Abstract

Our work presents detailed theoretical analysis of two-element optical systems of telephoto lenses and objectives of anallactic telescopes with internal focusing. The first element of such systems has positive optical power and the second element has negative optical power. This type of optical system is widespread in practice mainly in the field of photographic lenses and in surveying instruments (theodolites, leveling instruments, etc.) where the anallactic telescope with internal focusing is being used. In our work we propose methods to determine the basic parameters of such objectives, i.e., the focal lengths of both the elements of the objective lens and their mutual axial separation. Furthermore, the detailed analysis of aberration properties of such optical systems is performed and methods for measuring the focal lengths of individual elements and their mutual distance without the need for disassembling the investigated optical system are presented.

© 2012 Optical Society of America

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  14. R. Roelofs, “Fadendistanzmesser mit Innenfokussierung,” Z. Istrumentenkd. 61, 137–1947 (1941).
  15. S. V. Eliseev, Geodezicheskie Instrumenty a Pribory (Nedra, 1973).
  16. F. Deumlich, Instrumentenkunde der vermessungstechnik (VEB Verlag für Bauwesen, 1967).
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  22. D. Argentieri, Ottica Industriale (Hoepli, 1942).
  23. H. H. Hopkins and V. V. Rao, “The systematic design of two component objectives,” Opt. Acta 17, 497–514 (1970).
    [CrossRef]
  24. M. I. Khan, “Cemented triplets: a method for rapid design,” Opt. Acta 31, 873–883 (1984).
    [CrossRef]
  25. 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. 45, 2063–2084 (1998).
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  29. M. I. Khan and J. Macdonald, “Cemented doublets, a method for rapid design,” Opt. Acta 29, 807–822 (1982).
    [CrossRef]
  30. G. V. Kreopalova, N. L. Lazareva, and D. T. Puriajev, Optical Measurements (Maschinostroenie, 1987).
  31. J. Picht, Mess- und Prüfmethoden der optischen Fertigung (Akademie-Verlag, 1953).
  32. J. Flügge, Einführung in die Messung der optischen Grundgrössen (Verlag Braun, 1954).
  33. B. Dorband, H. Miller, and H. Gross, Handbook of Optical Systems, Vol. 5: Metrology of Optical Components and Systems (Wiley, 2012).
  34. D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, 2007).

2011 (1)

2008 (1)

2002 (1)

2001 (1)

1998 (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. 45, 2063–2084 (1998).

1996 (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]

1970 (1)

H. H. Hopkins and V. V. Rao, “The systematic design of two component objectives,” Opt. Acta 17, 497–514 (1970).
[CrossRef]

1962 (1)

1956 (1)

1941 (1)

R. Roelofs, “Fadendistanzmesser mit Innenfokussierung,” Z. Istrumentenkd. 61, 137–1947 (1941).

1936 (1)

H. Schulz, “Der anallaktische Punkt beim Fernrohr mit innerer Einstellinse,” Z. Istrumentenkd. 56, 357–360 (1936).

1932 (1)

W. Uhink, “Betrachtungen über Fernrohre mit Entfernungmessfäden,” Z. Istrumentenkd. 52, 435–442 (1932).

1929 (1)

O. Eggert, “Ein Beitrag zur Theorie des Fernrohrs mit Fokussierlinse,” Z. Vermessungswes. 23, 833–841 (1929).

1925 (1)

H. Wild, “Der neue Theodolit,” Schweiz. Z. Vermw. Kulturt. 23, 103–105 (1925).

1909 (1)

H. Wild, “Neue Nivellierinstrumente,” Z. Instrumentenkd. 29, 329–344 (1909).

Achtner, B.

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

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

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. 45, 2063–2084 (1998).

Chretien, H.

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

Deumlich, F.

F. Deumlich, Instrumentenkunde der vermessungstechnik (VEB Verlag für Bauwesen, 1967).

Dorband, B.

B. Dorband, H. Miller, and H. Gross, Handbook of Optical Systems, Vol. 5: Metrology of Optical Components and Systems (Wiley, 2012).

Eggert, O.

O. Eggert, “Ein Beitrag zur Theorie des Fernrohrs mit Fokussierlinse,” Z. Vermessungswes. 23, 833–841 (1929).

Eliseev, S. V.

S. V. Eliseev, Geodezicheskie Instrumenty a Pribory (Nedra, 1973).

Flügge, J.

J. Flügge, Einführung in die Messung der optischen Grundgrössen (Verlag Braun, 1954).

Gross, H.

B. Dorband, H. Miller, and H. Gross, Handbook of Optical Systems, Vol. 5: Metrology of Optical Components and Systems (Wiley, 2012).

