Abstract

Some of the questions concerning secondary chromatic aberration at both sides of the visible band of the spectrum are the following: (1) What is the bandwidth at different wavelengths, given the permissible chromatic aberration circle and the lens aperture? (2) What is the size of the chromatic aberration circle, given the wavelength, the bandwidth, and the lens aperture? The answers to these and other questions may be found with the new definitions of V-number and relative partial dispersion P based on infinitesimal bandwidths that we propose. In addition, an alignment chart for the secondary color of a normal glass doublet is presented, so fast answers to the questions posed above and to other questions concerned with secondary color can be found. In addition, a continual challenge in computer-aided lens design is the use of optical glasses as design parameters in simultaneous optimization of lens systems over various regions of the spectrum. This problem could be solved if we could find an ideal glass family, not too different from real glasses, such that, given the refractive index n and the V-number at any wavelength, the indices at all wavelengths could be determined. Therefore we derive a differential equation for normal glass dispersion and present a recursive solution.

© 1999 Optical Society of America

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References

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  1. J. Hoogland, “The design of apochromatic lenses,” in Recent Developments in Optical Design, R. A. Ruhloff, ed. (Perkin-Elmer, Norwalk, Conn., 1968), pp. 6–1 and 6–8.
  2. Laurin Publications, ed., The Photonics Design, Book 3 of The Photonics Design and Applications Handbook, 44th international ed. (Laurin Publications, Pittsfield, Mass., 1998), pp. H-384 and H-385.
  3. W. Smith, Modern Lens Design: A Resource Manual (McGraw-Hill, New York, 1992).
  4. W. Besenmatter, “How many glass types does a lens designer really need?” in International Optical Design Conference 1998, L. R. Gardner, K. P. Thompson, eds., Proc. SPIE3482, 294–305 (1998).
    [CrossRef]
  5. R. Bellman, Introduction to Matrix AnalysisMcGraw-Hill, New York, 1965), Spanish version, p. 208.

Bellman, R.

R. Bellman, Introduction to Matrix AnalysisMcGraw-Hill, New York, 1965), Spanish version, p. 208.

Besenmatter, W.

W. Besenmatter, “How many glass types does a lens designer really need?” in International Optical Design Conference 1998, L. R. Gardner, K. P. Thompson, eds., Proc. SPIE3482, 294–305 (1998).
[CrossRef]

Hoogland, J.

J. Hoogland, “The design of apochromatic lenses,” in Recent Developments in Optical Design, R. A. Ruhloff, ed. (Perkin-Elmer, Norwalk, Conn., 1968), pp. 6–1 and 6–8.

Smith, W.

W. Smith, Modern Lens Design: A Resource Manual (McGraw-Hill, New York, 1992).

Other (5)

J. Hoogland, “The design of apochromatic lenses,” in Recent Developments in Optical Design, R. A. Ruhloff, ed. (Perkin-Elmer, Norwalk, Conn., 1968), pp. 6–1 and 6–8.

Laurin Publications, ed., The Photonics Design, Book 3 of The Photonics Design and Applications Handbook, 44th international ed. (Laurin Publications, Pittsfield, Mass., 1998), pp. H-384 and H-385.

W. Smith, Modern Lens Design: A Resource Manual (McGraw-Hill, New York, 1992).

W. Besenmatter, “How many glass types does a lens designer really need?” in International Optical Design Conference 1998, L. R. Gardner, K. P. Thompson, eds., Proc. SPIE3482, 294–305 (1998).
[CrossRef]

R. Bellman, Introduction to Matrix AnalysisMcGraw-Hill, New York, 1965), Spanish version, p. 208.

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Figures (7)

Fig. 1
Fig. 1

Refractive index versus wavelength.

Fig. 2
Fig. 2

Normal line parameters A and B versus wavelength.

Fig. 3
Fig. 3

Differential relative partial dispersion λ versus differential V-number λ for four wavelengths.

Fig. 4
Fig. 4

(a) Primary color of a singlet given by Δf P , the axial distance between the foci of wavelengths λ1 and λ2. (b) Secondary color of achromatic doublets given by Δf S , the axial distance between the common paraxial focus of wavelengths λ1 and λ2 and the paraxial focus of wavelength λ0.

Fig. 5
Fig. 5

Hoogland’s apochromatic diagram.

