Abstract

The general equations for the point-spread function (PSF) and optical transfer function (OTF) are given for any pupil shape, and they are applied to optical imaging systems with circular and annular pupils. The symmetry properties of the PSF, the real and imaginary parts of the OTF, and the modulation transfer function (MTF) of a system with a circular pupil aberrated by a Zernike circle polynomial aberration are derived. The interferograms and PSFs are illustrated for some typical polynomial aberrations with a sigma value of one wave, and 3D PSFs and MTFs are shown for 0.1 wave. The Strehl ratio is also calculated for polynomial aberrations with a sigma value of 0.1 wave, and shown to be well estimated from the sigma value. The numerical results are compared with the corresponding results in the literature. Because of the same angular dependence of the corresponding annular and circle polynomial aberrations, the symmetry properties of systems with annular pupils aberrated by an annular polynomial aberration are the same as those for a circular pupil aberrated by a corresponding circle polynomial aberration. They are also illustrated with numerical examples.

© 2013 Optical Society of America

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References

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  1. V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd ed. (SPIE, 2011).
  2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  3. V. N. Mahajan, “Symmetry properties of aberrated point-spread functions,” J. Opt. Soc. Am. A 11, 1993–2003 (1994).
    [CrossRef]
  4. C.-J. Kim and R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering (Elsevier, 1987), Vol. X, pp. 193–221.
  5. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  6. V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed., 3rd ed. (Wiley, 2007), pp. 498–546.
  7. V. N. Mahajan, “Orthonormal polynomials in wavefront analysis,” in Handbook of Optics, V. N. Mahajan and E. V. Stryland, eds., 3rd ed. (McGraw-Hill, 2010), Vol. II, pp. 11.3–11.41.
  8. The isometric and contour plots of Zernike polynomial aberrations are also given (without mentioning the aberration value) in J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering (Elsevier, 1992), Vol. XI, pp. 1–53. These authors do not use polynomials in their orthonormal form, and they order them differently as well.
  9. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2004).
  10. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981).
    [CrossRef]
  11. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils (Errata),” J. Opt. Soc. Am. 71, 1408 (1981).
    [CrossRef]
  12. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 1, 685(1984).
    [CrossRef]
  13. V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Appl. Opt. 33, 8125–8127 (1994).
    [CrossRef]

1994

1984

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 1, 685(1984).
[CrossRef]

1981

1976

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Creath, K.

The isometric and contour plots of Zernike polynomial aberrations are also given (without mentioning the aberration value) in J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering (Elsevier, 1992), Vol. XI, pp. 1–53. These authors do not use polynomials in their orthonormal form, and they order them differently as well.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2004).

Kim, C.-J.

C.-J. Kim and R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering (Elsevier, 1987), Vol. X, pp. 193–221.

Mahajan, V. N.

V. N. Mahajan, “Symmetry properties of aberrated point-spread functions,” J. Opt. Soc. Am. A 11, 1993–2003 (1994).
[CrossRef]

V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Appl. Opt. 33, 8125–8127 (1994).
[CrossRef]

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 1, 685(1984).
[CrossRef]

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils (Errata),” J. Opt. Soc. Am. 71, 1408 (1981).
[CrossRef]

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981).
[CrossRef]

V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed., 3rd ed. (Wiley, 2007), pp. 498–546.

V. N. Mahajan, “Orthonormal polynomials in wavefront analysis,” in Handbook of Optics, V. N. Mahajan and E. V. Stryland, eds., 3rd ed. (McGraw-Hill, 2010), Vol. II, pp. 11.3–11.41.

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd ed. (SPIE, 2011).

Noll, R. J.

Shannon, R. R.

C.-J. Kim and R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering (Elsevier, 1987), Vol. X, pp. 193–221.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Wyant, J. C.

The isometric and contour plots of Zernike polynomial aberrations are also given (without mentioning the aberration value) in J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering (Elsevier, 1992), Vol. XI, pp. 1–53. These authors do not use polynomials in their orthonormal form, and they order them differently as well.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd ed. (SPIE, 2011).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

C.-J. Kim and R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering (Elsevier, 1987), Vol. X, pp. 193–221.

