1. INTRODUCTIONRecent developments in wave-front sensors for adaptiveoptics have mainly concentrated on applications in as-tronomy for which the most significant source of aberra-tion is atmospheric turbulence.1,2 The complex temporaland spatial nature of the optical effects of such turbulencecoupled with the unavoidably low photon flux with whichastronomers must work has led to the predominant use ofShack–Hartmann-type wave-front sensors or, to a lesserextent, zonal curvature sensing and shearing interferom-etry. Using real-time processing, one then extracts fromthese data the shape of the aberrated wave front or themagnitude of its modal aberration components, informa-tion that is used to drive the adaptive element.
It is generally considered that zonal wave-front sensingis preferable to modal sensing.1,2 Tyson points out that,theoretically, the required number of zones is roughlyequal to the number of modes that must be measured torepresent a wave front to the same degree of accuracy; inpractice, zonal methods are used, owing to the capabili-ties of available hardware. Since the correction of com-plex aberrations (e.g., due to atmospheric turbulence)would require the measurement of a large number of ab-erration modes, the higher modes would become difficultto measure directly and separation of the effects of, for ex-ample, astigmatism and coma would be problematic. Al-though for atmospheric turbulence it is true that a largenumber of aberration modes are required to represent thewave-front distortion accurately, in other applicationsthis may not be the case. In confocal microscopy, for ex-ample, it has been shown that only the lowest-orderZernike aberrations significantly affect the point-spreadfunction during focusing through a refractive-indexmismatch.3 This is due to a rapid decrease in the mag-nitude of the Zernike mode with increasing aberrationorder.3,4 In such a situation, modal sensing becomes a vi-able option, especially if used in conjunction with a modalmethod of correction (e.g., bimorph or membrane mir-rors).
In this paper we describe theoretically a novel wave-front sensor that measures directly the size of any chosenZernike mode present in a wave front. We show that theresponse of the sensor is linear in mode amplitude, a, overa region around a 5 0 and is thus ideally suited for in-corporation into a closed-loop adaptive system where thewave-front correction occurs within the loop. We alsodiscuss how the sensor can be implemented with a binaryoptical element and show how several of these sensorscan be combined in a multiplexed version that allows si-multaneous measurement of multiple different aberrationmodes.
2. CONCEPTUAL OPERATION OF THEWAVE-FRONT SENSORWe propose the wave-front sensor shown conceptually inFig. 1. We wish to measure the amount of a chosen ab-erration mode that is present in the unknown input wavefront. We can describe the input wave front as a uniformintensity but with a phase function, C(r), representingthe deviation of the phase of the aberrated wave frontfrom that of an unaberrated plane wave in the apertureA. This input beam is split into two identical beams thatthen pass through biasing phase plates placed in the backfocal plane of a lens. These transmissive phase platesare designed such that the first adds a bias aberration,F(r), to its input wave front and the second subtracts thesame aberration from its input wave front. The beamsare then focused by the lenses onto detector pinholesplaced in the Fourier plane such that the intensity in thedetector plane is given by the modulus squared of theFourier transform (FT) of the total optical field just afterthe bias plates. In other words, the intensity presentedto the first detector may be written as
I1~v! 5 uF $exp@ jC~r! 1 jF~r!#%u2 (1)
and to the second detector as
I2~v! 5 uF $exp@ jC~r! 2 jF~r!#%u2, (2)
2000 Optical Society of America
Neil et al. Vol. 17, No. 6 /June 2000/J. Opt. Soc. Am. A 1099
Fig. 1. Schematic description of the aberration sensor that uses biasing elements and Fourier transform lenses.
where F denotes the FT and v is the coordinate vector de-scribing the detector plane.
This sensor is a generalization of the curvature (defo-cus) sensor in which one detector pinhole is placed infront of the nominal focal plane of a lens and one behindit.1,5 For the special case in which the bias aberration ischosen such that F(r) } r2, the sensor of Fig. 1 is opti-cally identical to the curvature sensor.
