Strehl ratio and amplitude-weighted generalized orthonormal Zernikebased

Transcription

1 Strehl ratio and amlitude-weighted generalized orthonormal Zernikebased olynomials COSMAS MAFUSIRE* AND TJAART P. J. KRÜGER Deartment of Physics, Faculty of Natural and Agricultural Science, University of Pretoria, Private Bag X0, Hatfield 008, South Africa *Corresonding author: The concet of orthonormal olynomials is revisited by develoing a Zernike-based orthonormal set for a non-circular uil that is transmitting an aberrated, non-uniform field. We refer to this uil as a general uil. The rocess is achieved by using the QR form of the Gram Schmidt rocedure on Zernike circle olynomials and is interreted as a rocess of balancing each Zernike circle olynomial by adding those of lower order in the general uil, a rocedure which was reviously erformed using classical aberrations. We numerically demonstrate this concet by comaring the reresentation of hase in a square-gaussian uil using the Zernike-Gauss square and Zernike-circle olynomials. As exected, using the Strehl ratio, we show that only secific lower-order aberrations can be used to balance secific aberrations, for examle, tilt cannot be used to balance sherical aberration. In the rocess, we resent a ossible definition of the Maréchal criterion for the analysis of the tolerance of systems with aodized uils. OCIS codes: ( ) Aberration exansions; ( ) Phase measurement; ( ) Imaging systems; ( ) Numerical aroximation and analysis. 1. Introduction The hase of a weakly aberrated uniform otical field through a circular uil can be reresented by exansion of an infinite discrete set, known as the Zernike circle (ZC) olynomial set [1 5]. The ZC set is the only olynomial set derived by balancing Seidel aberrations that are normalized in a unit circle. For this reason, ZC exansion coefficients are associated with exeriencing secific otical effects by the field, and so they were initially alied to the roblem of diffraction theory of light beams carrying aberrations [6]. The advantages of exloiting the orthonormality of such olynomials include the following: (1) Each exansion coefficient is unique in that it is indeendent of the number of olynomials used in a articular exansion. () Only iston has a non-zero mean; therefore, the mean value of a linear combination of an orthonormal set is the iston coefficient, which defines the mean of the wavefront. (3) The wavefront variance is calculated from the sum of the squares of the non-iston terms. (4) Each coefficient, excet the iston, reresents the minimum variance associated with the aberration and, by extension, reresents the standard deviation of the resective aberration term. The last two statements have imlications in the design of systems with circular uils. In a secific hase exansion, removal of one ZC coefficient results in a lower wavefront error, making the hase less deformed. This is the basis of the oeration of adative otical systems, whether in large telescoe systems or in small-scale microscoy alications. They oerate by removing aberrations and taking wavefront error measurements until a selected value is achieved. [7]. It is aarent that the advantages of using orthonormal olynomials are lost if the uil is no longer circular. The case that has been extensively studied is the one that ertains to modern telescoe systems, which utilize non-circular mirrors, secifically annular, ellitical, hexagonal, square, rectangular and binocular shaes [8 14]. Polynomials that are orthogonal to these shaes have been available for at least ten years [8]. They were achieved using methods such as the recursive Gram Schmidt rocedure [8 15] and nonrecursive methods that include the Cholesky decomosition [16] and diffeomorhism [17]. Collectively, this work has shown that it is indeed ossible to generate a olynomial basis set orthonormal in a uil of any shae and thereby restore the advantages of orthonormality. Moreover, these olynomials are derived from the ZC set, meaning that the rocess creates the same aberrations, but they are normalized in the non-circular uil. For this reason, they 1

2 can be referred to as orthonormal Zernike-based (OZ) olynomials, of which the ZC set is a secial case. The whole rocess can be summarized as follows: ZC olynomials are derived by balancing classical Seidel aberrations by adding lower-order Seidel olynomials to minimize the variance in circular uils, whereas OZ olynomials are created by balancing ZC olynomials by adding lower-order ZC olynomials to balance them in non-circular uils. It is a one-to-one relationshi that is carried over from Seidel olynomials to any uil of choice, indicating that for every Seidel sherical aberration term, there is an equivalent ZC sherical aberration term and an OZ sherical aberration term. By extension, it is ossible to derive one OZ set from another, for examle, by deriving the Zernike annular set from the Zernike ellitical set. It is also ossible to determine an OZ set from the Seidel series [4]. However, minimal effort has been sent on non-uniform or aodized fields [18 0]. These fields are generally described as having an amlitude in which the energy is concentrated about the beam axis but decreases as one moves to the margins. Practical realization of these systems includes installing an aroriate absorbing filter at the uil or using laser transmitters. The finite size of the uil has the effect of truncating the energy of the field; the bigger the uil, the less the truncation such that if the