With, above’s paragraph, Descartes predicted the possibility of freeform lenses, more than three hundred years ago. Freeform optics comprise optical devices with at least one freeform surface which, has no translational or rotational symmetry about axes normal to the optical axis. In this chapter, we are going to generalise all the equations of chapter 2 to a three-dimensional model. We are going to get the general closed-form equation for freeform stig-matic lenses.
Figure 3.1: On-axis freeform stigmatic singlet. The first surface is given by (xa, ya, za) , and the second surface is given by (xb, yb, zb). The length between the first surface and the object is ta, the thickness at the centre of the lens is t, and the length between the second surface and the image is tb. ~v1 is unit vector of the incident ray, ~v2 is the unit vector of the refracted ray inside the lens, ~v3 is the output ~na is the normal vector of the first surface and ~nb is the normal vector of the second surface.
.
ˆ
v2 = 1
−sgn(ta)n[ˆna× (−ˆna× ˆv1)] − ˆna r
1 − 1
n2(ˆna× ˆv1) · (ˆna× ˆv1), (3.2) where ˆv1 is the unit vector of the incident ray; it travels from the image to the first surface.
ˆ
v2 is the unit vector of the refracted ray; it moves inside the lens. ˆv3 is the unit vector of output ray. Finally, ˆnais the normal vector of the first surface, and ˆnbis the normal vector of the second surface. Note −sgn(ta) comes from the fact that the object can be real or virtual.
Please see Fig. 3.1.
The unit vectors of the refraction at the first surface are,
ˆ
v1 = [xa, ya, (za− ta)]
px2a+ ya2+ (za− ta)2,
ˆ
v2 = [xb − xa, yb − ya, zb− za]
p(xb − xa)2+ (yb− ya)2+ (zb− za)2,
ˆ
na = [zax, zay, −1]
q
za2x + za2y + 1 ,
(3.3)
of the unknowns x0b, y0b, zb0 and the unknowns as well xb, yb, zb. We use the Fermat principle to avoid an algebraic differential equation system provided by the Snell’s law at the second surface.
where, zax ≡ ∂xza and zay ≡ ∂yzaare the partial derivatives of za(xa, ya) with respect to xa and ya.
The left side of Eq. (3.2) only depends on known parameters; actually, on the left side are the cosine directors of the refracted ray. Replacing the unit vectors of Eq. (3.3) on the left side of Eq. (3.2) we have,
℘x =
xa
z2ay + 1
− zax yazay+ ta− za
−sgn(ta)n
za2x + z2ay + 1
p(ta− za)2+ x2a+ ya2
(3.4)
− zax
r
1 −(yazax−xazay)2+[zax(za−ta)+xa]2+(zay(za−ta)+ya)2
n2(z2ax+zay2 +1)[(ta−za)2+x2a+ya2]
qza2x + z2ay + 1
,
℘y = ya za2x + 1 − zay(xazax + ta− za)
−sgn(ta)n
za2x + z2ay + 1
p(ta− za)2+ x2a+ ya2
(3.5)
− zay
r
1 − (yazax−xazay)2+[zax(za−ta)+xa]2+(zay(za−ta)+ya)2
n2(zax2 +z2ay+1)[(ta−za)2+x2a+ya2]
qza2x + za2y + 1
,
and,
℘z =
− (ta− za)
za2x + za2y
+ xazax + yazay
−sgn(ta)n
za2x + z2ay + 1
p(ta− za)2+ x2a+ ya2
+ (3.6)
r
1 −(yazax−xazay)2+[zax(za−ta)+xa]2+(zay(za−ta)+ya)2
n2(z2ax+zay2 +1)[(ta−za)2+x2a+ya2]
q
z2ax + za2y+ 1
,
where ℘x, ℘y, ℘z are the portion of the refracted ray in x, y and z respectively. Thus, ℘2x +
℘2y + ℘2z = 1.
Let ϑ be the distance that each ray travels inside the lens, hence,c ϑ ≡ p
(xb− xa)2+ (yb− ya)2+ (zb − za)2. (3.7) Now, Let’s replace ϑ, Eq. (3.7) and the cosine directors, Eq. (3.4), (3.5) and (3.6), in the vector form of the Snell’s law, Eq. (3.2),
xb− xa
ϑ = ℘x, (3.8)
yb− ya
ϑ = ℘y, (3.9)
cActually, ϑ gives the shape of the lens. For every ray, there is a corresponding ϑ.
and, zb − za
ϑ = ℘z, (3.10)
Solving for xb, yb and zb,
xb = xa+ ϑ℘x. yb = ya+ ϑ℘y. zb = za+ ϑ℘z,
(3.11)
The above’s equation is a nice way to write the solution; we only need to know the value of ϑ in terms of the known parameters xa, ya, za, ℘x, ℘y, ℘z, n, ta, t and tb. In the next section, we are going to get it, with the help of the Fermat principle.