The performance of the sensor depends on the apertureshape, the chosen bias aberration, and the size and shapeof the pinhole, all of which we may choose. In general,our aim is to obtain the magnitude of the various modesin an orthogonal modal expansion of the input phase ab-erration C(r). We now proceed to show that if we chooseour bias aberration to be exactly one of those orthogonalmodes, then, to first order, the sensor will be sensitive tothat mode in the input wave front while rejecting all oth-ers. Let us consider the case in which the detection pin-holes are infinitely small and positioned on the opticalaxis. We take our sensor output signal to be the differ-ence between the intensities at the two detectors. LetF(r) 5 afi(r) and C(r) 5 bfk(r), where fi and fk are or-thogonal functions representing the modal expansion ofthe phase aberration and a and b are representing themagnitude of the input aberration and the applied bias.The intensities at the two detectors are then given by
I1,2~0! 5 U EEA
exp~ jafk 6 jbfi!dAU2
. (3)
The sensor output is
DI 5 I1~0! 2 I2~0!, (4)
and we define the sensitivity, S, of the sensor as the gra-dient of DI with respect to a, the size of the input aberra-tion, at a 5 0:
S 5]DI
]aU
a50
. (5)
The first derivative of the intensities with respect to acan be shown to be
]I1,2
]a5 2 ImF EE
Aexp~ jafk 6 jbfi!dAEE
Afk
3 exp~ jafk 7 jbfi!dAG , (6)
and it follows that the sensitivity is given by
S 5 4 ImF EEA
exp~ jbfi!dAEEAfk exp~2jbfi!dAG . (7)
The two exponential terms can be expanded as a Maclau-rin series to separate the imaginary parts, giving
S 5 4S bEEAfidAEE
AfkdA 2 bEE
AfifkdA
3 EEAdA 1 ¯ D . (8)
So, if the bias b is sufficiently small that only first-orderterms are significant and the aberration modes fi and fkhave zero mean across the aperture, only the second termremains on the right-hand side of Eq. (8). It follows thatif fi and fk are members of a set of functions orthogonalover the aperture, A, then the sensitivity will be given by
S ' 24bAEEAfifkdA 5 24bACd ik , (9)
where d ik is a Kronecker delta and C is the constant oforthogonality. S will be nonzero only if fi and fk are thesame function. The sensor would therefore respond onlyto an input aberration mode that is identical to its biasmode. This result is valid for any aperture shape andany set of orthogonal functions with zero mean definedover that aperture.
For the more general case we would like to use detectorpinholes of finite size and a bias, b, of finite value in orderto maximize the sensitivity. Under these circumstancesthere will in general be some cross sensitivity betweenmodes. One particularly useful modal expansion is thatwhich uses Zernike polynomials to describe the phase ab-erration over a circular aperture. We now proceed toanalyze this system more fully, using Zernike modes asour orthogonal basis and finite-sized circularly symmetricdetector pinholes. We show that the sensor still pre-dominantly responds to its design mode but that there ex-ists some cross-sensitivity between certain modes.
1100 J. Opt. Soc. Am. A/Vol. 17, No. 6 /June 2000 Neil et al.
3. THEORETICAL TREATMENT OF AWAVE-FRONT SENSOR BASED ON ZERNIKEPOLYNOMIALSFor our analysis of the sensor, we consider circular pin-holes and apertures and use the Zernike circle polynomi-als as our orthonormal basis functions. We choose to usethe definition of Noll6 and the indexing scheme used byMahajan.7 This is described in Appendix A.
If we assume that only phase and not amplitude aber-rations are present, any circular input wave front can berepresented by the exponential of its phase function,C(r,u), which can be written as a series of Zernikepolynomials.6,8 The wave front may therefore be de-scribed by
exp@ jC~r,u!#circ~r ! 5 expF j(k50
`
akZk~r,u!Gcirc~r !,
(10)
where
circ~r ! 5 H 1 for r < 1
0 for r . 1, (11)
and Zk(r,u) is a Zernike polynomial. For simplicity,from now on we shall omit circ(r) in our descriptions.
Let us consider the case in which the input wave frontcontains only one Zernike term, which is denoted by thesubscript i, and the sensor is designed to detect theZernike mode denoted by the subscript k. Let theamount of input aberration be a and the size of the bias beb. Thus the total wave-front distortion in the first path isgiven by aZk(r,u) 1 bZi(r,u), whereas in the secondpath, where the bias is negative, the wave front is givenby aZk(r,u) 2 bZi(r,u). If we now introduce polar coor-dinates (n, j) in the detector plane, we can write the in-tensities incident on the detector plane more generallyfrom Eq. (3) as
I1,2~n,j! 51
8p3 U E0
2pE0
1
exp@ jaZk~r,u! 6 jbZi~r,u!#
3 exp@ jnr cos~u 2 j!#rdrduU2
. (12)
The factor of 1/8p3 ensures that the total summed powerin the two detector planes is equal to unity. For finite-sized circular detector pinholes of radius np , we integratethe intensity in the detector plane and obtain the detecteddifference signal (the signal from detector 1 minus thesignal from detector 2) as
DWik 5 E0
2pE0
np
I1~n, j!ndndj 2 E0
2pE0
np
I2~n, j!ndndj,
(13)
and again as above we define the sensitivity of a mode isensor to an input mode k as
Sik 5]DWik
]aU
a50
5 E0
npE0
2p ]I1
]aU
a50
2]I2
]aU
a50
djndn.
