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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
1. Introduction
Phasing the mirrors of a segmented primary telescope is a very involved task. For this, it is important to measure the phase error of each of the segments of the mirror. Based on the considerations made by Chanan, et al. [1], we can say that the measuring method (or a combination of different techniques) should be able to measure phase differences due to piston errors on the segments as large as 36 µm and as small as 100 nm or less. This task has been accomplished by Chanan, et al. [1], however, currently there is a search for alternative methods to bring independent confirmation and because, if the phase correction is to be performed frequently on the mirror set up, it is necessary to measure the phase in less time.
There are at least three different techniques for measuring the phase of segmented primary mirrors. One of them [2] is based on the combined analysis of two interference patterns; each of different wavelength. A plot of the irradiance obtained for each wavelength, I1 vs. I2, gives a point that moves on the plane because of the random phase shift produced by the air turbulence or other disturbing agents. This plot is part of a Lissajous curve whose specific shape depends on the constant phase difference produced by the piston error between the mirror segments. This technique has been shown only at laboratory level but it seems feasible for operation on a telescope. Another technique [3], used on the Keck Telescopes, is based on the analysis of the intrafocal and extrafocal images of an infrared light beam being reflected on the mirrors to be phased. This technique, however, is restricted to the measurement of piston errors as large as 400 nm. The third technique [1,4], also used on the Keck Telescopes, is based on the idea of observing the effect of the piston error between two adjacent mirror sections on the diffraction pattern of a circular aperture divided in two symmetric halves by the straight border between the two mirrors. The particular shape of the diffraction pattern is indicative of the piston error; then, a comparison between a set of 11 theoretically calculated shapes with the actual pattern, gives the amount of the piston error. A numerical correlation algorithm carries out the comparison by processing n2 pixels from the CCD, for each of the intersegment edges where n is the lateral size in pixels of each zonal image for evaluation of the diffraction patterns. For the Keck telescopes, n=33 and the number of intersegment edges is 84. Presently we do not know how long it takes to measure the relative phase of the segments, however, it must be clear that any reduction in the processing time would be valuable. Apart from computational matters, a reduction of the number of pixels to be processed should contribute to this task; this cannot be made by simply reducing the number of pixels for each image, otherwise resolution may be lost.
Following the latter technique described above, in 2001 we proposed the measurement of the piston error through the analysis of the diffraction patterns produced by the use of rectangular apertures instead of circular ones [5]. For each aperture, one side should be oriented parallel to the intersegment edge where the piston error is to be measured; in this way, the variation of the diffraction pattern along the parallel and orthogonal directions to the intersegment edge will be independent from each other. The one wavelength technique has been well established since then; for the two wavelength method, however, we proposed a spatial phase shift method which did not work properly. More recently Zou [6] and Schumacher, et al [7] have also incorporated rectangular apertures for phasing telescopes. Zou proposes to adjust the actuators of each segment till the single slit diffraction pattern is obtained. Schumacher, et al., measure the phase error by evaluation of the peak ratio, that is, the quotient of the intensities of the right over the left maximum at the center of the diffraction pattern. However, neither of these last two proposals makes explicit calculations to relate the variations in the diffraction pattern to the phase error.
In the present paper we describe a method for measuring the phase error based on location of the minima at the center of the diffraction pattern. One advantage of this proposal is that some initial calculations are easily carried out in a closed form, so its analysis is clearer.
The paper is arranged in the following way. We first deduce an expression for the diffraction pattern for an arbitrary value of the piston error. Next, we analyze the behavior of the pattern as the piston error changes and describe how this error can be evaluated. After that, we propose to use two wavelengths in order to increase the dynamic range of the measurement. Then some experimental results are shown, for one- and two- wavelength measurements.
2. Theory
2.1 Diffraction Pattern dependence on the height difference δ.
Following closely the calculations made by Chanan, et al. [1], we compute the diffraction pattern produced by a rectangular aperture divided by a straight line in two equal adjacent regions with a different uniform phase and no attenuation. The subaperture function is in this case given by
where ρ=(ξ, η), is the position vector on the pupil plane, the origin of coordinates is at the center of the slit; δ is the height (or piston) difference between the adjacent mirrors; the full optical path difference between the beams incident on different mirror segments is 2δ; 2a and 2b are the lengths of the sides of the aperture along the ξ and η directions respectively (see Fig. 1); k=2π/λ is the wavenumber as usual, and λ is the wavelength of the illuminating light (monochromatic).
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).