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

Hazra, L.

Hopkins, H. H.

H. H. Hopkins and V. V. Rao, “The systematic design of two component objectives,” Opt. Acta 17, 497–514 (1970).
[CrossRef]

Johnson, R. B.

R. Kingslage and R. B. Johnson, Lens Design Fundamentals (Elsevier, 2010).

Kazamaki, T.

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]

Kingslage, R.

R. Kingslage and R. B. Johnson, Lens Design Fundamentals (Elsevier, 2010).

Kondo, F.

Kreopalova, G. V.

G. V. Kreopalova, N. L. Lazareva, and D. T. Puriajev, Optical Measurements (Maschinostroenie, 1987).

Laikin, M.

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

Lazareva, N. L.

G. V. Kreopalova, N. L. Lazareva, and D. T. Puriajev, Optical Measurements (Maschinostroenie, 1987).

Macdonald, J.

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

Malacara, D.

D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, 2007).

Mikš, A.

Miller, H.

B. Dorband, H. Miller, and H. Gross, Handbook of Optical Systems, Vol. 5: Metrology of Optical Components and Systems (Wiley, 2012).

Novák, J.

Novák, P.

Picht, J.

J. Picht, Mess- und Prüfmethoden der optischen Fertigung (Akademie-Verlag, 1953).

Puriajev, D. T.

G. V. Kreopalova, N. L. Lazareva, and D. T. Puriajev, Optical Measurements (Maschinostroenie, 1987).

Rao, V. V.

H. H. Hopkins and V. V. Rao, “The systematic design of two component objectives,” Opt. Acta 17, 497–514 (1970).
[CrossRef]

Ray, S. F.

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

Roelofs, R.

R. Roelofs, “Fadendistanzmesser mit Innenfokussierung,” Z. Istrumentenkd. 61, 137–1947 (1941).

Schulz, H.

H. Schulz, “Der anallaktische Punkt beim Fernrohr mit innerer Einstellinse,” Z. Istrumentenkd. 56, 357–360 (1936).

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. 45, 2063–2084 (1998).

Smith, W.

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

Sussman, M. H.

Szulc, A.

Uhink, W.

W. Uhink, “Betrachtungen über Fernrohre mit Entfernungmessfäden,” Z. Istrumentenkd. 52, 435–442 (1932).

Welford, W. T.

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

Wild, H.

H. Wild, “Der neue Theodolit,” Schweiz. Z. Vermw. Kulturt. 23, 103–105 (1925).

H. Wild, “Neue Nivellierinstrumente,” Z. Instrumentenkd. 29, 329–344 (1909).

Appl. Opt. (3)

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. 45, 2063–2084 (1998).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Acta (3)

H. H. Hopkins and V. V. Rao, “The systematic design of two component objectives,” Opt. Acta 17, 497–514 (1970).
[CrossRef]

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]

Opt. Express (1)

Schweiz. Z. Vermw. Kulturt. (1)

H. Wild, “Der neue Theodolit,” Schweiz. Z. Vermw. Kulturt. 23, 103–105 (1925).

Z. Instrumentenkd. (1)

H. Wild, “Neue Nivellierinstrumente,” Z. Instrumentenkd. 29, 329–344 (1909).

Z. Istrumentenkd. (3)

W. Uhink, “Betrachtungen über Fernrohre mit Entfernungmessfäden,” Z. Istrumentenkd. 52, 435–442 (1932).

H. Schulz, “Der anallaktische Punkt beim Fernrohr mit innerer Einstellinse,” Z. Istrumentenkd. 56, 357–360 (1936).

R. Roelofs, “Fadendistanzmesser mit Innenfokussierung,” Z. Istrumentenkd. 61, 137–1947 (1941).

Z. Vermessungswes. (1)

O. Eggert, “Ein Beitrag zur Theorie des Fernrohrs mit Fokussierlinse,” Z. Vermessungswes. 23, 833–841 (1929).

Other (17)

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

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

S. V. Eliseev, Geodezicheskie Instrumenty a Pribory (Nedra, 1973).

F. Deumlich, Instrumentenkunde der vermessungstechnik (VEB Verlag für Bauwesen, 1967).

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

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

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

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

R. Kingslage and R. B. Johnson, Lens Design Fundamentals (Elsevier, 2010).

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

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

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

G. V. Kreopalova, N. L. Lazareva, and D. T. Puriajev, Optical Measurements (Maschinostroenie, 1987).

J. Picht, Mess- und Prüfmethoden der optischen Fertigung (Akademie-Verlag, 1953).

J. Flügge, Einführung in die Messung der optischen Grundgrössen (Verlag Braun, 1954).

B. Dorband, H. Miller, and H. Gross, Handbook of Optical Systems, Vol. 5: Metrology of Optical Components and Systems (Wiley, 2012).