beam is sufficiently small, the effects of truncation can be ignored. In 1995, Mahajan introduced the Zernike-Gauss circle and Zernike-Gauss annular olynomials and then investigated their use on weakly truncated, weakly aberrated Gaussian uils with circular and annular shaes, resectively [1]. In this aer, we extend Mahajan s model to the case of non-uniform fields with general non-circular shaes. Our aroach is based on diffraction theory to show the necessity of develoing an OZ set for the non-uniform fields in any aerture shae, which we collectively refer to as general uils. The model is extended to the case of strongly aberrated systems and analyzed by adding different amounts of ZC defocus and tilt, so that the conditions for maximizing the amlitude-weighted Strehl ratio of the system can be investigated. The aer is organized as follows: the Strehl ratio of a light field in a general uil is discussed in Section, with emhasis on weakly aberrated systems. In Section 3, a generalized olynomial set is derived, and its mathematical roerties are described. A relationshi between the orthonormal set and ZC olynomials is derived in Section 4 to emhasize the hysical meaning of such a set. In Section 5, we investigate Gaussian light fields in square aertures with the conclusion outlined in Section 6. Throughout this aer, bold small and caital letters are, resectively, used to reresent column vectors and matrices where t reresents transosition.. Amlitude-Weighted Strehl Ratio In this Section, a theory for the derivation of the Strehl ratio in general uils is outlined. The uil function for a system with a general uil may be resented by a uil function A ρ exfiϕ ρ g; inside the aerture E ρ 0; outside the aerture ; (1) where i ffiffiffiffiffi 1 and A ρ and ϕ ρ are the real amlitude and hase of the field, resectively, with ρ R a osition vector reresenting the transverse normalized satial coordinates of the field in the uil. The intensity of the focused field in the image lane at a distance R behind the uil can be calculated through the diffraction equation [4,] I r k ZZ πr A ρ exfiϕ ρ g ex ikρ r d ρ R where r R is a osition vector reresenting the transverse satial coordinates of the field in the image lane. In the context of imaging, the uil can be considered to be art of a centered otical system through which monochromatic light exits, as shown in Fig. 1. Behind the exit uil lies the Gaussian lane of a system at distance R away, such that one can imagine a Gaussian reference shere assing through the center of the uil, C, centered at r 0 at O on the Gaussian lane. It is also assumed that the image sace is of unit refractive index. In this case, aberrations can be defined as a deviation from this reference shere, and they are embodied in the function ϕ ρ and illustrated by the dotted line, which, incidentally, asses through O as well. The mean curvature of ϕ ρ is reresented by the observation shere also assing through C but centered at X, with a radius of curvature R x. The observation lane is now defocused from the Gaussian image lane and tilted from the axis, a result of the aberrations in the system. The focal sot generated by Eq. () is sometimes referred to as the incoherent oint sread function (PSF) of the imaging system and indicates that the weighting rovided by the amlitude of the field, the aberration comosition at the exit uil, and the shae all clearly influence the value. The resence of aberrations in an imaging system changes the osition of the focus from that at the Gaussian image oint Y. There are two main effects for which most basic imaging systems are designed to control, namely, field curvature and distortion. Field curvature is a result of the imaging system concentrating light on the Gaussian lane along the system axis at the center of the Gaussian reference shere. Deviation from the Gaussian shere is along the axis where the resultant focal lane is located in front of or behind Gaussian lane. The osition of the new foci can be calculated using [3]: Fig. 1. Schematic diagram of an imaging system focusing non-uniform light on a Gaussian lane at oint O. The resence of aberrations results in a new focal oint, X, on the defocused observation lane. ; ()

3 k RR x A ρ d ρ R x RR x x ϕ ρ A ρ d ρ ; R y k RR y A ρ d ρ RR y y ϕ ρ A ρ d ρ : (3) due to the hase ϕ ρ, where ρ ρ x;y. If the aberration consists only of system field curvature, then both R x and R y will reduce to R. Deviation from the Gaussian lane can be corrected by moving the observation lane to the new focal lane, a rocess called defocusing. Alternatively, distortion alters the direction of the roagation of light such that the centroid of the image sot moves transversely. The centroid coordinates at the image lane, hxi i ; hyi i, result in the beam ointing [4] α x hxi i R α y hyi i R RR x ϕ ρ A ρ d ρ RR A ρ d ; ρ RR y ϕ ρ A ρ d ρ RR A ρ d : (4) ρ Distortion can be corrected by tilting the image lane by an amount equal to the amount of distortion in the system. Note that for Eqs. (3) and (4), ϕ ρ should be a sum of the amount of field curvature in the system added to the extra hase function due to unwanted aberrations. It is ossible to defocus and tilt the image lane until the best image is acquired. The simlest control arameter is to measure the amount of light delivered to the center of the image lane. This is because the resence of aberrations in the system results in deflection of some of the light from the center of the PSF, the amount of which deends on the strength