Optical system including a distributed index optical element in combination with a lens having a homogeneous refractive index

An optical system includes a refractive index distribution type optical element which configuration is formed to be a parallel plane plate. The refractive index distribution is provided in the radial direction perpendicular to an optical axis of the optical system, in which the refractive index is increased and a color dispersion value .nu.d is decreased as a location on the optical element is farther away from the optical axis. The optical element is combined with a lens or a plurality of lenses, the refractive index of which is homogeneous, so that a chromatic aberration generated by the homogeneous refractive index lens can be compensated. When the refractive index distribution in the radial direction perpendicular to the optical axis is expressed by the following expression, EQU n=n.sub.0 i+n.sub.1 ih.sup.2 +n.sub.2 h.sup.4 where h is the height from the optical axis and i is the wave length respectively illustrated by lines d, F and C, then the following expression is satisfied, EQU .nu.dG=n.sub.1 d/(n.sub.1 F-n.sub.1 c)<20.

BACKGROUND OF THE INVENTION 
The present invention relates to a lens system including a lens element, 
the configuration of which is of a parallel plane plate, having a 
distribution of refractive index which is distributed in a radial 
direction perpendicular to the optical axis. More particularly, the 
present invention relates to a lens system in which chromatic aberration 
is compensated by the lens element. 
Conventionally, what is called a radial GRIN lens is variously utilized. In 
this case, the radial GRIN lens is defined as a lens element having a 
refractive index distribution in a radial direction perpendicular to the 
optical axis. 
Picture-taking lenses of simple construction for camera use having a 
refractive index distribution have been conventionally investigated. For 
example, the above picture-taking lenses are disclosed in the thesis by L. 
G. Atkinson ("Applied Optics" Vol. 21, No. 18, p 993-998) and the thesis 
by J. Brain Caldwell ("Applied Optics" Vol. 25, No. 18, p 3351-3355). 
Further, the technique is disclosed in Japanese Patent Publication Open to 
Public Inspection Nos. 42109/1992 and 221716/1985. 
Examples in which the radial GRIN lens is used as a picture-taking lens of 
a camera or a video camera, especially examples in which the radial GRIN 
lens is used as a zoom lens, can be seen in patents which have already 
been applied. The examples are Japanese Patent Publication Open to Public 
Inspection Nos. 911/1987, 38711/1989, 15610/1992, 79013/1990, and 
85819/1990. In any cases, the lens surface is not formed plane, but it is 
formed spherical or it is formed to be other curved surfaces. 
On the other hand, what is called a radial GRIN lens in which a lens 
element having a refractive index distribution in a radial direction 
perpendicular to the optical axis, is applied to an optical laser beam 
system. The radial GRIN lens can be seen in Japanese Patent Publication 
Open to Public Inspection Nos. 254923/1986, 91316/1985 and 124011/1988. 
The configuration of the surface of a picture-taking lens of the prior art 
having a refractive index distribution is formed as follows: 
Both surfaces of the lens having a refractive index distribution are not 
plane. At least, one of the lens surfaces is spherical or aspherical. 
Therefore, with respect to the monochromatic aberration, the degree of 
freedom of the lens is high compared with a radial GRIN rod lens, both 
surfaces of which are plane. Accordingly, the performance of the 
conventional lens can be highly improved. However, from the viewpoint of 
practical use, the conventional lens is inferior, because it is difficult 
to machine the lens, and the costs are high. 
In the conventional radial GRIN rod lens, both surface of which are formed 
plane in parallel, as it is applied to an optical laser beam system, 
consideration is given to the lens only when the refractive index 
distribution coefficient is a dominant wave length, and consideration is 
not given to the chromatic aberration of the optical system. 
Technique of achromatism is disclosed in Japanese Patent Publication Open 
to Public Inspection No. 124011/1988. However, in this case, only radial 
GRIN rod lenses are combined in this lens system, so that the lens 
thickness is increased 3 to 4 times as much as the lens diameter. For this 
reason, it is difficult to put the lens into practical use. 
SUMMARY OF THE INVENTION 
The first object of the present invention is to compensate the chromatic 
aberration of a lens or a lens group of a picture-taking lens of a camera 
or a video camera when the lens system or the lens group includes at least 
one piece of homogeneous lens and a radial GRIN rod lens. In the case of a 
radial GRIN rod lens, as will be described later, it is relatively easy to 
obtain the physical properties in which the color dispersion value .nu.d 
is approximately not more than 10. Therefore, in the color compensation of 
the lens system or the lens group, it is possible to reduce the power of 
the homogeneous lens. As a result, the entire optical system can be made 
compact, or the performance of the lens system can be improved under the 
condition that the dimensions are the same. 
An embodiment of the present invention includes a refractive index 
distribution type optical element characterized in that: the configuration 
is formed to be a parallel plane plate; the refractive index distribution 
is provided in the radial direction perpendicular to the optical axis, in 
which the refractive index is increased and the color dispersion value 
.nu.d is decreased as a location on the element is farther away from the 
optical axis; and the optical element is combined with a lens or a 
plurality of lenses, the refractive index of which is homogeneous, so that 
the chromatic aberration generated by the homogeneous refractive index 
lens can be compensated. 
When the refractive index distribution in the radial direction 
perpendicular to the optical axis is expressed by the following 
expression, 
EQU n=n.sub.0 i+n.sub.1 ih.sup.2 +n.sub.2 ih.sup.4 
where h is the height from the optical axis and i is the wave length 
respectively illustrated by lines d, F and C, then the following 
expression is satisfied. 
