## Abstract

We present simulated results on piston detection applying the classical Ronchi test to a segmented surface. We have found that a piston error in a test segment, induces a change in the transversal aberration, that can be analyzed by mutually comparing the fringes frequency in each segment. We propose that the piston term of the segmented surface can be recovered by geometrically relating the change in transversal aberration with the piston term. To test this, we have simulated some ronchigrams for a known piston error, and we have been able to recover this term for a dynamic range comprised among 57*nm* and 550 *µm*. For piston errors >550*µm* a change in the transversal aberration can be appreciated and measured in the ronchigrams although these large pistons are now classical defocusings. Thus we have demonstrated that the Ronchi test can be an alternative method for the piston detection with a large dynamic range.

© 2004 Optical Society of America

## 1. Introduction

At the present, ground-based optical and infrared telescopes of large diameter are being successfully manufactured. The cost ranges in the tens to hundred million dollars, scaling as a 2.5 power when the primary mirror diameter is larger than 4 meters, Schroeder [1]. The primary mirror cost can be reduced with segmented mirrors; however, this type of surfaces must correct errors in piston, tip/tilt, displacement and centering with precisions of a fraction of the wavelength, thereby allowing the segmented surface to perform as a monolithic surface. With this requirements, a reflected wavefront in a segmented surface will maintain its original shape in the optical system output.

It has been shown that a piston error less or equal to $\frac{\lambda}{10}$ is a reasonable criteria to consider that a segmented surface is aligned, Orlov [2]. This author, discusses the basic requirements that an optical test must meet to detect small piston errors: the test must use white light, since due to the spectral broad band of this illumination type the piston term can be recovered completely, the seeing must not induce large effects on the measurements, the test must use astronomical objects for the measurements and must be done in short time periods. These requirements could be fulfilled by the Teague test since the measurements are made in the exit pupil where the wavefront is a plane free of spherical aberration.

Voitsekhovich *et al*. [3], uses shearing interferometry, to recover piston errors of the order of a wavelength with random noise.

Zou [4], proposes a new calibration method, called sensor-by-sensor phase calibration method. They serially bring to phase the local areas at each sensor location and thus obtain the desired sensor readings one by one with an experimental accuracy better than 20-nm rms. Schumacher *et al*. [5], use a variation of the Shack-Hartman wavefront sensor in which the signal is the correlation between individual sub-images and simulated images and also discuss an alternative method to resolve the *λ* ambiguity in some problematic cases. Díaz-Uribe *et al*. [6]. show a simple approach for measuring the piston error between two adjacent segments based on the one dimension analysis of the diffraction pattern produced by a rectangular aperture, where they have obtained for one wavelength a precision of 3 nm and a dynamic range of 316 nm for a He-Ne laser. For two wavelength experiments a precision of 53 nm is obtained for 1670 nm dynamic range.

Other method for piston detection uses the technique of local curvature sensing, Chanan *et al*. [7]. Here, the authors compare real images in and out of focus of the exit pupil mirror. As this method uses a spherical wavefront the images contains spherical aberration that can reduce the accuracy of the measurements. With this test they have reduced a piston RMS error of 230 to 40 nm.

The main technique in piston detection is the broad band algorithm realized by Chanan *et al*. [8], where they have achieved the full diffraction limit in the infrared region for the Keck telescope by properly phasing the primary mirror. They use a variation of the Shack-Hartmann test by detecting a piston RMS error less than 200 nm. For detection of piston fractions, Chanan *et al*. [9] implemented the narrow band algorithm.

This work is a first approach in piston detection by means of the classical Ronchi test, Salinas [10–12]. The main idea is to show that the Ronchi test can detect a piston term despite the limitation that the technique can only detect the first derivative of the terms in the mathematical representation of a wavefront with primary Seidel aberrations (Malacara [13]) plus the piston term *δf* :*W*(*x,y*)=*A*(*x*
^{2}+*y*
^{2})^{2}+*By*(*x*
^{2}+*y*
^{2})+*C*(*x*
^{2}+3*y*
^{2})+*D*(*x*
^{2}+*y*
^{2})+*δf* ; where A, B, C and D are the coefficient for astigmatism, coma, spherical aberration, and defocusing respectively. As the piston term in this polynomial equation is a constant, its first derivative is zero and then the Ronchi test should not detect it. However, the piston term is also contained in the D term, the defocusing (Bai *et al*. [14]), so that when the Ronchi test is used for piston detection and a step or piston term is present in a segmented surface, a change in the fringes frequency for the segment with piston would be observed. So we can compare the Ronchi fringes frequency for a segment with piston to the Ronchi fringes frequency for a reference segment.