Solution
We recall the Fermat principle to solve for ϑ and complete the solution proposed in Eq. (3.1),
−ta+ nt + tb = − sgn(ta)p
x2a+ ya2+ (za− ta)2 + np
(xb− xa)2+ (yb− ya)2+ (zb− zb)2 + sgn(tb)
q
x2b + yb2+ (zb− t − tb)2, (3.1) Replacing Eq. (3.7) and (3.11) in above’s equation, Eq. (3.1),
−ta+ nt + tb = − sgn(ta)p
x2a+ y2a+ (za− ta)2+ nϑ (3.12) + sgn(tb)
q
(xa+ ϑ℘x)2+ (ya+ ϑ℘y)2+ (za+ ϑ℘z− t − tb)2, With the idea to simplify Eq. (3.12), we define the following variables that only depend on the known parameters xa, ya, za, ℘x, ℘y, ℘z, n, ta, t and tb,
f ≡ −ta+ nt + tb+ sgn(ta)p
x2a+ y2a+ (za− ta)2, (3.13) and,
τ ≡ za− t − tb. (3.14)
Substituting the new variables in Eq. (3.12) and squaring it,d
(f − nϑ)2 = (xa+ ϑ℘x)2+ (ya+ ϑ℘y)2+ (τ + ϑ℘z)2, (3.15) expanding, and tacking ℘2x+ ℘2y+ ℘2z = 1,
f2− 2f nϑ + n2ϑ2 = x2a+ 2xa℘xϑ + ℘2xϑ2 + y2a+ 2ya℘yϑ + ℘2yϑ2
+ τ2+ 2τ ℘rϑ + ℘2rϑ2, (3.16)
dEq. (3.15) still begin an implication of the Pythagoras theorem.
collecting for terms with ϑ,
ϑ2(1 − n2) + ϑ[2(f n + xa℘x+ ya℘y + τ ℘z)] + (x2a+ ya2+ τ2− f2) = 0.
(3.17) Recalling the quadratic formula and it solution,
ax2+ bx + c = 0, x = −b ±√
b2− 4ac
2a . (3.18)
Finally get ϑ in terms of xa, ya, za, ℘x, ℘y, ℘z, n, ta, t and tb, ϑ = −(f n + xa℘x+ ya℘y+ τ ℘z)
(1 − n2) (3.19)
± p(fn + xa℘x+ ya℘y+ τ ℘z)2− (1 − n2)(x2a+ ya2+ τ2− f2)
(1 − n2) .
With the expression of ϑ given by Eq. (3.19), we can write the solution as we proposed in the last section,
xb = xa+ ϑ℘x. yb = ya+ ϑ℘y. zb = za+ ϑ℘z,
(3.20)
Eq. (3.20) is the general solution to the problem of design a stigmatic freeform lens. The ϑ that should be used given by Eq. (3.19), and, the cosine directors are given by Eqs. (3.4), (3.5) and (3.6).
Eq. (3.20) is the most significant result of this chapter. It describes analytically the shape zb(xb, yb) of the output surface of the singlet lens in terms of the function za(xa, ya) of its freeform input surface and the design parameters (ta, tb, n, t). These expressions may look cumbersome, but it is quite exceptional that could be expressed in closed-form for an arbitrary freeform input surface. We recall that a prime requirement for the validity of Eq. (3.20) is that the surface normal should be perpendicular to the tangent plane to the input surface at the origin.
In the following section, we are going to show several exciting examples of explicitly using Eq. (3.20), without any optimisation process.
3.3.1 Illustrative examples
The generality of Eq. (3.20) allows us to show a large variety of interesting geometries of the singlet lens. In all examples, the input surface is freeform, and the user defines it.
The following gallery shows examples when the objects and images are real or virtual.
In all the cases the freeform singlets are free of spherical aberration.
Figure 3.2: Configuration settings: n = 1.7, ta = −50mm, t = 10mm, tb = 70mm, za = (x2a+ 4ya2)/130 and zb = Eq. (3.20).
Figure 3.3: Configuration settings: n = 1.9 ta = −50mm, t = 10mm, tb = 60mm, za = (x2a+ (ya− 1)2)/100 and zb = Eq. (3.20).
Figure 3.4: Configuration settings: n = 1.7 ta = −60mm, t = 10mm, tb = 70mm, za = (x2a+ 1.52ya2)/130 and zb = Eq. (3.20).
