Abstract

This work deals with an analysis of an influence of the third-order aberration design of an optical system for a specific position of the object on the diameter of the circle of confusion, the centroid of the spot diagram, and the position of the optimum image. Explicit analytical formulas are derived for the calculation of fundamental third-order aberration coefficients for an arbitrary value of the transverse magnification of the optical system. Analytical formulas are also derived for the calculation of the third-order aberrations of an optical system composed of several components.

© 2013 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  2. M. M. Rusinov, Technical Optics (Mashinostroenie, 1979).
  3. H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems, Volume 3: Aberration Theory and Correction of Optical Systems (Wiley-VCH, 2006).
  4. A. Miks, Applied Optics (Czech Technical University, 2009).
  5. W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic, 1974).
  6. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University, 1970).
  7. M. Herzberger, Strahlenoptik (Springer, 1931).
  8. M. Herzberger, “Theory of image errors of the fifth order in rotationally symmetrical systems I,” J. Opt. Soc. Am. 29, 395 (1939).
    [CrossRef]
  9. M. Herzberger, Modern Geometrical Optics (Interscience, 1958).
  10. C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. 65B, 429–437 (1952).
    [CrossRef]
  11. A. Walther, “Irreducible aberrations of a lens used for a range of magnifications,” J. Opt. Soc. Am. A 6, 415–422 (1989).
    [CrossRef]
  12. A. Miks and J. Novak, “Estimation of accuracy of optical measuring systems with respect to object distance,” Opt. Express 19, 14300–14314 (2011).
    [CrossRef]
  13. A. Miks and J. Novak, “Dependence of camera lens induced radial distortion and circle of confusion on object position,” Opt. Laser Technol. 44, 1043–1049 (2012).
    [CrossRef]
  14. J. B. Develis, “Comparison of methods for image evaluation,” J. Opt. Soc. Am. 55, 165–173 (1965).
    [CrossRef]
  15. R. Leach, Optical Measurement of Surface Topography (Springer, 2011).
  16. A. Miks, J. Novak, and P. Novak, “Analysis of imaging for laser triangulation sensors under Scheimpflug rule,” Opt. Express 21, 18225–18235 (2013).
    [CrossRef]
  17. D. Argentieri, Ottica industriale (Hoepli, 1942).
  18. H. Chretien, Calcul des Combinaisons Optiques (Masson, 1980).
  19. M. Berek, Grundlagen der Praktischen Optik (Walter de Gruyter & Co., 1970).
  20. M. I. Khan, “Cemented triplets: a method for rapid design,” Opt. Acta 31, 873–883 (1984).
    [CrossRef]
  21. C. H. Chen and S. G. Shiue, “Method of solving a triplet comprising a singlet and a cemented doublet with given primary aberrations,” J. Mod. Opt. 44, 1279–1292 (1997).
    [CrossRef]
  22. A. Szulc, “Improved solution for the cemented doublet,” Appl. Opt. 35, 3548–3558 (1996).
    [CrossRef]
  23. M. H. Sussman, “Cemented aplanatic doublets,” J. Opt. Soc. Am. 52, 1185–1186 (1962).
    [CrossRef]
  24. S. Banerjee and L. Hazra, “Experiments with a genetic algorithm for structural design of cemented doublets with prespecified aberration targets,” Appl. Opt. 40, 6265–6273 (2001).
    [CrossRef]
  25. M. I. Khan and J. Macdonald, “Cemented doublets, a method for rapid design,” Opt. Acta 29, 807–822 (1982).
    [CrossRef]
  26. A. Miks and J. Novak, “Design of a double-sided telecentric zoom lens,” Appl. Opt. 51, 5928–5935 (2012).
    [CrossRef]
  27. A. Miks, “Modification of the formulas for third-order aberration coefficients,” J. Opt. Soc. Am. A 19, 1867–1871 (2002).
    [CrossRef]
  28. A. Miks, J. Novak, and P. Novak, “Method of zoom lens design,” Appl. Opt. 47, 6088–6098 (2008).
    [CrossRef]
  29. A. Miks and J. Novak, “Third-order aberration coefficients of a thick lens,” Appl. Opt. 51, 7883–7886 (2012).
    [CrossRef]

2013 (1)

