Abstract

A simple approach for measuring the piston error between two adjacent segments in a primary mirror of a telescope, based on the one dimension analysis of the diffraction pattern produced by a divided slit, is proposed. Using two wavelengths allows an increase of the dynamic range of the measurement. The main advantages are that even maintaining the correlation based scheme used by other authors, the time of processing should be reduced. Some experimental results are presented which show that for one wavelength a precision of 3 nm and a dynamic range of 316 nm are feasible for the red line of a He-Ne laser. For the two wavelength experiments a precision of 53 nm is obtained for λeq/2=1670 nm dynamic range.

© 2004 Optical Society of America

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References

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    [CrossRef]
  2. Kjetil Doholen, Fredérique Décortiat, François Fresneau, and Patrick Lanzoni, �??A dual-wavelength, random phase-shift interferometer for phasing large segmented primaries,�?? in Advanced Technology Optical/IR Telescopes VI, Larry M. Stepp, Ed., Proceedings of the SPIE 3352, 551-559 (1998).
  3. Gary Chanan, Mitchell Troy, and Edwin Sirko, �??Phase discontinuity sensing: a method for sensing segmented mirrors in the infrared,�?? Appl. Opt. 38, 704-713 (1999).
    [CrossRef]
  4. Gary Chanan, Jerry Nelson, and Terry S. Mast, �??Segment Alignment for the Keck Telescope Primary Mirror�??, in Advanced Technology Optical telescopes III, L.D. Barr, ed., Proc. SPIE 628, 466-470 (1986).
    [CrossRef]
  5. A. Jiménez-Hernández, R. Díaz-Uribe, �??Medición de fase para óptica segmentada con patrones de difracción unidimensionales�?? (Phase Measurement for Segmented Optics with One Dimension Diffraction Patterns), presented at the Fourth Ibero American Meeting on Optics, together with the Seventh Latin American Meeting on Optics Lasers and its Applications (IV RIAO �?? VII OPTILAS), Tandil, Argentina, 3-7 Sept. 2001.
  6. Weiyao Zou, �??New phasing algorithm for large segmented telescope mirrors,�?? Opt. Eng. 41, 2338-2344 (2002).
    [CrossRef]
  7. Achim Schumacher, Nicholas Devaney, and Luzuma Montoya, �??Phasing segmented mirrors: a modification of the Keck narrow-band technique and its application to extremely large telescopes,�?? Appl. Opt. 41, 1297-1307 (2002).
    [CrossRef] [PubMed]
  8. R. Pastrana-Sánchez, G. Rodríguez-Zurita, J.F. Vázquez-Castillo, �??Phase-conjugate interferometer to estimate refractive index and thickness of transparent plane parallel plates,�?? Rev. Mex. Fís. 47, 142-147 (2001).

Appl. Opt. (3)

Opt. Eng. (1)

Weiyao Zou, �??New phasing algorithm for large segmented telescope mirrors,�?? Opt. Eng. 41, 2338-2344 (2002).
[CrossRef]

Rev. Mex. Fís. (1)

R. Pastrana-Sánchez, G. Rodríguez-Zurita, J.F. Vázquez-Castillo, �??Phase-conjugate interferometer to estimate refractive index and thickness of transparent plane parallel plates,�?? Rev. Mex. Fís. 47, 142-147 (2001).

SPIE (2)

Gary Chanan, Jerry Nelson, and Terry S. Mast, �??Segment Alignment for the Keck Telescope Primary Mirror�??, in Advanced Technology Optical telescopes III, L.D. Barr, ed., Proc. SPIE 628, 466-470 (1986).
[CrossRef]

Kjetil Doholen, Fredérique Décortiat, François Fresneau, and Patrick Lanzoni, �??A dual-wavelength, random phase-shift interferometer for phasing large segmented primaries,�?? in Advanced Technology Optical/IR Telescopes VI, Larry M. Stepp, Ed., Proceedings of the SPIE 3352, 551-559 (1998).

Other (1)

A. Jiménez-Hernández, R. Díaz-Uribe, �??Medición de fase para óptica segmentada con patrones de difracción unidimensionales�?? (Phase Measurement for Segmented Optics with One Dimension Diffraction Patterns), presented at the Fourth Ibero American Meeting on Optics, together with the Seventh Latin American Meeting on Optics Lasers and its Applications (IV RIAO �?? VII OPTILAS), Tandil, Argentina, 3-7 Sept. 2001.

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Figures (12)

Fig. 1.
Fig. 1.

(a) Slit aperture. The dark region is opaque while the clear slit is divided in two regions (I and II); each region has a different uniform phase, as shown in (b).

