determination of the wavefront errors of an astronomical telescope

1997MNRAS.284..655H

Mon. Not. R. Astron. Soc. 284, 655-668 (1997)

Determination of the wavefront errors of an astronomical telescope using lateral shearing interferometry

c. M. Humphries, J. W. Harris and E. Atad

Royal Observatory, Blackford Hil4 Edinburgh EH9 3HJ

Accepted 1996 September 3. Received 1996 August 28; in original form 1996 May 10

ABSTRACT Practical aspects of using lateral shearing interferometry for the determination of wavefront errors in astronomical optical systems are discussed, as well as underlying theory. The relationship between polynomial descriptions of the sheared and unsheared wavefronts is reconsidered, and a single-step wavefront reconstruction process is derived. The particular reconstruction matrix tabulated in this paper allows solutions for Zernike polynomials up to fifth radial degree. Some results of applying this method to the determination of the wavefront errors of the UK Infrared Telescope are presented.

Key words: techniques: interferometric - telescopes.

1 INTRODUCTION

With the advance of adaptive optical techniques to compensate for the effects of atmospheric seeing, it has become increasingly necessary to provide corrections also for the wavefront errors that originate at the telescope itself. This requires measurement of the telescope optical performance to detect variations with time for active control of the optical support systems. The three methods that have been particularly used in recent years for wavefront sensing on astronomical telescopes are (i) Shack-Hartmann sampling, (ii) curvature sensing and (iii) shearing interferometry. In this paper we describe some practical aspects of using lateral shearing interferometry; we also reconsider the relationship beween polynomial descriptions of the sheared and unsheared wavefronts in the process of wavefront reconstruction, and we give some results of applying this method to the determination of the wavefront errors of the UK Infrared Telescope (UKIRT).

Applications to astronomical telescopes include measurements by Saunders (1964), who first used lateral shearing interferometry at the O.66-m refractor of the Leander McCormick Observatory, and later applied the same method to determine the wavefront errors of the 2.1-m telescope at Kitt Peak National Observatory (Saunders & Bruening 1968). Brown (1982) discussed the degradation in image quality of astronomical telescopes as detected with shearing interferometry and, in particular, considered how the wavefront errors due to distant and local atmospheric seeing could be separated from each other by using shearing interferometric exposures of different duration. Such measurements were made in 1979 at the former coude focus of

©1997 RAS

UKIRT. Dunlop & Major (1988) and Dunlop, Haman & Major (1989) used a Jamin-based shearing interferometer having a pair of prisms with anamorphic demagnification (Brown & Scaddan 1979) to produce a range of shear values on each interferogram; with this instrument attached to a 30-cm Polaris-pointing telescope at La Palma, atmospheric seeing modulation transfer functions (MTFs) were derived.

Serabyn, Phillips & Masson (1991) used lateral shearing interferometry at millimetre wavelengths to determine the surface figure of the segmented primary reflector of the lOA-m telescope at Caltech Submillimeter Observatory; to do this, they measured the focal plane diffraction pattern of the telescope using a double mirror interferometric arrangement that, in effect, imaged light simultaneously from two directions, each laterally displaced with respect to the other on the primary. A measurement accuracy of A/115 (9 11m) was achieved.

Lateral shearing interferometry was employed as the wavefront sensor in one of the first practical systems for making real-time adaptive corrections for the atmosphere (Hardy, Lefebvre & Koliopoulos 1977; see also Hardy 1978). This used a crossed double-frequency grating, as described by Wyant (1973, 1974, 1975). Kibblewhite et al. (1992) chose two lateral shearing interferometers mounted orthogonally for the primary wavefront sensor of the Chicago Adaptive Optics System. Although a conventional Shack-Hartmann sensor is available also in this system, shearing interferometry was chosen for its 'linearity, better signal processing requirements, and direct control of dynamic range'. Additional factors that may influence choosing lateral shearing interferometry for wavefront sens-

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1997MNRAS.284..655H

656 C. M. Humphries, 1. W Harris and E. Atad

ing include convenience of size, stability with respect to thermal and mechanical disturbances, insensitivity to stray light, large dynamic range, and capability of high spatial frequency sampling.

The review by Murty (1978) describes the various instrumental means of obtaining lateral shear interferograms, and discusses also some of the underlying theory and the wavefront reconstruction method. Perhaps the least convenient requirement of the method is the need to have orthogonal pairs of interferograms for wavefront reconstruction. If it is necessary to obtain these pairs simultaneously rather than sequentially, then this may mean using two array detectors as in the Chicago system mentioned above, or the Wyant double-frequency grating method (Rimmer & Wyant 1975). For active telescope control, however, sequential measurements can usually be tolerated, although this does require incorporation of moving parts into the mechanical design of the interferometer.

2 THE INTERFEROMETER

2.1 Instrumental

The particular type of interferometer discussed here is a Jamin system in which two parallel plates are used to divide and recombine the input light (Brown 1954,1955). We have briefly described the instrument previously when used with an astronomical telescope (Atad et al. 1990), but a description of its properties will also be useful here.

Fig. 1 (inset) shows a plan view ofthe separate paths with a convergent beam entering the instrument. A and B are the parallel plates at which beam division and recombination take place. Between the first and second plates a small prism (P) may be inserted in one of the beams so that the divergent output cone is rotated about the projected focus, whereas the other beam passes through a compensating plate (C) and is undeviated.