2012 (3)

2011 (1)

2008 (1)

2002 (1)

2001 (1)

1997 (1)

C. H. Chen and S. G. Shiue, “Method of solving a triplet comprising a singlet and a cemented doublet with given primary aberrations,” J. Mod. Opt. 44, 1279–1292 (1997).
[CrossRef]

1996 (1)

1989 (1)

1984 (1)

M. I. Khan, “Cemented triplets: a method for rapid design,” Opt. Acta 31, 873–883 (1984).
[CrossRef]

1982 (1)

M. I. Khan and J. Macdonald, “Cemented doublets, a method for rapid design,” Opt. Acta 29, 807–822 (1982).
[CrossRef]

1965 (1)

1962 (1)

1952 (1)

C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. 65B, 429–437 (1952).
[CrossRef]

1939 (1)

Argentieri, D.

D. Argentieri, Ottica industriale (Hoepli, 1942).

Banerjee, S.

Berek, M.

M. Berek, Grundlagen der Praktischen Optik (Walter de Gruyter & Co., 1970).

Blechinger, F.

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems, Volume 3: Aberration Theory and Correction of Optical Systems (Wiley-VCH, 2006).

Born, M.

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

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University, 1970).

Chen, C. H.

C. H. Chen and S. G. Shiue, “Method of solving a triplet comprising a singlet and a cemented doublet with given primary aberrations,” J. Mod. Opt. 44, 1279–1292 (1997).
[CrossRef]

Chretien, H.

H. Chretien, Calcul des Combinaisons Optiques (Masson, 1980).

Develis, J. B.

Gross, H.

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems, Volume 3: Aberration Theory and Correction of Optical Systems (Wiley-VCH, 2006).

Hazra, L.

Herzberger, M.

M. Herzberger, “Theory of image errors of the fifth order in rotationally symmetrical systems I,” J. Opt. Soc. Am. 29, 395 (1939).
[CrossRef]

M. Herzberger, Strahlenoptik (Springer, 1931).

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

Khan, M. I.

M. I. Khan, “Cemented triplets: a method for rapid design,” Opt. Acta 31, 873–883 (1984).
[CrossRef]

M. I. Khan and J. Macdonald, “Cemented doublets, a method for rapid design,” Opt. Acta 29, 807–822 (1982).
[CrossRef]

Leach, R.

R. Leach, Optical Measurement of Surface Topography (Springer, 2011).

Macdonald, J.

M. I. Khan and J. Macdonald, “Cemented doublets, a method for rapid design,” Opt. Acta 29, 807–822 (1982).
[CrossRef]

Miks, A.

Novak, J.

Novak, P.

Peschka, M.

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems, Volume 3: Aberration Theory and Correction of Optical Systems (Wiley-VCH, 2006).

Rusinov, M. M.

M. M. Rusinov, Technical Optics (Mashinostroenie, 1979).

Shiue, S. G.

C. H. Chen and S. G. Shiue, “Method of solving a triplet comprising a singlet and a cemented doublet with given primary aberrations,” J. Mod. Opt. 44, 1279–1292 (1997).
[CrossRef]

Sussman, M. H.

Szulc, A.

Walther, A.

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic, 1974).

Wolf, E.

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

Wynne, C. G.

C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. 65B, 429–437 (1952).
[CrossRef]

Zügge, H.

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems, Volume 3: Aberration Theory and Correction of Optical Systems (Wiley-VCH, 2006).

Appl. Opt. (5)

J. Mod. Opt. (1)

C. H. Chen and S. G. Shiue, “Method of solving a triplet comprising a singlet and a cemented doublet with given primary aberrations,” J. Mod. Opt. 44, 1279–1292 (1997).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Opt. Acta (2)

M. I. Khan and J. Macdonald, “Cemented doublets, a method for rapid design,” Opt. Acta 29, 807–822 (1982).
[CrossRef]

M. I. Khan, “Cemented triplets: a method for rapid design,” Opt. Acta 31, 873–883 (1984).
[CrossRef]

Opt. Express (2)

Opt. Laser Technol. (1)

A. Miks and J. Novak, “Dependence of camera lens induced radial distortion and circle of confusion on object position,” Opt. Laser Technol. 44, 1043–1049 (2012).
[CrossRef]

