Abstract

To represent the freeform surface shape, the axially asymmetric quadric and a new set of the orthogonal polynomials are introduced. In this representation, surface tilt, paraxial properties, and higher order surface shape are clearly separated. With this representation, the optimization process can be simple and efficient.

© 2012 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon, 1975).
  2. G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express 20, 2483–2499 (2012).
    [CrossRef]
  3. P. Jester, C. Menke, and K. Urban, “B-spline representation of optical surfaces and its accuracy in a ray trace algorithm,” Appl. Opt. 50, 822–828 (2011).
    [CrossRef]
  4. K. H. Fuerschbach, K. P. Thompson, and J. P. Rolland, “A new generation of optical systems with ϕ-polynomial surfaces,” Proc. SPIE 7652, 76520C (2010).
    [CrossRef]
  5. K. H. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Design with ϕ-polynomial surfaces,” Proc. SPIE 8167, 81670Z (2011).
    [CrossRef]
  6. A. Yabe, “Sensitivity control to surface irregularity,” Proc. SPIE 6342, 634225 (2006).
    [CrossRef]
  7. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15, 5218–5226 (2007).
    [CrossRef]
  8. R. N. Youngworth, “Tolerancing Forbes aspheres: advantage of an orthogonal basis,” Proc. SPIE 7433, 74330H (2009).
    [CrossRef]
  9. A. Yabe, “Construction method of axially asymmetric lenses,” Appl. Opt. 50, 3369–3374 (2011).
    [CrossRef]
  10. A. Yabe, “Method to allocate freeform surfaces in axially asymmetric optical systems,” Proc. SPIE 8167, 816703 (2011).
    [CrossRef]

2012

2011

K. H. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Design with ϕ-polynomial surfaces,” Proc. SPIE 8167, 81670Z (2011).
[CrossRef]

A. Yabe, “Method to allocate freeform surfaces in axially asymmetric optical systems,” Proc. SPIE 8167, 816703 (2011).
[CrossRef]

P. Jester, C. Menke, and K. Urban, “B-spline representation of optical surfaces and its accuracy in a ray trace algorithm,” Appl. Opt. 50, 822–828 (2011).
[CrossRef]

A. Yabe, “Construction method of axially asymmetric lenses,” Appl. Opt. 50, 3369–3374 (2011).
[CrossRef]

2010

K. H. Fuerschbach, K. P. Thompson, and J. P. Rolland, “A new generation of optical systems with ϕ-polynomial surfaces,” Proc. SPIE 7652, 76520C (2010).
[CrossRef]

2009

R. N. Youngworth, “Tolerancing Forbes aspheres: advantage of an orthogonal basis,” Proc. SPIE 7433, 74330H (2009).
[CrossRef]

2007

2006

A. Yabe, “Sensitivity control to surface irregularity,” Proc. SPIE 6342, 634225 (2006).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1975).

Forbes, G. W.

Fuerschbach, K. H.

K. H. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Design with ϕ-polynomial surfaces,” Proc. SPIE 8167, 81670Z (2011).
[CrossRef]

K. H. Fuerschbach, K. P. Thompson, and J. P. Rolland, “A new generation of optical systems with ϕ-polynomial surfaces,” Proc. SPIE 7652, 76520C (2010).
[CrossRef]

Jester, P.

Menke, C.

Rolland, J. P.

K. H. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Design with ϕ-polynomial surfaces,” Proc. SPIE 8167, 81670Z (2011).
[CrossRef]

K. H. Fuerschbach, K. P. Thompson, and J. P. Rolland, “A new generation of optical systems with ϕ-polynomial surfaces,” Proc. SPIE 7652, 76520C (2010).
[CrossRef]

Thompson, K. P.

K. H. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Design with ϕ-polynomial surfaces,” Proc. SPIE 8167, 81670Z (2011).
[CrossRef]

K. H. Fuerschbach, K. P. Thompson, and J. P. Rolland, “A new generation of optical systems with ϕ-polynomial surfaces,” Proc. SPIE 7652, 76520C (2010).
[CrossRef]

Urban, K.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1975).

Yabe, A.

A. Yabe, “Construction method of axially asymmetric lenses,” Appl. Opt. 50, 3369–3374 (2011).
[CrossRef]

A. Yabe, “Method to allocate freeform surfaces in axially asymmetric optical systems,” Proc. SPIE 8167, 816703 (2011).
[CrossRef]

A. Yabe, “Sensitivity control to surface irregularity,” Proc. SPIE 6342, 634225 (2006).
[CrossRef]

Youngworth, R. N.