D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, 2007).

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Figures (7)

Fig. 1.
Fig. 1.

Optical scheme of two-element optical system (telephoto lens, objective of anallactic telescope).

Fig. 2.
Fig. 2.

Dependence of the length D of the objective on the power φ 1 of the first element for S IV = 0 .

Fig. 3.
Fig. 3.

Dependence of M ¯ 1 , M ¯ 2 , N ¯ 1 , N ¯ 2 on the power φ 1 of the first element of the objective for S I = S II = S III = S IV = S V = 0 .

Fig. 4.
Fig. 4.

Wave aberration and Strehl definition for axial point for wavelengths λ d , λ C and λ F . Image plane is shifted by the value s 0 = 0.12 mm from paraxial image plane.

Fig. 5.
Fig. 5.

Spot diagram for axial point. Image plane is shifted by the value s 0 = 0.12 mm from paraxial image plane. Thin circle shows the Airy disk for wavelength λ d . R g ( d ) , R g ( C ) and R g ( F ) are radii of gyration for wavelengths λ d , λ C and λ F .

Fig. 6.
Fig. 6.

Spot diagram for field angle 2 w = 12 ° . Image plane is shifted by the value s 0 = 0.12 mm from paraxial image plane. Thin circle shows the Airy disk for wavelength λ d . R g ( d ) , R g ( C ) and R g ( F ) are radii of gyration for wavelengths λ d , λ C and λ F .

Fig. 7.
Fig. 7.

Measurement of the position of the focal points and EFLs.

Tables (2)

Tables Icon

Table 1. Parameters of the First Element of the Objective

Tables Icon

Table 2. Parameters of the Second Element of the Objective

Equations (33)