For a prism of refractive index n and angle ex in a beam from a system of focal length f, the linear shift of the sheared

r-------------------------

'X' I

pupil is ex(n -1)[, and the value ofthe shear S expressed as a fraction of the pupil radius is 2ex(n -l)F, where F is the focal aperture ratio. A selection of prisms giving a range of angular deviations from 0.2 to 60 mrad allows choice of a fixed, accurately measured, shear suitable for most applications. Each prism and its compensating plate are mounted in a frame so that the pair is introduced simultaneously into the respective paths. Alternatively, if the prism/compensator pair is removed, then zero-shear interferograms are obtained and these provide a reference measure of the magnitudes and directions of the chosen fringe tilt as well as a photometric reference for the measurement of fringe visibilities.

Tilt fringes are introduced into the shearing interferogram by a continuously variable control that rotates the second Jamin plate about the axis X-X' in Fig. 1, located mid-way between the top and bottom faces of the plate. In the absence of defocus or other aberrations, this produces tilt fringes parallel to the direction of shear. Fig. 1 also shows the measurement system, comprising reference wavefront source (see Section 7), interferometer, beam-alignment mirrors, imaging and relay lenses, filters, image intensifier with fibre optic output faceplate, and frame transfer CCD camera (SpectraSource Lynxx) with shutter for varying exposure times down to 0.01 s. The interferometer has a motor drive that allows it to be rotated by 90° to provide x- and y-shear interferograms; the two beam-alignment mirrors allow the divergent cone of the emerging unsheared wavefront to be coaxial with the entrant converging cone (so that when the interferometer is rotated the xand y-shear interferograms remain centred on the CCD detector). The pupil lens re-images the telescope pupil (i.e., the second mirror in the case of the UKIR T) on to the input face of the image intensifier. CCD frames are recorded by a framegrabber and then stored to disc for subsequent processing. Fig. 2 (opposite p. 656) shows the assembly mounted on a rig that interfaces at a Cassegrain telescope focal plane.

--------~--"\'--------- pupil . field

~!~~~?~~ nnager collimator lens

----·-·r-·-!-·a ! ~·-·+·t·--·-·~·-·-·-·-·a·+·-·-·--·-~·-·D r-=J.., . ~._L'-I ~ ~ L ___ ~: _____ ..: filter . ima~fie phosphor c~~

LED source (reference wavefront)

mtensl er

Figure 1. Measurement system including interferometer layout (inset).

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1997MNRAS.284..655H

Opposite p. 656, MNRAS, 284

Figure 2. Assembly comprising interferometer (A), image intensifier (B), and camera (C).

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1997MNRAS.284..655H

2.2 Coherence requirements

For monochromatic fringe formation, spatial coherence is required across the pupil shear distance SR, where S is the l~near shear expressed as a fraction of the pupil radius R. Thus, if the angular diameter of the source or atmospheric seeing disc is 0, then the requirement for obtaining fringes is O;::;J../SR. When the input is not monochromatic, the spectral coherence requirement is that fringes due to extreme wavelengths (J.. - AJ../2) and (J.. + AJ../2) are separated by much less than a half-wavelength across the pupil radius, i.e., AJ.. « )J2p, where J.. is the central wavelength, and p is the number vf fringes across the pupil radius. In our case, using a filter at J..=546 nm (AJ..= 10 nm FWHM) on a 3.8-m telescope with 0.5-arcsec FWHM atmospheric seeing, typically 6-10 fringes of acceptable visibility across the pupil are used for the interferograms with a shear value S::;; 0.04 (equivalent to 7.6 cm at the primary mirror of a 3.8-m telescope ).

3 COORDINATE SYSTEM

The coordinate system used here is right-handed Cartesian, centred at the pupil from which the wavefront originates (Fig. 3). Fig. 3 also defines the directions of positive shear along x and y, expressed as fractions of the pupil radius (here 'l~<;umed normalized to unity), where the measured optical p.:h differences referred to the original coordinates are AW -' W(x-S,y) - W(x,y) for points separated by the shear distance S along x, and A Uj, = W (x, y - T) - W (x, y) for points separated by the shear distance Talongy. Usually S = T. With this convention, the small-shear approximation becomes AJV.= -(oW(x,y)/ox)S and AUj,= -(oW(x,y)/ oy)T. Thus, in the absence of added instrumental tilt, the fringes in x-shear interferograms are formed along contours of equal x-slope, each separated by slope increments of J../S (or NT for the y-slopes in y-shear interferograms).

Wavefront aberrations are discussed here in terms of Zemike circle polynomials. Table 1 gives these for terms up to fifth radial degree, where An is the Zemike coefficient of polynomial n using the ordering given by Noll (1976) but with the cosine term preceeding the sine term for each pair

positive x- shear

•

Wavefront errors of an astronomical telescope 657

of given azimuthal and radial degree, and without Noll's normalization. Shown in Table 1 are (i) geometric blur circle dimensions (100 per cent encircled energy diameters), given in angular units of J..ID rad, where D is the telescope aperture, and (ii) the rms dispersion radii about the image centroids (for the coma polynomials the separate x and y dispersions are given). The blur diameters were obtained from the maximum and minimum wavefront slopes of each aberration polynomial. The rms dispersions were derived by taking the difference of the polynomial first differential (wavefront slope) from its mean, squaring, integrating over a circle of unit radius, and normalizing by the area. The column in Table 1 giving diffraction-limited An values is discussed later in Section 7.3.