EQU .nu.dG=n.sub.1 d/(n.sub.1 F-n.sub.1 c)&lt;20 
When the chromatic aberration compensation element is applied to a 
conventional optical system composed of the picture-taking lens of a 
single lens, a doublet or a zoom lens in which the chromatic aberration is 
conventionally difficult to be compensated, excellent effects can be 
provided. 
In the case where a single lens or a doublet is used for the picture-taking 
lens, it is preferable that the lens described below is disposed on the 
same side as that of a lens of the homogeneous refractive index and an 
aperture stop. The above lens is described as follows: the configuration 
is a parallel plane plate; the refractive index is increased and the color 
dispersion value .nu.d is decreased as a location on the lens is farther 
away from the optical axis; and the refractive index distribution is 
directed in the radial direction perpendicular to the optical axis. 
Especially, the lens is preferably disposed on the object side. 
In the case of a zoom lens, it is preferable that at least one lens group 
among a plurality of lens groups includes: one or plural lenses of the 
homogeneous refractive index; and a lens element for compensating 
chromatic aberration caused in the aforementioned lens, and the 
configuration of the lens element is a parallel plane plate, having a 
refractive index distribution in the radial direction perpendicular to the 
optical axis. Especially, it is preferable that a lens group having a 
negative refraction includes: a lens element for compensating chromatic 
aberration caused by one or a plurality of lenses of the homogeneous 
refractive index, wherein the configuration of the lens element is a 
parallel plane plate, and the refractive index distribution of the lens 
element is directed in the radial direction perpendicular to the optical 
axis, and further the lens element has the function of a positive lens. 
In the case where the lens system is composed of thin contact lenses, 
investigations are made as follows. 
When the synthetic focal distance of a lens having a homogeneous refractive 
index is fH and the equivalent .nu. value of the synthetic system is 
.nu.dH, and also when the focal distance of the radial GRIN rod lens is fG 
and the equivalent .nu. value is .nu.dG, the conditions of achromatism is 
expressed as follows. 
EQU (1/fH.multidot..nu.dH)+(1/fG.multidot..nu.dG)=0 (1) 
In this connection, when a plurality of lenses having a homogeneous 
refraction index are used, that is, when k pieces of lenses having a 
homogeneous refraction index are used, each focal distance fj (j=1 to k) 
and color dispersion value .nu.dj (j=1 to k) can be expressed as follows. 
[Expression] 
##EQU1## 
where 
EQU .nu.dj=(n.sub.dj -1) / (n.sub.Fj -n.sub.cj) 
When the refractive index distribution of the radial GRIN rod lens is 
expressed by the following expression 
EQU n=n.sub.0 i+n.sub.1 ih.sup.2 . . . (4) 
where H: height from the optical axis, and i: d, F and C line, the 
equivalent .nu. value .nu.dG of the radial GRIN rod lens can be given by 
the following expression. 
EQU .nu.dG=n.sub.1 d/(n.sub.1 F-n.sub.1 c) . . . (5) 
In this connection, in the case of optical material having a homogeneous 
refractive index such as glass and plastics, the actual color dispersion 
value .nu.d satisfies the following inequalities. 
In the case of nd=1.5, .nu.d&gt;50. 
In the case of nd=1.7, .nu.d&gt;30. 
In the case of nd&gt;1.9, .nu.d&gt;20. 
However, when consideration is given to the equivalent value .nu. of the 
radial GRIN rod lens, for example, in the case where nd=1.52206 and 
.nu.d=43 on the optical axis, and also in the case where nd=1.56602 and 
.nu.d=29 at the end of the effective diameter, the equivalent .nu. value 
.nu.dG becomes as follows. 
EQU .nu.dG=6 
That is, material of extremely high color dispersion can be relatively 
easily obtained. 
When the material of extremely high color dispersion is applied to the 
radial GRIN rod lens, the focal distance of the lens having a homogeneous 
refractive index can be extended from the achromatic condition described 
before. 
For example, when the focal distance of a lens system is f and achromatism 
is carried out using a convex lens having a homogeneous refractive index 
in which the focal distance is f.sub.1 and .nu.d=50, and also using a 
concave lens in which the focal distance is f.sub.2 and .nu.d=25, the 
following expression is satisfied. 
EQU 1/f=(1/f.sub.1)+91/f.sub.2) 
Also, from the expression (1), the following relation is satisfied. 
EQU f.sub.1 /f.sub.2 =25/-50=-1/2 
Accordingly, the following results are obtained. 
EQU f.sub.1 =0.5f f.sub.2 =-f 
A convex lens of a homogeneous refractive index, the focal distance of 
which is f.sub.1, and the value .nu.d of which is .nu.d =50, and a radial 
GRIN rod lens which functions as a concave lens, the focal distance of 
which is f.sub.2, and the value .nu.d of which is .nu.d=10, are combined, 
in the same manner, the following expression is satisfied. 
EQU f.sub.1 /f.sub.2 =-1/5 
Accordingly, the following results are obtained. 
EQU f.sub.1 =0.8f f.sub.2 =-4f 
As explained above, the refraction of a lens having a homogeneous 
refractive index can be lowered. Therefore, not only the aberration can be 
reduced, but also the thickness of the center of a concave lens, or the 
thickness of the edge of a convex lens can be reduced. As a result, the 
overall lens length can be reduced. The refraction of a radial GRIN rod 
lens obtained by the refractive index distribution is limited. Therefore, 
when f.sub.2 is increased, it is possible to suppress the increase of the 
thickness of the radial GRIN rod lens, which is very advantageous. 