Figure 3.5: Configuration settings: n = 2 ta = 80mm, t = 10mm, tb = 70mm, za = (x2a+ 1.52ya2)/100 and zb = Eq. (3.20).
Figure 3.6: Configuration settings: n = 1.9 ta = 50mm, t = 10mm, tb = −80mm, za = (x2a+ 4ya2)/100 and zb = Eq. (3.20).
3.4 The freeform collector lens
The freeform collector lens is a lens with a freeform shape such as it collects all the rays, that comes from a single point object at menus infinity, in a single point image.
At the moment, we have an expression for freeform stigmatic lenses for finite object and images. The only thing that we need is to apply the limit when ta → −∞ over some parameters inside Eq. (3.20). e
Let’s recall these parameters of Eqs. (3.4), (3.5), (3.6), and (3.11)
℘x =
xa
za2y + 1
− zax yazay + ta− za
−sgn(ta)n
za2x + za2y + 1
p(ta− za)2+ x2a+ ya2
(3.4)
− zax
r
1 −(yazax−xazay)2+[zax(za−ta)+xa]2+(zay(za−ta)+ya)2
n2(z2ax+z2ay+1)[(ta−za)2+x2a+y2a]
qz2ax + za2y+ 1
,
℘y = ya za2x + 1 − zay(xazax + ta− za)
−sgn(ta)n
za2x + za2y + 1
p(ta− za)2+ x2a+ ya2
(3.5)
− zay
r
1 −(yazax−xazay)2+[zax(za−ta)+xa]2+(zay(za−ta)+ya)2
n2(z2ax+zay2 +1)[(ta−za)2+x2a+ya2]
qza2x + z2ay + 1
,
℘z =
− (ta− za)
za2x + za2y
+ xazax + yazay
−sgn(ta)n za2
x + z2a
y + 1
p(ta− za)2+ x2a+ ya2
+ (3.6)
r
1 −(yazax−xazay)2+[zax(za−ta)+xa]2+(zay(za−ta)+ya)2
n2(z2ax+zay2 +1)[(ta−za)2+x2a+ya2]
qz2ax + za2y+ 1
,
and,
f = −ta+ nt + tb+ sgn(ta)p
x2a+ y2a+ (za− ta)2, (3.11) and, let’s applied the limit when ta→ −∞ over them,
talim→−∞(℘x) =
zaxq
(n2− 1)(za2x + z2ay) + n2− 1 n
za2x + za2y + 1 , (3.21)
talim→−∞(℘y) =
zayq
(n2− 1)(za2
x+ za2
y) + n2 − 1 n
za2x+ za2y+ 1
, (3.22)
eThe equation for the finite object finite image case.
talim→−∞(℘z) =
z2ax + za2y+q
(n2− 1)(z2ax + za2y) + n2 n
za2x + za2y+ 1 , (3.23) and,
talim→−∞(f ) = nt + tb− za, (3.24) Thus, ϑ when ta → −∞ is,
talim→−∞(ϑ) =
−β ± v u u
tβ2 − (1 − n2) (
x2a+ ya2+ τ2 −
talim→−∞(f )
2)
(1 − n2) ,
(3.25) where,
β =
talim→−∞(f )
n + xa
talim→−∞(℘x)
+ ya
talim→−∞(℘y)
+ τ
talim→−∞(℘z)
,
(3.26) Finally, we have the same structure for the second surface, but with the limit when ta → −∞
applied.
talim→−∞(xb) = xa+
talim→−∞(ϑ)
talim→−∞(℘x)
,
talim→−∞(yb) = ya+
talim→−∞(ϑ)
talim→−∞(℘y)
,
talim→−∞(zb) = za+
talim→−∞(ϑ)
talim→−∞(℘z)
.
(3.27)
Eq. (3.27) is the equation we were looking for; it describes the shape of the second surface, such as the freeform singlet is a collector lens.
Eq. (3.27) preserves a one o one relationship with the points of the first surface and the points of the second surface since it is derived from Eq. (3.20). In Eq. (3.27), it can be seen the one o one relation, since for every refracted ray by the first surface there is a point in the second surface such as the singlet is stigmatic. Therefore, Eq. (3.27) works only for lenses such as the rays inside the singlets do not cross each other.
To use Eq. (3.27) we need to use Eqs. (3.21), (3.22), (3.23), (3.24), (3.26) and (3.25).
3.4.1 Illustrative examples
In this section, we implemented Eq. (3.27) to design a stigmatic collector freeform lens. The ray tracing is present and in the bottom in the caption are the parameters to reproduce it.
Figure 3.7: Configuration settings: n = 1.5, t = 10mm, tb = 60mm, za = −300 + p3002− x2a− 9ya2and zb = Eq. (3.27).