The Fraunhofer diffraction pattern is given by the Fourier transform of the subaperture function f, as follows
where ω=(x, y), is the position vector on the observing plane (the focal plane of the lenslet array in a Shack-Hartmann camera), and the normalizing factor 1/4ab, is to have unit intensity at the principal maximum when δ=0. Actually, x and y are angular variables and are measured in radians. The resulting diffracted field is then
The behavior of such a function with δ has many similarities to the function obtained by Chanan1 for a circular aperture. From Eq. (3), however, one important feature of this function is readily evident: the pattern is affected by the piston error δ, only along the y direction; the x direction remains unchanged no matter what value of δ we have. So, for the following discussion only the behavior of the diffraction pattern along the y direction will be considered. For δ=0, the pattern is the same as for a uniform aperture without phase changes. Equation (3) reduces to the typical expression for a rectangular aperture with lengths 2a and 2b, equal to the product of two sinc functions, one for each coordinate on the observing plane. It is easy to see that, as δ is increased, the principal maximum and all the secondary ones are displaced from the center to negative values of y (see Fig. 2). For the zeros, however, only the odd numbered are moved; the even ones always remain at the same position. In this process, the principal maximum is reduced, sharing part of its energy with the plus one secondary maximum.
Another interesting feature of the pattern is that the second main intensity peak not only grows but also has a π phase change in reference to the original main peak; for δ=0 and δ=λ/2, the intensity patterns are exactly the same, but the amplitude exhibits a phase change. This must be true even for a circular subaperture, but it is not evident from the paper of Chanan, et. al.
Fig. 2. Diffracted field amplitude for the aperture in Fig. 1. Several cases are shown for phase differences kδ between 0 and π/2.
2.2 Principal maximum displacement
From Fig. 2 is clear that not only the principal maximum is displaced from its original position; all the maxima and minima for the field (maxima, for intensity), are displaced. Their positions are given by
where β=kby/2 and Δ=kδ. The value β=0 is a solution only when Δ=nπ/2 (n=0, ±1, ±2, …). A graphical solution to Eq. (4) is shown in Fig. 3. Then, one can try to measure the phase difference Δ by measuring the displacement of the maxima. There are at least two reasons why this is not the best choice: first, even though there is a way to solve Eq. (4) for Δ in a closed form, the resulting equation is non-linear. Second, for different values of Δ the intensity maxima can have large variations, making it highly probably that the CCD will become saturated, making it difficult to find the precise location of the maxima.
2.3 Zero intensity points
The position of the zeros for the intensity, however, are affected in a different way: the odd numbered zeros have positions given by the following linear relationship:
with N=0, ±1, ±2, …, while the even zeros maintain their original positions
with N≠0, in spite of the value of Δ; this is also shown in Fig. 3. The zeros are no longer equally spaced as for the case Δ=Nπ(N=0, ±1, ±2, …). In general, their spacing is given by
The plus sign stands for the spacing between odd and even zeros going from the negative to the positive values of y for N positive. From Eq. (5.a) the phase difference between the two beams passing through each half of the diffraction aperture is obtained from Eq. (5.a) as
where N is an unknown integer. Then the phase difference 2Δ can be determined with a Δo=(2N+1)π ambiguity (or δo=(2N+1)λ/2 in the piston error), as is usual in one wavelength interferometry. In other words, the piston error can be measured with the present method only between -λ/2 and +λ/2, assuming that the mirrors are very close (N=0); if the piston error is higher, the method is not able to discriminate the correct value for N.
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).
2.4 Two wavelengths patterns
In order to increase the dynamic range of the measurement it is convenient to use two wavelengths. Each wavelength produces its own diffraction pattern independent from the other wavelength pattern; they are mutually incoherent, so they do not interfere. The total intensity of a two wavelength pattern is the incoherent addition of two one-wavelength patterns as described by Eq. (3); in this case however we have to include a different maximum intensity for each wavelength, obtaining the following expression
For δ=0 the intensity pattern is almost the same as for the one wavelength pattern; as δ is increased the pattern is significantly different, essentially, the intensity of the principal maximum decreases while the intensity of the secondary maxima increase. This makes the visibility of the fringes smaller, until a certain value of δ for which the visibility reaches its minimum value. After that point, the effect is reversed; the visibility is again increased until it reaches a maximum value equal to one, and a new cycle is started. It is worth recognizing here that due to the fact that the width of the diffraction pattern is comparable to the width of the interference fringes, the variation of the visibility is not determined only for the piston error 2δ, but also for the width 2b of the slit. Thus, measuring the visibility is not a convenient way to determine δ; we propose instead the following procedure. By taking separate images for each wavelength, it is possible to make two independent measurements of δ according to Eq. (7), so we have
and
where y1 and y2 are the angular positions of some odd zero for each wavelength pattern.
Solving for δ, we obtain the following equation
where λeq is given by
The second term of Eq. (10) can be evaluated without ambiguity; the first term, as for Eq. (7), cannot be known. The difference between one- and two- wavelength interferometry is that for the latter the piston error can be evaluated between 0 and λeq, increasing the dynamic range as expected. By properly choosing the values of both wavelengths the value of λeq can be selected more or less at will.