In the present invention, the right side of the expression (1) which shows 
the condition of achromatism, is not necessarily zero, and it is possible 
to permit an allowable range. The expression (1) can be deformed as 
follows. 
EQU -fG/fH=.nu.dH/.nu.dG 
Here, an allowable range is permitted as follows. 
EQU -fG/fH=k (.nu.dH/.nu.dG) (6) 
where k is a constant. Consequently, chromatic aberration can be 
compensated in a range satisfying the following inequality. 
EQU 0.4&lt;k&lt;2.5 
That is, 
EQU 0.4(.nu.dH/.nu.dG)&lt;-fG&lt;2.5(.nu.dH/.nu.dG) (7) 
Further, when the material satisfying the following inequality, achromatism 
can be carried out with positive and negative lenses having low power. 
EQU .nu.dG&lt;20 (8) 
The foregoing is one of the characteristics of the radial GRIN rod lens. 
Since the value of .nu.dG can be reduced, the following inequality can be 
satisfied. 
EQU (.nu.dH/.nu.dG)&gt;4 . . . (9) 
Due to the foregoing, achromatism can be carried out with positive and 
negative lenses having low power in the same manner. 
"The parallel plane plate" described in this specification permits a small 
radius of curvature and a small error in parallelism that are caused in 
the process of machining.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
From the viewpoint of cost reduction, a single lens or a doublet, the 
F-number of which is large, is used for the picture-taking lens of a 
camera of relatively low price. Accordingly, chromatic aberration can not 
be sufficiently compensated. 
According to the present invention, when a radial GRIN rod lens is 
combined, the configuration of which is a parallel plane plate, the radial 
GRIN rod lens having a refractive index distribution in the radial 
direction perpendicular to the optical axis, chromatic aberration on the 
axis and chromatic aberration of magnification generated in a lens system, 
the refractive index of which is homogeneous, can be compensated. 
Specifically, the lens system is constructed as follows: One piece of 
radial GRIN rod lens is provided on the object side, and at least one 
piece of lens, the refractive index of which is homogeneous, is provided 
on the image side. 
In the case where two lenses, the refractive index of which is homogeneous, 
are provided, at least one of them is preferably made of glass, and the 
other is preferably made of plastics. At least one of the surfaces of the 
lens, the refractive index of which is homogeneous, may be aspherical, and 
at least one piece of radial GRIN rod lens may be provided respectively 
before and behind the aperture stop. 
The radial GRIN rod lens may be made of plastics, and the refractive index 
of the lens is increased a location on the lens is farther away from the 
optical axis, and the value .nu.d of color dispersion of the lens is 
decreased. 
Specifically, the refractive index nd(1) of d-line and the color dispersion 
.nu.d(1) on the optical axis are expressed as follows. 
EQU 1.4&lt;nd(1)&lt;1.6 (10) 
EQU 30&lt;.nu.d(1)&lt;70 . . . (11) 
Further, the refractive index nd(2) of d-line and the color dispersion 
.nu.d(2) at the end of the effective diameter of the lens are expressed as 
follows. 
EQU 1.5&lt;nd(2)&lt;1.7 . . . (12) 
EQU 15&lt;.nu.d(2)&lt;55 . . . (13) 
where 
EQU .nu.d(1)={nd(1)-1}/{nF(1)-nc(1)} 
EQU .nu.d(2)={nd(2)-1}/{nF(2)-nc(2)} 
nF (1) , nc (1): the refractive indexes of F-line and C-line on the optical 
axis 
nF (2) , nc (2): the refractive indexes of F-line and C-line at the end of 
the effective diameter 
Further, the refractive index nd and color dispersion .nu.d satisfy the 
following inequality. 
EQU 0&lt;{nd(2)-nd(1)}/{.nu.d(1)-.nu.d(2)}&lt;0.05 . . . (14) 
It is preferable that the following lenses are arranged in the optical 
system by at least one with respect to each lens: one is a lens formed 
into the configuration of a parallel plane plate, the refractive index of 
which is increased as a location on the lens is farther away from the 
optical axis, and the lens has a refractive index distribution in the 
radial direction perpendicular to the optical axis; and the other is a 
lens formed into the configuration of a parallel plane plate, the 
refractive index of which is decreased as a location on the lens is 
farther away from the optical axis, and the lens has a refractive index 
distribution in the radial direction perpendicular to the optical axis. 
Alternatively, it is preferable that the following lenses are arranged on 
the same side with respect to the aperture stop in the optical system: one 
is a convex lens of the homogeneous refractive index; and the other is a 
lens formed into the configuration of a parallel plane plate, the 
refractive index of which is increased and the color dispersion .nu.d of 
which is decreased as it is separate from the optical axis, and the lens 
has a refractive index distribution in the radial direction perpendicular 
to the optical axis. 
When the radial GRIN rod lenses are respectively disposed on the front and 
rear sides of the stop in the lens system having the inner stop, it is 
possible to compensate only ON-axis chromatic aberration, and it is also 
possible to remove the remaining chromatic aberration of magnification 
when the thicknesses of the radial GRIN lenses on the front and rear sides 
are adjusted. 
In the case of a lens of which the F-number is relatively bright, it is 
possible to compensate the remaining ON-axis chromatic aberration and 
chromatic aberration of magnification by the radial GRIN rod lens when two 
pieces of homogeneous lenses including an aspherical surface lens are 
used. 