of the aberrations. This henomenon is characterized by the Strehl ratio [3,4], which can be exressed as a measure of the erformance of an otical system in image formation, and it is defined, in our case, as a ratio of the PSF at r 0 with and without aberrations. After making the necessary adjustments, the amlitude-weighted Strehl ratio, S, in the image lane is cast into S I RR ϕ r 0 I ϕ 0 r 0 A ρ exfiϕ ρ gd ρ RR A ρ d ρ jhexfiϕ ρ gij ; (5) also defined as the squared modulus of the amlitude-weighted average of exfiϕ ρ g in a general uil. If the aberrations are small enough, the Strehl ratio can be recast into any one of the forms [4]: S S 1 1 h Δϕ ρ i; Nijboera roximation; S S 1 1 ; i h Δϕ ρ MarOchal aroximation; S S 3 exf h Δϕ ρ ig; Mahajan aroximation; (6) with S 3 generally considered the most accurate [4]. The term h Δϕ ρ i is the amlitude-weighted wavefront error, h Δϕ ρ i h ϕ ρ hϕ ρ i i; (7) where the angled brackets denote the amlitude-weighted satial average value of the nth ower of the hase, RR ϕ hϕ n n ρ A ρ d ρ ρ i RR A ρ d : (8) ρ When n 1 and n, Eq. (8) reresents the amlitudeweighted mean and the mean of the square of the aberration function, resectively. It is obvious that, in each case, the definition of the Strehl ratio, as defined by Eq. (6), monotonically increases as h Δϕ ρ i is decreased, excet for S for large values of the variance. The Maréchal criterion states that a wavefront of a uniform field is regarded as diffraction-limited if its Strehl ratio, s 1 1 h Δϕ c i 0.8, where h Δϕ c i is the variance of a uniform field in a circular uil. For a given amount of the wavefront variance, the Strehl ratio of a nonuniform uil is generally higher than that of a uniform uil [4] by an amount ΔS S s > 0. Therefore, the Maréchal criterion for a non-uniform field to be regarded as diffractionlimited is that its Strehl ratio fulfils the following condition, S s ΔS 0.8 ΔS. As a result, the aberration tolerance for a non-uniform uil is higher than that of a uniform beam for a given Strehl ratio. 3. Zernike-Based Orthonormal Polynomial Set The urose of the Zernike-based orthonormal olynomial formulation is to analyze weakly aberrated systems with general uils by maximizing the Strehl ratio, a condition that is achieved by minimizing the wavefront error. The set needs to meet three additional conditions: it needs to be normalized, linearly indeendent, and related to classical aberrations. For a unit circle, the ZC set was found to fulfil all three conditions in a circular uil and so forms the basis for the derivation of the OZ set in general uil. The ordered OZ set is given by a column vector, z ρ, whose elements form the basis in the general uil, such that the hase can be reresented by an exansion ϕ ρ c t z ρ ; (9) where the column vector c contains an ordered set of exansion coefficients. c t z ρ is the hase exansion as a sum exressed as an inner roduct of the two vectors in Euclidean sace. It is exected that the OZ set is a comlete, orthonormal set which obeys the following equation: hz ρ z t ρ i I : (10) The matrix on the left-hand side of Eq. (10) is referred to as a Gram matrix [5]. If it is an identity matrix, the imlication is that the OZ set is orthonormal with its linear indeendence being roven by the Gram matrix s non-singularity thereby fulfilling the first two conditions. Therefore, the combination of Eqs. (9) and (10) can be used to generate the exansion coefficient vector c hϕ ρ z ρ i: (11) The average value of the hase can be calculated from Eq. (8) for n 1, hϕ ρ i c t e 1 ; (1) where e 1 is the first element of the standard basis in an Euclidean lane [5]. In short, the right-hand side in the above equation reresents the iston coefficient. The average value of 3

4 the hase for n in Eq. (8) is the sum of the square of the coefficients hϕ ρ i c t c: (13) Maximizing the Strehl ratio is achieved by minimizing the wavefront variance, which, incidentally, is given by h Δϕ ρ i c t I e 1 e t 1 c: (14) 4. Derivation of Orthonomal Polynomials from the ZC Set Having established the mathematical roerties of the OZ olynomials, we now roceed to derive a relationshi between these olynomials and the ZC set. This way the new set can be related to the classical aberrations, thereby fulfilling the third and last condition to be fulfilled by these olynomials, as mentioned in Section. The hase can now be resented as a linear combination of the ZC set: z ρ; θ fz m n ρ; θ g j 1 fz j ρ; θ g j 1 ; (15) which form the basis vectors in the circular uil and where j establishes the ZC set as an ordered set as defined by Noll [], where Z m n ρ; θ 0, when n jmj is either odd or negative. Thus, the hase function in a circular uil can be reresented as a linear combination of the elements of the set as shown by ϕ c ρ; θ b t z ρ; θ : (16) The orthonormality and linear indeendence of the ZC set inr a unit circle are verified through the Gram matrix relation, 1 π R 10 π 0 z ρ; θ z t ρ; θ ρdρdθ I. The exansion coefficients can be R calculated as elements of a vector equation, b 1 π R 10 π 0 ϕ c ρ; θ z ρ; θ ρdρdθ. The first 15 ZC olynomials corresonding to 1 j 15 are listed in Table 1. The QR form of the Gram Schmidt rocedure states that OZ olynomials can be exressed as a linear combination of the ZC set. The rocedure then takes the form Table 1. List of Normalized Zernike Polynomials U to the Fourth Order Name of aberration j n m Z j Z m n ρ;θ Piston Z 1 1 x-tilt 1 1 Z ρ Cos θ y-tilt Z 3 ρ Sin θ Defocus 4 0 Z 4 ffiffi 3 ρ 1 y-astigmatism 5 Z 