For astronomical telescopes with a central obscuration in the emerging wavefront, fully orthogonal annular polynomials should properly be used, and these have been investigated by Swantner & Lawrey (1980), Wang & Silva (1980) and Mahajan (1981) (see also Schroeder 1987). However, for systems where the obscuration ratio e is small, the errors introduced by not using annular polynomials are also small. For example, in a system having e < 0.5 and containing third-order astigmatism or coma, the error introduced by using circle instead of annular polynomials amounts to a few per cent of the rms wavefront error at the aberrated diffraction focus. With third-order spherical aberration the error introduced by not using annular polynomials is higher, but for astronomical telescopes with e ::;; 0.25 the rms wavefront deformation due to Au calculated causing circle polynomials is at most 13 per' cent higher than that obtained using the appropriate annular polynomial. For convenience, we have chosen to keep the use of Zemike circle polynomials here.

4 OPTICAL TRANSFER FUNCTIONS DERIVED FROM LATERAL SHEAR INTERFEROGRAMS

Lateral shearing interferometry yields the x and y components of the pupil phase-difference map Ac5 (x, y) = k A W (x, y) for points separated by shear S along x or T along y, where k=21t/J... Expressing the optical transfer function

x

positive y- shear r Figure 3. Coordinate system used.

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Table 1. Blur diameters (100 per cent encircled energy, expressed in units of )JD rad), rms dispersion radii about the image centroid, and diffraction-limited values of the Zemike polynomial coefficients

An'

n Aberration

piston: 1

2 x tilt: p cos6

3 y tilt: p sin6

4 defocus: 2p2 - 1

5 0°-astigmatism: p2 cos26

6 45° astigmatism: p2 sin26

7 x coma: (3p3_2p)cos6

8 y coma: (3 p3_2p)sin6

9 0° trefoil: p3 cos36

10 30° trefoil: p3 sin36

11 spherical aberration: 6p4_6p2 + 1

12 0° astigmatism (5th order): (4p4 - 3p~ cos26

13 45° astigmatism (5th): (4p4 - 3p~ sin26

14 0° quatrefoil: p4 cos46

15 22.5" quatrefoil: p4 sin46

16 x coma (5th): (10ps-12p3+3p)cose

17 ycoma(5th): (10ps-12 p3+3p)sin6

18 0° trefoil (7th): (S pS - 4 p3) cos36

19 30° trefoil (7th): (5 pS - 4p3) sin36

20 cinquefoil: pS cos 56

21 cinquefoil: p5 sin 56

(OTF) 7: as the autocorrelation ofthe wavefront, and ordering the autocorrelation integral appropriately, we obtain, for shear along x,

(1)

a(x - Afx, y)a (x, y) exp [ - i(b(x - Afx, y) - b(x, y)] dx dy

[00 [00 a 2(x,y)dxdy

where a (x, y) is the aperture function, and the spatial frequencies for a pupil of unit radius arefx=S/A and multiples thereof up to the cut-off frequency 2/A. In units of cycle rad- 1 for a pupil of radius R, the frequencies are SRIA, etc. For a uniformly illuminated pupil, and for a sufficiently dense sampling of phase difference kAW"" over the interferogram so that the integrals can be replaced by summations, (1) becomes

100% blur RMSradial Diffraction

diameter dispersion limited An

(lID radians) (lID radians) value (waves)

16A4 41"2A4 0.12

RAs 21"2As 0.17

RA6 21"2A6 0.17

1RA7,12A7 31"2A7,1"6A7 0.20

12Ag,IRAg l"6Ag,31"2Ag 0.20

12AjI 213Ag 0.20

12AiO 213A IO 0.20

4RAu 41"6AIJ 0.16

40A12 21i.0A12 0.23

40A 13 2v'iOA13 0.23

16A14 4A14 0.23

16A1S 4A1S 0.23

41A I6,32A16 413A16,4A 16 0.25

32A17,41AI7 4Am 4,13A17 0.25

S2A1g 21i.3A 1g 0.25

52A19 2v'i3A19 0.25

20A20 2.rsA20 0.25

20A21 2.rsA21 0.25

N

7:(fx)=(lIN) I [cos (kAW",,)-i sin (kAW",,)]. (2) n=l

7: (t) is obtained similarly from the y-shear interferogram. When the radial portion in the Zemike polynomial

description of the pupil phase function tJ is even, so that the difference function AtJ is odd, the sine term in (2) has a zero sum and the OTF is then entirely real (Hopkins 1962; see also O'Neill 1963) and identical to the modulation transfer function (MTF), i.e., in such cases one-dimensional MTFs may be obtained simply from the N-sample mean of the cosines of the measured phase differences. This includes defocus, astigmatism and spherical aberration. It may apply also when the phase has a distribution of random amplitudes with equal + / - probability about the mean, for example with random polishing errors produced during manufacture of a mirror or with atmospheric seeing phase fluctuations that are sufficiently well averaged spatially or temporally. In other cases the imaginary part of (2) has to be included in the determination of the transfer function.

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Table 2. Aberration contributions forming linear fringes in sheared interferograms; i andj are unit vectors along coordinate axes x and y, respectively.

