Abstract

The geometrical propagation of a beam of light is treated in this paper using aberration theory. Stop shifting on the aberration function and on the irradiance function provides insight into how the geometrical field changes as it propagates in free space. The formulae discussed in this paper give as a function of the field and aperture of an optical system, and to the sixth order of approximation, the wavefront deformation and, to the fourth order, the irradiance at the exit pupil plane of the system. The use of the formulae is illustrated with a lens design for uniform illumination that is nearly insensitive to the location of the illuminated surface.

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References

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  1. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Robert & Company, 2005).
  2. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978).
  3. J. Sasian, Introduction to Aberrations in Optical Imaging Systems (Cambridge University, 2013).
  4. J. Sasian, “Theory of sixth-order wave aberrations,” Appl. Opt. 49, D69–D93 (2010).
    [Crossref]
  5. A. E. Conrady, Applied Optics and Optical Design, Part II (Dover, 1985).
  6. D. Reshidko and J. Sasian, “Geometrical irradiance changes in a symmetrical optical system,” Opt. Eng. 56, 015104 (2017).
    [Crossref]
  7. W. R. Hamilton, “Theory of systems of rays,” Trans. R. Irish Acad. 15, 69–174 (1828).
  8. W. R. Hamilton, “Supplement to an essay on the theory of systems of rays,” Trans. R. Irish Acad. 16, 1–61 (1830).
  9. M. P. Rimmer, “Optical aberration coefficients,” M.S. thesis (University of Rochester, 1963).

2017 (1)

D. Reshidko and J. Sasian, “Geometrical irradiance changes in a symmetrical optical system,” Opt. Eng. 56, 015104 (2017).
[Crossref]

2010 (1)

1830 (1)

W. R. Hamilton, “Supplement to an essay on the theory of systems of rays,” Trans. R. Irish Acad. 16, 1–61 (1830).

1828 (1)

W. R. Hamilton, “Theory of systems of rays,” Trans. R. Irish Acad. 15, 69–174 (1828).

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design, Part II (Dover, 1985).

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Robert & Company, 2005).

Hamilton, W. R.

W. R. Hamilton, “Supplement to an essay on the theory of systems of rays,” Trans. R. Irish Acad. 16, 1–61 (1830).

W. R. Hamilton, “Theory of systems of rays,” Trans. R. Irish Acad. 15, 69–174 (1828).

Reshidko, D.

D. Reshidko and J. Sasian, “Geometrical irradiance changes in a symmetrical optical system,” Opt. Eng. 56, 015104 (2017).
[Crossref]

Rimmer, M. P.

M. P. Rimmer, “Optical aberration coefficients,” M.S. thesis (University of Rochester, 1963).

Sasian, J.

D. Reshidko and J. Sasian, “Geometrical irradiance changes in a symmetrical optical system,” Opt. Eng. 56, 015104 (2017).
[Crossref]

J. Sasian, “Theory of sixth-order wave aberrations,” Appl. Opt. 49, D69–D93 (2010).
[Crossref]

J. Sasian, Introduction to Aberrations in Optical Imaging Systems (Cambridge University, 2013).

Appl. Opt. (1)

Opt. Eng. (1)

D. Reshidko and J. Sasian, “Geometrical irradiance changes in a symmetrical optical system,” Opt. Eng. 56, 015104 (2017).
[Crossref]

Trans. R. Irish Acad. (2)

W. R. Hamilton, “Theory of systems of rays,” Trans. R. Irish Acad. 15, 69–174 (1828).

W. R. Hamilton, “Supplement to an essay on the theory of systems of rays,” Trans. R. Irish Acad. 16, 1–61 (1830).

Other (5)

M. P. Rimmer, “Optical aberration coefficients,” M.S. thesis (University of Rochester, 1963).

A. E. Conrady, Applied Optics and Optical Design, Part II (Dover, 1985).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Robert & Company, 2005).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978).