5 ffiffiffi 6 ρ Sin θ x-astigmatism 6 Z 6 ffiffiffi 6 ρ Cos θ y-coma Z 7 ffiffi 8 3ρ Sin θ x-coma Z 8 ffiffi 8 3ρ Cos θ y-trefoil Z 9 ffiffiffi 8 ρ 3 Sin 3θ x-trefoil Z 10 ffiffi 8 ρ 3 Cos 3θ Sherical aberration Z 11 ffiffiffi 5 6ρ 4 6ρ 1 x-secondary astigmatism 1 4 Z 1 ffiffiffiffiffi 10 4ρ 4 3ρ Cos θ y-secondary astigmatism 13 4 Z 13 ffiffiffiffiffi 10 4ρ 4 3ρ Sin θ x-quadrafoil Z 14 ffiffiffiffiffi 10 ρ 4 Cos 4θ y-quadrafoil Z 15 ffiffiffiffiffi 10 ρ 4 Sin 4θ z ρ Hz ρ; θ ; (17) where H is a lower triangular matrix. Since the elements of vector z ρ are linearly indeendent, H is nonsingular. H is the transformation matrix associated with change of basis from the circular uil to the uil reresented by the old basis vector set z ρ; θ to the general uil reresented by the new basis vector set z ρ. It hels carry over the roerties of the ZC to the OZ set. H can be calculated by eliminating z ρ from Eqs. (10) and (17) and resulting in the following equation: H t H hz ρ; θ z t ρ; θ i 1 : (18) The matrix on the right-hand side is symmetrical and ositive definite, which means that the lower triangular matrix H can be calculated using the Cholesky decomosition [15]. This matrix can be used to relate the exansion coefficients when using either the ZC set or the OZ set to define a general uil. The resulting OZ vector is obtained by combining Eqs. (15) and (17): z ρ fz m n ρ g j 1 fz j ρ g j 1 : (19) Equation (19) indicates that the olynomial sets have a oneto-one relationshi. Since H is a lower triangular matrix, Eq. (17) reveals an imortant roerty: every term in Eq. (19) is calculated from the resective term in Eq. (15) lus a few of the lower-order terms, a rocess called balancing. In other words, the ZC set, which is already balanced in the circular uil, is balanced in the general uil by adding a number of lower-order ZC terms to generate the OZ olynomials. In a discrete form, Eq. (17) becomes Z j ρ P j j 0 1 h jj 0Z j 0 ρ; θ where h jj 0 H. Equation (0) can be written in the form Z j ρ h jj Z j ρ; θ Xj 1 X jj 0Z j 0 ρ; θ ; (0) j 0 1 where X jj 0 h jj 0 h jj is the amount of the aberration Z j 0 ρ; θ added er unit of Z j ρ; θ to balance Z j 0 ρ; θ, with h jj as a normalizing coefficient. 5. Numerical Results for a Square-Gausian Puil It is generally assumed that high ower laser sources and diode lasers have square rofiles. However, a much more suitable model for such systems is a Gaussian beam going through a square aerture. This reresents a rime alication for the model develoed in this aer, because it reresents a noncircular amlitude-weighted uil. For the choice of the aerture, the unit square inscribed in a unit circle [8] will be reresented by the field function exf γ E x;y x y gexfiϕ x;y g; 1 ffiffi x;y 1 ffiffi 0; 1 ffiffi >x;y> 1 ffiffi ; (1) where x and y are normalized at the edge of the circular uil, which has a radius a. The Gaussian beam truncation ratio is then given by γ a ω, where ω is the 1 e beam radius. The larger γ is, the less the truncation and the greater the transmitted energy, leading to stronger aodization. For a square-gaussian uil, 4

5 the Strehl ratio calculated by substituting Eq. (1) into Eq. (5) is given by γ S γ πerf γ ffiffiffi Z 1ffi Z 1ffi exf γ x y exfiϕ x;y gdxdy : 1 ffi 1 ffi () The exansion coefficients of the first 15 square-gaussian olynomials are shown in Table, each for four cases, γ 0, 1,, and 3. The results for γ 0 are for a uniform square uil and match those introduced in Ref. [8]. The results for the other truncation arameters are introduced in this aer and are resented in the last three columns of Table in numerical form. (We considered the analytical forms too comlex to resent here.) There is a general trend where the overall value of h jj 0 increases with γ. It is obvious that each OZ olynomial deends on the same ZC olynomials regardless of the value of γ. For examle, OZ sherical aberration deends on the following ZC aberrations: sherical aberration, x- and y-astigmatism, defocus, and iston, but it is indeendent of x- and y-coma, x- and y-tilt, and x- and y-triangular astigmatism for all γ. In short, each OZ term is built u by adding lower ZC order terms to the ZC term of the same order as the OZ term, all weighted by the aroriate h coefficients. If a articular aberration does not contribute to a secific OZ term, then the resective h becomes 0. An alternative way of investigating this roblem is to add an aroriate number of lower-order ZC terms until S γ is maximized, as we will see in Section 5.B. A. Strehl Ratio for Zernike-Gauss Square Polynomials For weakly aberrated systems, the Strehl ratio deends only on aberration variance, regardless of the tye of aberration in the system. This is only ossible for minimum values of the Strehl Table. Order Non-Zero Elements of H for the Square-Gaussian Puil for the Four Truncation Parameters U to the Fourth h jj 0 Z j ρ n m Z j ρ;θ n 0 m 0 γ 0 γ 1 γ γ 3 Z Z ffiffiffiffiffiffiffi Z 1 1 Z ffiffiffiffiffiffiffi Z Z ffiffiffiffiffiffiffiffiffiffi Z 4 0 Z ffiffiffiffiffiffiffi Z ffiffiffiffiffiffiffi Z 5 Z ffiffiffiffiffi Z 6 Z Z Z ffiffiffiffiffiffiffiffiffiffiffiffi Z ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi Z Z ffiffiffiffiffiffiffiffiffiffiffiffi Z ffiffiffiffiffiffiffiffiffiffiffiffi Z Z Z ffiffiffiffiffiffiffiffiffiffi Z ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi Z Z ffiffiffiffiffiffiffiffiffiffi Z Z ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi Z Z ffiffiffiffiffiffiffiffiffiffi Z Z ffiffiffiffiffi Z 1 4 Z ffiffiffi Z 6 45 ffiffiffi ffiffiffiffiffiffiffi Z 13 4 Z Z 5 3 ffiffiffi Z Z ffiffiffiffiffiffiffi Z ffiffiffiffiffiffiffiffiffiffiffiffi Z ffiffiffiffiffiffiffiffiffiffiffiffi Z ffiffiffiffiffiffiffi ffiffiffiffiffiffiffi Z Z