R. N. Youngworth, “Tolerancing Forbes aspheres: advantage of an orthogonal basis,” Proc. SPIE 7433, 74330H (2009).
[CrossRef]

Appl. Opt.

Opt. Express

Proc. SPIE

R. N. Youngworth, “Tolerancing Forbes aspheres: advantage of an orthogonal basis,” Proc. SPIE 7433, 74330H (2009).
[CrossRef]

A. Yabe, “Method to allocate freeform surfaces in axially asymmetric optical systems,” Proc. SPIE 8167, 816703 (2011).
[CrossRef]

K. H. Fuerschbach, K. P. Thompson, and J. P. Rolland, “A new generation of optical systems with ϕ-polynomial surfaces,” Proc. SPIE 7652, 76520C (2010).
[CrossRef]

K. H. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Design with ϕ-polynomial surfaces,” Proc. SPIE 8167, 81670Z (2011).
[CrossRef]

A. Yabe, “Sensitivity control to surface irregularity,” Proc. SPIE 6342, 634225 (2006).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics (Pergamon, 1975).

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Figures (3)

Fig. 1.
Fig. 1.

Starting lens of the example.

Fig. 2.
Fig. 2.

Tilt of the screen and induced keystone distortion.

Fig. 3.
Fig. 3.

Result of the optimization.

Tables (2)

Tables Icon

Table 1. Combination of m and n up to the Eighth Order

Tables Icon

Table 2. Specification of the Example

Equations (25)

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f(x,y)=c(x2+y2)1+1Kc2(x2+y2)+i,j0i+jNaijxiyj.
x=rsinθ,y=rcosθ,andρ=r/r0,
Vnm(ρ,θ)=Pnm(ρ)cos(mθ)andVnm(ρ,θ)=Pnm(ρ)sin(mθ),
f(r,θ)=cr21+1Kc2r2+n=0Nk=0nbnn2kVnn2k(r/r0,θ),
axxx2+2axyxy+ayyy2+(zR)2=R2.
cxx=axx/R,cxy=axy/R,cyy=ayy/R,andγ=1/R,
z=cxxx2+2cxyxy+cyyy21+1γ(cxxx2+2cxyxy+cyyy2).
cxx=cyy,cxy=0andγ=Kcxx=Kcyy.
u=ρsinθandv=ρcosθ,
Wnm(u,v)=Qnm(ρ)cos(mθ),andWnm(u,v)=Qnm(ρ)sin(mθ),
01Qnm(ρ)Qnm(ρ)ρdρ=12(n+1)δnn,
Qn0(ρ)=Qn2(ρ)=Pn4(ρ),Qn1(ρ)=Pn3(ρ).
Qnm(ρ)=Pnm(ρ).
P44(ρ)=ρ4,P64(ρ)=6ρ65ρ4,P84(ρ)=28ρ842ρ6+15ρ4.
(W31W33)=(1131)(u2vv3),(W31W33)=(1113)(u3uv2),
(W40W42W44)=(121101161)(u4u2v2v4),(W42W44)=(2244)(u3vuv3),
(W51W53W55)=(5105151055101)(u4vu2v3v5)+(4412400)(u2vv3),
(W51W53W55)=(5105510151105)(u5u3v2uv4)+(4441200)(u3uv2),
(W60W62W64W66)=(6181866666630306115151)(u6u4v2u2v4v6)+(51055055305000)(u4u2v2v4),
(W62W64W66)=(122412240246206)(u5vu3v3uv5)+(1010202000)(u3vuv3),
(W71W73W75W77)=(216363216310521213535637735211)(u6vu4v3u2v5v7)+(30603090603030606000)(u4vu2v3v5)+(101030100000)(u2vv3),
(W71W73W75W77)=(216363212121105637633535121357)(u7u5v2u3v4uv6)+(30603030609066030000)(u5u3v2uv4)+(101010300000)(u3uv2),
(W80W82W84W86W88)=(2811216811228285605628281122801122881120112812870281)(u8u6v2u4v4u2v6v8)+(4212612642424242424221021042710510570000)(u6u4v2u2v4v6)+(15301515015159015000000)(u4u2v2v4),
(W82W84W86W88)=(56168168561121121121124811211248856568)(u7vu5v3u3v5uv7)+(841688416801684214042000)(u5vu3v3uv5)+(303060600000)(u3vuv3).
f(x,y)=cxxx2+2cxyxy+cyyyy1+1γ(cxxx2+2cxyxy+cyyyy)+n=0Nk=0nbnn2kWnn2k(x/r0,y/r0).

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