It is also possible to compensate not only chromatic aberration but also 
coma when two pieces of radial GRIN rod lenses, the configuration of which 
is a parallel plane plate, are provided in one lens system, wherein on 
lens functions as a concave lens and the other lens functions as a convex 
lens. 
Methods for effectively arranging the radial rod GRIN lenses are described 
as follows. 
(1) When the radial rod GRIN lense is arranged at a position separate from 
the stop, chromatic aberration of magnification can be effectively 
removed, and also coma caused in the radial GRIN rod lens can be 
suppressed. 
(2) When one radial GRIN rod lens is arranged on the front side of the stop 
and the other radial GRIN rod lens is arranged on the rear side of the 
stop, while coma caused in the radial GRIN rod lenses is compensated by 
the radial GRIN rod lenses themselves, chromatic aberration can be 
compensated. 
(3) When a radial GRIN rod lens functioning as a concave lens and a radial 
GRIN rod lens functioning as a convex lens are arranged, coma can be 
compensated. 
In this connection, in the case where the radial GRIN rod lens is made of 
plastics, in general, the refractive index of plastics satisfies the 
following inequalities. 
EQU 1.4&lt;n&lt;1.7, 15&lt;.nu.d&lt;70 
In this case, there is a tendency that the higher the refractive index, the 
lower the color dispersion .nu.d. Accordingly, from the viewpoint of 
manufacture, it is easy to obtain lenses in which the expressions (10) to 
(14) are satisfied. Both ON-axis chromatic aberration and chromatic 
aberration of magnification can be compensated when the radial GRIN rod 
lense and the convex lens of the homogeneous refractive index are disposed 
on the same side with respect to the stop. 
Examples of the present invention will be described as follows. Characters 
shown in the following tables are defined as follows. 
r: Radius of curvature of the refractive surface 
d: Interval of surfaces 
nd: Refractive index of d-line of lens material 
.nu.d: Abbe's number of lens material 
f: Focal distance 
F: F-number 
.omega.: Half field angle 
Refractive index distribution function 
ni=n.sub.0 i+n.sub.1 ih.sup.2 +n.sub.2 ih.sup.4 
h: Height from the optical axis 
i: Wave length of each of d-line, F-line and C-line In this case, an 
aspherical surface is expressed by the expression 3. 
##EQU2## 
K: Circular cone constant Ai: Each aspherical surface coefficient 
For comparison, a sectional view of the conventional lens is shown in FIG. 
11, the aberration diagram is shown in FIG. 12, and the lens data is shown 
in the following table. 
______________________________________ 
f = 28.8 F9.8 .omega. = 37.degree. 
Surface 
No. r d nd .nu.d 
______________________________________ 
1 5.095 0.9 1.492 57 
2 7.492 1.8 
3 Off-axis luminous 
0.8 
flux stop (.phi.3.8) 
4 Aperture stop (.phi.2.5) 
0.2 
5 Off-axis luminous 
flux stop (.phi.2.5) 
______________________________________ 
Next, the first example is shown as follows. The lens data of the first 
example is shown in the following table. The sectional view of the lens of 
the first example is shown in FIG. 1, and the aberration diagram is shown 
in FIG. 2. 
______________________________________ 
f = 29.4 F9.68 .omega. = 37.degree. 
Surface 
No. r d nd .nu.d 
______________________________________ 
1 .infin. 1.0 Radial GRIN rod lens 
2 .infin. 0.2 
3 (Aspherical surface) 
1.0 1.492 57 
5.685 
4 9.470 2.1 
5 Off-axis luminous 
0.5 
flux stop (.phi.3.6) 
6 Aperture stop (.phi.2.6) 
0.2 
7 Off-axis luminous 
flux stop (.phi.2.6) 
______________________________________ 
Refractive index distribution function 
n.sub.0 i 
n.sub.1 i 
______________________________________ 
d 1.52206 0.17584 .times. 10.sup.-2 
F 1.53056 0.19648 .times. 10.sup.-2 
C 1.51850 0.16804 .times. 10.sup.-2 
______________________________________ 
Aspherical surface 
k = 0.044154 
A.sub.4 = A.sub.6 = A.sub.8 = A.sub.10 = 0 
______________________________________ 
Effective radius of the radial GRIN rod lens is 5 mm. 
nd(1) = 1.52206 
.nu.d(1) = 43 
nd(2) = 1.56602 
.nu.d(2) = 29 
{nd(2) - nd(1)}/{.nu.d(1) - .nu.d(2)} = 0.0031 
______________________________________ 
The second example is shown as follows. The lens data of the second example 
is shown in the following table. The sectional view of the lens of the 
second example is shown in FIG. 3, and the aberration diagram is shown in 
FIG. 4. 
______________________________________ 
f = 35.0 F8.24 .omega. = 30.degree. 