We propose to use the Ronchi test to measure the piston term in a segmented surface, through the defocusing term by considering the piston term as a small defocusing in the order of the work wavelength. So in our work the phrase “co-phasing of an optical segmented surface”, means the action of bringing together all the segments to a position where their behavior is close to a monolithic surface performance, that is when the transversal aberration in each segment is the same. For piston detection with the Ronchi test we use monochromatic light at a wavelength reference of *λ*=632.8 nm.

This work is organized as follows: First, we analyze the light behavior at a segmented surface from the point of view of geometric optics. Next, we show how the transversal aberration that rules the Ronchi test is related geometrically to the piston term. Followed by the analysis of some critical cases to be considered before using the Ronchi test for piston detection. Furthermore, we show some numerical simulated cases of co-phasing ending with a discussion of the piston detection range that can be achived with this test.

## 2. Theoretical basis

According to geometrical optics when an spherical monolithic surface is tested with a point source located in its local center of curvature the beam returns in the same path. However, if the surface under test is segmented and a relative piston error is present, we will have multiple convergence points for each segment onto the optical axis. This can be analyzed by means of the Gaussian formula for the lenses in the mirror case, Hecht-Zajac [15],

For a concave mirror *R*<0, so

where *s _{o}* is the object position,

*s*is the image position and R is the curvature radius measured from the parent surface vertex under test. If we take a reference segment, its image point position is found by setting

_{i}*s*=

_{o}*R*; therefore the position of its image plane is in

*s*=

_{i}*R*. Fig. 1.

As the segment is displaced by a piston term *δ _{f}* along the optical axis, its position would be at

whereas its image plane would placed at,

For the case when *δ _{f}*≪

*R*, that is when the piston error is on the order of a fraction of a wavelength, the term

Thus, the image plane position would be at *s _{i}*=

*R*+

*δ*. This means that the image plane position changes linearly with the piston error whenever $\frac{R}{R+2{\delta}_{f}}\approx 1$ or

_{f}*δ*≪

_{f}*R*. Now we can estimate values of the defocusing term for which

*δ*can be considered a piston term, see Table 1. For values of piston less than

_{f}*λ*/10 the factor $\frac{R}{R+2{\delta}_{f}}\approx 1$, but for values larger than 600

*µ*m the piston term has a non linear behavior in the image plane.

On the other hand, Hopkins [16], established that if a constant term is added to a wavefront, it induces a change in the focal plane position by producing an angular aberration that can be neglected. For our work this means that a defocusing can only be a piston term when *δ _{f}*≪

*R*. Then the linearity condition obtained by geometrical optics and the Hopkins theory constitute the theoretical basis that allows considering a small defocusing as a piston term in a segmented surface.

## 3. Relation between the transversal aberration and the piston term

Figure 1 shows a difference in the transversal aberration Δ*T* produced by the piston term *δ _{f}* at a segmented surface, shown there for an on-axis source. However the case of off-axis sources is also important. If the source is placed way out off-axis it introduces aberrations that can produce strong effects in the piston measurement. For example, when the off-axis source position in the x direction is >3.0 cm (perpendicular to the fringes), or when the off-axis position in the y direction is >8

*λ*Fig. 2.

In the case that other aberrations are present, the segments with piston will show ronchigrams with very particular patterns, as can be appreciated in Fig. 3, where the relative piston is maintained at 30*λ*, while different amounts of sphericity, coma and astigmatism are introduced. Also if air turbulence is present the ronchigrams suffer a degradation effect, Fig. 3. Therefore, for piston detection with the Ronchi test is necessary to take into account these problems.