Wavefront tilt components

Aberration term x- shear y- shear

- 4A4Tj

2AsTj

- 2A6Ti

Defocus: Ai~ + 2y - 1)

O· astigmatism: At..r -y2)

45· astigmatism: 2A~

Interferometer wavefront tilt:

- 4Afli

-2A~i

- 2AeSj

tj -ti

Sum: - (4A4+2AS)Si - (2AeS- t)j

When the UKIRT 3.8-m primary mirror was being manufactured by NEI Parsons Ltd. (formerly Sir Howard Grubb Parsons Ltd), Newcastle-upon-Tyne, under the direction of the late Dr D. S. Brown, the final optical performance was specified contractually as an MTF requirement. Whereas the early concept for UKIRT was that of a low-cost infrared flux collector with only modest imaging resolution, a second contractual stage of surface polishing was undertaken to give a much improved performance. Thus, in 1977 March the 80 and 90 per cent encircled energy diameters achieved were 1.6 and 2.4 arcsec, respectively (MTF=0.21 at 0.44 cycles arcsec 1, A=578 nm), whereas the original requirement was for 90 and 98 per cent energy diameters of these dimensions; instead of proceeding towards that goal, however, it was agreed that the effort should be concentrated first on reducing astigmatic errors, and that a more ambitious stage of final polishing should then be attempted with the final imaging performance specified as an MTF. This MTF requirement was set at 0.80 (at 0.44 cycle arcsec-l, A = 578 nm, corresponding to a 5 cm shear at the primary mirror), and the value achieved in 1977 October was 0.92 (equivalent to a 98 per cent encircled energy diameter of 0.6 arcsec). The MTFs were measured by the contractor using shearing interferometry and calculated as described above. For contractual purposes, the data were first corrected for defocus and coma introduced by the measurement system, and for astigmatism contributed by the mirror support system, and the final MTF values used were the means of the values derived separately from the x and y interferograms. The MTF values cited above include these corrections. One of several advantages of using MTF-based measurements in this case was that the performance, including the effect of aperture diffraction, could be readily converted from that at the measurement wavelength to that at the telescope operational wavelength (A=2.2 ~m).

5 LINEAR FRINGE ANALYSIS

The second-degree polynomials for defocus and third-order astigmatism each give rise to linear fringes when laterally sheared and, when present in the wavefront being measured, produce a change in the observed numbers and/or directions of the tilt fringes in the x- and y-shear interferograms compared with the unaberrated case. Although for many applications the restriction of the analysis to linear sheared fringes only would be of little interest, there are some practical applications where it can be very useful. One

© 1997 RAS, MNRAS 284, 655-668

such case is the optical surface testing of an astronomical mirror during the stages of polishing leading towards the final figure, where it is important to check at each stage that astigmatism is not being introduced by imperfect support conditions or tool settings. Similarly, the parallelism of the instrumental tilt fringes to the x- or y-axis of the interferometer is sensitive to the amount of defocus in setting the axial position ofthe interferometer, and this can be used for accurate determination of the focal length of the mirror being tested.

For the interferometer considered here, the instrument tilt fringes are initially parallel to the direction of shear, the slope vector being orthogonal to this. Assume that the sign of this slope vector is positive along + y for shear along + x, and therefore negative along + x when rotated counterclockwise to produce shear in the + y direction. The unsheared instrument tilt components for x shear and for y shear are then + tj and - Ii, where i andj are unit vectors alongx andy, respectively, and t is the value of the selected instrumental wavefront tilt (obtained by plate rotation about axis X-X' in Fig. 1) expressed in wavelengths per pupil radius.

Table 2 shows the tilt components in the x- and y-shear interferograms for defocus and astigmatism, as well as the instrumental wavefront tilt. Taking defocus as an example, the OPDs resulting from x shear are A W = -(0 W/ax) S = - 4A4Sx, so that the x slope of the defocus fringes is - 4A4Si. The final row in Table 2 gives the sum of the slope components.

The resultant slopes of the fringes inx- andy-shear interferograms of a wavefront containing defocus and astigmatism are obtained from the final row of Table 2 as

w;=(4A4 + 2AS)2S2 + (2A6S _t)2,

W~=(2A6T +t)2 + (4A4 - 2AS)2T2,

(3a)

(3b)

where Wx and Wy are expressed in wavelengths per unit radius for x and y shear, respectively. The position angles of the x- and y-shear tilt vectors, measured counterclockwise from the x-axis, are

ex=tan- 1 [(2A6S - t)/(4A4 + 2As)S],

ey=tan-1 [(4A4 - 2As) T/(2A6T + t)].

Putting S = T, (3a), (3b), (4a) and (4b) yield

A4.= - (wx cos ex + Wy sin ey)/8S,

As = - (wx cos ex - Wy sin ey)/4S,

(4a)

(4b)

(Sa)

(5b)


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660 C. M. Humphries, 1. W. Harris and E. Atad

A6= - (cox cos ex + COy sin ey)/4S,

t = (cox cos ex - COy sin ey)/2.

(5c)

(5d)

A 1t ambiguity in each of ex and ey results from not knowing the sign of the fringe slope components without further information. This gives four possible solutions to (5a)-(5d), the correct one of which is determined if the signs of the tilt and defocus are known (in the latter case, for example, by translating the interferometer axially and observing the direction of fringe rotation). In general, if either A4 or As is non-zero, then the fringes both in x-shear and in y-shear interferograms are rotated from their initial (zero-shear) positions. For the special case 2A4 = ±As, the fringes rotate in only one of the orthogonal shearing interferograms but not in the other. From (3a) and (3b), if at least one ofA4,As or A6 is non-zero, then the fringe spacings change from the initial setting determined by the chosen value of t.