Proc. Phys. Soc. (1)

C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. 65B, 429–437 (1952).
[CrossRef]

Other (12)

R. Leach, Optical Measurement of Surface Topography (Springer, 2011).

D. Argentieri, Ottica industriale (Hoepli, 1942).

H. Chretien, Calcul des Combinaisons Optiques (Masson, 1980).

M. Berek, Grundlagen der Praktischen Optik (Walter de Gruyter & Co., 1970).

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

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

M. M. Rusinov, Technical Optics (Mashinostroenie, 1979).

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems, Volume 3: Aberration Theory and Correction of Optical Systems (Wiley-VCH, 2006).

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

W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic, 1974).

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University, 1970).

M. Herzberger, Strahlenoptik (Springer, 1931).

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Figures (2)

Fig. 1.
Fig. 1.

Dependence of the circle of confusion diameter on transverse magnification m for an optical system corrected for magnifications m=1 and m=0.

Fig. 2.
Fig. 2.

Dependence of circle of confusion diameter on transverse magnification m for an optical system corrected for magnification m=1 and axial point (y=0mm) and off-axis point (y=10mm).

Tables (1)

Tables Icon

Table 1. Two-Element Optical System (f=1/φ=1, s1=)

Equations (38)

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δy=12ng[S1A3sinφS2A2(1+2sin2φ)tanω+3S3Asinφtan2ω+S4n2Ay2sinφS5tan3ω],δx=12ng[S1A3cosφS2A2sin2φtanω+S3Acosφtan2ω+S4n2Ay2cosφ],
S1=g4SI,S2=g3gPSII,S3=g2gP2SIII,S4=SIV,S5=ggP3SV,
S=AS0,
S=(S1S2S3S4S5)=(g4SIgPg3SIIgP2g2SIIISIVgP3gSV),A=(a11a12a13a14a15a16a17014a1212a1312a1434a15a16a270016a13012a15a16a370001000000014a15a160),S0=(SISIISIIISIVSVSVI1),G=(g400000gPg300000gP2g200000100000gP3g),S0=(SISIISIIISIVSV),S=GS0.
a11=(gPg)4,a15=4(gPg)gP3,a12=4(gPg)3gP,a16=gP4,a13=6(gPg)2gP2,a17=nf(gPg)[(gPg)23gP(gPg)+3(gP21)],a14=2n2f2(gPg)2,a27=nf(gPg)[gP(gPg)+2(gP21)],a37=nf(gPg)(gP21),
h1=s1/g,σ1=1/g,hP1=sP1/gP,σP1=1/gP,
h1=f,σ1=0,hP1=sP1/gP,σP1=1/gP,
δx=0,δy=12gP(g2SIIAM2+gP2SVtan2w)tanω,
AM=12F0(gg)P,tanω=yf(gPg)=ygf(gPg),
dc=3AM6g3AM4S12+16AM2S1S3tan2ω+8AM2S1S4y2+8AM2S22tan2ω+30S32tan4ω+24S3S4y2tan2ω+6S42y4.
Dopt=2AM2S1+6S3tan2ω+3S4y26g2=2AM2S1+[6S3+3S4f2(gPg)2]tan2ω6g2.