V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed., 3rd ed. (Wiley, 2007), pp. 498–546.

V. N. Mahajan, “Orthonormal polynomials in wavefront analysis,” in Handbook of Optics, V. N. Mahajan and E. V. Stryland, eds., 3rd ed. (McGraw-Hill, 2010), Vol. II, pp. 11.3–11.41.

The isometric and contour plots of Zernike polynomial aberrations are also given (without mentioning the aberration value) in J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering (Elsevier, 1992), Vol. XI, pp. 1–53. These authors do not use polynomials in their orthonormal form, and they order them differently as well.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2004).

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Figures (10)

Fig. 1.
Fig. 1.

Isometric plot, interferogram, and PSF for some typical Zernike circle polynomial aberrations with a sigma value of 1 wave. The fringes formed by a positive (negative) aberration are indicated by a plus (minus) sign. The piston aberration Z1, for which σ=0, yields the aberration-free case.

Fig. 2.
Fig. 2.

3D PSF plots for Zernike circle polynomial aberrations as in Fig. 1, but for a sigma value of 0.1 wave.

Fig. 3.
Fig. 3.

3D, tangential or along x axis (in long red dashes), sagittal or along y axis (in medium green dashes), and at 45° from the x axis (in small blue dashes) MTF plots for Zernike circle polynomial aberrations as in Fig. 1, but for a sigma value of 0.1 wave. The solid curve represents the aberration-free MTF. The spatial frequency v is normalized by the cutoff frequency 1/λF.

Fig. 4.
Fig. 4.

Real and imaginary parts of the OTF for a Zernike polynomial aberration with a sigma value of 0.1 wave. (a) Z8 (primary coma) showing the even and odd symmetry of the real and imaginary parts. (b) Z10 showing the sixfold symmetry of the real part and the threefold symmetry of the imaginary part, in addition to their even and odd symmetry, respectively. The thick and thin contours of the imaginary part in both cases represent its positive and negative values, respectively.

Fig. 5.
Fig. 5.

Strehl ratio for Zernike circle polynomial aberrations with a sigma value of 0.1 wave, shown on a nominal scale as well as on an expanded scale.

Fig. 6.
Fig. 6.

Isometric plot, interferogram, and PSF for some typical annular polynomial aberrations for ϵ=0.5 and a sigma value of 1 wave. The fringes formed by a positive (negative) aberration are indicated by a plus (minus) sign. The piston aberration A1, for which σ=0, yields the aberration-free case.

Fig. 7.
Fig. 7.

3D PSF plots for annular polynomial aberrations as in Fig. 6, but for a sigma value of 0.1 wave.

Fig. 8.
Fig. 8.

3D, tangential or along x axis (in long red dashes), sagittal or along y axis (in green medium dashes), and at 45° from the x axis (in small blue dashes) MTF plots for annular polynomial aberrations for ϵ=0.5 and a sigma value of 0.1 wave. The solid curve represents the aberration-free MTF. The spatial frequency v is normalized by the cutoff frequency 1/λF.

Fig. 9.
Fig. 9.

Real and imaginary parts of the OTF for an annular polynomial aberration for ϵ=0.5 and a sigma value of 0.1 wave. (a) A8 (primary coma) showing the even and odd symmetry of the real and imaginary parts. (b) A10 showing the sixfold symmetry of the real part and the threefold symmetry of the imaginary part, in addition to their even and odd symmetry, respectively. The thick and thin contours of the imaginary part represent its positive and negative values, respectively.

Fig. 10.
Fig. 10.

Strehl ratio for annular polynomial aberrations for ϵ=0.5 and a sigma value of 0.1 wave, shown on a nominal scale as well as on an expanded scale.