6 ratio, below which each aberration begins to deviate as the wavefront error increases. In this case, the orthonormal coefficient of each ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aberration is reresented by the standard deviation, h Δϕ m n i. Figure shows how S γ varies with OZ x-tilt, defocus, x-astigmatism, x-coma, x-triangular astigmatism, and sherical aberration. The grahs of S 1, S, and S 3 are added ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in each case for comarison. Evidently, S 1 <S < S 3 when h Δϕ m n i < 0.5, and the three aroximations are virtually identical for Strehl ratio less than or aroximately equal to 0.8 regardless of γ and aerture shae and not just of tyes of aberrations. However, for strong aberrations, the aroximations differ substantially. S γ was calculated using Eq. (). The grahs were calculated for the four values of the truncation factor from Table ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for selected OZ olynomials. In all cases, for each value of h Δϕ m n i, s γ increases with increases in γ, with S 3 reresenting the best aroximation. In the case of Z 1 1, Fig. (a) shows that S 3 slightly overestimates s γ for the selected values of γ, with increasingly better fit achieved as γ aroaches 3. In the cases of Z o and Z, as seen in Figs. (b) and (c), resectively, the situation is the oosite in that S 3 underestimates S γ with the best fit achieved for a uniform field that gets worse as γ increases in size excet for γ 0 and 1, where S 3 estimates S γ fairly accurately. However, Figs. (d) (f) show that S 3 underestimates S γ for all values of γ with underestimation increasing with decreasing truncation. However, Fig. (f) shows that S γ for Z o 4 are fairly close together and, in fact, at the standard deviation of 0.5λ, S γ aroaches the same value of 0.5 at the four values of γ. Figure indicates that both the tye of aberration and the truncation ratio aear to be indeendent of S γ, as exected for near-diffraction-limited systems. The minimum value of S γ for which the indeendence can be achieved is different for different aberrations and values of γ. As exlained in Section, S can be exressed in terms of the Maréchal criterion as having a minimum value of 0.8 ΔS. For nonuniform and, for that matter, non-circular uniform uils, the minimum S γ is greater than or equal to this value, deending on γ. Here, we resent numerical results for ΔS to show that otical systems with general uils have a lower tolerance comared to those with circular uils. A method we roose in this aer involves going back to the Maréchal criterion. As mentioned in Section, this criterion was devised for circular uils in which a minimum Strehl ratio of 0.8 is set, which corresonds to a standard deviation of λ 14. This standard deviation is selected as a maximum for any uil from which the corresonding minimum ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Strehl ratio is calculated. The S γ corresonding to h Δϕ m n i < λ 14 can be read from Fig.. It follows that S , S , and S The corresonding values for S γ for the square-gaussian uils are listed in Table 3. The results show a clear trend in which S γ increases with γ, an indication that truncation decreases as a result of the otical system having lower tolerance. B. Strehl Ratio for ZC Aberrations in a Square- Gaussian Puil The efficacy of using ZC olynomials in general uils is now illustrated, using Eq. () for secific amounts of selected ZC aberrations and varying amounts of a lower-order aberration. In rincile, which aberration combinations lead to a maximum S γ needs to be determined, articularly for cases where the aberrations are strong. This imlies that this is not an attemt to just use the numerical results to derive OZ olynomials but rather to evaluate the extent to which these olynomials can be used in strong aberration regimes. The results from the following numerical model can assist in determining which aberration can be added to a system to control the effects of aberrations already in the system, but it does not rovide a method to execute. In short, it is necessary to know which otical effects are associated with which aberrations. To determine the effects, Eqs. (3) and (4) are used where hase is defined as ϕ ϕ sys ϕ ext. The system generates a hase of value ϕ sys to the light beam such that it generates the system field curvature R x Ry R if ϕ ext 0 is entered into Eq. (3) and centroid values of hx i i hi y i 0 if entered into Eq. (4). The extra hase term ϕ ext reresents Table 3. Maréchal Criterion Shows the Tolerances of an Imaging System with Square-Gaussian Puils for the Selected Aberrations Presented, S γ, Corresonding to a Standard Deviation of λ 14 Fig.. Deendence of the Strehl ratio on the standard deviation for the aberrations (a) Z 1 1, (b) Z 0, (c) Z, (d) Z 1 3, (e) Z 3 3, and (f) Z 0 4, all in a square-gaussian uil. In each grah, S 1, S, and S 3 are added for imroved ersective. γ Z 1 1 Z 0 Z Z 1 3 Z 3 3 Z

