Abstract

Formulas for a minimum of wave aberration variance and a maximum of the Strehl ratio in the optimal image point are derived using the third- and fifth-order aberration theory. Moreover, relations for the calculation of the optimal value of f-number of the optical system were derived, which enabled us to theoretically analyze real optical systems and their image quality. The optimal f-number corresponds to such a value of f-number when the image quality of a real optical system is comparable to an aberration-free optical system. This value may also serve as an auxiliary criterion of the image quality of the optical system, for example, in photography.

© 2012 Optical Society of America

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References

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  1. A. Maréchal, Imagerie Géométrique Aberrations (Revue d’Optique, 1952).
  2. H. H. Hopkins, Wave Theory of Aberrations (Clarendon Press, 1950).
  3. A. Miks, Applied Optics (Czech Technical University Press, 2009).
  4. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  5. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, 1963).
  6. W. T. Welford, Aberrations of Optical Systems (Hilger, 1986).
  7. S. F. Ray, Applied Photographic Optics (Focal Press, 2002).
  8. J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread function” Prog. Opt.   51, 349–468 (2008).
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    [CrossRef]
  13. J. B. DeVelis, “Comparison of methods for image evaluation,” J. Opt. Soc. Am. 55, 165–173 (1965).
    [CrossRef]
  14. A. van den Bos, “Aberration and the Strehl ratio,” J. Opt. Soc. Am. A 17, 356–358 (2000).
    [CrossRef]
  15. A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus of high-numerical-aperture beams,” J. Eur. Opt. Soc. Rap. Public. 2, 07008 (2007).
    [CrossRef]
  16. A. J. E. M. Janssen, S. van Haver, P. Dirksen, and J. J. M. Braat, “Zernike representation and Strehl ratio of optical systems with variable numerical aperture,” J. Mod. Opt. 55, 1127–1157 (2008).
    [CrossRef]
  17. W. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2007).
  18. M. Laikin, Lens Design, 4th ed. (CRC Press, 2006).
  19. H. Gross, F. Blechinger, and B. Achtner, Survey of Optical Instruments, Volume 4 of Handbook of Optical Systems(Wiley-VCH, 2008).
  20. I. Powell, “Pupil exploration and wave-front-polynomial fitting of optical systems,” Appl. Opt. 34, 7986–7997 (1995).
    [CrossRef]
  21. W. B. King, “A direct approach to the evaluation of the variance of the wave aberration,” Appl. Opt. 7, 489–494 (1968).
    [CrossRef]
  22. W. B. King and J. Kitchen, “The evaluation of the variance of the wave-aberration difference function,” Appl. Opt. 7, 1193–1197 (1968).
    [CrossRef]
  23. M. Herzberger, Modern Geometrical Optics (Interscience, 1958).
  24. D. Malacara, Optical Shop Testing (Wiley, 2007).
  25. H. A. Buchdahl, Optical Aberration Coefficients (Dover, 1968).

2008 (2)

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread function” Prog. Opt.   51, 349–468 (2008).

A. J. E. M. Janssen, S. van Haver, P. Dirksen, and J. J. M. Braat, “Zernike representation and Strehl ratio of optical systems with variable numerical aperture,” J. Mod. Opt. 55, 1127–1157 (2008).
[CrossRef]

2007 (1)

A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus of high-numerical-aperture beams,” J. Eur. Opt. Soc. Rap. Public. 2, 07008 (2007).
[CrossRef]

2000 (1)

1995 (1)

1991 (1)

1983 (1)

1982 (1)

1968 (3)

1965 (1)

Achtner, B.

H. Gross, F. Blechinger, and B. Achtner, Survey of Optical Instruments, Volume 4 of Handbook of Optical Systems(Wiley-VCH, 2008).

Blechinger, F.

H. Gross, F. Blechinger, and B. Achtner, Survey of Optical Instruments, Volume 4 of Handbook of Optical Systems(Wiley-VCH, 2008).

Born, M.

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

Braat, J. J. M.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread function” Prog. Opt.   51, 349–468 (2008).

A. J. E. M. Janssen, S. van Haver, P. Dirksen, and J. J. M. Braat, “Zernike representation and Strehl ratio of optical systems with variable numerical aperture,” J. Mod. Opt. 55, 1127–1157 (2008).
[CrossRef]

A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus of high-numerical-aperture beams,” J. Eur. Opt. Soc. Rap. Public. 2, 07008 (2007).
[CrossRef]

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, 1968).