(14)
We can represent the sensitivities by a matrix, S, inwhich each row represents a sensor designed for a par-ticular Zernike mode and each column represents the in-put wave-front mode. If we take a general input wavefront containing several Zernike modes of amplitude akand express it as a vector a, then the matrix vector prod-uct w 5 Sa gives us the signals detected on the output ofthe sensor (to a linear approximation). In general use,however, we detect the signals w and would like to knowwhich Zernike modes are present in the input wave front.This can be done simply by computing the inverse of S,and then the amplitudes of the Zernike modes are givenby a 5 S21w. In the ideal case, S would be diagonalwith all of the off-diagonal elements identically zero. Inpractice, this does not hold, but we will show that the ma-trix S is indeed sparse. Most elements will be shown tobe necessarily zero, and general conditions will be derivedto determine which elements are nonzero. First, we takeEq. (12) and write it as
I1,2~n, j! 51
8p3 uF @exp~ jaZk 6 jbZi!#u2
51
8p3 F @exp~ jaZk 6 jbZi!#
3 F @exp~ jaZk 6 jbZi!#* , (15)
where * denotes the complex conjugate. Differentiationby parts with respect to a gives the derivative at a 5 0 as
]I1,2
]aU
a50
51
4p3 Im$F @exp~6jbZi!#F @Zk exp~7jbZi!#%,
(16)
where we define
Zi~r,u! 5 Zi~r,u 1 p!. (17)
Let us now take the case in which Zi(r,u)5 Ri(r)cos(mi u) and use the following expansion of theexponential in terms of Bessel functions,9
exp~ jz cos f! 5 (p52`
`
jpJp~z !cos~ pf!, (18)
where Jp( • ) is the pth-order Bessel function of the firstkind. The first FT in Eq. (16) becomes
F @exp~6jbZi!# 5 (p52`
`
jpF @Jp~6bRi!cos~ pmiu!#.
(19)
In this summation the functions are of the formf(r)cos(gu). The properties of the FT’s of such functionsare summarized in Appendix B and further allow us towrite,
F @exp~6jbZi!# 5 2p (p52`
`
jp~mi11 !~6!p cos~ pmij!
3 E0
1
Jp~bRi!Jpmi~nr !rdr, (20)
Neil et al. Vol. 17, No. 6 /June 2000/J. Opt. Soc. Am. A 1101
and, similarly choosing Zk(r,u) 5 Rk(r)cos(mk u), we canwrite the second FT in Eq. (16) as
F @Zk exp~7jbZi!#
5 p (q52`
`
jq~7!pH jqmi1mk cos@~qmi1mk!j8#
3 E0
1
RkJq~bRi!Jqmi1mk~nr !rdr
1 jqmi2mk cos@~qmi 2 mk!j8#
3 E0
1
RkJq~bRi!Jqmi2mk~nr !rdrJ , (21)
where j8 5 j 1 p. The differential intensity signal atthe detector plane is therefore given by
]I1
]aU
a50
2]I2
]aU
a50
51p
ImX ((p52` q52`
~ p1q ! odd
` `
~21 !qj ~ p1q !~mi11 !
3 H jmk cos~ pmij!cos@~qmi1mk!j8#
3 E0
1
Jp~bRi!Jpmi~nr !rdr
3 E0
1
RkJq~bRi!Jqmi1mk~nr !rdr
1 j2mk cos~ pmij!cos@~qmi 2 mk!j8#
3 E0
1
Jp~bRi!Jpmi~nr !rdr
3 E0
1
RkJq~bRi!Jqmi 2 mk~nr !rdrJ C, (22)
where the condition that ( p 1 q) must be odd is appliedto the summations. To calculate the detected power sen-sitivity Sik , we must now integrate over the detector pin-holes according to Eq. (14). At this point we can use thestandard orthogonality relationships between sines andcosines for the integral over j. In particular, we now notethat if we had chosen Zk to have a negative azimuthal in-dex, then the terms in Eq. (22) would all be of the formcos(A). sin(B) and when integrated over j would give ex-actly zero sensitivity. This implies that a sensor for amode of positive azimuthal order is insensitive to modesof negative azimuthal order. Likewise, by a simple coor-dinate transform (rotating by p/2 about the origin) we cansimilarly deduce that a sensor for a mode of negative azi-muthal order is insensitive to modes of positive azimuthalorder.