J. Sasian, Introduction to Aberrations in Optical Imaging Systems (Cambridge University, 2013).

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Figures (5)

Fig. 1.
Fig. 1. Wavefront deformation forms of orders zeroth, second, fourth, and sixth order, with respect to the reference sphere.
Fig. 2.
Fig. 2. Ray in bold is described at the old pupil by ${y}_{\rm old}\vec{\rho}+{\bar{y}}_{\rm old}\vec{H}$ and the aberration function $W(\vec{H},\frac{{y}_{\rm old}\vec{\rho}+{\bar{y}}_{\rm old}\vec{H}}{{y}_{\rm old}})$ gives its optical path difference. At the new exit pupil the ray in bold is described by $\vec{\rho}$ and its optical path difference is given by ${W}^{\ast}(\vec{H},\vec{\rho})$. The difference between the aberration functions is the wavefront deformation from free space propagation ${\Delta}_{Z}W(\vec{H},\vec{\rho})$.
Fig. 3.
Fig. 3. Graphical display of the irradiance aberrations as they depend on the aperture vector.
Fig. 4.
Fig. 4. Lens singlet for uniform illumination at a target surface 25 mm from the lens.
Fig. 5.
Fig. 5. Relative illumination vs. position at several screen distances from the lens: 25 mm, 50 mm, 75, and 100 mm.

Tables (6)

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Table 1. Stop Shifting Formulas for Fourth-Order Aberration Coefficients

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Table 2. Recursive Stop Shifting Formulas for Fourth-Order Aberration Coefficients

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Table 3. Recursive Stop Shifting Formulas for Sixth-Order Aberration Coefficients

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Table 4. Second- and Fourth-Order Terms of the Irradiance Function at the Exit Pupil

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Table 5. Second-Order Terms of the Irradiance Function with Stop Shifting

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Table 6. Lens for Uniform Illumination Prescription

Equations (45)