Surface 
No. r d nd .nu.d 
______________________________________ 
1 .infin. 1.1 Radial GRIN rod lens 
2 .infin. 0.2 
3 8.000 1.1 1.7725 49.6 
4 11.342 2.6 
5 Off-axis luminous 
1.6 
flux stop (.phi.4.0) 
6 Aperture stop (.phi.3.5) 
0.2 
7 Off-axis luminous 
flux stop (.phi.3.5) 
______________________________________ 
Refractive index distribution function 
n.sub.0 i 
n.sub.1 i 
______________________________________ 
d 1.52206 0.17584 .times. 10.sup.-2 
F 1.53056 0.19648 .times. 10.sup.-2 
C 1.51850 0.16804 .times. 10.sup.-2 
______________________________________ 
Effective radius of the radial GRIN rod lens is 5 mm. 
nd(1) = 1.52206 
.nu.d(1) = 43 
nd(2) = 1.56602 
.nu.d(2) = 29 
{nd(2) - nd(1)}/{.nu.d(1) - .nu.d(2)} = 0.0031 
______________________________________ 
The third example is shown as follows. The lens data of the third example 
is shown in the following table. The sectional view of the lens of the 
third example is shown in FIG. 5, and the aberration diagram is shown in 
FIG. 6. 
______________________________________ 
f = 35.0 F5.6 .omega. = 30.degree. 
Surface 
No. r d nd .nu.d 
______________________________________ 
1 .infin. 1.5 Radial GRIN rod lens 
2 .infin. 0.1 
3 6.355 1.3 1.7725 49.6 
4 11.726 0.2 
5 (Aspherical surface) 
1.0 1.492 57 
13.338 
6 (Aspherical surface) 
2.0 
6.817 
7 Aperture Stop (.phi.5.0) 
______________________________________ 
Refractive index distribution function 
n.sub.0 i 
n.sub.1 i 
______________________________________ 
d 1.52206 0.17584 .times. 10.sup.-2 
F 1.53056 0.19648 .times. 10.sup.-2 
C 1.51850 0.16804 .times. 10.sup.-2 
______________________________________ 
Aspherical surface coefficient 
Fifth surface 
K = 4.0143 
A.sub.4 = 0.90352 .times. 10.sup.-3 
A.sub.6 = -0.27578 .times. 10.sup.-4 
A.sub.8 = 0.29421 .times. 10.sup.-5 
A.sub.10 = -0.15117 .times. 10.sup.-6 
Sixth surface 
K = 3.6795 
A.sub.4 = 0.61862 .times. 10.sup.-3 
A.sub.6 = -0.20367 .times. 10.sup.-3 
A.sub.8 = 0.36071 .times. 10.sup.-4 
A.sub.10 = -0.40855 .times. 10.sup.-5 
______________________________________ 
Effective radius of the radial GRIN rod lens is 4.4 mm. 
nd(1) = 1.52206 
.nu.d(1) = 43 
nd(2) = 1.55610 
.nu.d(2) = 32 
{nd(2) - nd(1)}/{(.nu.d(1) - .nu.d(2)} = 0.0031 
______________________________________ 
The fourth example is shown as follows. The lens data of the fourth example 
is shown in the following table. The sectional view of the lens of the 
fourth example is shown in FIG. 7, and the aberration diagram is shown in 
FIG. 8. 
______________________________________ 
f = 30.39 F8 .omega. = 37.degree. 
Surface 
No. r d nd .nu.d 
______________________________________ 
1 .infin. 0.80 Radial GRIN rod lens 1 
2 .infin. 0.10 
3 (Aspherical surface) 
1.72 1.492 57 
6.660 
4 17.566 0.39 
5 Aperture Stop (.phi.3.2) 
0.70 
6 -5.097 1.56 1.583 30 
7 (Aspherical surface) 
0.10 
-6.098 
8 .infin. 0.80 Radial GRIN rod lens 2 
9 .infin. 
______________________________________ 
Refractive index distribution function 
(common between 1 and 2) 
n.sub.0 i 
n.sub.1 i 
______________________________________ 
d 1.51544 0.13133 .times. 10.sup.-2 
F 1.52410 0.14555 .times. 10.sup.-2 
C 1.51181 0.12581 .times. 10.sup.-2 
______________________________________ 
Aspherical surface coefficient 
Third surface 
K = 0.34708 .times. 10.sup.-2 
A.sub.4 = 0.88764 .times. 10.sup.-3 
A.sub.6 = -0.12139 .times. 10.sup.-4 
A.sub.8 = 0.11303 .times. 10.sup.-4 
A.sub.10 = -0.20622 .times. 10.sup.-6 
Seventh surface 
K = -0.35023 
A.sub.4 = 0.73951 .times. 10.sup.-3 
A.sub.6 = -0.45899 .times. 10.sup.-4 
A.sub.8 = 0.15859 .times. 10.sup.-4 
A.sub.10 = -0.60102 .times. 10.sup.-6 
______________________________________ 
Effective radius of the radial GRIN rod lens 1 is 3.64 mm. 
nd(1) = 1.51544 
.nu.d(1) = 42 
nd(2) = 1.53284 
.nu.d(2) = 36 
{nd(2) - nd(1)}/{.nu.d(1) - .nu.d(2)} = 0.00029 
Effective radius of the radial GRIN rod lens 2 is 3.46 mm. 
nd(1) = 1.51544 
.nu.d(1) = 42 
nd(2) = 1.53116 
.nu.d(2) = 36 
{nd(2) - nd(1)}/{.nu.d(1) - .nu.d(2)} = 0.00026 
______________________________________ 
The fifth example is shown as follows. The lens data of the fifth example 
is shown in the following table. The sectional view of the lens of the 
fifth example is shown in FIG. 9, and the aberration diagram is shown in 
FIG. 10. 
______________________________________ 
f = 35.04 F8 .omega. = 30.degree. 