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

a F = f ( 1 d / f 2 ) , a F = f ( 1 d / f 1 ) .
f = f 1 f 2 f 1 + f 2 d ,
m = y y = q f = f q ,
a 1 = f ( 1 m 1 + d f 2 ) , a 2 = f ( 1 m d f 1 ) .
δ y = y P ( y P 2 + x P 2 ) 2 ( s 1 s ¯ 1 ) 3 u 1 3 u K S I + y 1 ( 3 y P 2 + x P 2 ) 2 ( s 1 s ¯ 1 ) 3 u 1 2 u K u ¯ 1 S II y 1 2 y P 2 ( s 1 s ¯ 1 ) 3 u 1 u K u ¯ 1 2 ( 3 S III + I 2 S IV ) + y 1 3 2 ( s 1 s ¯ 1 ) 3 u K u ¯ 1 3 S V , δ x = x P ( y P 2 + x P 2 ) 2 ( s 1 s ¯ 1 ) 3 u 1 3 u K S I + 2 y 1 y P x P 2 ( s 1 s ¯ 1 ) 3 u 1 2 u K u ¯ 1 S II y 1 2 y P 2 ( s 1 s ¯ 1 ) 3 u 1 u K u ¯ 1 2 ( S III + I 2 S IV ) ,
I = h 1 h ¯ 1 ( 1 s 1 1 s ¯ 1 ) = u 1 h ¯ 1 u ¯ 1 h 1 .
h ¯ 1 = s 1 s ¯ 1 s ¯ 1 s 1 ,
h ¯ j = h j ( h ¯ 1 + i = 2 j d i 1 h i 1 h i ) ,
S I = i = 1 K h i 4 M i , S II = i = 1 K h i 3 h ¯ i M i + i = 1 K h i 2 N i , S III = i = 1 K h i 2 h ¯ i 2 M i + 2 i = 1 K h i h ¯ i N i + i = 1 K φ i , S IV = i = 1 K p i φ i , S V = i = 1 K h i h ¯ i 3 M i + 3 i = 1 K h ¯ i 2 N i + i = 1 K h ¯ i h i ( 3 + p i ) φ i .
( S IV ) i = φ i / n i = j = 1 L i φ j / n j = j = 1 L i p j φ j = ( j = 1 L i φ j ) / n i ,
n i = j = 1 L i φ j j = 1 L i φ j / n j .
C I = i = 1 K h i 2 φ i ν i P i λ ,
C II = i = 1 K h i h ¯ i φ i ν i P i λ ,
ν = n d 1 n F n C , P = λ n F n λ n F n C ,
φ 1 = φ 2 , d = 1 / φ 1 2 , M 1 = 2 φ 1 3 ( φ 1 1 ) 2 ( p + 3 ) , N 1 = φ 1 2 ( φ 1 1 ) ( p + 3 ) , M 2 = 2 φ 1 7 ( p + 3 ) / ( φ 1 1 ) 2 , N 2 = φ 1 4 ( p + 3 ) / ( φ 1 1 ) .
φ = 1 , d = ( φ 1 + φ 2 φ ) / ( φ 1 φ 2 ) = 1 / φ 1 2 , h 1 = 1 , h 2 = 1 d φ 1 , a 2 = h 2 , h ¯ 1 = 0 , h ¯ 2 = d , or h ¯ 1 = d / ( 1 d φ 1 ) , h ¯ 2 = 0 .
D = d + a 2 = 1 + d ( 1 φ 1 ) = 1 + ( 1 φ 1 ) / φ 1 2 .
M i = φ i 3 [ M ¯ i + 2 N ¯ i Y i + ( 3 / 4 + p i / 2 ) Y i 2 ] , N i = φ i 2 [ N ¯ i + ( 1 + p i / 2 ) Y i ] , M ¯ i = f i 3 M i 2 f i 2 N i Y i + ( 5 / 4 + p i / 2 ) Y i 2 , N ¯ i = f i 2 N i ( 1 + p i / 2 ) Y i .
M i = φ i 3 [ M ¯ i + 2 N ¯ i Y i + 1.06 Y i 2 ] , N i = φ i 2 ( N ¯ i + 1.31 Y i ) , M ¯ i = f i 3 M i 2 f i 2 N i Y i + 1.56 Y i 2 , N ¯ i = f i 2 N i 1.31 Y i ,
Y i = s i + s i s i s i = m i + 1 m i 1 = 1 2 s i φ i , Y i + 1 = h i φ i h i + 1 φ i + 1 ( Y i 1 ) 1 ,
M ¯ 1 = 7.24 φ 1 ( φ 1 1 ) + 1.56 , N ¯ 1 = 3.62 φ 1 2.31 , M ¯ 2 = 7.24 φ 1 3 ( φ 1 1 ) 5.68 φ 1 2 + 3.12 φ 1 + 1.56 ( φ 1 1 ) 2 , N ¯ 2 = 3.62 φ 1 2 1.31 ( φ 1 + 1 ) ( φ 1 1 ) .
R g = i = 1 Q ( δ x 2 + δ y 2 ) Q ,
f = y y f K ,
Measured values: s F , s F , f . Calculations: α = a F / f s F / f , β = a F / f s F / f , d = f ( 1 + α β ) , f 1 = d / ( 1 α ) , f 2 = d / ( 1 + β ) .
a F = f ( 1 d / f 1 ) , f = f 1 f 2 f 1 + f 2 d .
δ = f I s F II f II s F I + e f I f II ( s F I s F II ) f I f II .
a I = a F I = s F I δ , a II = a F II = s F II δ ,
A = a I a II f I / f II , d = e ( f I a I ) A , f 1 = e f I A , f 2 = e a I e A .
Δ d = ( D 1 Δ f I ) 2 + ( D 2 Δ f II ) 2 + ( D 3 Δ e ) 2 + ( D 4 Δ a I ) 2 + ( D 5 Δ a II ) 2 ,
Δ f 1 = ( D 6 Δ f I ) 2 + ( D 7 Δ f II ) 2 + ( D 8 Δ e ) 2 + ( D 9 Δ a I ) 2 + ( D 10 Δ a II ) 2 ,
Δ f 2 = ( D 11 Δ f I ) 2 + ( D 12 Δ f II ) 2 + ( D 13 Δ e ) 2 + ( D 14 Δ a I ) 2 + ( D 15 Δ a II ) 2 ,
B = e ( a I f I ) / A , C = 1 / ( e A ) D 1 = d f I = e A a II B f II A , D 2 = d f II = B ( a I / A 1 ) f II , D 3 = d e = f I a I A , D 4 = d a I = B e A , D 5 = d a II = f I B f II A , D 6 = f 1 f I = 1 + f I a II / ( f II A ) A e , D 7 = f 1 f I = ( 1 a I / A ) e f I f II A , D 8 = f 1 e = f I A , D 9 = f 1 a I = e f I A 2 , D 10 = f 1 a II = e f I 2 f II A 2 , D 11 = f 2 f I = e a I a II C 2 f II , D 12 = f 2 f II = e a I C f II [ 1 + ( a I e ) C ] , D 13 = f 2 e = a I C ( 1 e C ) , D 14 = f 2 a I = e C ( 1 + a I C ) , D 15 = f 2 a II = e a I f I C 2 f II .
Δ d = 14.3 Δ , Δ f 1 = 17.3 Δ , Δ f 2 = 12.4 Δ .

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