For cases where mi is zero or positive, then the onlyway that terms in the summation integrate to nonzero re-sults are when
pmi 5 6~qmi 1 mk! or pmi 5 6~qmi 2 mk!.(23)
If mi 5 0 then nonzero sensitivity is obtained only whenmk 5 0, thus a sensor for a circularly symmetric mode isonly sensitive to other circularly symmetric modes andlikewise a sensor for a noncircularly symmetric mode isinsensitive to circularly symmetric modes. Applying thiscondition of mi 5 mk 5 0 to Eq. (22) and integratingover j according to Eq. (14) gives
Sikumi5mk50 5 4 ImH ((p52` q52`
~p1q ! odd
` `
~21 !q~ j !~ p1q !
3 E0
npF E0
1
Jp~bRi!J0~nr !rdr
3 E0
1
RkJq~bRi!J0~nr !rdrGndnJ5 4 ImH E
0
npF E0
1
exp~ jbRi!J0~nr !rdr
3 E0
1
Rk exp~2jbRi!J0~nr !rdrGndnJ .
(24)
Turning to now to the case of modes of positive azi-muthal order, we note that the conditions set by Eq. (23)along with the condition that ( p 1 q) must be odd resultsin the fact that mk /mi must itself be an odd integer ifnonzero terms are to exist. Thus a sensor for a mode of agiven azimuthal order is insensitive to other modes whoseazimuthal order is not an odd multiple of that sensor azi-muthal order. Applying this condition to Eq. (22) andagain integrating according to Eq. (14) therefore gives
Sikumk/mi 5 odd 5 2~21 !mk 1 mi
2mi
3 E0
npH (p52`
` E0
1
Jp~bRi!Jpmi~nr !rdr
3 E0
1
Rk@Jp 1
mk
mi
~bRi!
2 Jp2
mk
mi
~bRi!#Jpmi~nr !rdrJ ndn.
(25)
Equation (25) can be further simplified as terms in thesummation for negative values of p are identical to thosefor positive values of p. This fact, along with the fact thesummation converges very quickly for low and moderatevalues of b and np , makes Eq. (25) particularly suitablefor numerical computation of sensitivity matrix coeffi-cients. We should also note that the same result wouldbe obtained here had we chosen Zi(r,u)5 Ri(r)sin(2mi u) and Zk(r,u) 5 Rk(r)sin(2mk u), sincethis simply corresponds to a change of coordinate system.
1102 J. Opt. Soc. Am. A/Vol. 17, No. 6 /June 2000 Neil et al.
These results show that most elements of the sensitiv-ity matrix, S, are identically zero and that the matrix isalways sparse. Moreover, this result is independent ofthe nature of the radial components of Zi and Zk andwould apply to other choices of radial functions. There-fore this result also holds for annular apertures and pin-holes. In this case the Zernike circle polynomials are re-placed by Zernike annular polynomials,10 which areclosely related to the circle polynomials as given in Eq.(A1), but the radial function is replaced by
RnmF S r2 2 e2
1 2 e2 D 1/2G , (26)
where e is the inner radius of the annulus. Since the azi-muthal component retains the same form, we again findthat the elements of S are nonzero only if mk is an oddmultiple of mi .
There are two parameters with which we can tune theperformance of the sensor: the magnitude of the bias, b,and the size of the pinhole radius, np . Ideally, we re-quire a sensor for which Sii is as large as possible whileall Sik are zero or very small when i Þ k. Consider thesensitivity when the pinhole is very small, np → 0. Thiscan be derived from Eq. (24) for the case when mi 5 mk5 0 as
Sikumi5mk50 5 2np2 ImF E
0
1
exp~ jbRi!rdr
3 E0
1
Rk exp~2jbRi!rdrG . (27)
It can also be seen that for sufficiently small bias, b,such that b2 and higher terms are insignificant, the sen-sitivity becomes
Sikumi5mk50 5 2bnp2S E
0
1
Ri rdrE0
1
Rk rdr
2 E0
1
rdrE0
1
RkRi rdr D ; (28)
however, in this case the first two integrals are zero andthe last integral is equal to d ik/2 from the orthogonalityrelationship [Eq. (A4)], so the sensitivity becomes
Sik 5 2b
2np
2d ik . (29)
For the remaining cases, when mi 5 mk Þ 0, we canuse the fact that for a small argument a Bessel function isgiven approximately by the first term in its Maclaurinpower series expansion11 such that as n → 0:
Jpmi~nr ! '
1
~ pmi!!S nr
2 D pmi
. (30)
It follows that the terms in the summation of Eq. (25) arevanishingly small unless p 5 0, leaving
Sikumk/mi 5 odd ' ~21 !mk1mi
2mk np2E
0
1
J0~bRi!rdr
3 E0
1
Rk@Jmk
mi
~bRi!2J 2
mk
mi
~bRi!#rdr.