Surface 
No. r d nd .nu.d 
______________________________________ 
1 12.5 4.0 1.834 37.2 
2 23.960 0.5 
3 .infin. 2.0 Radial GRIN rod lens 1 
4 .infin. 0.1 
5 .infin. 2.0 Radial GRIN rod lens 2 
6 .infin. 0.2 
7 Aperture Stop (.phi.3.6) 
2.05 
8 Off-axis luminous 
flux stop (.phi.4.0) 
______________________________________ 
Refractive index distribution function 
Radial GRIN rod lens 1 
n.sub.0 i 
n.sub.1 i 
______________________________________ 
d 1.52206 0.01425 
F 1.53056 0.01490 
C 1.51850 0.01400 
______________________________________ 
Radial GRIN rod lens 2 
n.sub.0 i 
n.sub.1 i 
______________________________________ 
d 1.56602 -0.01175 
F 1.57968 -0.01217 
C 1.56051 -0.01159 
______________________________________ 
Effective radius of the radial GRIN rod lens 1 is 3.3 mm. 
nd(1) = 1.52206 
.nu.d(1) = 42 
nd(2) = 1.67724 
.nu.d(2) = 31 
{nd(2) - nd(1)}/{.nu.d(1) - .nu.d(2)} = 0.014 
Effective radius of the radial GRIN rod lens 2 is 2.6 mm. 
nd(1) = 1.56602 
.nu.d(1) = 29 
nd(2) = 1.48659 
.nu.d(2) = 32 
{nd(2) - nd(1)}/{.nu.d(1) - .nu.d(2)} = 0.026 
______________________________________ 
In this example, the radial GRIN rod lens 1 functions as a concave lens, 
and the radial GRIN rod lens 2 functions as a convex lens. 
An example in which the present invention is applied to a zoom lens is 
shown here. For example, in a zoom lens in which the lens groups are 
disposed from the object side in the order of a positive group, negative 
group, stop, positive group and positive group, it is possible to reduce 
the front lens diameter when the radial GRIN rod lens is used for the 
second lens group. 
Recently, there is a tendency that the size of charge coupled device (CCD) 
is reduced from 1/3 inch to 1/4 inch. In accordance with the reduction of 
the size, the radius of curvature of the lens is also reduced. Therefore, 
it becomes difficult to machine the lens. When the optical system 
described above is subjected to achromatism by the radial GRIN rod lens, 
the radius of curvature of the homogeneous lens combined with it can be 
made gentle, so that the difficulty can be reduced in the process of 
machining. 
In the case of a zoom lens, a large number of lenses are arranged for the 
purpose of achromatism. Therefore, the radial GRIN rod lens is 
advantageously used. In the case where monochromatic aberration can not be 
sufficiently compensated by this radial GRIN rod lens, an aspherical 
surface lens may be used for the homogeneous lens for the purpose of 
compensation. 
The lens data of the sixth example is shown in the following table. The 
sectional views of the lens of this example are shown in FIGS. 13(a) to 
13(c), and the aberration diagrams are shown in FIGS. 14 to 16. 
______________________________________ 
f = 5.2-13.7 F2.8 .omega. = 25.degree.-9.6.degree. 
Surface 
No. r d nd .nu.d 
______________________________________ 
1 34.464 0.70 1.84666 23.8 
2 16.154 2.50 1.72600 53.5 
3 -1702.755 0.10 
4 10.687 2.70 1.77250 49.6 
5 26.524 Var- 
iable 
6 100.184 0.50 1.88300 40.8 
7 5.005 1.20 
8 -15.944 0.50 1.77250 49.6 
9 25.295 0.50 
10 .infin. 1.20 Radial GRIN rod lens 
11 .infin. Var- 
iable 
12 26.389 1.88 1.69680 55.5 
13 -17.378 Var- 
iable 
14 9.803 0.60 1.92286 20.9 
15 5.172 0.25 
16 5.887 3.50 1.72916 54.7 
17 -12.304 Var- 
iable 
18 .infin. 3.68 1.51633 64.1 
19 .infin. 2.50 
20 .infin. 0.80 1.51633 64.1 
21 .infin. 
______________________________________ 
Surface interval 
Wide angle Intermediate 
Telephoto 
______________________________________ 
d.sub.5 
0.5 3.2 5.0 
d.sub.11 
6.75 4.05 2.25 
d.sub.13 
5.47 4.78 4.75 
d.sub.17 
2.43 3.12 3.15 
______________________________________ 
Refractive index distribution 
coefficient of radial GRIN rod lens 
n.sub.0 i 
n.sub.1 i 
______________________________________ 
d 1.56602 -0.70336 .times. 10.sup.-2 
F 1.57968 -0.78592 .times. 10.sup.-2 
C 1.56051 -0.67216 .times. 10.sup.-2 
______________________________________ 
Concerning the second group 
.nu.dG = 6.2 
fG/fH = -14.6 .nu.dH/.nu.dG = 7.0 
Consequently -fG/fH = 2.1 .nu.dH/VdG 
______________________________________ 
The lens data of the seventh example is shown in the following table. The 
sectional views of the lens of this example are shown in FIGS. 17(a) to 
17(c), and the aberration diagrams are shown in FIGS. 18 to 20. 
______________________________________ 
f = 5.2-13.7 F2.8 .omega. = 25.degree.-9.8.degree. 