3. Experimental results
3.1 One wavelength measurements
In order to test the proposal, the experimental set up shown in Fig. 4 was built. A red He-Ne (λ=632.8 nm), laser beam was used as a light source; the beam was filtered and collimated. A square aperture of 1 mm on each side was used to diffract the light and a positive lens was used to project the Fraunhofer diffraction pattern onto a CCD camera. The phase shift was produced in a controlled way by using a parallel plate of glass [8]; only a half of the cross section of the collimated beam was transmitted through the plate, whereas the other half propagated freely without being affected in phase or intensity. By rotating the plate an angle θ, the phase difference ϕ between the two halves of the beam was modified according to
where n is the refractive index of the plate and d is its thickness. It is worth stating here that the phase change ϕ introduced by the glass plate is equal to the phase difference 2Δ [Eq. (7)], introduced on a diffracting light beam by two mirrors with a piston error equal to δ.
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.
Figure 5 shows a set of four diffraction patterns for θ=0, 1, 2, and 3 degrees. Between 0 and 1 degrees, the phase shift is so small that the vertical displacement of the pattern is almost imperceptible. The change for 2 and 3 degrees is clearly seen. Along the horizontal direction, however, the pattern is not changed at all; the maxima and minima are always located at the same points. In Fig. 6 a composition of 79 images corresponding to the central column of pixels at the images obtained for different plate angles θ (from 0.1 to 7.9 degrees in 0.1° steps), is shown. From Fig. 6, the effect of a phase change on the diffraction pattern is evident. The displacement of the minima is not uniform because the phase difference is not linear with θ as described by Eq. (12). For small angles the phase difference can be approximated by
where
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.
Then, for small angles, the phase difference produced by the glass plate is proportional to the square of the plate rotation angle; this explains the curves described by the minima of the diffraction patterns in Fig. 6.
In order to evaluate the phase step from a diffraction pattern we choose to use the position of the intensity minima because they are linearly related (Eq. (6)), instead of the maxima positions because their relationship to the phase step is a little more complicated. In addition, the maxima are often obscured by the saturation of the CCD sensor. We have developed an algorithm to measure the position of the diffraction pattern minima. This includes the following procedures:
a) A selection of the central part of the pattern. This is important because the even minima are always located at the same position, so they do not give information of the phase step at all. When an odd minimum is close to an even minimum, the maximum in between is so dim that the CCD sensor cannot distinguish the structure of the pattern and only sees a large minimum; this fact makes the evaluation of the position of the odd minima more uncertain. To avoid this situation, we select the central part of the pattern where there are no even minima.
b) A smoothing of the pattern. Due to noise, the intensity pattern shows small random oscillations causing many maxima and minima not related to the phase step, then we smooth the pattern by adding all the gray values along a row of pixels in the image; this procedure does not affect the diffraction minima positions, because they are located at the same height (y-coordinate), no matter what the horizontal position (x-coordinate) is.
c) A numerical differentiation procedure. As is usual, the minima are located by finding the points where the intensity derivative is null, and the derivative goes from negative to positive values.
The resulting points of this algorithm are highlighted as superimposed color spots on Fig. 7; the agreement is evident, except for some artifacts that must be deleted manually. These are some points incorrectly evaluated because the smoothing procedure does not eliminate all the oscillations described in b) above.
As a quantitative proof of this result, in Fig. 8 the positions of the minima are plotted versus θ. The vertical axis units are pixels; they are linearly related to the phase difference between the beam passing through the rotated glass plate and the beam freely propagated. Fig. 8 represents the wrapped phase difference. In Fig. 9 the phase is unwrapped showing a quadratic dependence with the angle, but the vertex of the parabola is far from the origin. This is due to two facts: first, for 0° the absolute phase change of the beam passing through the glass plate is unknown (unless the refractive index is known in advance), so the parabola is arbitrary located in the vertical direction; it depends on the plate width d and on its refractive index n. Second, the initial alignment had an angular error of about 1°, as is evident from Figs. 6, 7 and 8; this causes a horizontal displacement of the parabola. With this, Eq. (13) must be modified as
or
where
and
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. 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. 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.
Solving Eq. (17.a) for n, we have an expression to measure the refractive index as
Fitting Eq. (16) to the data, we found these values: a=1054.6, b=-39.907, and c=0.3877; the glass thickness was d=1.18 mm, and the wavelength λ=632.8 nm. With these values and using Eqs. (16) and (17) we get the glass refractive index as n=1.562, the initial plate angle θo=1.084 degrees, and the initial phase difference ϕo=-1.017 radians.
