Tutorial on Strehl ratio, wavefront power series expansion, Zernike polynomials expansion in small aberrated optical systems By Sheng Yuan

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1 Tutoial on Stel atio, wavefont powe seies expansion, Zenike polynomials expansion in small abeated optical systems By Seng Yuan. Stel Ratio Te wave abeation function, (x,y, is defined as te distance, in optical pat lengt (poduct of te efactive index and pat lengt, fom te efeence spee to te wavefont in te exit pupil measued along te ay as a function of te tansvese coodinates (x,y of te ay intesection wit a efeence spee centeed on te ideal image point. It is not te wavefont itself but it is te depatue of te wavefont fom te efeence speical wavefont (OPD as indicated in Figue. y Exit Pupil x z ave Abeation (x,y Actual Ray (nomal to Abeated avefont Abeated avefont Refeence Speical avefont Image Plane Figue. ave Abeation Function fo a distant point object Fo small abeations, te Stel atio is defined as te atio of te intensity at te Gaussian image point (te oigin of te efeence spee is te point of maximum intensity in te obsevation plane in te pesence of abeation, divided by te intensity tat would be obtained if no abeation wee pesent, S = I(0 Φ / I(0 Φ= 0, ee Φ is pase abeation. Stel atio is a vey impotant figue of meit in system wit small abeation, i.e., astonomy system wee abeation is almost always well coected, tus a good

2 undestand of te elationsip between Stel atio and abeation vaiance is absolutely necessay. Now we switc to pola exit pupil coodinates, fom te definition of Stel atio, π i (, S= e ρ ρdρd π ( 0 0 By expanding te complex exponential of equation ( into a powe seies and keep te fist tems only fo small abeation, te auto deived te following appoximated expession fo Stel atio (Maecal fomula, σ S (, σ is te standad deviation of pase abeation ( Now neglecting te σ tem in fomula (, te auto povided anote famous appoximated expessions fo Stel atio, S σ, σ is te vaiance of pase abeation acoss te exit pupil ( σ = π π 0 0 [ ( ρ, ] ρdρd = < Φ > <Φ > ( Fom Equation (-(, we can see fo small abeated system, we can maximize te Stel ation by minimize te wavefont vaiance. So we need to figue out a standad way to minimizeσ Φ fo any given wave abeation function (x,y.. Powe seies expansion of ( ρ, A standad way of descibing te wave abeation is to use a Taylo expansion polynomial in field (object eigt and pupil coodinates. (, = cos ( 0 (ige ode tems klm ae te waveabeation coefficients fo te vaious tems o modes is te eigt of te object and, ae te pola coodinates in te pupil plane Defocus Speical Abeation Coma cos ( Astigmatis m Pimay o Seidel Abeations Field Cuvatue Distotion ige ode tems Seconday,Tetiay, etc., Abeations

3 Fom equation ( above, we ae able to see wavefont vaiance is integation ove te pupil function, so we can suppess te image eigt by absobing it into te wave abeation coefficients. Te wave abeation polynomial is also typically expessed in tems of te nomalized pupil adius, ρ =, wee a is te exit pupil adius. So we can a ewite te powe seies expansion as: ( ρ, = a ρ a ρ a ρ cos ( a ρ a ρ cos ( a ρ... (5 0 0 Howeve, te tems in te Taylo seies do not fom an otogonal set of basis functions, tus if we put (5 into (, we will get many tems and make ou calculation complex. Teefoe powe seies expansion is not ecommended fo data fitting and descibing expeimental measuements of wavefont abeations..y Use Zenike Polynomials? Optical system abeations ave istoically been descibed, caacteized, and catalogued by powe seies expansions, wee te wave abeation is expessed as a weigted sum of powe seies tems tat ae functions of te pupil coodinates. Eac tem is associated wit a paticula abeation o mode. Fo example, speical abeation, coma, astigmatism, field cuvatue, distotion, and ote ige ode modes. Many optical systems ave cicula pupils. So many analyses and calculations (e.g. diffaction will involve te integation of te pupil function and wave abeation function ove a cicula pupil. Expeimental measuements will also be pefomed ove a cicula pupil and will commonly equie some fom of data fitting. It is, teefoe, convenient to expand te wave abeation in tems of a complete set of basis functions tat ae otogonal ove te inteio of a cicle. Expeimental data can be fit to a weigted sum of tese otogonal basis functions. Zenike polynomials fom a complete set of functions o modes tat ae otogonal ove a cicle of unit adius and ae convenient fo seving as a set of basis functions. Tey ae unique in tat tey ae te only polynomials in two vaiables ρ and, wic (a ae otogonal ove a unit cicle,(b ae invaiant in fom wit espect to otation of te coodinate axes about te oigin, and (c include a polynomial fo eac pemissible pai of n and m values. Tis makes tem suitable fo accuately descibing wave abeations as well as fo data fitting. Tey ae usually expessed in pola coodinates, and ae eadily convetible to Catesian coodinates. Tese polynomials ae mutually otogonal, and ae teefoe matematically independent, making te vaiance of te sum of modes equal to te sum of te vaiances of eac individual mode. Tey can be scaled so tat non-zeo ode modes ave zeo mean and unit vaiance. Tis puts all modes in a common efeence fame tat enables meaningful elative compaison between tem.