(31)
Using the relationship J2n(.) 5 (2)nJn(.) [Ref. 11] andEq. (30), we find that for small b then terms in b existonly when mi 5 mk and we obtain
Sikumi/mk 5 1 ' 2bnp2E
0
1
rdrE0
1
RkRirdr 5 2b
2np
2d ik .
(32)
In addition to the above, we find that for the cases ofoff-diagonal sensitivity that are not identically equal tozero (i.e., when i Þ k and mk /mi is odd), then the re-sidual sensitivity from Eqs. (24) and (25) is at mostO(bnp
4) 1 O(b3np2). Thus, to summarize, we find that the
terms in the sensitivity matrix as the bias and detectorpinhole radius tend to zero are given by
Sik 5 52
b
2np
2 i 5 k
<O~b3np2! 1 O~bnp
4! i Þ k,mk
miis odd
0 otherwise
.
(33)
4. SENSITIVITY MATRIXThe above analysis shows that the sensitivity matrix pro-duced for this type of sensor is indeed sparse but thatthere is also a trade-off between maximizing the size ofthe on-diagonal elements while minimizing the size of theoff-diagonal elements. In particular, Eq. (33) shows thatto make the on-diagonal elements as large as possible, thepinhole size and the bias should be increased, whereas tominimize the relative size of the off-diagonal elements,the pinhole size and bias should be kept small. A furthercomplication arises in that beyond the limits of validity ofthe approximations made in Eqs. (29) and (32) the on-diagonal sensitivity itself saturates and reaches a maxi-mum before declining at higher values of bias and pinholesize.
As an example, we present results for a sensor de-signed to operate with near-maximum on-diagonal sensi-tivity alone. We find that a fairly broad maximum sen-sitivity peak is achieved for all modes in the region of b5 0.7 and np 5 p. Table 1 shows the matrix, S, formodes i,k 5 4 to 19, obtained with these values of biasand pinhole size. Each row represents a sensor designedfor mode i; each column represents an input mode k. Wedo not include tip, tilt, and piston, since piston has no ef-fect and we consider it better to use other methods to de-tect tip and tilt. It can be seen that elements of S arezero unless mk is an odd multiple of mi and the matrix is
Neil et al. Vol. 17, No. 6 /June 2000/J. Opt. Soc. Am. A 1103
Fig. 2. Output signal from wave-front sensors for Zernikemodes 4–10. The bias b 5 0.7, and the pinhole radius np 5 p.
dominated by the diagonal terms. However, for each ab-erration sensor there is significant sensitivity to othermodes; for example, the defocus sensor will also detectspherical aberration. We also note that sensitivity be-tween modes of different azimuthal order (mi Þ mk), al-though not actually zero, is significantly less than that be-tween modes of the same azimuthal order (mi 5 mk).The inverse matrix S21 is given in Table 2.
Figure 2 shows the output signal of the sensor calcu-lated with Eq. (13) when the input and the bias aberra-tions are the same, for Zernike modes 4–10. Each modeshows a similar response curve, which is linear in a re-
1104 J. Opt. Soc. Am. A/Vol. 17, No. 6 /June 2000 Neil et al.
gion around a 5 0. The curves that overlap exactly arewhere the aberrations that are being sensed have thesame azimuthal order but opposite sign. Even then thesimilarities between the different curves are notable.