Fig. 6
Fig. 6

Alignment chart for the secondary color of a normal glass doublet.

Fig. 7
Fig. 7

Glass triangle for the differential I/ V-number at several wavelengths over the IR-to-near-UV spectrum.

Tables (4)

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Table 1 Parameters A(λ) and B(λ) of Abbe’s Line Normal for 256 Glasses Computed by Least Squares

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Table 2 Statistics of 286 Glass Types at 13 Wavelengths

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Table 3 Fractional Secondary Color Δf/f for Various Types of Lens Systems Designed with Normal Glasses in the Spectral Band F′–C

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Table 4 Comparison of the Refractive Index and its First Derivative Evaluated from the Sellmeier Formulas and the Recursive Solution

Equations (86)

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Vd=nd-1nF-nC,  PF,d=nF-ndnF-nC,
Ve=ne-1nF-nC,
Px,y=nx-nynF-nC.
Px,y=ax,yVd+bx,y.
V=n0-1n1-n2,  P=n1-n0n1-n2,
n1=n0-dndλ Δλ+12!d2ndλ2Δλ2-13!d3ndλ3Δλ3+,
n2=n0+dndλ Δλ+12!d2ndλ2Δλ2+13!d3ndλ3Δλ3+;
n1-n0=-dndλ Δλ+12!d2ndλ2Δλ2-13!d3ndλ3Δλ3+,
n1-n2=-2 dndλ Δλ-23!d3ndλ3Δλ3+.
V=-12n0-1dndλ Δλ.
Vˆλ=-12n0-1dndλ;
V1Δλ Vˆλ.
P=-dndλ Δλ+12!d2ndλ2Δλ2-13!d3ndλ3Δλ3+-2 dndλ Δλ-23!d3ndλ3Δλ3-.
P=12-14d2ndλ2dndλ Δλ+.
Pˆλ=-14d2ndλ2dndλ,
P12+PˆλΔλ.
P=aV+b.
Pˆλ=aΔλ2 Vˆλ+2b-12Δλ.
Aλ=limΔλ0aΔλ2,  Bλ=limΔλ02b-12Δλ.
aAλΔλ2,  b½+BΔλ.
Pˆλ=AλVˆλ+Bλ.
ΔfP=fV,  ΔfPfVˆλ Δλ
DCDP=HCAV,  DCDPHCAVˆλ Δλ.
ΔλVˆλDCDPHCA.
KαVα+KβVβ=0.
ΔfS=Pα-PβVα-Vβ f;
ΔfSPˆλ,α-Pˆλ,βVˆλ,α-Vˆλ,β fΔλ2,
ΔfSf=aAλΔλ2.
DCDSHCA=|a||Aλ|Δλ2.
Δλ=DCDHCA|Aλ|1/2.
Pˆλ,α-Pˆλ,βVˆλ,α-Vˆλ,β=0.
Aˆλ=0.
Pˆλ,α=Pˆλ,β,  Vˆλ,αVˆλ,β,
xα=Vα,  yα=Pα,
y=ax+b.
x*=1x,  y*=1xy-k,
x=1x*,  y=y*x*+k,
y*=a+bx*.
y*=a.
xα*=1Vα, yα*=1VαPα-b,  yα*xα*=Pα-b.
x*=1Vˆλ,  y*=Pˆλ-BλVˆλ.
ΔPˆα=Pˆα-AVˆα+B.
ε=AVˆλ+B-PˆλA2+1;
 εi2=min.
d2nλdλ2+4Bλdnλdλ-2nλ-1Aλ=0
d2nλdλ2+4Bλdnλdλ-2Aλnλ=-2Aλ.