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G ( H , ρ ) = I ¯ 0 I ¯ ( H , ρ ) exp { i 2 π λ [ S ( H , ρ ) + W ( H , ρ ) ] } ,
S ( H , ρ ) = R 0 cos ( θ ) R 0 2 + y ¯ i 2 ( H H ) ,
S ( H , ρ ) = 1 2 Ж ( u u ¯ ) ( ρ ρ ) Ж ( H ρ ) 1 2 Ж ( u ¯ u ) ( H H ) ,
I ¯ ( H , ρ ) = cos 4 ( θ ) = 1 2 u ¯ 2 ( H H ) 2 u 2 ( ρ ρ ) 4 u u ¯ ( H ρ ) + 3 u ¯ 4 ( H H ) 2 + 12 u 3 u ¯ ( H ρ ) ( ρ ρ ) + 12 u 2 u ¯ 2 ( H ρ ) 2 + 6 u 2 u ¯ 2 ( H H ) ( ρ ρ ) + 12 u u ¯ 3 ( H H ) ( H ρ ) + 3 u 4 ( ρ ρ ) 2 + .
W ( H , ρ ) = W 000 + W 200 ( H H ) + W 111 ( H ρ ) + W 020 ( ρ ρ ) + W 040 ( ρ ρ ) 2 + W 131 ( H ρ ) ( ρ ρ ) + W 222 ( H ρ ) 2 + W 220 ( H H ) ( ρ ρ ) + W 311 ( H H ) ( H ρ ) + W 400 ( H H ) 2 + W 240 ( H H ) ( ρ ρ ) 2 + W 331 ( H H ) ( H ρ ) ( ρ ρ ) + W 422 ( H H ) ( H ρ ) 2 + W 420 ( H H ) 2 ( ρ ρ ) + W 511 ( H H ) 2 ( H ρ ) + W 600 ( H H ) 3 + W 060 ( ρ ρ ) 3 + W 151 ( H ρ ) ( ρ ρ ) 2 + W 242 ( H ρ ) 2 ( ρ ρ ) + W 333 ( H ρ ) 3 ,
Δ Z W ( H , ρ ) = 1 2 S ¯ Ж ρ W ( H , ρ ) ρ W ( H , ρ ) ,
S ¯ = y ¯ n e w y ¯ o l d y = u ¯ n e w u ¯ o l d u .
W ( H , ρ ) = W ( H , ρ + S ¯ H ) + 1 2 S ¯ Ж ρ W ( H , ρ + S ¯ H ) ρ W ( H , ρ + S ¯ H ) .
I ¯ ( H , ρ ) = I ¯ 000 + I ¯ 200 ( ρ ρ ) + I ¯ 111 ( H ρ ) + I ¯ 020 ( H H ) + I ¯ 040 ( H H ) 2 + I ¯ 131 ( H H ) ( H ρ ) + I ¯ 222 ( H ρ ) 2 + I ¯ 220 ( H H ) ( ρ ρ ) + I ¯ 311 ( H ρ ) ( ρ ρ ) + I ¯ 400 ( ρ ρ ) 2 ,
d Φ 2 = L o n 2 dA dS R 0 2 cos 4 ( θ ) ,
d I ¯ ( H , ρ ) = d Φ 2 dS = L o n 2 dA R 0 2 cos 4 ( θ ) = L o dA n 2 R 0 2 dA dA cos 4 ( θ ) = d I ¯ 0 dA dA cos 4 ( θ ) ,
y ¯ i ( H + Δ H ) = y ¯ i ( ( H r + Δ H r ) r + ( H t + Δ H t ) t ) ,
J ( H , ρ ) = y ¯ i 2 y ¯ o 2 ( 1 + Δ H r H r + Δ H t H t + Δ H r H r Δ H t H t Δ H r H t Δ H t H r ) .
I ¯ ( H , ρ ) = cos 4 ( θ ) = 1 2 u ¯ 2 ( H H ) 2 u 2 ( ρ ρ ) 4 u u ¯ ( H ρ ) + 3 u ¯ 4 ( H H ) 2 + 12 u 3 u ¯ ( H ρ ) ( ρ ρ ) + 12 u 2 u ¯ 2 ( H ρ ) 2 + 6 u 2 u ¯ 2 ( H H ) ( ρ ρ ) + 12 u u ¯ 3 ( H H ) ( H ρ ) + 3 u 4 ( ρ ρ ) 2 + . . . .
I ¯ 200 = 2 u 2 4 Ж W 131 = 0 ,
I ¯ 400 = 3 u 4 6 Ж W 151 + 3 Ж W 131 u 2 + 3 Ж 2 W 131 W 131 = 0 ,
W 151 = W 131 = u = 0.
Δ W 060 = 1 2 S ¯ Ж ( 16 W 040 W 040 ) ,
Δ W 151 = 1 2 S ¯ Ж ( 24 W 040 W 131 ) ,
Δ W 242 = 1 2 S ¯ Ж ( 16 W 040 W 222 + 8 W 131 W 131 ) ,
Δ W 240 = 1 2 S ¯ Ж ( 16 W 040 W 220 + W 131 W 131 ) ,
Δ W 333 = 1 2 S ¯ Ж ( 8 W 131 W 222 ) ,
Δ W 331 = 1 2 S ¯ Ж ( 8 W 040 W 311 + 4 W 131 W 222 + 12 W 131 W 220 ) ,
Δ W 422 = 1 2 S ¯ Ж ( 4 W 131 W 311 + 4 W 222 W 222 + 8 W 222 W 220 ) ,