7 the aberrations from the samle or subject being imaged and generated along the ath of the beam as it is transferred from the subject to the entrance uil. By definition, the size of ϕ ext is beyond the control of the imaging system, at least not directly; indirectly, imact on the imaging rocess can be minimized by adjusting the observation lane accordingly. In that case, ϕ results in R x and R y not necessarily equal to R and nonzero hxi i and/or hyi i. This state of affairs is resolved by moving the observation lane longitudinally away from the Gaussian lane, which is called defocusing, and tilting the observation lane, to comensate for the resective changes in field curvature and distortion. Substituting ϕ as defined by ϕ sys ϕ ext into Eqs. (3) and (4) can be used to determine whether field curvature and distortion is due to ϕ ext. Let us assume that ϕ ext is due to each aberration listed in Table 1. Ifϕ is inserted into Eqs. (3) and (4), then it is now ossible to classify each aberration as field curvature/defocus, distortion/tilt, or neither. The result of this exercise is given in Table 4 where Z 0, Z, Z 0 4, Z 4, and Z 4 4 focus the field. The astigmatic aberrations Z and Z 4 also focus the field ositively in the x-axis and negatively in the y-axis and vice-versa. For this reason, these two cannot be balanced by defocusing and tilting as discussed so far, but balancing them will be discussed later. In the second column, the aberrations Z 1 1, Z 1 1, Z 1 3, Z 1 3, Z 3 3 and Z 3 3 can all alter the ointing of a light beam. The third column contains all aberrations that do not have either effect on a beam, and those effects are outside the scoe of this aer. The hase distribution to be investigated is of the form ϕ ext x;y C 0 4 Z 0 4 x;y C0 Z 0 x;y, where each coefficient C m n c. The challenge is to establish how much C 0 should be added to the system for a fixed amount of C 0 4 associated with the unwanted field curvature to maximize S γ. The hase function can then be recast into the form, ϕ ext x;y C 0 4 Z x;y C C 0 Z 0 : x;y (3) 4 Figure 3 shows S γ for selected values of C 0 4 between 0 and 0.5λ in stes of 0.15λ, with each balanced by varying amounts of C 0. As exected, increasing the amount of C 0 4 when C 0 0 results in a dro in S γ for all values of γ, as observed along the vertical axis. The reason for this is that ositive C 0 4 ushes the focal lane away from the uil lane, thereby delivering less light to the center of the image in the observation lane, effectively lowering S γ. This situation can be resolved by moving the observation lane to the new focal lane. The larger C 0 4 is, the greater the amount of C 0 required for comensation. In the case of negative C 0 4, the moment of the focal lane is toward the uil requiring negative defocus to comensate for the movement. Table 4. Classification of Aberrations According to Effects on the Image Plane Field curvature/defocus Distortion/Tilt Otherwise Z 0, Z, Z 0 4, Z 4, Z 4 4 Z 1 1, Z 1 1, Z 1 3, Z 1 3, Z 3 3, Z 3 3 Z, Z 4, Z 4 4 Fig. 3. Maximizing S γ for systems with C 0 4 by adding varying C 0 for γ, equal to (a) 0, (b) 1, (c), and (d) 3. As Fig. 3 shows, the effect of defocusing the system results in an increase in S γ. The increase in C 0 4 means that maximum S γ is achieved when using larger C 0, though the maximum S γ actually decreases. A summary of the results is shown in Table 5. Included in this table are results from the theoretical model for weak Z 0 4, where X 11;4 h 11;4 h 11;11 takes the values 0.9, 1.187, 1.73, and.38 for the γ values 0, 1,, and 3, resectively. Values of C 0 added to maximize S γ for each of the investigated values of C 0 4 are shown, and the last set is of the ratio C 0 C 0 4, which can be comared with X 11;4. The results for C 0 C 0 4 that come close to the low aberration model of Table 3 are those for which S γ is above 0.8; in fact, the higher S γ is, the closer to X 11;4 the result is, which indicates that the system is aroaching the weak aberration conditions. Note that in the case of negative C 0 4, the grahs would mirror those shown in Fig. 3 about the C 0 0 axis. Next, otical systems with varying amounts of x-comainduced distortion were investigated. For these systems, C 1 1 can be used to increase S γ, using the following hase function: ϕ ext x;y C 1 3 Z x;y C 1 C 1 Z 1 1 : x;y (4) 3 The effect of having this kind of distortion is that, instead of shifting the image longitudinally like for C 0 4, the image is shifted transversely. Positive C 1 1 has the effect of shifting the center of the image in the ositive x-direction, which imlies that ositive C 1 1 is required to move the observation lane by the same amount to comensate. Negative C 1 3 and negative C 1 1 should behave similarly but with a shift in the negative x-direction. Table 5. Values C 0 C0 4 for Maximum S γ for γ 0, 1,, and 3 0 C C 0 4 for Maximum S γ γ X 11;

8 The numerical results are shown in Fig. 4. As exected, S γ for ϕ ext x;y C 1 3 Z 1 3 x;y dros below 1 as soon as C 1 3 becomes finite and continues to dro as C 1 3 increases. However, addition of a sufficient C 1 1 raises S γ, as was observed for sherical aberration. Once again, the maximum S γ dros with increase in C 1 3. The numerical results for C 1 3 confirm this observation, though the value C 1 1 C 1 3 is shown to increase with increase in C 1 3, as was observed with C 0 4 and C 0. The results, in this case, are summarized in Table 6, which also show that the results closest to X 8; are those for which S γ aroaches 1. For comleteness, we now investigate the case of adding an aberration to a system that is already aberrated. The additional aberration will not increase S γ but rather causes it to dro immediately. We illustrate this using C 1 1 for increasing S of a system with sherical aberration, as