6 WAVEFRONT RECONSTRUCTION

A general method for two-dimensional wavefront reconstruction from laterally sheared interferograms was first given by Rimmer & Wyant (1975), and is described with further detail in the chapter by Murty (1978) in Optical Shop Testing (ed. D. Malacara), First Edition, and also by Mantravadi (1992) in the Second Edition. Essentially, the method involves first transforming Zemike polynomials fitted to the observed interferograms from polar coordinates (with coefficients A ') to Cartesian coordinates (polynomial terms with coefficients B') through the matrix operation B' = HA', then equating the elements of B' with analytic expressions for the individual Cartesian term contributions incorporating the appropriate powers of Sand T [since the polynomials describing L1 W (x, y) contain powers of (x - S) and (y - T)]; this gives the Cartesian polynomial term coefficients of the original wavefront which are finally converted back to Zernike coefficients using the inverse matrix H- 1• Note that in Malacara's Optical Shop Testing the ordering of the Zernike polynomials is not the same as that used in the present paper, and that several errors in the tabulated values for the elements of the matrices Hand H- 1

in the First Edition (tables A2.4 and A2.5) have since been corrected in the Second Edition (Chapter 13 by D. Malacara & S. L. DeVore, tables 13.4 and 13.5). Thus the reduction procedure consists of analysing the fringes in the sheared interferograms by fitting Zernike polynomials to the wavefront difference contours A W (x, y), and then transforming the coefficients of these polynomials to those of the original wavefront W(x,y).

A more direct method of performing the wavefront reconstruction can be derived without the need each time for transforming between polar coordinates and Cartesian polynomial term coefficients using the Hand H -I matrices, as is now shown. Just as the derivatives of Zernike polynomials may be expressed as linear combinations of Zemike polynomials (Noll 1976), so the Zemike polynomial coefficients (A':') that fit the x- and y-shear interferograms are given by

N

A':"= I CmnAn, (6) n=2

N

A':'y = I 3i'mnA n, (7) n=2

where the matrix elements C mn and 3i'mn (Tables 3 and 4) are functions only of the shear values Sand T, respectively, An is the coefficient of the nth aberration polynomial in the wavefront being measured, and N is the maximum n for which the wavefront reconstruction is required. Tables 3 and 4 show explicitly the powers of Sand T that operate between the coefficients of the unsheared W (x, y) and the sheared L1 W (x, y) Zernike polynomials. Provided that N includes all polynomials of a chosen maximum radial degree together with those of lower degree, and if M is the maximum value of m for which the A':" or A':'y rows have non-zero contributions in Tables 3 and 4, (6) and (7) give 2M equations that can be solved to obtain N - 1 Zemike coefficients An (excluding piston) of the original wavefront. In practice, the piston terms are often not measured in shearing interferograms, in which case the tilt terms in the original wavefront cannot be recovered and there are then 2M - 2 equations for N - 3 Zernike coefficients.

Some properties of the matrices C and 3i' may be noted at this stage. For example, each column in Table 3 gives the values of the polynomial coefficients for the x-shear interferogram resulting from an original wavefront having a Zernike coefficient An of unity - similarly for the Tdependent terms in each column of Table 4 for the y-shear interferogram. Thus, for a wavefront with spherical aberrationA 11 = 1, the Zernike coefficients for the x-sheared interferogram areA{.=6S4 + 6S2, A;' = -24S3_4S,A;x=A;x= 12S2, andA;x= - 8S; for the y-sheared interferogram they are A:y=6T4+6T2, A;y= -24T3-4T, A;y=12T2, A;y = -12 T2, and A;y = - 8 T. Other coefficients for the shearing interferograms in this example are zero. For other input values of An, the entries in Tables 3 and 4 scale accordingly.

The elements of C and 3i' are derived by taking the Cartesian form of the Zemike polynomial for each aberration in turn and expressing AU'" and L1Uj as Taylor-series expansions, giving

L1Wx=W(x-S,y) - W(x,y)= - (oW(x,y)/ox)S

02W(X, y)/ox2 2 03W(x, y)/OX3 3 (8) + S- S+ ... , 2! 3!

L1Uj=W(x,y -T) - W(x,y) = - (oW(x,y)/ox)T

02W(X,y)/oy2 2 02W(X,y)/oy3 3 (9) + T- T+ ... ,

2! 3!

By inserting the Zernike polynomial differentials into (8) and (9) and rearranging the resulting expressions as linear combinations of lower order Zemike polynomials, the mUltiplying factors thus derived give the elements C mn and 3i'mn for individual columns of Tables 3 and 4. C and 3i' can also be produced by direct substitution of each Zernike polynomial into L1U',,=W(x-S, y)- W(x, y) and L1Uj=W(x,y-T)- W(x,y), followed again by rearrangement into linear combinations of lower order polynomials. In either case, C and 3i' are obtained without assuming the small-shear approximation. For S or T« 1, however, the

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. M

atri

x el

emen

ts :

F mn

' sho

win

g re

lati

onsh

ip b

etw

een

shea

red

(A~y

) an

d un

shea

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Zer

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pol

ynom

ial

coef

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n =

A2

to A

21) .

Al

A3

A,

A,

A.

A7

AB

A.

AIO

A

ll

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A1

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u

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A16

A'IY

I

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21

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1"-2

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1997MNRAS.284..655H

linear terms in Tables 3 and 4 predominate, and when these are divided by S or T the factors that are obtained are those in tables II and III of Noll (1976) for the matrix elements that express the first derivatives of Zeroike polynomials as linear combinations of Zeroike polynomials (taking into account the different polynomial ordering and normalization used by Noll, as well as a change in algebraic sign due to the definitions of AJ¥" and MVy).

Referring back to (6) and (7), the 2M equations obtained are greater in number than the required N - 1 Zernike coefficients. Methods for the solution of such a set of overdetermined linear equations in adaptive optical systems have been discussed by Tyson (1991). As presented in Tables 3 and 4, matrixes tf and !F are each singular and together do not possess an inverse that can be used to solve for An without first reducing the number of equations used. When the measured parameters are selected to match the N - 1 coefficients, however, the column vector An is then given by

[An] = [tf, !F]-I [A;, Ay1, (10)

where [tf, !Frl is the inverse of a square matrix formed initially by stacking !F below tf and then selecting to remove row redundancies, and [A;,Ay1 is a correspondingly ordered vector of the Zeroike coefficients that fit the shearing interferograms.

