication and Testing of Optics and Large Optics, V. J. Doherty, ed., Proc. SPIE1994, 90–101 (1993).

W. S. Meyers, H. P. Stahl, “Contouring of a free oil surface,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y’Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 84–94 (1992).

Orozco, B.

Pedrayes, M. H.

Percino-Zacarías, M. E.

Quiroz, F.

Ronchi, V.

V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437–451 (1964).
[CrossRef]

V. Ronchi, “Le frange di combinazioni nello studio delle superficie e dei sistemi ottici,” Riv. Ottica Mecc. Precis. 2, 9 (1923).

Ruiz, E.

Salas, L.

Sohn, E.

Stahl, H. P.

W. S. Meyers, H. P. Stahl, “Sensitivity of two-channel Ronchi test to grating misalignment,” in Fabrication and Testing of Optics and Large Optics, V. J. Doherty, ed., Proc. SPIE1994, 90–101 (1993).

K. Stultz, H. P. Stahl, “A discussion of techniques that separate orthogonal data produced by Ronchi cross grating patterns,” in Current Developments in Optical Design and Optical Engineering, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2263, 226–232 (1994).

W. S. Meyers, H. P. Stahl, “Contouring of a free oil surface,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y’Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 84–94 (1992).

Stultz, K.

K. Stultz, H. P. Stahl, “A discussion of techniques that separate orthogonal data produced by Ronchi cross grating patterns,” in Current Developments in Optical Design and Optical Engineering, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2263, 226–232 (1994).

Valdez, J.

Vázquez-Montiel, S.

Zárate, S.

A. Cordero, E. Luna, S. Zárate, O. Harris, “Evaluación de la calidad de la imagen del telescopio de 2.1 m,” (Instituto de Astronómia, UNAM, Mexico City, 1995).

Zárate-Vázquez, S.

Appl. Opt. (7)

V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437–451 (1964).
[CrossRef]

A. Cordero-Dávila, A. Cornejo Rodríguez, O. Cardona-Nuñez, “Polynomial fitting of interferograms with Gaussian errors on the fringe coordinates. I. Computer simulations,” Appl. Opt. 33, 7339–7342 (1994).

A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, R. Díaz-Uribe, “Polynomial fitting of interferograms with Gaussian errors on the fringe coordinates. II. Analytical study,” Appl. Opt. 33, 7343–7349 (1994).
[CrossRef]

L. Salas, “Variable separation in curvature sensing: fast method for solving the irradiance transport equation in the context of optical telescopes,” Appl. Opt. 35, 1593–1596 (1996).
[CrossRef] [PubMed]

L. Salas, L. Gutierrez, M. H. Pedrayes, J. Valdez, C. Carrasco, M. Carrillo, B. Orozco, B. Garcia, E. Luna, E. Ruiz, S. Cuevas, A. Iriarte, A. Cordero, O. Harris, F. Quiroz, E. Sohn, L. A. Martinez, “Active primary mirror support for the 2.1-m telescope at the San Pedro Martir Observatory,” Appl. Opt. 36, 3708–3716 (1997).
[CrossRef] [PubMed]

A. Cordero-Dávila, E. Luna-Aguilar, S. Vázquez-Montiel, S. Zárate-Vázquez, M. E. Percino-Zacarías, “Ronchi test with a square grid,” Appl. Opt. 37, 672–675 (1998).
[CrossRef]

A. Cordero-Dávila, O. Cardona-Nuñez, A. Cornejo-Rodríguez, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. III. Nonlinear solution,” Appl. Opt. 37, 7983–7987 (1998).
[CrossRef]

Astron. J. (2)

W. H. Jefferys, “On the method of least squares,” Astron. J. 85, 177–181 (1980).
[CrossRef]

W. H. Jefferys, “On the method of least squares. II,” Astron. J. 86, 149–155 (1981).
[CrossRef]

Riv. Ottica Mecc. Precis. (1)

V. Ronchi, “Le frange di combinazioni nello studio delle superficie e dei sistemi ottici,” Riv. Ottica Mecc. Precis. 2, 9 (1923).

Other (5)

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 13.

K. Stultz, H. P. Stahl, “A discussion of techniques that separate orthogonal data produced by Ronchi cross grating patterns,” in Current Developments in Optical Design and Optical Engineering, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2263, 226–232 (1994).

W. S. Meyers, H. P. Stahl, “Contouring of a free oil surface,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y’Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 84–94 (1992).

W. S. Meyers, H. P. Stahl, “Sensitivity of two-channel Ronchi test to grating misalignment,” in Fabrication and Testing of Optics and Large Optics, V. J. Doherty, ed., Proc. SPIE1994, 90–101 (1993).