4 Diffeent Zenike polynomial definitions ae cuently in use. Te convention adopted by te OSA as x oizontal, y vetical, and is measued counte-clockwise fom x- axis (i.e. igt-anded coodinate system. Moe taditional notation measues clockwise fom y-axis. Tee is te Otogonal type wee te polynomials ae nomalized to ave unity magnitude at edge of pupil. Tee is also te Otonomal type wee te tems ae nomalized so tat te coefficient of a paticula tem o mode is te RMS contibution of tat tem. Hee I am going to adopt Pof. Maaja s definition.. Zenike Polynomials expansion of ( ρ, Fo detail of Zenike polynomials, please efe to Pof. Maajan s publised pape am going to list te esults ee only. 9. I Te otonomal Zenike polynomials and te names associated wit some of tem wen identified wit abeations ae listed in table blow fo n 8. Te numbe of Zenike (o otogonal abeation tems in te expansion of an abeation function toug a cetain ode n is given by

5 Table. Otonomal Zenike cicle polynomials 5

6 Conside a typical Zenike abeation tem: Tus, eac expansion coefficient, wit te exception of c 00, epesents te standad deviation of te coesponding abeation tem. Te vaiance of te abeation function is accodingly given by: Tus by using Zenike polynomials expansion, te vaiance of te abeation function becomes a simple adding of squaed expansion coefficients, wic geatly educed ou calculation of wavefont vaiance in te calculation of Stel ation. Refeence:. V. N. Maajan, Stel atio fo pimay abeations: some analytical esults fo cicula and annula pupils, J. Opt.Soc. Am. 7, (98, Eata, 0, 09 (99; Stel atio fo pimay abeations in tems of tei abeation vaiance, J. Opt. Soc. Am. 7, (98.. V. N. Maajan, Symmety popeties of abeated point-spead functions, J. Opt. Soc. Am. A, (99.. V. N. Maajan, Line of sigt of an abeated optical system, J. Opt. Soc. Am., 8 86 (985.. V. N. Maajan, Zenike annula polynomials fo imaging systems wit annula pupils, J. Opt. Soc. Am. 7, 75 85(98; 7, 08 (98;, 685 (98; Zenike annula polynomials and optical abeations of systems wit annula pupils, Appl. Opt., ( V. N. Maajan, Unifom vesus Gaussian beams: a compaison of te effects of diffaction, obscuation, and abeations, J. Opt. Soc. Am. A, ( V. N. Maajan, Zenike cicle polynomials and optical abeations of systems wit cicula pupils, Appl. Opt.,8 8 (99; and Zenike polynomials and optical abeations, Appl. Opt., (995. 6

7 7. V. N. Maajan, Zenike-Gauss polynomials fo optical systems wit Gaussian pupils, Appl. Opt., ( V. N. Maajan, Optical Imaging and Abeations, Pat II: ave Diffaction Optics, SPIE Pess, Bellingam,asington (00.Poc. of SPIE Vol V. N. Maajan, Zenike polynomials and abeation balancing, Appl. Opt., (995. Poc. of SPIE Vol. 57 7