A particular solution for the wavefront reconstruction matrix [tf, !F]-I is shown in Table 5, where the ordering has been chosen so that (i) the upper left 9 x 9 submatrix provides the reconstruction for Zernike polynomials up to third radial degree (A2 to A IO), (ii) the upper left 14 x 14 submatrix gives the reconstruction for polynomials up to fourth radial degree (A2 toA1s), and (iii) the entire matrix is used for reconstructing polynomials up to fifth radial degree (A2 to A 21). Although the redundancies in A':" and A':'y allow other particular forms for [tf, !F]-I, i.e. by selecting other combinations of equations to be represented by [tf, !F], the results are numerically identical in the absence of experimental errors. If it is required to spread the effect of errors over all of the measured parameters, then combinations of these solutions may be used. Using Table 5 in equation (10) to perform the wavefront reconstruction is a single-step process that is easier than continually switching between polar and Cartesian polynomial terms through the Hand H -I matrices. Also, Tables 3 and 4 are useful in showing clearly the powers of Sand T that relate the individual Zernike coefficients of the sheared and unsheared wavefronts. Table 5 does not possess a column headedA;y, and so does not include the second-degree polynomial solution of Section 5 as an independent upper left-hand submatrix. Such a solution does exist (e.g., by including A;y instead of A~x), but the individual terms in some of the rows become more complicated, and some of the simplicity is lost.

The considerations so far have involved derivations based on theory for AJ¥" and AWy , with the resulting polynomial expressions formulated always in terms of a circle of unit radius in the coordinate system with origin at the centre of the original unsheared pupil. Measurement of the shearing interferograms in the pupil overlap region, and the fitting of polynomials to these, proceeds on a different basis, however. For small shear, the pupil overlap region is close in position and size to that of the original pupil, but for larger

© 1997 RAS, MNRAS 284, 655-668


shear values (S > 0.1) corrections are required for the changed geometry in which the fringes are produced and measured. This is the 'expand and shift' method described by Atad et al. (1990), in which the fitted polynomial coefficients are measured first within the circle centred on the overlap area of each interferogram, of radius r such that the circle just touches the overlap cusps. These coefficients are then modified by referring the polynomials back to the unit circle represented by the original pupil of radius R. Thus the required corrections are (a) a geometric expansion in which the derived coefficients are each multiplied by a factor that includes the magnification m =Rlr, of powers thereof, and (b) a translation from the centre of the overlap region by an amountx= -S12 (ory= - TI2). For the present considerations with shear values S and T < 0.1, these corrections are negligible.

Analysis of the observed fringe systems to obtain the coefficients of the shared Zeroike polynomials is usually performed by commercial software packages. For pupils with a central obscuration this does not always proceed automatically, since fringe-following algorithms usually become confused in the region of the central obscuration and some degree of manual interaction is then required. An alternative method that avoids this is to Fourier-transform the sheared interferograms, as described by Roddier & Roddier (1987). Yet another method for small shears is to fit directly the observed wavefront differences A W by the first derivatives of the Zernike polynomials.

7 ERRORS AND MEASUREMENT ACCURACY REQUIREMENTS

7.1 Error sources

Each row of Tables 3 and 4 contains at least one term of first degree in S or T contributing to A':" and A':'y, and for S = T < 1 it is these that dominate in equations (6) and (7) compared with terms of higher degree. Solutions satisfying (10) contain reciprocals of S, from which it follows that A n (1±BA)OCS- I(l±Bs ), where BA and Bs are fractional errors. Thus, to first-order approximation, fractional errors in the reconstructed polynomial coefficients become inversely proportional to the value of shear used, and directly proportional to the fractional errors Bs and BT in the assumed values of shear (as previously noted by Korwan 1983). Fig. 4 shows a plot of rms wavefront error for individual third-order aberrations against percentage error in the shear values S or T.

A particular source of error that can occur in the fringe analysis is when there is a systematic x/y distortion in the measurement of the fringes over a nominal unit circle. This may arise from the camera imaging system, or from software routines used in capturing and displaying the frames, and it results in an astigmatic contribution that may be falsely attributed to the wavefront being measured. It can be removed by calibration using a reference wavefront, as described below. The fiducial box in which the unit circle is fitted should be of identical pixel dimensions both for the xand y-shear interferograms of the wavefront being measured, and for the x- and y-shear interferograms of the reference wavefront. In this way any false astigmatic content of the former is compensated by subtraction of an identical amount in the latter.


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1997MNRAS.284..655H

7.2 Removal of systematic errors: subtraction of reference wavefront

Since the interferometer plates are each at 4So with respect to the optical axis, and the transmitted portions of the beams are either convergent or divergent, astigmatism is introduced within the instrument itself. This affects slightly both the observed fringe tilt and the position angle of the fringes in the sheared interferograms. However, the instrumental astigmatism is removed in the reduction process, since the combined instrumental tilt and astigmatism fringes are merely rotated from the x- to the y-shear interferogram, whereas astigmatism in the beam being measured produces a difference in the fringe tilts and position angles (apart from the 90° rotation) for the x- and y-shear interferograms. Astigmatism still exists in the light diverging from the interferometer but, by using pupil imaging optics between the interferometer and the detector, elongation in the final image presented to the CCD can be minimized.