Tables (5)

Tables Icon

Table 1. Orthonormal Zernike Circle Polynomials Zj(ρ,θ) for Imaging Systems with Circular Pupils

Tables Icon

Table 2. P-V Numbers (in Units of Wavelength) of Orthonormal Zernike Polynomial Aberrations for a Sigma Value of 1 Wave

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Table 3. Orthonormal Annular Polynomials Aj(ρ,θ;ϵ) for Imaging Systems with Annular Pupils of Obscuration Ratio ϵ

Tables Icon

Table 4. P-V Numbers (in Units of Wavelength) of Orthonormal Annular Polynomials for ϵ=0.5 and a Sigma Value of 1 Wave

Tables Icon

Table 5. Symmetry of Interferogram, PSF, Real and Imaginary Parts of OTF, and MTF for m-fold Symmetric Circle or Annular Polynomial Aberration Varying as cosmθ or sinmθ

Equations (69)

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I(r⃗)=A2λ2R2|exp[iΦ(r⃗p)]exp(2πiλRr⃗p·r⃗)dr⃗p|2,
S=Ia(0)Iu(0)=1Sex2|exp[iΦ(r⃗p)]dr⃗p|2,
Sexp(σΦ2).
τ(v⃗)=I(r⃗)exp(2πiv⃗·r⃗)dr⃗,
τ(v⃗)=Sex1exp[iQ(r⃗p)]dr⃗p,
Q(r⃗p;v⃗)=Φ(r⃗p)Φ(r⃗pλRv⃗)
I(r⃗)=τ(v⃗)exp(2πiv⃗·r⃗)dv⃗.
S=Reτa(v⃗)dv⃗τu(v⃗)dv⃗.
I(r,α)=1π2|0102πexp[iΦ(ρ,θ)]exp[πiρrcos(αθ)]ρdρdθ|2,
I(r)=[2J1(πr)/πr]2,
SI(0)=1π2|0102πexp[iΦ(ρ,θ)]ρdρdθ|2.
I(r)=4|01exp[iΦ(ρ)]J0(πrρ)ρdρ|2,
SI(0)=4|01exp[iΦ(ρ)]ρdρ|2.
S=[sin(3σd)3σd]2,
σd=πD283λ|1z1R|,
Φ(ρ)=πD24λ(1z1R)ρ2.
τ(v)=(2/π)[cos1vv(1v2)1/2],0v1,
01τ(v)vdv=1/8.
τ(0)=4/π,
Zevenj(ρ,θ)=2(n+1)Rnm(ρ)cosmθ,m0,
Zoddj(ρ,θ)=2(n+1)Rnm(ρ)sinmθ,m0,
Zj(ρ,θ)=n+1Rn0(ρ),m=0,
Rnm(ρ)=s=0(nm)/2(1)s(ns)!s!(n+m2s)!(nm2s)!ρn2s,
0102πZj(ρ,θ)Zj(ρ,θ)ρdρdθ/0102πρdρdθ=δjj.
W(ρ,θ)=j=1JajZj(ρ,θ),
aj=1π0102πW(ρ,θ)Zj(ρ,θ)ρdρdθ,
σ2=W2(ρ,θ)W(ρ,θ)2=j=2Jaj2,
I(r,α+2πk/m)=1π2|0102πexp[iΦ(ρ,θ)]exp[πiρrcos(θα2πk/m)]ρdρdθ|2.
Φ(ρ,θ2πk/m)cos[m(θ2πk/m)]=cos(mθ2πk)=cosmθΦ(ρ,θ).