DeVelis, J. B.

Dirksen, P.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread function” Prog. Opt.   51, 349–468 (2008).

A. J. E. M. Janssen, S. van Haver, P. Dirksen, and J. J. M. Braat, “Zernike representation and Strehl ratio of optical systems with variable numerical aperture,” J. Mod. Opt. 55, 1127–1157 (2008).
[CrossRef]

A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus of high-numerical-aperture beams,” J. Eur. Opt. Soc. Rap. Public. 2, 07008 (2007).
[CrossRef]

Gross, H.

H. Gross, F. Blechinger, and B. Achtner, Survey of Optical Instruments, Volume 4 of Handbook of Optical Systems(Wiley-VCH, 2008).

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Clarendon Press, 1950).

Janssen, A. J. E. M.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread function” Prog. Opt.   51, 349–468 (2008).

A. J. E. M. Janssen, S. van Haver, P. Dirksen, and J. J. M. Braat, “Zernike representation and Strehl ratio of optical systems with variable numerical aperture,” J. Mod. Opt. 55, 1127–1157 (2008).
[CrossRef]

A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus of high-numerical-aperture beams,” J. Eur. Opt. Soc. Rap. Public. 2, 07008 (2007).
[CrossRef]

King, W. B.

Kitchen, J.

Laikin, M.

M. Laikin, Lens Design, 4th ed. (CRC Press, 2006).

Mahajan, V. N.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 2007).

Maréchal, A.

A. Maréchal, Imagerie Géométrique Aberrations (Revue d’Optique, 1952).

Martial, G.

Miks, A.

A. Miks, Applied Optics (Czech Technical University Press, 2009).

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, 1963).

Powell, I.

Ray, S. F.

S. F. Ray, Applied Photographic Optics (Focal Press, 2002).

Smith, W.

W. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2007).

van den Bos, A.

van Haver, S.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread function” Prog. Opt.   51, 349–468 (2008).

A. J. E. M. Janssen, S. van Haver, P. Dirksen, and J. J. M. Braat, “Zernike representation and Strehl ratio of optical systems with variable numerical aperture,” J. Mod. Opt. 55, 1127–1157 (2008).
[CrossRef]

A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus of high-numerical-aperture beams,” J. Eur. Opt. Soc. Rap. Public. 2, 07008 (2007).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Hilger, 1986).

Wolf, E.

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

Appl. Opt. (3)

J. Eur. Opt. Soc. Rap. Public. (1)

A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus of high-numerical-aperture beams,” J. Eur. Opt. Soc. Rap. Public. 2, 07008 (2007).
[CrossRef]

J. Mod. Opt. (1)

A. J. E. M. Janssen, S. van Haver, P. Dirksen, and J. J. M. Braat, “Zernike representation and Strehl ratio of optical systems with variable numerical aperture,” J. Mod. Opt. 55, 1127–1157 (2008).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

Prog. Opt. (1)

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread function” Prog. Opt.   51, 349–468 (2008).

Other (13)

A. Maréchal, Imagerie Géométrique Aberrations (Revue d’Optique, 1952).

H. H. Hopkins, Wave Theory of Aberrations (Clarendon Press, 1950).

A. Miks, Applied Optics (Czech Technical University Press, 2009).

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

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, 1963).

W. T. Welford, Aberrations of Optical Systems (Hilger, 1986).

S. F. Ray, Applied Photographic Optics (Focal Press, 2002).

W. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2007).

M. Laikin, Lens Design, 4th ed. (CRC Press, 2006).

H. Gross, F. Blechinger, and B. Achtner, Survey of Optical Instruments, Volume 4 of Handbook of Optical Systems(Wiley-VCH, 2008).

M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

D. Malacara, Optical Shop Testing (Wiley, 2007).

H. A. Buchdahl, Optical Aberration Coefficients (Dover, 1968).

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Figures (1)

Fig. 1.
Fig. 1.

Dependence of longitudinal spherical aberration on pupil radius (F=2.8).