nλi=124Bi-12+2Ai-11/2exp λ1i-1-exp λ2i-1×dni-1dλ+Bi-14Bi-12+2Ai-11/2exp λ1i-1-exp λ2i-1+12exp λ1i-1+exp λ2i-1ni-1-Bi-14Bi-12+2Ai-11/2exp λ1i-1-exp λ2i-1+12exp λ1i-1+exp λ2i-1+1,
dnλidλ=-Bi-14Bi-12+2Ai-11/2exp λ1i-1-exp λ2i-1+12exp λ1i-1+exp λ2i-1dni-1dλ+Ai-14Bi-12+2Ai-11/2exp λ1i-1-exp λ2i-1ni-1-Ai-14Bi-12+2Ai-11/2×exp λ1i-1-exp λ2i-1,
λ1i-1=λi-λi-1-2Bi-1+4Bi-12+2Ai-11/2,
λ2i-1=λi-λi-1-2Bi-1-4Bi-12+2Ai-11/2.
nλ1=1.72273,  dndλλ=λ1=-0.601849.
x1λ=nλ,
x2λ=dnλdλ;
x2λ=x˙1λ.
x˙2λ=-4Bλx2λ+2Aλx1λ-2Aλ.
x˙2λx˙1λ=-4Bλ 2Aλ1   0x2λx1λ+-2Aλ0.
x˙λ=Hλxλ+fλ,
Hλ=-4Bλ 2Aλ1   0.
xλ=Φλ, λ0xλ0+λ0λ Φλ, τfτdτ,
Φλ, λ0=limk Mkλ,
M0=I,
Mkλ=I+λ0λ HσMk-1σdσ,
Hλλ0λ Hτdτ=λ0λ HτdτHλ,
Φλ, λ0=expλ0λ Hσdσ.
Φλ, λi-1=expHλi-1λ-λi-1,  λλi-1, λi
xλ=expHλi-1λ-λi-1xλi-1+λi-1λ expHλi-1×λ-τfλi-1dτ.
xixλi=expHλi-1λi-λi-1xλi-1+λi-1λ expHλi-1λi-τfλi-1dτ=expHλi-1λi-λi-1xλi-1+expHλi-1×λi-τ|λi-1λi-H-1λi-1fλi-1.
xi=expHλi-1λi-λi-1xλi-1+expHλi-1λi-λi-1-IH-1λi-1fλi-1.
H-1=011/2Aλ2Bλ/Aλ.
x2λix1λi=exp-4Bλi-1λi-λi-12Aλi-1λi-λi-1λi-λi-10x2λi-1χ1λi-1+exp-4Bλi-1λi-λi-12Aλi-1λi-λi-1λi-λi-10-1001011/2Aλi-12Bλi-1/Aλi-1-2Aλi-10.
dndλλinλi=exp-4Bλi-1λi-λi-12Aλi-1λi-λi-1λi-λi-10dndλλi-1nλi-1+exp-4Bλi-1λi-λi-12Aλi-1λi-λi-1λi-λi-10-1001011/2Aλi-12Bλi-1/Aλi-1-2Aλi-10.
Aλi-1=Ai-1,  Bλi-1=Bi-1,  Hλi-1=Hi-1.
detH-λI=0.
T-1HT=λ100λ2=D.
H=-4B2Ac0,
A=λ-λi-1Ai-1=cAi-1, B=λ-λi-1Bi-1=cBi-1.
λ1=c-2B+4B2+2A1/2,
λ2=c-2B-4B2+2A1/2.
Hw¯=λw¯.
w¯1=w11w21=-2B+4B2+2A1/21+8B2+2A-4B4B2+2A1/21/2×11+8B2+2A-4B4B2+2A1/21/2,
w¯2=w12w22=-2B-4B2+2A1/21+8B2+2A+4B4B2+2A1/21/2×11+8B2+2A+4B4B2+2A1/21/2.
T=w¯1, w¯2.
expH=expTDT-1=I+i=1TDT-1ii!=I+i=1TDiT-1i!=Ti=0Dii!T-1=TexpDT-1.
expH=Texp λ100exp λ2T-1=w11w12w21w22exp λ100exp λ2w11w12w21w22-1.
expHi-1λi-λi-1=-Bi-14Bi-12+2Ai-11/2exp λ1i-1-exp λ2i-1+½exp λ1i-1+exp λ2i-12Ai-124Bi-12+2Ai-11/2exp λ1i-1-exp λ2i-1124Bi-12+2Ai-11/2exp λ1i-1-exp λ2i-1Bi-14Bi-12+2Ai-11/2exp λ1i-1-exp λ2i-1+½exp λ1i-1+exp λ2i-1.
λ1i-1=λi-λi-1-2Bi-1+4Bi-12+2Ai-11/2, λ2i-1=λi-λi-1-2Bi-1-4Bi-12+2Ai-11/2.

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