5. IMPLEMENTATION IN BINARY OPTICSAn alternative implementation that is more versatilethan that of Fig. 1 is to use binary phase optics to gener-ate the positive and negative bias by use of a single dif-fractive element. The method, which originated incomputer-generated holography,12 allows the generationof focal spots containing arbitrary amounts of anyaberration.13
Suppose we take a wave front f(x, y) 5 exp$ j@f(x, y)1 t (x, y)#%, including a desired bias aberration, f(x, y)and a linear phase tilt, t (x, y). It can be shown that bi-narizing this wave front by taking the sign of the real partof f(x, y) produces a binary wave front g(x, y) with am-plitude 11 or 21 that can be described by the Fourier se-ries expansion as
g~x, y ! 52
pH exp@ j~ f 1 t!# 1 exp@2j~ f 1 t!#
21
3exp@ j3~ f 1 t!# 2
1
3exp@2j3~ f 1 t!#
1 ¯J . (34)
In the Fourier plane of a lens the individual compo-nents in the expansion are now diffracted to spatiallyseparated positions because each generated componenthas a different overall tilt. This results in diffracted or-ders positioned at relative displacements of $11, 21, 13,23, 15, 25 . . . % from the axis with relative powers of $1,1, 1/9, 1/9, 1/25, 1/25, . . . %. Each order itself carries acorrespondingly scaled analogue phase modulation$11,21,13,23,15,25 . . . % 3 f(x, y). Clearly, the 11and 21 diffraction orders here correspond to our positivebias and negative bias, respectively. In the configurationof Fig. 3, the aberrated input wave is incident on such abinary phase plate that lies in the back focal plane of thelens. The intensity distribution in the Fourier plane ofthe lens then contains all of the above-mentioned diffrac-tion orders separated spatially. The 11 and 21 ordersare identical to the intensity distributions at detectors 1and 2 in Fig. 1 but reduced in magnitude by a factor 8/p2,or ;81%. Suitably placed pinhole detectors obscure the
Fig. 3. Schematic description of the aberration sensor that usesa binary phase optical element to produce the two ‘‘biased’’ spots.
light from the other orders. This method of using a bi-nary phase element is thus not only convenient in that itmultiplexes both bias elements in one but also in that itprovides a highly accurate method of defining the re-quired bias patterns with a digital binary element.
If the binary element were reconfigurable, for example,if it were a spatial light modulator, it would be possible tochange the bias aberration so that the sensor could se-quentially measure each Zernike mode. It would, how-ever, be more desirable to have a single fixed element thatcould allow us to measure the different modes simulta-neously. For a multiplexed filter of this kind the binaryoptical element produces a number of symmetrical pairsof spots in different positions in the focal plane, each paircorresponding to a different Zernike mode. A suitablydesigned array of pinhole detectors can then be used toderive the output signal for each mode.
Such a mask can be designed with standard optimiza-tion techniques (e.g., direct binary search) where the bi-nary phase element starts as a grid of random binary pix-els and we cycle through each pixel in a pseudorandomsequence, choosing the optimum value that maximizes
Fig. 4. (a) Optimized multiplexed wave-front sensing mask(transmission: white 5 1, black 5 21, and gray 5 0), (b)simulated output intensity in focal plane for plane wave-front in-put (logarithmic scale over 20-dB range).
Neil et al. Vol. 17, No. 6 /June 2000/J. Opt. Soc. Am. A 1105
some cost function associated with the intensity sensitiv-ity to each mode at the center of each spot according tothe equation
] I~n,j!
] akU
ak50
5 21
4p3 ReH E0
1E0
2p
U~r,u!
3 exp@ jrn cos~u 2 j!#durdr
3 E0
1E0
2p
jZkU* ~r,u!
3 exp@2jrn cos~u 2 j!#durdrJ ,
(35)
where U(r,u) is the binary optical element transmissionand Zk is the aberration to be sensed. One of these bi-nary phase masks of size 128 3 128 pixels and its corre-sponding detector plane intensity pattern are shown inFig. 4. The characteristic point-spread function corre-sponding to the aberration to be sensed can clearly beseen in each spot pair.
Calculation of the detected power is best achieved bynoting that the power falling onto a detector of a fixed sizepositioned anywhere in the output plane is given by theconvolution of the intensity pattern with that detectorpattern. Thus we can get the differential power sensitiv-ity by performing a convolution of the differential inten-
Fig. 5. Sensitivity matrices: (a) time multiplexed, (b) spatiallymultiplexed, optimized for sensitivity with suppression of certainoff-diagonal elements and weighting of diagonal elements for bet-ter uniformity.
sity pattern with the detector pattern, which can beachieved by simply using standard fast Fourier transformtechniques. Figure 5 shows diagrammatic representa-tions of the sensitivity matrices shown for (a) the previ-ously described time-multiplexed sensor (Table 1) and (b)the optimized spatially multiplexed sensor. For the lat-ter we again use the value of pinhole size np 5 p but relyon the optimization procedure to maximize the on-diagonal sensitivity to a uniform level and simultaneouslyto suppress off-diagonal sensitivity where its value isknown to be exactly zero in the nonmultiplexed case.The impulse height in these diagrams represents the sen-sitivity matrix element values, with the top corner beingthe (0, 0) element of each matrix.