I(r,α+2πk/m)=1π2|0102πexp[iΦ(ρ,θ2πk/m)]exp[πiρrcos(θα2πk/m)]ρdρdθ|2=I(r,α).
I(r,α±πj/m)=1π2|0102πexp[iΦ(ρ,θ)]exp[πiρrcos(θαπj/m)]ρdρdθ|2.
Φ(ρ,θ±πj/m)cos[m(θ±πj/m)]=cos(mθ±πj)={cosmθforevenjcosmθforoddj{Φ(ρ,θ)forevenjΦ(ρ,θ)foroddj.
I(r,α±πj/m)=1π2|0102πexp[iΦ(ρ,θ)]exp[πiρrcos(θαπj/m)]ρdρdθ|2
={I(r,α)forevenjI(r,α+π)I(r⃗)foroddj,
τ(v⃗)=Reτ(v⃗)+iImτ(v⃗),
Reτ(v⃗)=I(r⃗)cos(2πv⃗·r⃗)dr⃗
Imτ(v⃗)=I(r⃗)sin(2πv⃗·r⃗)dr⃗,
Reτ(v,ϕ)=I(r,α)cos[2πvrcos(αϕ)]rdrdα
Imτ(v,ϕ)=I(r,α)sin[2πvrcos(αϕ)]rdrdα.
Reτ(v,ϕ+πj/m)=I(r,α)cos[2πvrcos(αϕπj/m)]rdrdα.
Reτ(v,ϕ+πj/m)=I(r,απj/m)cos[2πvrcos(αϕπj/m)]rdrdα=Reτ(v,ϕ).
I(r,απj/m)=I(r,α+π)
Reτ(v,ϕ+πj/m)=I(r,α+π)cos[2πvrcos(α+πϕπj/m)]rdrdα=I(r,απj/m)cos[2πvrcos(αϕπj/m)]rdrdα=Reτ(v,ϕ).
Imτ(v⃗)=I(r⃗)sin(2πv⃗·r⃗)dr⃗.
Imτ(v,ϕ+πj/m)=Imτ(v,ϕ).
Imτ(v,ϕ+πj/m)=I(r,α)sin[2πvrcos(αϕπj/m)]rdrdα=I(r,α+πj/m)sin[2πvrcos(α+πϕπj/m)]rdrdα=Imτ(v,ϕ).
W(ρ,θ)=ajZevenj(ρ,θ)+bjZoddj(ρ,θ)=2(n+1)Rnm(ρ)(ajcosmθ+bjsinmθ)=2(n+1)Rnm(ρ)aj2+bj2cos{m[θ(1/m)tan1(bj/aj)]}.
I(r,α)=1π2(1ϵ2)2|ϵ102πexp[iΦ(ρ,θ)]exp[πiρrcos(αθ)]ρdρdθ|2.
I(r;ϵ)=1(1ϵ2)2[2J1(πr)πrϵ22J1(πϵr)πϵr]2.
SI(0)=1π2(1ϵ2)2|ϵ102πexp[iΦ(ρ,θ)]ρdρdθ|2.
I(r)=(21ϵ2)2|ϵ1exp[iΦ(ρ)]J0(πrρ)ρdρ|2
SI(0)=(21ϵ2)2|ϵ1exp[iΦ(ρ)]ρdρ|2,
S={sin[3σd]3σd},
σd=πD283λ|1z1R|(1ϵ2)
τ(v;ϵ)=11ϵ2[τ(v)+ϵ2τ(v/ϵ)τ12(v;ϵ)],0v1,
τ12(v;ϵ)=2ϵ2,0v(1ϵ)/2
=(2/π)(θ1+ϵ2θ22vsinθ1),(1ϵ)/2v(1+ϵ)/2
=0,otherwise.
cosθ1=4v2+1ϵ24v
cosθ2=4v21+ϵ24ϵv,
01τ(v;ϵ)vdv=(1ϵ2)/8.
τ(0;ϵ)=4/π(1ϵ),
Aevenj(ρ,θ;ϵ)=2(n+1)Rnm(ρ;ϵ)cosmθ,m0,
Aoddj(ρ,θ;ϵ)=2(n+1)Rnm(ρ;ϵ)sinmθ,m0,
Aj(ρ,θ;ϵ)=n+1Rn0(ρ;ϵ),m=0,
AjAj=1π(1ϵ2)ϵ102πAjAjρdρdθ=δjj.
W(ρ,θ;ϵ)=j=1JajAj(ρ,θ;ϵ),
aj=1π(1ϵ2)ϵ102πW(ρ,θ;ϵ)Aj(ρ,θ;ϵ)ρdρdθ,
σ2=W2(ρ,θ;ϵ)W(ρ,θ;ϵ)2=j=2Jaj2,

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