Remaining systematic errors, produced either in the interferometer itself or introduced by the fringe measurement and reduction process, are removed by subtraction of a sheared reference wavefront. This is achieved by presenting a spherical wavefront to the interferometer from a point source - in our case this consists of a green LED source and a SO-Ilm pinhole, both mounted in a IS-mm diameter copper tube that fits the entrance aperture of the interferometer housing when required. The normal focus of the beam entering the interferometer is at the centre of the prism but, since the calibration source cannot reach this position, sheared defocus fringes are produced. However, these do not impair the results in any way and, for the aberrations of interest, the Zemike terms from the wavefront reconstruction using the reference point source are subtracted from those of the beam being measured.

In practice, the known algebraic sign and magnitude of the defocus using the point source are useful in checking the reconstructed wavefront, particularly in setting up a new experimental system when unintentional sign inversions may occur. Thus the reconstructed reference wavefront may be checked to ensure that it contains the known defocus in Zemike coefficient A 4 • Similarly, the sign of the selected interferometer wavefront tilt is known, and this also can be

0.08,---.--------r-----,-------,

0.06 RMSerror

(wavelengths)

0.04

0L----SL-------~10--------1~S------~20-

Percentage error in S (or 1')

Figure 4. Rms wavefront error versus percentage error in shear value S (input Zemike coefficients of 1 wave assumed).

© 1997 RAS, MNRAS 284, 655-668


used for checking the reconstructions for the wavefront being measured, as well as that from the point source.

Apart from the defocus and astigmatism already mentioned, and a O.I-wave trefoil component, all other aberrations in the reconstructed reference wavefront were small and, once the subtraction described above had been applied, Zemike coefficient residuals were found to be typically less than 0.01 wave (A=S46 nm) due to random experimental errors.

7.3 Measurement accuracy requirements

Table 1 (final column) gave diffraction-limited values for the Zemike polynomial coefficients. These were obtained by equating the rms wavefront error for each aberration with the diffraction-limited A/14 requirement {rms= L [eml2 (n + 1) f12 Am, where n is the radial power of the polynomial considered, and em = 2 for the rotationally symmetric aberrations; otherwise em = 1 (Rimmer & Wyant 1975)}. Thus individual aberrations need measurement accuracies better than 0.1-0.2A if they are close to diffraction-limited, and proportionately less when several aberrations are present.

8 ASTIGMATIC CONTRIBUTIONS FROM SEPARATE OPTICAL ELEMENTS

A common situation in measuring the wavefront from a two-mirror telescope is where there are separate astigmatic contributions from the primary and the secondary mirrors. First, there may be practical reasons that make it necessary for the shearing interferometric data to be obtained with the measurement coordinate system x', y' rotated with respect to the required reference system x, y. If the angular rotation from x, y to x' , y' is ¢ counterclockwise, then a wavefront with third-order astigmatism is measured as

W=A;p2 cos 28' +A:p2 sin 28'

=A;p2 cos 2(8 - ¢) +A:p2 sin 2(8 - ¢), (11)

where A; and A: are the 0°-90° and 4SO -13So astigmatism components, respectively, and 8 and p are azimuthal and radial coordinates. Trigonometric rearrangement of (11) gives the required coefficients in the reference coordinate system as

A5=A;cos2¢-A~sin2¢, (12a)

A6 =A; sin 2¢ + A~ cos 2¢, (12b)

from which the magnitudeA5,6 and position angle 85,6 of the resultant, also measured counterclockwise from the x-axis, are

A 5,6 = (A; +A~)I12,

85,6=0.5 tan- 1 (A6IA 5)'

(13a)

(13b)

An astigmatic wavefront reflected from the primary may be represented by

(14)

where 0 ~ IXp < 1t is the position angle ofthe maximum ( +z) wavefront displacement measured counterclockwise from the x-axis, and Ap is the overall Zernike astigmatic vector


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1997MNRAS.284..655H

666 C. M. Humphries, J W. Harris and E. Atad

from the primary, given by

Ap=(A;p +A~p)I/2,

CXp = 0.5 tan- l (A6piASP)'

(15a)

(15b)

Similar expressions apply for the astigmatic contribution (A" cx,) to the wavefront from the secondary mirror. Combining these, we obtain for the overall astigmatism measured at a Cassegrain focus:

App2 cos 2(8 - cxp) + A,p2 cos 2(8 - cx,) =AI p2 cos 2(8 - cx l ),

(16)

whereA l and CX l denote the magnitude and position angle of the resultant. If the secondary mirror is now rotated to a new orientation, and the 0° -90° and 45°-135° astigmatism is remeasured, then the separate contributions from the primary and secondary can be derived. For example, if the

(a)

secondary is rotated counterclockwise by 90° viewed from the focal surface towards the pupil, then

App2 cos 2(8 - cxp) +Asp2 cos 2(8 - CXs -nI2)

=A2P2 cos 2(8 - cx2),

and (16) and (17) are together satisfied if

2Ap cos 2cxp =Al cos 2cxI + A2 cos 2cx2 ,

2Ap sin 2cxp =Al sin 2cxI + A2 sin 2cx2.

These give for the primary mirror component:

Ap =0.5 [A~ +A~ + 2AIA2 cos 2(cx I - cx2)],

(b)

(17)

(18a)

(18b)

(19a)

(19b)

Figure 5. (a) and (b): 0.5-s integration interferograms showing complex fringe structure due to the atmosphere.

(a) (b)

Figure 6. (a) y-shear interferogram obtained at Cassegrain focus, showing a band of low fringe visibility due to thermal effects (shear va!ue=0.0438), (b) interferogram showing improved visibility for reduced shear (S=0.015); 20-s integration in each case.