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

W=k,l,mWklmykrlcosmφ,
W=W000+(W111y+W311y3+W511y5)rcosφ+(W020+W420y4)r2+(W222y2+W422y4)r2cos2φ+(W131y+W331y3)r3cosφ+(W040+W240y2)r4+W333y3r3cos3φ+W242y2r4cos2φ+W151yr5cosφ+W060r6,
W=W00+W11rcosφ+W20r2+W22r2cos2φ+W31r3cosφ+W40r4+W33r3cos3φ+W42r4cos2φ+W51r5cosφ+W60r6,
W11=W111y+W311y3+W511y5,W20=W020+W420y4,W22=W222y2+W422y4,W40=W040+W240y2,W60=W060,W42=W242y2,W31=W131y+W331y3,W33=W333y3,W51=W151y,
δx=RnWX,δy=RnWY,
W=A00+A11rcosφ+A20r2+p2(A22Ssin2φ+A22Tcos2φ)r2+pA31r3cosφ+A40r4,
A20=s08F2,A11=δyDp3y02F,A22S=δsS8F2,A22T=δsT8F2,A31=δyKS2F,A40=δsK16F2,
Fm=12nsinσm=R2nrm,
F0=12nsinσ0=R2nr0.
F0=Fm/(r0/rm).
S.D.=Imax(W0)Imax(W=0)=|1πrm202π0rmexp[ikW(r,φ)]rdrdφ|21(2πλ)2E0,
E0=W¯2W¯2,W¯=1πrm202π0rmW(r,φ)rdrdφ,W¯2=1πrm202π0rmW2(r,φ)rdrdφ.
E0=9W602112rm12+(W40W606+W51212+W60W4212)rm10+(3W20W6020+4W40245+4W40W4245+3W60W2240+3W33W5120+W31W515+17W422360)rm8+(W20W406+W20W4212+W40W2212+W11W514+5W22W4248+5W33264+3W33W3116+W3128)rm6+(W20212+W20W2212+W11W313+W22216+W11W334)rm4+W1124rm2.
E0W20=0,E0W11=0.
W11=23W31rm212W33rm212W51rm4,W20=12W22W40rm212W42rm2910W60rm4.
(S.D.)max1(2πλ)2E0min,
E0min=9W602700rm12+(W51248+W40W6060+W60W42120)rm10+(W402180+W40W42180+19W422720+W31W5130+W51W3340)rm8+(W31272+W22W4216+W31W3348+W33264)rm6+W22224rm4.
W111=12W151rm4(23W131+23W331y2+12W333y2)rm2W311y2W551y4,W020=W040rm2910W060rm4.
E0min=9W0602700rm12+W040W06060rm10+W0402180rm8+y2(W151248+W240W06060+W060W242120)rm10+y2(4W240245y2+17W2422360y2+W040W24090+W040W242180+W131W15130+W331W15130y2+4W240W24245y2+W333W15140y2)rm8+y2(W131272+W331272y4+W333264y4+W131W33136y2+W222W24012y2+W131W33348y2+5W222W24248y2+W240W4206y4+W240W42212y4+W420W24212y4+5W242W42248y4+W331W33348y4)rm6+y4(W222216+W222W42012y2+W222W4228y2+W420212y4+W420W42212y4+W422216y4)rm4.
E0=445A402rm8+(A3128p2+A20A406+A40A22S12p2+A40A22T12p2)rm6+(A20212+A20A22S12p2+A20A22T12p2+(A22S)216p4A22SA22T24p4+(A22T)216p4+A11A313p)rm4+A1124prm2.
A11=23A31prm2,A20=A22S+A22T2p2A40rm2.
E0min=A402180rm8+p2A31272rm6+p4(A22TA22S)224rm4.
A11=23A31prm2,A20=A40rm2,E0min=A402180rm8+p2A31272rm6+p42[(A22T)2+(A22S)2]+(A22TA22S)248rm4.
E0(λ2π)2(1S.D.)(λ14)2
E0(λ2π)2(1S.D.),
E0min=9W602700ropt12+W40W6060ropt10+W402180ropt8=(λ2π)2(1S.D.),
W60=Δsk24Fm2(1q0)=Δsext6Fm2q02,W40=32q0W60,
(r1)min=f2(n+2)(n1)n(2n+1),(r2)min=f2(n+2)(n1)n(2n1)4.
δsK=12fH2(SI)min=12fH2n(4n1)4(n+2)(n1)2=1.025H2f,
δyKS=H22ftanwK(SII)min=H22ftanwK12(n+2)=0.0711H2ftanwK,
δsT=12ftan2wK(3SIII+SIV)=1.83ftan2wK,δsS=12ftan2wK(SIII+SIV)=0.83ftan2wK,
δsK=1.025mm,δyKS=0.0125mm,δsT=5.690mm,δsS=2.580mm.
M=FmFopt

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