(dc)min=AM6gAM4S12+24AM2S22tan2ω+18S32tan4ω.
S=BS+b,
B=(a11a12a13a14a15014a1212a1312a1434a150016a13012a1500010000014a15),b=(a16SVI+a17a16SVI+a27a16SVI+a370a16SVI),S=(SISIISIIISIVSV).
S=B1(Sb).
B1=(1/(ggP)44/(ggP)46/(ggP)42f2/(ggP)24/(ggP)401/gP(ggP)33/gP(ggP)3f2/gP(ggP)3/gP(ggP)3001/gP2(ggP)202/gP2(ggP)20001000001/gP3(ggP)).
SI=2S4f2(ggP)2+fg(g24ggP+6gP21)+fgP(13gP2)+SVIgP4+S14(S2+S5)+6S3(ggP)4,SII=S4f2(ggP)2+fg(1+ggP3gP2)+fgP(2gP21)SVIgP4+S23(S3S5)gP(ggP)3,SIII=f(gP21)(ggP)+SVIgP4+S32S5gP2(ggP)2,SIV=S4,SV=S5SVIgP4gP3(ggP).
X(i)=[A(i)3cosφ,A(i)2sin2φtanω(i),A(i)cosφtan2ω(i),A(i)y(i)2cosφ,0],Y(i)=[A(i)3sinφ,A(i)2(1+2sin2φ)tanω(i),3A(i)sinφtan2ω(i),A(i)y(i)2sinφ,tan3ω(i)],
δx(i)=X(i)S(i)2g(i)=S(i)TX(i)T2g(i),δy(i)=Y(i)S(i)2g(i)=S(i)TY(i)T2g(i),
A(i+1)=g(i)A(i),tanω(i+1)=gP(i)tanω(i),y(i)=f(i)(gP(i)g(i))tanω(i),y(i+1)=y(i)/g(i),
δx=i=1N1δx(i)q=i+1Nm(q)+δx(N),δy=i=1N1δy(i)q=i+1Nm(q)+δy(N),
δx=δx(1)m(2)m(3)+δx(2)m(3)+δx(3),δy=δy(1)m(2)m(3)+δy(2)m(3)+δy(3).
U=(cosφ00000sin2φ00000cosφ00000cosφ000000),
V=(sinφ0000012sin2φ000003sinφ00000sinφ000001),
C(i)=(A(i)3A(i)2tanω(i)A(i)tan2ω(i)A(i)y(i)2tan3ω(i))=(A(1)3j=1i1g(j)3A(1)2tanω(1)j=1i1g(j)2gP(j)A(1)tan2ω(1)j=1i1g(j)gP(j)2A(1)y(1)2j=1i11/g(j)tan3ω(1)j=1i1gP(j)3),
X(i)T=UC(i),Y(i)T=VC(i),
δx(i)=S(i)TUC(i)2g(i),δy(i)=S(i)TVC(i)2g(i).
SI=m(2)4SI(1)+SI(2),SII=m(2)3mP(2)SII(1)+SII(2),SIII=m(2)2mP(2)2SIII(1)+SIII(2),SIV=SIV(1)+SIV(2),SV=m(2)mP(2)3SV(1)+SV(2),
M(i)=k=1K(i)(σk+1σk1/nk+11/nk)2(σk+1nk+1σknk),N(i)=k=1K(i)(σk+1σk1/nk+11/nk)(σk+1nk+1σknk),
h1(i)=f(i),σ1(i)=0,hP1(i)=sP1(i)/gP(i),σP1(i)=1/gP(i).
SI(i)=f(i)M(i),SII(i)=1gP(i)(sP1(i)M(i)+f(i)N(i)),SIII(i)=(sP1(i)gP(i))2(M(i)/f(i)+2N(i)/sP1(i)+f(i)/sP1(i)2),SIV(i)=1f(i)k=1K(i)nk+1σk+1nkσknknk+1,SV(i)=(sP1(i)gP(i))3(M(i)/f(i)2+3N(i)/f(i)sP1(i)+f(i)SIV/sP1(i)2+3/sP1(i)2),SVI(i)=(sP1(i)gP(i))4(M(i)/f(i)3+4N(i)/f(i)2sP1(i)+2SIV/sP1(i)2+3/f(i)sP1(i)21/f(i)2sP1(i)).
δs=H22fM,δym=32(Hf)2tanω(sPM+fN),
nj+1σj+1njσj=hj(nj+1nj)/rj,hj+1=hjdjσj+1,j=1,2,
M=(nn1)2(n+2nσ222n+1nσ2+1),N=(nn1)(1n+1nσ2).
1r1=φn1nσ2,1r2=φn1(nσ21),
S0=Dq+d,
D=(1f13(f11)4f1304(f11)3f131f10(f11)f12(f11)3f12f1022f10f130f12(3f13)0),d=((f11)2(2pf1+4)f13(f11)2(p+2)f120f1(f11)2(p+3)),
M1=2(p6f12f1p+f12p+3f12+3)/f12,M2=(2p12f14f1p+2f12p+4f12+f13+6)/(f12(f11)2),N1=(3f1p+f1p3)/f1,N2=(2f1p+f1p3)/(f1f12).

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