As a measure of the uncertainty obtained in the evaluation of the phase, we can say that the maximum difference between the data and the least squares fit is equal to λ/32 (δ≈9 nm), the rms difference value for all the evaluated data is λ/136 (δ≈3 nm). Of course, these are results obtained in laboratory conditions, for phasing mirrors the uncertainty would be larger due to noise, vibration or even air turbulence.
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).
3.2 Two-wavelength measurements
For the two wavelength pattern an additional green laser diode (λ=532 nm) was used together with the original red He-Ne laser in the same optical set up of Fig. 4, with the additions shown in dotted lines. The green beam was introduced along the path of the red beam by using a beamsplitter. A color CCD camera was used for recording the patterns. Some of the colored patterns obtained for several values of the parallel plate angle θ are show in Fig. 10. It is clear that each color beam is affected in a different way by the phase change introduced by the inclined parallel plate.
In order to show the procedure proposed in Section 2.4, we processed the color images as follows. Each image was separated into the three basic channels, Red, Green and Blue (see Fig 11). The blue channel was not used at all, while the red and green channels were considered as the monochromatic patterns for the red and green laser beams. Each color was processed as was described in the monochromatic case (Section 3.1) to obtain the minima at the central part of the pattern.
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
At this point it must be noted that the phase is a concept directly related to the wavelength, so for two wavelengths the phase change ϕ introduced by the glass plate [Eq. (12)] is not an adequate quantity to be evaluated. We evaluate instead the equivalent piston error δ introduced by the plate [Eq. (10)], as if the phase change was produced by the piston error between two mirrors; this figure is independent of the wavelength. The corresponding results are shown in Fig. 12. From this figure, it can be seen that the value of δ (the full height difference for the piston error between two adjacent mirrors) jumps between different parabolas (dashed lines); these parabolas are given by δ=kϕ/2+Nλ eq/2, with ϕ given by Eq. (16) with different values of parameter c. Then, each parabola is displaced vertically from the next a constant value given by λeq/2=1670 nm; i.e., the equivalent piston error is evaluated except for a Nλeq/2 uncertainty, where N is an integer. Lacking any other information, it must be assumed that 0≤δ≤λeq/2, so N has to be chosen accordingly in order to obtain a definite figure for the piston error.
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.
For the last part of the plot, between 5 and 6 degrees of rotation angle of the glass plate, some of the evaluated data are very different from the parabolas because some of the green light is present on the red image. This is particularly important when a green maximum falls where a red minimum is present (see Fig. 11.b); for this case locating the minima of the red pattern gives a wrong result, affecting the evaluation of the piston error. Due to this fact, the proposal of using the red and green channel of an RGB color image is not good. For this case the maximum difference between the experimental data and the fit is 156 nm (9.3% of λeq/2), while the rms value is 53 nm (3.2 % of λeq/2); for these results we used only the data between 0.1 and 5.1 degrees of the glass plate angle θ. Although the results are not as good as for one-wavelength experiments, we decided to present them here to show the principle of the two-wavelength experiment. A better approach for this experiment would be to split the diffracted beam in two different, separated, red and green images and acquire each image through a different Black and White CCD camera, in this way the intensity minima could be obtained for each beam independently.
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 λeq/2 between two adjacent parabolas.
4. Conclusions
We have proposed a slight modification to the method for measuring the piston error proposed by Chanan, et al. [1], by changing the circular aperture to a rectangular one. This proposal allows a simpler analysis of the diffraction pattern for obtaining the piston error. The first evident advantage is that the mathematical theory becomes extremely simple as compared to the theory involved in the circular-aperture method. The second advantage is that all the information of the piston error is displayed on the diffraction pattern along a direction parallel to one of the symmetry axes of the rectangular aperture; in consequence, the analysis of the pattern must be made only along this direction whereas the orthogonal direction can be neglected. Two experiments were conducted, with one and two wavelengths respectively, in order to show the feasibility of the proposal in laboratory conditions. For the first experiment the results were obtained with high precision, 3 nm (rms differences), whereas for the second the particular way in which the images were obtained increased the uncertainty in the location of the diffraction pattern minima, raising the final uncertainty on the piston error to 53 nm (rms). The dynamic range, however, is λ/2=316 nm for the first case, while for the second case it is λ eq/2=1670 nm, for λ 1=632.8 nm and λ2=533 nm; for the latter method, the dynamic range can be modified by a proper choice of the two wavelengths.
Acknowledgments
Part of this work was performed during a sabbatical leave of RDU, from the Laboratorio de Optica Aplicada, Centro de Instrumentos, UNAM. For this leave special support was obtained from the Dirección General de Asuntos del Personal Académico (DGAPA), UNAM, from the Consejo Nacional de Ciencia y Tecnología, SEP, México, and from the Optical Sciences Center, U. of A. Recent work on this subject has been supported by DGAPA-UNAM under a PAPIIT grant No. IN115000.
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