Optical: systems and elements – Lens – With graded refractive index
Reexamination Certificate
1999-10-07
2001-05-08
Epps, Georgia (Department: 2873)
Optical: systems and elements
Lens
With graded refractive index
C359S653000, C359S654000
Reexamination Certificate
active
06229654
ABSTRACT:
The present invention relates in general to (GRIN, graded-index) transparent articles with radial gradient refractive-index, for example optical lenses and optical fibers, and more particularly transparent articles of this type which have little chromatic aberration.
An optical article is characterized by its geometry, its thickness and its refractive index, the latter often being uniform. A gradient refractive index in an optical material provides an additional degree of freedom regarding the use of the article. This is because the gradient index makes it possible to vary the optical path of the rays independently of the geometry of the article.
Distinction is generally made between four types of gradient refractive index, each depending on the symmetry present in the article.
axial gradient index: the index varies along a given axial direction of the article; it is uniform in any plane perpendicular to this direction;
radial gradient index: the index varies as a function of the distance from a given axis; it is uniform on any cylindrical surface of given radius having the same axis as the gradient. If the index decreases from the center of the article toward the periphery (positive GRIN), a positive lens effect (convergence) is obtained. Conversely, if the index increases from the center of the article toward the periphery (negative GRIN), a negative lens effect (divergence) is obtained;
spherical gradient index: the index varies as a function of the distance from a given point; the surfaces of equal index are spherical.
Invariant index through a translatory motion, i.e. the index varies in a plane surface perpendicular to a defined axial direction of the article.
An optical equivalence has been demonstrated between a plano-convex (or plano-concave) lens with uniform index and a lens with radial parabolic gradient index, the index of which decreases from the center toward the edge (or from the edge toward the center).
In particular, the purpose of producing optical articles with a gradient refractive-index is to design simpler optical systems with performance equal to that obtained with systems consisting of optical elements having a uniform index. For example, production of this type makes it possible to manufacture multi-element optical systems in which the number of elements would thus be reduced, or to produce corrective glasses or lenses which are less thick and/or have a simpler geometry.
Furthermore, optical articles with a radial or spherical gradient index offer many applications in optoelectronics or in telecommunications when the index distribution is quasi-parabolic or parabolic. Very short-distance light focusing properties are obtained, which are highly desired in photocopiers, laser disk drives and players or optical fibers.
Gradient-index lenses and their manufacturing processes are described, amongst other things, in documents EP-0,407,294, EP-0,504,011 and FR-9,502,266.
A process for manufacturing optical fibers with a radial index gradient is described in the article by Oktsuka Koike (Applied Optics, 24 (24), pages 4316 to 4320 (1985)).
In what is following, we will focuse particularly on articles having a gradient index obtained by radial diffusion of a monomer precursor of a polymer 2 (refractive index n
2
, Abbe number &ngr;
2
) in a polymer matrix
1
(refractive index n
1
, Abbe number &ngr;
1
).
The chromatism of a transparent article is linked not only to the geometry of the article but also to the fact that the refractive index of a transparent material varies as a function of the wavelength of the light which passes through it.
In general, the refractive index n of a transparent material increases when the wavelength of the light decreases, that is to say the refractive index of the material is higher for blue than for red.
In general, the measure of the light-dispersion properties of a homogeneous thin lens satisfies the equation:
δ
f
f
+
δ
n
n
-
1
=
0
(
1
)
where
f is the focal length,
n is the refractive index, and
&dgr;n is the difference in refractive index between 2 wavelengths of light.
The term
δ
n
n
-
1
is referred to as the dispersion factor and is related to the inverse of the Abbe number &ngr;
D
.
Classically, this Abbe number is evaluated using refractive index of the material for different wavelengths, by the equation
1
ν
D
-
n
F
-
n
C
n
D
-
1
where
n
F
is the refractive index of the material for light with wavelength &lgr;=486 nm (blue)
n
C
is the refractive index of the material for light with wavelength &lgr;=656 nm (red)
n
D
is the refractive index of the material for sodium D line.
Furthermore, the power of a gradient-index transparent article depends partly on the geometrical shape of the article as in a homogeneous lens (Ph) and partly on the gradient index (PGrin), that is to say
P total=P
H
+P
GRIN
An Abbe number can therefore be defined which characterizes the chromatism of the power linked with the gradient index, and the total chromatism of the article is the sum of the two contributions:
axial
chromatism
≃
P
H
ν
D
+
P
GRIN
ν
GRIN
The longitudinal paraxial chromatic aberration (PAC) of a lens is defined by the equation:
PAC
=
h
2
u
′
(
Ph
ν
A
+
P
GRIN
ν
GRIN
)
(
2
)
where
h is the height of the marginal paraxial ray,
u′ is the exit angle of the paraxial ray,
P
H
is the power of a homogeneous lens with refractive index n
DA
,
n
DA
is the refractive index for Dline at the center of the lens,
&ngr;
A
is the Abbe number related to index n
DA
,
P
GRIN
is the power of the gradient index of the lens, and
&ngr;
GRIN
is the Abbe number of the GRIN lens.
From equation (2) it follows that reducing the value of the factor
P
GRIN
ν
GRIN
reduces the PAC value.
In the case of a lens with a quasi parabolic type radial gradient index, the refractive index as a function of the radius of the lens can be expanded as a polynomial:
n
&lgr;
(
r
)=
n
1&lgr;
+N
1&lgr;
r
2
+N
2&lgr;
r
4
+
where
r is the radius of the lens
n
&lgr;
(r) is the refractive index as a function of the radius at a particular wavelength
n
1
is the refractive index at r=0 i.e. on the optical axis of the lens.
For reasons of simplifications, we consider that n
1&lgr;
is the refractive index at the particular wavelength of the transparent polymer constituting the matrix, that is to say n
1&lgr;
=n
DA
in the previous formula (2).
N
i&lgr;
are constants which describe the gradient index at the particular wavelength.
In the case of paraxial rays, that is to say ones which make a small angle, generally ≦15° with respect to the axis of the lens, it is possible, as an approximation, to consider the gradient index as parabolic and express the refractive index as a function of radius by the equation:
n
&lgr;
(
r
)=
n
1&lgr;
+N
1&lgr;
r
2
(3)
i.e. for F line
n
F
(
r
)=
n
1f
+N
1F
r
2
(3′)
for C line
n
C
(
r
)=
n
1C
+N
1C
r
2
(3″)
for D line
n
D
(
r
)=
n
1D
+N
1D
r
2
(3′″)
In the paraxial range and for a lens having a thickness e, the power of the gradient index is expressed by the equation:
P
GRIN
=−2
N
1D
e
(4)
It is deduced from equations (1) and (4) that
δ
f
f
=
δ
N
1
N
1
D
=
N
1
F
-
N
1
C
N
1
D
=
1
ν
GRIN
(
5
)
where N
1F
, N
1C
and N
1D
are the constant factor of r
2
in equations 3′, 3″, 3′″.
The GRIN Abbe number is deduced by combining equations (3), (3′), (3″), (3′″) and equation (5) and assuming that the concentration of the material
2
(material diffused into the material
1
of the matrix) is 100% at the maximum radius, that is to say at the radial extremity of the lens, the Abbe number of the gradient index can be expressed by the equation:
ν
GRIN
=
(
n
2
D
-
n
1
D
)
Cabeza Stéphane
Grand-Clement Didier
Koike Yasuhiro
Epps Georgia
Essilor International (Compagnie Generale d'Optique)
Fulbright & Jaworski L.L.P.
Lester Evelyn A.
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