A. Cordero, E. Luna, S. Zárate, O. Harris, “Evaluación de la calidad de la imagen del telescopio de 2.1 m,” (Instituto de Astronómia, UNAM, Mexico City, 1995).

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Figures (6)

Fig. 1
Fig. 1

Experimental setup for testing a reflecting telescope.

Fig. 2
Fig. 2

Detected bironchigram for the 0.84-m reflecting telescope of the National Astronomical Observatory, Universidad Autónoma de México.

Fig. 3
Fig. 3

Superposition of the experimental and the estimated (after fitting) points for the bironchigram obtained with the 0.84-m reflecting telescope and a polynomial fitting of 2°.

Fig. 4
Fig. 4

Superposition of the experimental and the estimated (after fitting) points for the bironchigram obtained with the 0.84-m reflecting telescope and a polynomial fitting of 6°.

Fig. 5
Fig. 5

Simulated bironchigrams: (a) ideal, (b) shifted 0.1 cm in the X direction.

Fig. 6
Fig. 6

Simulated bironchigrams: (a) rotated 3°, (b) shifted 0.1 cm in the X direction, shifted 0.3 cm in the Y direction and rotated 3°, simultaneously.

Tables (3)

Tables Icon

Table 1 Comparison of Evaluation Obtained with the New Algorithm and Traditional Evaluationa

Tables Icon

Table 2 Evaluation Obtained with Our Algorithm for the Bironchigram Shown in Fig. 3 with a 6° Polynomial

Tables Icon

Table 3 Evaluation Obtained with Our Algorithm for a Simulated Bironchigram of a Spherical Mirror for Several Degrees of Misalignment Shown in Figs. 5 and 6

Equations (66)