Surface 
No. r d nd .nu.d 
______________________________________ 
1 24.751 0.70 1.84666 23.8 
2 13.790 2.50 1.72600 53.5 
3 422.058 0.10 
4 9.079 2.10 1.77250 49.6 
5 18.395 Var- 
iable 
6 39.731 0.50 1.88300 40.8 
7 3.893 1.20 
8 -83.471 0.80 1.77250 49.6 
9 -184.467 0.50 
10 .infin. 1.20 Radial GRIN rod lens 1 
11 .infin. Var- 
iable 
12 88.129 1.50 1.69680 55.5 
13 -8.370 Var- 
iable 
14 .infin. 1.00 Radial GRIN rod lens 2 
15 .infin. 0.25 
16 10.304 2.00 1.72916 54.7 
17 -24.846 Var- 
iable 
18 .infin. 3.68 1.51633 64.1 
19 .infin. 0.42 
20 .infin. 0.80 1.51633 64.1 
21 .infin. 
______________________________________ 
Surface interval 
Wide angle Intermediate 
Telephoto 
______________________________________ 
d.sub.5 
0.5 3.2 5.0 
d.sub.11 
6.75 4.05 2.25 
d.sub.13 
0.53 0.83 1.99 
d.sub.17 
2.47 2.17 1.01 
______________________________________ 
Refractive index distribution coefficient 
Radial GRIN rod lens 1 
n.sub.0 i 
n.sub.1 i 
______________________________________ 
d 1.56602 -0.70336 .times. 10.sup.-2 
F 1.57968 -0.78592 .times. 10.sup.-2 
C 1.56051 -0.67216 .times. 10.sup.-2 
______________________________________ 
Radial GRIN rod lens 2 
n.sub.0 i 
n.sub.1 i 
______________________________________ 
d 1.52206 +0.70336 .times. 10.sup.-2 
F 1.53056 +0.78592 .times. 10.sup.-2 
C 1.51850 +0.67216 .times. 10.sup.-2 
______________________________________ 
Concerning the second group 
.nu.dG = 6.2 
fG/fH = -12.4 .nu.dH/.nu.dG = 6.6 
Consequently -fG/fH = 1.9 .nu.dH/.nu.dG 
Concerning the fourth group 
.nu.dG = 6.2 
fG/fH = -6.9 .nu.dH/.nu.dG = 8.8 
Consequently -fG/fH = 0.78 .nu.dH/.nu.dG 
______________________________________ 
The lens data of the eighth example is shown in the following table. The 
sectional views of the lens of this example are shown in FIGS. 21(a) to 
21(c), and the aberration diagrams are shown in FIGS. 22 to 24. 
______________________________________ 
f = 5.3-13.9 F2.8 .omega. = 25.degree.-9.5.degree. 
Surface 
No. r d nd .nu.d 
______________________________________ 
1 23.964 0.70 1.84666 23.8 
2 13.291 2.50 1.72600 53.5 
3 500.678 0.10 
4 9.078 2.10 1.77250 49.6 
5 17.709 Var- 
iable 
6 41.492 0.50 1.88300 40.8 
7 3.879 1.20 
8 .infin. 1.20 Radial GRIN rod lens 1 
9 .infin. Var- 
iable 
10 51.405 1.50 1.69680 55.5 
11 -7.792 Var- 
iable 
12 .infin. 1.20 Radial GRIN rod lens 2 
13 .infin. 0.25 
14 10.787 2.00 1.72916 54.7 
15 -19.667 Var- 
iable 
16 .infin. 3.68 1.51633 64.1 
17 .infin. 0.50 
18 .infin. 0.80 1.51633 64.1 
19 .infin. 
______________________________________ 
Surface interval 
Wide angle Intermediate 
Telephoto 
______________________________________ 
d.sub.5 
0.5 3.2 5.0 
d.sub.9 
6.75 4.05 2.25 
d.sub.11 
0.98 1.31 2.55 
d.sub.15 
2.02 1.69 0.45 
______________________________________ 
Refractive index distribution coefficient 
Radial GRIN rod lens 1 
n.sub.0 i 
n.sub.1 i 
______________________________________ 
d 1.56602 -0.70336 .times. 10.sup.-2 
F 1.57968 -0.78592 .times. 10.sup.-2 
C 1.56051 -0.67216 .times. 10.sup.-2 
______________________________________ 
Radial GRIN rod lens 2 
n.sub.0 i 
n.sub.1 i 
______________________________________ 
d 1.52206 +0.70336 .times. 10.sup.-2 
F 1.53056 +0.78592 .times. 10.sup.-2 
C 1.51850 +0.67216 .times. 10.sup.-2 
______________________________________ 
Concerning the second group 
.nu.dG = 6.2 
fG/fH = -12.2 .nu.dH/.nu.dG = 6.6 
Consequently -fG/fH = 1.8 .nu.dH/.nu.dG 
Concerning the fourth group 
.nu.dG = 6.2 
fG/fH = -6.0 .nu.dH/.nu.dG = 8.8 
Consequently -fG/fH = 0.68 .nu.dH/.nu.dG 
______________________________________ 
The lens data of the ninth example is shown in the following table. The 
sectional views of the lens of this example are shown in FIGS. 25(a) to 
25(c), and the aberration diagrams are shown in FIGS. 26 to 28. 
______________________________________ 
f = 5.2-13.7 F2.8 .omega. = 25.degree.-9.6.degree. 