shown in Fig. 5. For a uniform field, γ 0 [in Fig. 5(a)] and the maximum S γ is achieved when C for all amounts of C 0 4, meaning that any transverse movement of the observation lane results in an automatic dro in light transmitted to the center of the observation lane. An excetion emerges, as observed in Figs. 5(b) 5(d), when C , in which case, S γ is very low, just less than 0.1. These results indicate that it is ossible to tilt the observation lane of the system to increase S γ on systems with unwanted field curvature. However, this is only ossible in rare circumstances, and the increase in S γ achieved is generally very small, in this case the largest achieved is about 0.16 for γ 3 indicating the relative useleness of using tilt to correct for systems with sherical aberration. Fig. 5. Maximizing S γ for systems with fixed C 0 4 by adding varying C 1 1 for γ, equal to (a) 0, (b) 1, (c), and (d) 3. An interesting roblem concerns ZC x-astigmatism, Z x;y, which, according to Table, does not require addition of any other ZC aberrations to increase S γ. However, insection of Table 4 tells us that when ϕ ext x;y C Z x;y ffiffi 6 C x y, field curvature would increase in the x-axis and decrease in the y-axis and vice-versa, deending on whether C is ositive or negative. The indeendence of the two foci means that it is ossible to increase S γ by using a cylindrical lens. The extra hase in this case is given by ϕ ext x;y C Z x;y C0 ;x Z 0 x, where the extra term actually reresents a one-dimensional Zernike defocus in the x-lane, Z 0 x ffiffiffi 3 x 1. Adding various amounts of C 0 ;x increases S γ, which is not ossible to achieve with ZC defocus. The numerical results in Fig. 6 illustrate the outcome, which shows that for all γ, an increasingly negative C 0 ;x results in an increase in S γ, leading to a maximum. As was observed with sherical aberration in Fig. 3 and with x-coma in Fig. 4, the maximum achieved S γ decreases with increasing C. The effect of Z x;y can be comletely ffiffi removed from the system if y-defocus, Z 0 y 3 y 1, is also added. The aberration function then becomes ϕ ext x;y C Z x;y C0 ;x Z 0 x C ffiffiffi 0 ;y Z 0 y, which can be simlified using the identity Z x;y Z 0 x Z 0 y to become Fig. 4. Maximizing S γ for systems with fixed C 1 3 by adding varying C 1 1 for γ, equal to (a) 0, (b) 1, (c), and (d) 3. Table 6. Values of C 1 1 C1 3 for Maximum S γ for γ Equal to 0, 1,, and 3 1 C 1 C 1 3 for Maximum S γ γ X 8; Fig. 6. Maximizing S γ for systems with fixed C by adding varying C 0 ;x for γ, equal to (a) 0, (b) 1, (c), and (d) 3. 8

9 Fig. 7. Maximizing S γ for systems with fixed C by adding varying C 0 ;x;y for γ, equal to (a) 0, (b) 1, (c), and (d) 3. ϕ ext x;y 1 ffiffiffi C fz 0 x Z 0 y g C 0 ;x Z 0 x C0 ;y Z 0 y : (5) If C 0 ;y C 0 ;x C 0 ;x;y is selected, this imlies that ϕ ext x;y 0 when C 0 ;x;y 1 ffiffiffi C : (6) Folding the observation lane along the x-axis and in the oosite sense along the y-axis with the axes erendicular and indeendant of each other creates a toric-shaed lane. Using such an observation lane to conjugate the two field curvatures comletely comensates for the x-astigmatism, leading to S γ 1 as shown in Fig. 7. Note that the value of C 0 ;x;y for which S γ is maximized is the same for all grahs and can be calculated from Eq. (6). An easier alternative would be to insert a toric lens between the uil and the lanar image lane. This is a lens with different otical ower and focal length in two orientations erendicular to each other. This exlains why toric lenses are used by ohthamologists to treat atients with astigmatism [6]. 6. Conclusions ZC olynomials have been the standard in wavefront analysis since they were introduced in 193, and the main limitation is that they are only orthonormal in circular uils. This has been overcome in the last four decades with the develoment of other Zernike-based olynomial sets that are orthonormal in noncircular uils for use in ground-based astronomical systems, which was achieved using the Gram Schmidt rocedure. An imortant advance was the introduction of the Cholesky decomosition, which is comutationally cheaer and less error-rone. The urose of this aer was to contribute to this rogress by investigating imaging systems that use customized light beams as long as the aroriate inner roduct can be found. This aer emhasizes the imortance of linear algebra to wavefront analysis where an otical uil is resented as a vector sace and the OZ set constitutes the basis vectors. The core of the model is the Gram matrix, which is used to define orthonormality when this matrix is an identity matrix and to define linear indeendence if it is nonsingular. This aroach could be used to normalize any olynomial set, and it has been alied, with great success, in a recently submitted aer in which orthonormal vector olynomials in general uils were derived by normalizing the Cartesian gradient of the ZC set [7]. The model was then used to generate a Zernike-based olynomial set orthonormal in square-gaussian uils, which serves as a ossible theoretical reresentation of the uil of a diode or high owered laser source. The numerical calculation of the Strehl ratio confirmed that only secific ZC aberrations of a lower order can be used to balance a selected ZC aberration in such a way that addition maximizes the Strehl ratio. Those which do not contribute result in an immediate dro in this arameter. A good examle is how, in a square-gaussian uils, ZC defocus increases the Strehl ratio in an