Δ W 420 = 1 2 S ¯ Ж ( 4 W 220 W 220 + 2 W 311 W 131 ) ,
Δ W 511 = 1 2 S ¯ Ж ( 4 W 222 W 311 + 4 W 220 W 311 ) ,
Δ W 600 = 1 2 S ¯ Ж ( W 311 W 311 ) .
V ( H , ρ ) = S ( H , ρ ) + W ( H , ρ ) + cte ,
L = V ( H , ρ ) n x and M = V ( H , ρ ) n y .
y s ρ x + V ( H , ρ ) n x R and y s ρ y + V ( H , ρ ) n y R ,
ε x = y s ρ x + V ( H , ρ ) n x R y ¯ i H x , ε y = y s ρ y + V ( H , ρ ) n y R y ¯ i H y ,
ε = y s ρ y ¯ i H + R n ( ρ W ( H , ρ ) + ρ S ( H , ρ ) ) .
R = R 0 2 + ( y ¯ i H y + ε y y s ρ y ) 2 + ( y ¯ i H x + ε x y s ρ x ) 2 R 0 + R 0 2 ( u 2 ( ρ ρ ) + 2 u u ¯ ( H ρ ) + u ¯ 2 ( H H ) + 2 n u ρ W ( H ρ ) ( u ρ + u ¯ H ) ) = R 1 + R 2 ,
R 1 = R 0 + R 0 2 ( u 2 ( ρ ρ ) + 2 u u ¯ ( H ρ ) + u ¯ 2 ( H H ) ) ,
R 2 = R 0 n u ρ W ( H ρ ) ( u ρ + u ¯ H ) .
ε = y s ρ y ¯ i H + R 1 n ρ S ( H , ρ ) + R 1 + R 2 n ρ W ( H , ρ ) + R 2 n ρ S ( H , ρ ) .
ε = y s ρ y ¯ i H + R 1 n ρ S ( H , ρ ) = 0 ,
ρ W ( H , ρ ) = 1 R 0 u ρ W 6 ( H , ρ ) + 1 R 0 u ρ W 4 ( H , ρ ) ,
1 n ρ S ( H , ρ ) = u ρ + u ¯ H ,
ε = 1 n u ρ W 6 ( H , ρ ) + 1 n u ρ W 4 ( H , ρ ) + O 5 ( H , ρ ) ,
O ( 5 ) ( H , ρ ) = 1 2 n u ( u 2 ( ρ ρ ) + 2 u u ¯ ( H ρ ) + u ¯ 2 ( H H ) ) ρ W 4 ( H , ρ ) + 1 n u ( ρ W 4 ( H ρ ) ( u ρ + u ¯ H ) ) × ( u ρ + u ¯ H ) .
ε ( 3 ) = 1 n u ( 4 W 040 ( ρ ρ ) ρ + W 131 ( ρ ρ ) H + 2 W 131 ( H ρ ) ρ + 2 W 222 ( H ρ ) H + 2 W 220 ( H H ) ρ + W 311 ( H H ) H ) ,
ε ( 5 ) = 1 n u ( 6 W 060 ( ρ ρ ) 2 ρ + W 151 ( ρ ρ ) 2 H + 4 W 151 ( H ρ ) ( ρ ρ ) ρ + 3 W 333 ( H ρ ) 2 H + W 331 ( H H ) ( ρ ρ ) H + 2 W 331 ( H H ) ( H ρ ) ρ + 2 W 242 ( H ρ ) 2 ρ + 2 W 242 ( H ρ ) ( ρ ρ ) H + 4 W 240 ( H H ) ( ρ ρ ) ρ + 2 W 422 ( H H ) ( H ρ ) H + 2 W 420 ( H H ) 2 ρ + W 511 ( H H ) 2 H + O ( 5 ) ( H , ρ ) ) ,
O ( 5 ) ( H , ρ ) = { ( 4 u u ¯ W 040 + 1 2 u 2 W 131 ) ( ρ ρ ) 2 H + ( 4 u u ¯ W 131 + 4 u ¯ 2 W 040 + u 2 W 222 ) ( H ρ ) ( ρ ρ ) H + ( 4 u u ¯ W 222 + 2 u ¯ 2 W 131 ) ( H ρ ) 2 H + ( 2 u u ¯ W 220 + 3 2 u ¯ 2 W 131 + 1 2 u 2 W 311 ) ( H H ) ( ρ ρ ) H + ( 2 u u ¯ W 311 + 2 u ¯ 2 W 220 + 3 u ¯ 2 W 222 ) ( H H ) ( H ρ ) H + 3 2 u ¯ 2 W 311 ( H H ) 2 H + 6 u 2 W 040 ( ρ ρ ) 2 ρ + ( 4 u 2 W 131 + 8 u u ¯ W 040 ) ( H ρ ) ( ρ ρ ) ρ + ( 2 u 2 W 222 + 4 u u ¯ W 131 ) ( H ρ ) 2 ρ + ( 3 u 2 W 220 + u u ¯ W 131 + 2 u ¯ 2 W 040 ) ( H H ) ( ρ ρ ) ρ + ( u 2 W 311 + 4 u u ¯ W 220 + u ¯ 2 W 131 + 2 u u ¯ W 222 ) ( H H ) ( H ρ ) ρ + ( u u ¯ W 311 + u ¯ 2 W 220 ) ( H H ) 2 ρ } .
ε = y ¯ i Ж Δ H = A H + B ρ .

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