Figure 5(a) has been scaled to that of Fig. 5(b), assum-ing a theoretical 81% efficiency obtained by use of a bi-nary optical element, and a factor of 16 because the sen-sor must be time multiplexed 16 times. Even with thisnormalization, the on-diagonal elements of the spatiallymultiplexed device are still lower than those of the tem-porally multiplexed device. Apart from the inevitablepenalty of producing a multiplexed element, this can inpart be explained by the fact that the spatially multi-plexed device uses a significant part of the space band-width available on the 128-pixel-diameter element: Fulluse of the space bandwidth would incur a reduction of ef-ficiency to 60% (owing to the pixelation, the output inten-sity pattern is effectively apodized by a sinc function).Increasing the number of pixels in the element would re-duce this effect significantly. One advantage of the opti-mized element is that additional flexibility can be intro-duced, for example, to equalize the height of on-diagonalterms or possibly to further suppress off-diagonal terms.
6. CONCLUSIONSIn this paper we have presented a theoretical analysis ofa new modal wave-front sensor intended for use in aclosed-loop adaptive system. Conceptually, the sensorconsists of two beam paths: In the first path a bias ab-erration is added to the input wave front; in the secondpath the same bias aberration is subtracted from the in-put wave front. Aberrated focal spots are produced by fo-cusing these beams onto two pinhole photodetectors.The output signal from the sensor is taken as the differ-ence between the two photodetector signals. WhenZernike circle polynomials are used as the bias modes, theresponse of the sensor can be summarized by a sparsesensitivity matrix that is easily inverted to allow calcula-tion of the magnitude of each Zernike component in theinput wave front. We have shown that most elements ofthe matrix are necessarily zero. In the limiting case,when the detector pinholes and the bias are small, thesensitivity matrix is diagonal. We have also shown thatother choices of radial polynomials and detector geom-etries will yield similar results.
The individual sensors can be temporally or spatiallymultiplexed to allow sequential or simultaneous measure-ment, respectively, of each aberration mode. A numeri-cally optimized binary optical element may be used togenerate the focal spot pattern required for the spatiallymultiplexed sensor. We have shown how the optimiza-
1106 J. Opt. Soc. Am. A/Vol. 17, No. 6 /June 2000 Neil et al.
tion process may be used to tailor the properties of thesensitivity matrix, for example, to suppress cross sensitiv-ity between modes or to equalize the on-diagonal ele-ments of the matrix.
In a future paper we shall consider the application ofthis new sensor to confocal microscopy and discuss theo-retically its operation for different imaging modes.
Znm~r,u! 5 H A2~n 1 1 !Rn
2m~r !sin~2mu! m , 0
An 1 1Rn0~r ! m 5 0
A2~n 1 1 !Rnm~r !cos~mu! m . 0
,
(A1)
with
Rnm~r ! 5 (
s50
~n2umu!/2 F ~21 !s~n 2 s !!
s!@~n 1 m !/2 2 s#!@~n 2 m !/2 2 s#!rn22sGcirc~r !. (A2)
APPENDIX A: ZERNIKE CIRCLEPOLYNOMIALSWe have chosen to use a single index as described byMahajan7 to refer to the Zernike circle polynomials.Many authors (e.g., Ref. 6) use two indices n and m thatrelate to the order of the radial and azimuthal variations,respectively. This method, although less compact, ismore intuitive than a single index, so it is useful to beable to switch between the two systems. A small alter-ation of Mahajan’s system allows each polynomial to beuniquely defined by either the single index, i, or the dualindex (n, m). We refer to those polynomials containing acosine term as having a positive value of m, whereasthose with a sine term take negative m. n may take thevalue of any positive integer or zero; m is restricted to val-ues given by the conditions that n > umu and n 2 umu iseven. The single index i starts at 1 for n 5 0 and rises ininteger steps, first with rising n, then with rising magni-tude of m, and finally with the sign of m. The polynomi-als may then be defined as
Table 3 shows the first 22 Zernike modes thus defined,and the terms used to describe these modes as commonaberrations where appropriate.
The orthogonality relationship is given by
1
pE
0
1E0
2p
Znm~r,u!Zn8
m8~r,u!durdr 5 dnn8dmm8 , (A3)
where d ij denotes the Kronecker delta. We have chosento use the definition of Zn
m(r,u) given by Noll6; the nor-malization is such that, for n Þ 0, the mean of each poly-nomial Zn
m(r,u) over the unit circle is zero and the vari-ance is p (for n 5 0, the mean is 1 and the variance is 0).