With these efects in mind, we will develope the following theoretical model. We will start by considering the case of applying the Ronchi test to a monolithic surface, as a basis for the analysis of the Ronchi test of a segmented surface.

Let *W*(*x,y*) be a mathematical representation of an aberrated wavefront, Malacara [14],

where (*x,y*) are the coordinates on the monolithic surface, and the coefficients *A, B* and *C* are the Seidel or third-order aberrations: spherical aberration, coma and astigmatism. The last term, the defocusing *D*, is the key factor for our analysis, since it is detected by the Ronchi ruling position which is very close to the surface curvature center.

The transversal aberration, TA(x), is found according to the Rayces formula, Malacara [14],

where *R* is the curvature radius of the surface under test, and ∇*W*(*x,y*) is the wavefront first derivative detected by the Ronchi test. Equation 7 can be expressed in components as,

Now, if we replace Eq. (8) in Eq. (6), for the reference segment we get

On the other hand, from geometrical relationships between each segment, Fig. 1, we can get for the reference segment,

Solving for the transversal aberration *T*
_{1} in the reference segment, we have

In Equation (11), the transversal aberration is a straightforward function of the Ronchi ruling position, Δ*F*, and of the position on the mirror and it is reciprocal to the curvature radius.

If we equal Eq. (9) and Eq. (11), the defocusing term *D* detected for the ruling depends on the Ronchi ruling position, Δ*F*, and it is reciprocal to the quadratic curvature radius (O’Neill [17]).

The Ronchi test applied to a segmented surface allows the comparison between the reference segment fringes and the test segment fringes. Both segments share some of the tests parameters, such as the position of the source and of the Ronchi ruling, Fig. 8. A segmented surface will perform as a monolithic surface when the same transversal aberration is detected in each segment. Thereby, in a segmented surface each segment will detect half the transversal aberration of a monolithic surface and from the geometry of the Fig. 1 we have,

Solving for *T*
_{1}, we obtain the transversal aberration for the reference segment as follows,

Next we can get the parameters for the transversal aberration *T*
_{2} for a displaced segment as function of the piston error, *δ _{f}*, from Fig. 1, where the segment has been displaced along of optical axis by

*δ*, and its radius of curvature has also been displaced,

_{f}from which,

By using the expressions for *T*
_{1}, *T*
_{2}, we can obtain the difference in the transversal aberration, Δ*T*=*T*
_{2}-*T*
_{1} detected by the Ronchi test in a segmented surface. After some simplifications we have,

and solving for the piston term, *δ _{f}*,

This expression provides a geometrical relation between the transversal aberration and piston term when the Ronchi test is used for piston detection. If the piston term were constant for the whole surface, the factor $\frac{\Delta T}{x}$ would also be constant. In other words, if G is a constant,

then the equation for the piston term is,

Table 2 shows a numerical example for the G constant behavior in some points on a spherical segmented surface for its center of fringes loci.

## 4. Piston term simulation

We have simulated a Ronchi test of a segmented surface numerically by exact ray tracing. The reference segment was kept at a fixed position and the test segment was piston displaced by adding a constant *z*
_{0} of the order of the wavelength to its sagitta (Fig. 10).

Then the piston term was recovered by the analysis of the simulated ronchigram in the way prescribed in Eq. (18). The change in the transversal aberration Δ*T* is evaluated in the observation plane in a certain point x on the surface, and is calculated as the difference of the transversal aberrations *T*
_{1} and *T*
_{2}, (for the reference and test segment respectively) in the x direction for the central maximums in the Ronchi fringes, Luna E.[18].

Equation 18 must consider an amplification factor, *ESC*, since the piston term is obtained in a plane nearby the curvature center of the surface. This plane is obtained for the different convergence image points of each segment. Thus, the width of a Ronchi fringe *f* must be compared with the transversal aberration of the test segment, *T*
_{2} then

Here the width *f* of a fringe is obtained from the following,

where *NLP* is the number of lines per inch of the Ronchi ruling. In this way we evaluate the piston term in a simulated ronchigram from the Ronchi fringes central maximums by means of

and we find an excellent agreement of the recovered piston *δ _{f}* and the simulated introduced piston

*z*

_{0}, as will be seen next, Table 3.