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Wx, y=k=0NR akzk,
T=-rpWx, -rpWy,
-rpWx=mxδx,
-rpWy=myδy,
-rpk=1NR akzxk-mxδx=0,
-rpk=1NR akzyk-myδy=0,
RT=X1, Y1, X2, Y2,, XN, YN,
Xi=R2i1,  Yi=R2i.
σ=diagσ0, σ0,, σ0.
ET=E1, E2,, E2N.
r=R+E,
aT=a1, a2,, aNR,
Δr, a=-rpk=1NR akzxk1+mx1δx/rpk=1NR akzyk1+my1δy/rpk=1NR akzxk2+mx2δx/rpk=1NR akzyk2+my2δy/rpk=1NR akzxkN+mxNδx/rpk=1NR akzykN+myNδy/rp.
Δrˆ=rpTxx1Txy10000Txy1Tyy1000000Txx2Txy20000Txy2Tyy2000000000000000000TxxNTxyN0000TxyNTyyN,
Txxik=1NR ak2zkx2xi,yi,
Txyik=1NR ak2zkxyxi,yi,
Tyyik=1NR ak2zky2xi,yi.
Δa=-rpzx11zx21zx31zx41zxNR1zy11zy21zy31zy41zyNR1zx12zx22zx32zx42zxNR2zy12zy22zy32zy42zyNR2zx1Nzx2Nzx3Nzx4NzxNRNzy1Nzy2Nzy3Nzy4NzyNRN,
zxkL=zkxxL,yL.
W-1=σ0rpTxx12+Txy12Txy1Txx1+Tyy10000Txy1Txx1+Tyy1Txy12+Tyy12000000Txx22+Txy22Txy2Txx2+Tyy20000Txy2Txx2+Tyy2Txy22+Tyy22000000TxxN2+TxyN2TxyNTxxN+TyyN0000TxyNTxxN+TyyNTxyN2+TyyN2.
W=1σ0rp2Ta1Tb10000Tb1Td1000000Ta2Tb20000Tb2Td2000000TaNTbN0000TbNTdN,
deti=Txyi2-TxxiTyyi2,
Tai=Txyi2+Tyyi2deti,
Tbi=-TxyiTxxi+Tyyideti,
Tdi=Txxi2+Txyi2deti.
ϕˆ=-rpk=1NR akzxk1+mx1δx/rp-Txx1xˆ10-x1-Txy1yˆ10-y1k=1NR akzyk1+my1δy/rp-Txy1xˆ10-x1-Tyy1yˆ10-y1k=1NR akzxk2+mx2δx/rp-Txx2xˆ20-x2-Txy2yˆ20-y2k=1NR akzyk2+my2δy/rp-Txy2xˆ20-x2-Tyy2yˆ20-y2k=1NR akzxkN+mxNδx/rp-TxxNxˆN0-xN-TxyNyˆN0-yNk=1NR akzykN+myNδy/rp-TxyNxˆN0-xN-TyyNyˆN0-yN,
Eˆ0=xˆ10-x1yˆ10-y1xˆ20-x2yˆ20-y2xˆN0-xNyˆN0-yN.
Aδˆ=B,
A=ΔaˆTr, aWΔaˆr, a,  B=-ΔaˆTr, aWϕˆ,
δˆ=δˆ1δˆ2δˆ3δˆ4δˆNR.
Aξη=i=1NzxηiTaizxξi+Tbizyξi+zyηiTbizxξi+Tdizyξi.
Bη=-i=1NϕxiTaizxηi+Tbizyηi+ϕyiTbizxηi+Tdizyηi.
aˆnew=aˆ+δˆ.
Eˆnew=-Txx1Ta1+Txy1Tb1k=1NR zxk1δk+ϕ1x+Txx1Tb1+Txy1Td1k=1NR zyk1δk+ϕ1yTxy1Ta1+Tyy1Tb1k=1NR zxk1δk+ϕ1x+Txy1Tb1+Tyy1Td1k=1NR zyk1δk+ϕ1yTxx2Ta2+Txy2Tb2k=1NR zxk2δk+ϕ2x+Txx2Tb2+Txy2Td2k=1NR zyk2δk+ϕ2yTxy2Ta2+Tyy2Tb2k=1NR zxk2δk+ϕ2x+Txy2Tb2+Tyy2Td2k=1NR zyk2δk+ϕ2yTxxNTaN+TxyNTbNk=1NR zxkNδk+ϕNx+TxxNTbN+TxyNTdNk=1NR zykNδk+ϕNyTxyNTaN+TyyNTbNk=1NR zxkNδk+ϕNx+TxyNTbN+TyyNTdNk=1NR zykNδk+ϕNy,
rˆnew=R+Eˆnew.
xˆi*=xi-TxxiTai+TxyiTbik=1NR zxkiak+δk-Txxixˆi0-xi-Txyiyˆi0-yi-TxxiTbi+TxyiTdik=1NR zykiak+δk-Txyixˆi0-xi-Tyyiyˆi0-yi,
yˆi*=yi-TxyiTai+TyyiTbik=1NR zxkiak+δk-Txxixˆi0-xi-Txyiyˆi0-yi-TxyiTbi+TyyiTdik=1NR zykiak+δk-Txyixˆi0-xi-Tyyiyˆi0-yi.
Δr+E, a=0,
r=r+E,
S0=1/2Eˆtσ-1Eˆ,
S0=1/2Eˆtσ-1Eˆ+Δtrˆ, aˆμˆ,
σ-1Eˆ+ΔrˆTrˆ, aˆμˆ=0,
ΔaTrˆ, aˆμˆ=0,
Δrˆ, aˆ=0,
σ-1Eˆ+εˆ+ΔrˆTμˆ=0,
ΔaˆTμˆ=0,
Δ+Δrˆεˆ+Δaˆδˆ=0,
ΔaTWΔaˆδˆ=-ΔaˆTWϕˆ,
ϕˆ=Δˆ-ΔrˆEˆ,
W=ΔrˆδΔrˆT-1.
aˆnew=aˆ+δˆ
Eˆnew=-σΔrˆTWϕˆ+Δaˆδˆ.
rˆnew=rˆ+Eˆnew,
Mxiδx=-R j=1Nt ajZxjxi, yi,
Myiδy=-R j=1Nt ajZyjxi, yi.
χi=Mxiiδx+R j=1Nt ajZxjXi, yi,
ηi=Myiiδy+R j=1Nt ajZyjXi, Yi,
Sx2=i=1N χi2,
Sy2=i=1N ηi2,
j=1Nti=1NZxjXi, YiZxkXi, Yiaj=-δxRi=1N MxiZxkXi, Yi,
j=1Nti=1NZyjXi, YiZykXi, Yiaj=-δyRi=1N MyiZykXi, Yi,
S2=i=1Nχi2+ηi2,
S2=i=1NMxiiδx+R j=1Nt ajZxjXi, Yi2+Myiiδy+R j=1Nt ajZyjXi, Yi2,
j=1Nti=1NZxjXi, YiZxkXi, Yi+ZyjXi, YiZykXi, Yiaj=-δxRi=1N MxiZxkXi, Yi-δyRi=1N MyiZykXi, Yi.
X=X+SX* cosRot+Y+SY* sinRot,
Y=-X+SX* sinRot+Y+SY* cosRot,

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