Fig. 2.
Fig. 2.

Diffracted field amplitude for the aperture in Fig. 1. Several cases are shown for phase differences between 0 and π/2.

Fig. 3.
Fig. 3.

Graphical solution of Eq. (4) for the phase step Δ, as a function of displacement of the maxima (red), odd minima (blue), and even minima (purple).

Fig. 4.
Fig. 4.

Experimental setup for observing the diffraction pattern of a rectangular aperture with a phase step. The dotted lines show the additional setup used for two wavelengths.

Fig. 5.
Fig. 5.

Four different diffraction patterns for different phase steps, given by the parallel plate rotated at θ=0,1, 2 and 3 degrees respectively.

Fig. 6.
Fig. 6.

Composite image showing the experimental intensity variation along the y axis for different values of the rotation angle θ of the parallel plate.

Fig. 7.
Fig. 7.

Composed image for the center of the diffraction pattern for 79 different plate angles θ(=0.1, 0.2, 0.3, …, 7.9 degrees). The superimposed color spots are the pixels where the minima were found through the algorithm described in the text. The color spots located far from the minima, are artifacts resulting from the algorithm.

Fig. 8.
Fig. 8.

Plot of the minima y-position (in pixels) vs. the glass plate rotation angle θ (in degrees). The plot is inverted in reference to the images in Figs. 6 and 7, because for an image the pixels are numbered from top to bottom. The y position is proportional to the phase step given by the glass plate.

Fig. 9.
Fig. 9.

Unwrapped phase difference as measured by the one-wavelength procedure. The red line is the best quadratic fit as given by Eq. (16).

Fig. 10.
Fig. 10.

Composite image showing the two wavelength patterns for inclinations of the glass plate at 0°, 1°, 2°, 3°, 4°, 5°, and 6°, respectively

Fig. 11.
Fig. 11.

Separation of the Red, Green and Blue channels of the RGB images shown in Fig. 10, at (a) 0° and (b) 6°, of inclination of the glass plate.

Fig. 12.
Fig. 12.

Equivalent piston error produced by rotating the glass plate and evaluated by the two-wavelength technique. The dotted lines represent different plots of the theoretical equivalent piston error vertically displaced λ eq /2 between two adjacent parabolas.

Equations (23)

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f ( ρ ; k δ ) = { exp ( i k δ ) b η 0 ; a ξ a exp ( i k δ ) 0 > η b ; a ξ a 0 η > b ; ξ > a }
f ̂ ( ω ; k δ ) = 1 4 ab a a d ξ { b 0 exp ( i k δ ) + 0 b exp ( + i k δ ) } d η exp ( i k ρ · ω ) ,
f ̂ ( x , y ; k δ ) = sinc ( k a x ) sinc ( k b y 2 ) cos [ k ( δ + by 2 ) ] .
d f ̂ ( y ; k δ ) dy = kb 2 [ cos ( Δ + 2 β ) β sin ( β ) cos ( β + Δ ) β 2 ] = 0
β + Δ = ( 2 N + 1 ) π 2 ,
k b y 2 = N π
β = ( π ± Δ ) .
Δ = π ( N + 1 2 by λ ) ,
I T ( x , y ; k 1 , k 2 , δ ) = I 01 sinc ( k 1 ax ) sinc ( k 1 by 2 ) cos [ k 1 ( δ + by 2 ) ]
+ I 02 sinc ( k 2 ax ) sinc ( k 2 by 2 ) cos [ k 2 ( δ + by 2 ) ] .
Δ 1 = π ( N 1 + 1 2 b y 1 λ 1 ) ,
Δ 2 = π ( N 2 + 1 2 b y 2 λ 2 ) ,
δ = ( N 2 N 1 ) λ eq 2 b 2 ( y 2 λ 1 y 1 λ 2 λ 1 λ 2 ) ,
λ eq = λ 1 λ 2 λ 1 λ 2
ϕ = 2 π λ d [ ( n 2 sin 2 θ ) 1 2 cos θ n + 1 ] ,
ϕ π d ( n 1 ) λ n θ 2 ϕ o ,
ϕ o = 2 π λ d ( n 1 ) .
ϕ ϕ o π d ( n 1 ) λ n ( θ θ o ) 2 ,
ϕ a θ 2 + b θ + c ,
a = π d ( n 1 ) λ n ,
b = 2 π d ( n 1 ) λ n θ o = 2 a θ o ,
c = π d ( n 1 ) λ n θ o 2 ϕ o = a θ o 2 ϕ o .
n = π d π d λ a .

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