## 5. Dynamic range of the test

In the Ronchi test, the detection range changes in accordance to the line frequency of the Ronchi ruling and the piston error value to be measured. A piston term of the order of fractions of a wavelength can only be measured with a high density Ronchi ruling, up to the limit where diffraction dominates. On the other hand, a piston term of the order of micrometers, can be detected with a low density ruling.

In the case of a small relative piston, the transversal aberration is almost the same in all the segments. In Fig. 6(a) we show an example of this. A small piston term 0.09*λ* (≈57nm) results in seven fringes. Random noise of 10% has been added to the simulation to make it more real. Here, we just barely miss to observe a discontinuity in the ronchigrams and their fringes are aligned to each other perfectly. Then the minimum limit of piston detection is obtained by the resolving power of each Ronchi ruling. In Fig. 6(b) we show a large change in the fringes frequency as a result of a large piston term (note that according to Eq. (3) this is still a piston term). In this case (for a surface with R=1200 mm, and D=200 mm) the upper limit of the detection range is 550*µ*m. The upper limit changes in accordance with the radius of curvature of the surface. In Fig. 6(c), we show the effect of piston values larger than 550 *µ*m in the surface plane that can still be seen with the Ronchi test, although this case rapidly becomes non linear as can be seen from Fig. 1 and Eq. (3). That is, the Ronchi test can continue to be used to detect lack of co-phasing, although it may be that the actual amount cannot be correctly evaluated for large piston errors.

Figures 7 to 10 show the central maximums of the Ronchi fringes for different cases of piston errors in the dynamic range from 550 *µ*m to 57 nm for seven and three fringes in each segment. We have used a reference wavelength of 632.8 nm.

The numerical results that we obtained are summarized in Table 3, for various piston errors and for rulings of 500 and 1000 lines per inch. We have not taked into account the diffraction effects in the piston measurements. Random noise of 10% has been added to the simulations. The recovered piston *δ _{f}* agrees with the introduced piston

*z*

_{0}to better than 1/1000. The knowing of

*δ*will allow the active correction of the piston term by bringing equal the two transversal aberrations

_{f}*T*

_{1}and

*T*

_{2}.

## 6. Conclusions

The Ronchi test is one of the simplest and most powerful methods to evaluate and measure an optical system. We have shown that this test can be used to measure the piston term in a segmented surface.

We have also found the following advantages:

• For piston detection the Ronchi test does not depend on the wavelength and can use any illumination type.

• Measurement of the phase with the Ronchi test is critical, The off-axis position of the source does not contribute in the piston measurements whenever this position is offset less that 3.0 cm.

• The Ronchi test has a very large dynamic range. It can measure relative pistons from fractions to multiples of *λ* with a same Ronchi ruling and with the same measurement algorithm, Salinas [19].

• If only piston detection of the order of micrometers is required, a single low density Ronchi ruling can be ussed. Otherwise, for the detection of small pistons a ruling of high frequency is required. However, for the high frequency rulings (larger than 1000 lines per inch), diffraction effects contribute significantly in the piston measurements.

It is important to mention that in the case where relative piston exists, the fringes in each segment have different frequencies. However, one may always keep the central fringe perfectly aligned at each segment, Fig. 4(b). This has the additional advantage that the test can also be sensitive to detect inclinations, because the central fringe in each segment should not remain aligned by tip/tilts. A segment inclination will be seen as a shifting of the fringes. These effects will be further studied in the future.

## Acknowledgments

The authors want to express our gratitude to the project Telescopio Infrarrojo Mexicano, (TIM), UNAM. This research was also supported by Consejo Nacional de Ciencia y Tecnologia (CONACYT) to one of the authors with register number of Ph.D. scholarship 114488/115164.

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**19. **IAUNAM, OAN, Apdo. Postal 877, Ensenada B. C. México, c.p. 22830 and Javier Salinas-Luna et al. are preparing a manuscript to be called “The classical Ronchi test for piston detection:experimental part.”