otical system with ZC sherical aberration but ZC tilt does not. This rocess demonstrates that calculating the Strehl ratio can be used to numerically derive OZ olynomials, in addition to the Gram Schmidt rocedure. The aberrations used to balance the selected aberrations can be associated with secific effects, namely, defocus, and tilt. This means that it is ossible to design an exeriment with an aroriate diffraction otical element or using a satial light modulator to generate secific ZC aberration and then maximize the Strehl ratio by defocusing the imaging lane by an amount that can be easily measured. This amount of defocus can be related to the sherical aberration, as redicted by the numerical results. The same result is exected using ZC x-tilt to balance ZC x-coma, in which the imaging lane can be tilted by an angle that can also be measured in the lab. This imlies that the numerical model discussed in this aer can be used to design an exeriment to measure OZ exansion coefficients, a concet which is a subject of future research. Finally, a method to determine the Maréchal criterion for the analysis of aodized systems is roosed. We assert that for any uil the wavefront standard deviation maximum should be fixed at λ 14, but the tolerance be the corresonding Strehl ratio allowed to change with aerture shae and aodization. Our results show that the Strehl ratio for aodized uils is greater than 0.8, with the value increasing with increased aodization thereby imlying a decrease in tolerance. The work resented here can be alied to the analysis of aberrated light beams, in articular, in settings where the beam quality requires imrovement. This would be imortant for laser-based free sace otical communication systems and fiber communication systems. The model resented is also useful for develoing more efficient Zernike-based methods to control aberrations in laser-based adative otics methods in microscoy and otical communication systems. In addition, imaging systems that use laser radiation, such as suer-resolution imaging, are now being augmented by using adative systems [8]. For all these cases, adative comensation has been the tool of choice, though ZC olynomials are used. It is the intention of the authors of this aer to romote the use of OZ olynomials through the resented numerical results. 9

10 References 1. F. Zernike, Diffraction theory of knife-edge test and its imroved form, the hase contrast method, Mon. Not. R. Astron. Soc. 94, (1934).. R. J. Noll, Zernike olynomials and atmosheric turbulence, J. Ot. Soc. Am. 66, (1976). 3. M. Born and E. Wolf, Princiles of Otics: Electromagnetic Theory of Proagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999). 4. V. N. Mahajan, Otical Imaging and Aberrations, Part II: Wave Diffraction Otics, nd ed. (SPIE, 011). 5. V. Lakshminarayanan and A. Fleck, Zernike olynomials: a guide, J. Mod. Ot. 58, 1678 (011). 6. B. R. A. Nijboer, The diffraction theory of aberrations, Ph. D. Thesis (University of Groningen, 194). 7. R. K. Tyson, Princiles of Adative Otics 3rd ed. (CRC Press, 011). 8. V. N. Mahajan and G.-M. Dai, Orthonormal olynomials in wavefront analysis: analytical solution, J. Ot. Soc. Am. A 4, 994 (007). 9. V. N. Mahajan, Zernike annular olynomials and otical aberrations of annular uils, Al. Ot. 33, (1994). 10. W. Swanter and W. W. Chow, Gram Schmidt orthonormalization of Zernike olynomials for general aerture shaes, Al. Ot. 33, (1994). 11. R. Uton and B. Ellerbroek, Gram Schmidt orthonormalization of the Zernike olynomials on aertures of arbitrary shae, Ot. Lett. 9, (004). 1. V. Mahajan and G.-M. Dai, Orthonormal olynomials for hexagonal uils, Ot. Lett. 31, (006). 13. H. Lee, Use of Zernike olynomials for efficient estimation of orthonormal aberration coefficients over variable noncircular uils, Ot. Lett. 35, (010). 14. R. J. Mathar, Orthogonal set of basis functions over the binocular uil, arxiv: [hysics.otics], D. Watkins, Fundamentals of Matrix Comutations (Wiley, 00). 16. G.-M. Dai and V. Mahajan, Nonrecursive determination of orthonormal olynomials with matrix formulation, Ot. Lett. 3, (007). 17. R. Navarro, J. L. Loez, J. A. Diaz, and E. P. Sinusia, Generalization of Zernike olynomials for regular ortions of circles and ellises, Ot. Exress, 163 (014). 18. J. Schwiegerling, Gaussian weighting of ocular wave-front measurements, J. Ot Soc. Am. A 1, (004). 19. J. Nam and J. Rubinstein, Weighted Zernike exansion with alications to the otical aberration of the human eye, J. Ot. Soc. Am. A, (005). 0. K. Liao, Y. Hong, and W. Sheng, Wavefront aberrations of x-ray dynamical diffraction beams, Al. Ot. 53, (014). 1. V. Mahajan, Zernike Gauss olynomials and aberrations of systems with Gaussian uils, Al. Ot. 34, (1995).. J. W. Goodman, Introduction to Fourier Otics, 3rd ed. (Roberts & Comany, 004). 3. C. Mafusire and A. Forbes, Mean focal length of an aberrated lens, J. Ot. Soc. Am. A 8, (011). 4. V. N. Mahajan, Strehl ratio of a Gaussian beam, J. Ot. Soc. Am. A, (005). 5. J. T. Scheick, Linear Algebra with Alications (McGraw-Hill, 1997). 6. H. Momeni-Moghaddam, S. A. Naroo, F. Askarizadeh, and F. Tahmasebi, Comarison of fitting stability of the different soft toric contact lenses, Contact Lens & Anterior Eye 37, (014). 7. C. Mafusire and T. P. J. Krüger, Orthonormal vector olynomials in general uils, J. Ot. (Manuscrit under review). 8. M. Booth, D. Andrade, D. Burke, B. Patton, and M. Zurauskas, Aberrations and adative otics in suer-resolution microscoy, Microscoy 64, 5 61 (015). 10