© 1997 RAS, MNRAS 284, 655-668


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1997MNRAS.284..655H

Similar expressions are obtained for As and IX" but with A2 replaced by -A2•

9 APPLICATIONS: MEASUREMENT OF WAVEFRONT ERRORS AT THE UK INFRARED TELESCOPE

The optical performance of UKIRT has been measured on two occasions using the lateral shearing interferometer operated in a passband centred at ,1,=546 nm with a bandwidth of 9.2 nm. Interferograms of the wavefront at the f/36 Cassegrain focus were obtained over three nights in 1991 (October 1-3) and two nights in 1993 (June 13-14). The 1991 observations also included measurements at prime focus using a flat mirror of interferometric quality to replace the usual convex secondary mirror, thus folding the beam and making the prime focus accessible; Fig. 5 shows two short-exposure interferograms in this configuration.

The first interferograms at Cassegrain focus, with lateral shear S = 0.04, were found to have fringes of low contrast that varied in shape from one exposure of 10 or 20 s to another. Particular regions within the pupil had very low contrast, and these moved around the pupil as the telescope pointing direction was changed and as air within the telescope path was shifted by wind gusts. These effects evidently were caused by thermal convection of plumes of warm air rising through the central hole of the primary mirror and spreading across the pupil (Fig. 6). A comparison of the fringe visibilities in the interferograms obtained at shear values S = 0.04 and 0.015 indicated that such local thermal effects were limiting the optical performance by giving an effective coherence length '0 in the range 4-8 cm, far smaller than the value of '0=20 cm estimated from images obtained at the CFH telescope at the same time. The effect on image size was that 0.5-arcsec FWHM images at 546 nm, limited by external seeing, were being degraded to 1.25-arcsec FWHM, limited by effects within the dome. Other conclusions obtained from the observations were that (a) the spherical aberration of the telescope was found to be small, giving A 11 values less than + 0.24:,1, for all pointing directions investigated, and (b) the dominant aberration was astigmatism in the primary mirror, with peak-to-valley deformations of up to 6,1" depending on zenith angle.

For the 1993 observations, the telescope aberrations were measured with the same interferometric system and, simultaneously, with a wavefront curvature sensor provided by the resident UKIRT team; this was located also at the Cassegrain focus, but received the light reflector from the dichroic beamsplitter instead of the transmitted portion used by the interferometer. Measurements were made to include the following:

(i) telescope pointing to zenith in its normal working configuration;

(ii) rotation of the secondary mirror by 90°; (iii) variations in the axial support forces on the primary

mirror, and (iv) tilt of the secondary mirror.

Integration times in the range 5 to 30 s for the interferometer and 20 to 40 s for the curvature sensor were used with stellar sources brighter than mv=5. For each set of

© 1997 RAS, MNRAS 284, 655-668


measurements a pair of x and y interferograms was used for the wavefront reconstruction with the interferometer, and a pair of in-focus and out-of-focus images were used for the curvature sensor.

Fig. 7 shows the measured interferometric third-order Zernike coefficients As, A 6 , A 7 , As, An for (a) the nominal telescope configuration (measurements 1, 3 and 7), (b) the secondary mirror rotated by 90° (measurement 2), (c) the secondary mirror tilted by 1 arcmin (measurement 8), and (d) removal of pressure on individual axial support pads of the primary (measurements 4, 5 and 6). Spherical aberration at the Cassegrain focus, indicating the amount of mismatch between the conic constants of the primary and secondary mirrors as well as the effects of the respective support systems, was again found to be small. As in 1991, astigmatism was the dominant aberration. Relatively small comatic contributions showed that alignment of the primary and secondary mirrors was satisfactory. In order to calibrate the astigmatism as a function of the applied forces at individual axial support pads of the primary mirror, pneumatic pressure was removed first from a pad located in the outer ring of the three-ring pneumatic support system, and then

.5r---~---r--~----r---~---r---'

Z~e ~ coeffi. / " (waves atl= / '. 546nm) All ~ ...... <>0. ••••• '"

o ~'-::'-:;r.:,""'''''... .' A8·· ... ······· .. "'~~ ;.; ..... ; /.,- ... - - 7 .... A1 0 ____ 0-.- __ 11'" .~ . ~---- ,

v/ '\,i'l6 " ,.' V', ~- .. _.

. '" As', /' '. ",.,

", / ", /., ..........

-5

-10~1--~2----3~---4~--~5----~6--~7~--~8 Sequence of measurements

Figure 7. Third-order Zemike coefficients As, A 6 , A 7 , As, A11 for the sequence of measeurements (see text) at the //36 Cassegrain focus of UKIRT in 1993.

0.5 arcsec

Figure 8. Contour plot of the point-spread function at A=21lm for the image corresponding to measurement #2 in Fig. 7.


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1997MNRAS.284..655H


successively from one in the middle ring and one in the inner ring.

For the best telescope configuration at the time of measurement in 1993 (measurement 2 in Fig. 7), the 50 per cent encircled energy diameter contributed by the telescope aberrations was 0.45 arcsec compared with the diffraction limited value at A = 2 Jlm of 0.11 arcsec. The contour plot of the point-spread function is shown in Fig. 8, and was computed using all of the measured Zemike coefficients up to fifth degree. The three-lobed image structure here was due to a particular combination of third-order astigmatism and coma, and not the presence of trefoil aberrations. The optical performance of the telescope in the period described here is currently being improved through the UKIRT Upgrades project; this includes the provision of an adaptive tip/tilt secondary mirror with x/y translation capability, active control of the primary mirror, and a substantial redesign of the thermal environment of the telescope and dome (Hawarden et al. 1994).

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