Using the single index scheme, we refer to a Zernikepolynomial as Zi(r,u) 5 Ri(r)cos(miu) or Zi(r,u)5 Ri(r)sin(2miu) where appropriate. In Eq. (A3), theintegral in u, as a consequence of the orthogonality of thetrigonometric functions, gives rise to the second Kro-necker delta, dmm8 , which is zero unless the azimuthal or-ders are identical, in which case the integral in r givesrise to dnn8 . The following properties of the radial poly-nomials can thus be determined:
E0
1
Ri~r !Rk~r !rdr 5 H d ik/2 mi 5 mk 5 0
d ik mi 5 mk Þ 0. (A4)
Moreover, it can also be shown that
E0
1
Ri~r !rdr 5 0 mi 5 0. (A5)
APPENDIX B: FOURIER TRANSFORMS OFPOLAR FUNCTIONSConsider the FT of a function Z(r,u) 5 f(r)cos(mu). Thisis defined as
F $Z~r,u!% 5 E0
2pE0
1
f~r !cos~mu!
3 exp@ jnr cos~u 2 j!#rdrdu. (B1)
If we let f 5 u 2 j and rearrange the integration we findthat
Neil et al. Vol. 17, No. 6 /June 2000/J. Opt. Soc. Am. A 1107
F$ Z~r,u!% 5 E0
1
f~r !F cos~mj!E0
2p
cos~mf!
3 exp~ jnr cos f!df 2 sin~mj!
3 E0
2p
sin~mf!exp~ jnr cos f!dfGrdr.
(B2)
Using symmetry considerations, the second integral in fis found to be equal to zero and the first integral can bewritten analytically with Ref. 11,
Jm~nr ! 5j2m
2pE
0
2p
cos~mf!exp~ jnr cos f!df, (B3)
to give the result
F $Z~r,u!% 5 2pjm cos~mj!E0
1
f~r !Jm~nr !rdr. (B4)
The equivalent process for a function Z(r,u)5 f(r)sin(mu) yields
F $Z~r,u!% 5 2pjm sin~mj!E0
1
f~r !Jm~nr !rdr. (B5)
REFERENCES1. J. W. Hardy, Adaptive Optics for Astronomical Telescopes
(Oxford U. Press, Oxford, UK, 1998).
2. R. K. Tyson, Principles of Adaptive Optics (Academic, Lon-don, 1991).
3. M. J. Booth, M. A. A. Neil, and T. Wilson, ‘‘Aberration cor-rection for confocal imaging in refractive index mismatchedmedia,’’ J. Microsc. 192, 90–98 (1998).
4. P. Torok, P. Varga, and G. Nemeth, ‘‘Analytical solution ofthe diffraction integrals and interpretation of wave-frontdistortion when light is focused through a planar interfacebetween materials of mismatched refractive indices,’’ J.Opt. Soc. Am. A 12, 2660–2671 (1995).
5. F. Roddier, ‘‘Curvature sensing and compensation: a newconcept in adaptive optics,’’ Appl. Opt. 27, 1223–1225(1988).
6. R. J. Noll, ‘‘Zernike polynomials and atmospheric turbu-lence,’’ J. Opt. Soc. Am. 66, 207–211 (1976).
7. V. N. Mahajan, ‘‘Zernike circle polynomials and optical ab-errations of systems with circular pupils,’’ Eng. Lab. Notesin Opt. Photon. News, Aug. 1994 [Appl. Opt. 33, 8121–8124(1994)].
8. M. Born and E. Wolf, Principles of Optics (Pergamon, Ox-ford, UK, 1975).
9. A. Gray and G. B. Matthews, A Treatise on Bessel Functionsand Their Applications to Physics (Dover, New York, 1966).
10. V. N. Mahajan, ‘‘Zernike annular polynomials and opticalaberrations of systems with annular pupils,’’ Eng. Lab.Notes in Opt. Photon. News, Nov. 1994 [Appl. Opt. 33,8125–8127 (1994)].
11. M. Abramovitz and I. A. Stegun, Handbook of Mathemati-cal Functions (Dover, New York, 1965).
12. W. H. Lee, ‘‘Computer-generated holograms: techniquesand applications,’’ in Progress in Optics, E. Wolf, ed.(Elsevier, New York, 1978), Chap. 3.
13. M. A. A. Neil, M. J. Booth, and T. Wilson, ‘‘Dynamic wave-front generation for the characterization and testing of op-tical systems,’’ Opt. Lett. 23, 1849–1851 (1998).
