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Figure 2. Geometrical design layout of the freeform lens.
the entrance surface, the unit vector of the incident ray can be
written as
I =
(
sin ϕ cos θ,sin ϕ sin θ,cos ϕ
)
(1)
Then, we know that the position vector of point P is given by
P = ρ × I (2)
Taking the first partial derivative of P with respective to θ and ϕ
yields two tangent vectors at point P along θ and ϕ directions. The
cross product of these two tangent vectors gives us the normal to
the freeform surface at point P, which can be written as
N =
P
ϕ
× P
θ
P
ϕ
× P
θ
(3)
where P
ϕ
and P
θ
are the two tangent vectors at point P. We as-
sume that O is the unit vector of the outgoing ray. It is apparent
that the three vectors I, O and N should obey Snell’s law, which
means the three vectors should satisfy Equation (4)
O = nI + P
1
N (4)
where n is the refractive index of the freeform lens and the pa-
rameter P
1
is given by
P
1
=
1 − n
2
+ n
2
(
I · N
)
2
− n
(
I · N
)
Here, the black dot in the equation above denotes the dot prod-
uct of two vectors. Equations (3) and (4) show clearly that the three
components of O given by Equation (4) are functions of ρ, θ , ϕ,
Figure 3. A variable separation mapping. A prescribed uniform rectangu-
lar illumination is defined on the domain [−X, X] × [−Y, Y ]. X = 5 mm, Y =
10 mm and the lighting distance between the source and the illumination
plane equals 10 mm. The refractive index of the lens equals 1.59. The light
rays from a Lambertian point source are directed by the freeform lens to
produce the prescribed illumination.
ρ
θ
and ρ
ϕ
.Hereρ
θ
and ρ
ϕ
are the first-order derivatives of ρ with
respective to θ and ϕ. It should be noted that ρ
θ
and ρ
ϕ
are also
the functions of ρ, θ and ϕ. After we calculate the vector O,we
can get the x and y coordinates of point T, t
x
and t
y
. Likewise, t
x
and t
y
are also functions of ρ, θ, ϕ, ρ
θ
and ρ
ϕ
. We assume that t
x
and t
y
can be written as
t
x
= f
1
ρ,θ,ϕ,ρ
θ
,ρ
ϕ
t
y
= f
2
ρ,θ,ϕ,ρ
θ
,ρ
ϕ
(5)
Equation (5) tells us that the location of the intersection point
of the outgoing ray on the illumination plane is fully determined
by the surface profile. Solving Equation (5) for ρ
θ
and ρ
ϕ
gives us
ρ
θ
and ρ
ϕ
ρ
θ
= g
1
ρ,θ,ϕ,t
x
, t
y
ρ
ϕ
= g
2
ρ,θ,ϕ,t
x
, t
y
(6)
Since t
x
and t
y
are functions of θ and ϕ, the two partial
differential equations in Equation (6) become solvable only if
the relationship between point T and the vector I is known.
That is to say we need to choose a ray mapping which specifies
where an incident ray should go after refraction by the freeform
lens. Choosing a ray mapping is the key step in the ray mapping
method.
[50,51,54]
A ray mapping can be determined by the conser-
vation law of energy and this usually yields an infinite number
of possible ray mappings. An easy mapping to compute (but that
will not lead to a desirable solution) is obtained if a separation
of variables is applied.
[50,51,54]
Figure 3 gives a variable separation
mapping used to design a freeform lens which produces a uni-
form rectangular illumination on a target plane. The ray mapping
Laser Photonics Rev. 2018, 1700310
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2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim1700310 (4 of 18)