Surface 
No. r d nd .nu.d 
______________________________________ 
1 .infin. 1.50 Radial GRIN rod lens 1 
2 .infin. 0.10 
3 20.958 1.70 1.77250 49.6 
4 808.153 0.10 
5 9.251 1.80 1.77250 49.6 
6 16.978 Var- 
iable 
7 43.232 0.50 1.88300 40.8 
8 3.780 1.10 
9 .infin. 1.50 Radial GRIN rod lens 2 
10 .infin. Var- 
iable 
11 147.543 1.50 1.77250 49.6 
12 -8.070 Var- 
iable 
13 .infin. 1.50 Radial GRIN rod lens 3 
14 .infin. 0.25 
15 12.080 2.00 1.77250 49.6 
16 -19.706 Var- 
iable 
17 .infin. 3.68 1.51633 64.1 
18 .infin. 0.62 
19 .infin. 0.80 1.51633 64.1 
20 .infin. 
______________________________________ 
Surface interval 
Wide angle Intermediate 
Telephoto 
______________________________________ 
d.sub.6 
0.5 3.2 5.0 
d.sub.10 
6.75 4.05 2.25 
d.sub.12 
0.60 0.95 2.31 
d.sub.16 
2.10 1.75 0.39 
______________________________________ 
Refractive index distribution coefficient 
Radial GRIN rod lens 1 
n.sub.0 i 
n.sub.1 i 
______________________________________ 
d 1.52206 +0.17584 .times. 10.sup.-2 
F 1.53056 +0.19648 .times. 10.sup.-2 
C 1.51850 +0.16804 .times. 10.sup.-2 
______________________________________ 
Radial GRIN rod lens 2 
n.sub.0 i 
n.sub.1 i 
______________________________________ 
d 1.56602 -0.70336 .times. 10.sup.-2 
F 1.57968 -0.78592 .times. 10.sup.-2 
C 1.56051 -0.67216 .times. 10.sup.-2 
______________________________________ 
Radial GRIN rod lens 3 
n.sub.0 i 
n.sub.1 i 
______________________________________ 
d 1.52206 +0.70336 .times. 10.sup.-2 
F 1.53056 +0.78592 .times. 10.sup.-2 
C 1.51850 +0.67216 .times. 10.sup.-2 
______________________________________ 
Concerning the first group 
.nu.dG = 6.2 
fG/fH = -14.7 .nu.dH/.nu.dG = 8.0 
Consequently -fG/fH = 1.8 .nu.dH/.nu.dG 
Concerning the second group 
.nu.dG = 6.2 
fG/fH = -10.1 .nu.dH/.nu.dG = 6.6 
Consequently -fG/fH = 1.5 .nu.dH/.nu.dG 
Concerning the fourth group 
.nu.dG = 6.2 
fG/fH = -4.8 .nu.dH/.nu.dG = 8.0 
Consequently -fG/fH = 0.6 .nu.dH/.nu.dG 
______________________________________ 
The lens data of the tenth example is shown in the following table. The 
sectional views of the lens of this example are shown in FIGS. 29(a) to 
29(c), and the aberration diagrams are shown in FIGS. 30 to 32. 
______________________________________ 
f = 5.7-64.8 F1.89-3.05 .omega. = 30.degree.-2.6.degree. 
Surface 
No. r d nd .nu.d 
______________________________________ 
1 31.380 0.75 1.84666 23.8 
2 18.278 4.50 1.69680 55.5 
3 172.697 0.70 
4 23.590 2.32 1.69680 55.5 
5 64.456 Var- 
iable 
6 45.673 0.55 1.77250 49.6 
7 8.261 2.20 
8 -15.005 0.55 1.69350 53.2 
9 (Aspherical surface) 
1.00 
18.501 
10 .infin. 2.50 Radial GRIN rod lens 
11 .infin. Var- 
iable 
12 19.052 2.15 1.60311 60.7 
13 -158.743 1.00 
14 -12.945 1.30 1.49200 57.0 
15 (Aspherical surface) 
Var- 
-14.015 iable 
16 (Aspherical surface) 
1.30 1.49200 57.0 
16.088 
17 16.323 0.35 
18 24.455 0.55 1.84666 23.8 
19 8.846 3.50 1.69680 55.5 
20 -19.959 Var- 
iable 
21 -19.335 1.40 1.49200 57.0 
22 (Aspherical surface) 
3.67 
-14.078 
23 .infin. 4.90 1.51633 64.1 
24 .infin. 
______________________________________ 
Aspherical surface coefficient 
Ninth surface K = 5.42190 .times. 10.sup.-1 
A.sub.4 = -8.02320 .times. 10.sup.-5 
A.sub.6 = -3.52530 .times. 10.sup.-6 
A.sub.8 = 7.83780 .times. 10.sup.-7 
A.sub.10 = -2.63160 .times. 10.sup.-8 
15th surface K = 7.07610 .times. 10.sup.-1 
A.sub.4 = 4.34240 .times. 10.sup.-5 
A.sub.6 = 8.36450 .times. 10.sup.-6 
A.sub.8 = -3.85290 .times. 10.sup.-7 
A.sub.10 = 6.34440 .times. 10.sup.-9 
16th surface K = -6.94640 .times. 10.sup.-1 
A.sub.4 = -8.52630 .times. 10.sup.-5 
A.sub.6 = 8.04260 .times. 10.sup.-6 
A.sub.8 = -3.36690 .times. 10.sup.-7 
A.sub.10 = 4.95380 .times. 10.sup.-9 
22nd surface K = -5.60460 .times. 10.sup.-1 
A.sub.4 = -1.77190 .times. 10.sup.-3 
A.sub.6 = 1.19010 .times. 10.sup.-4 
A.sub.8 = -4.75240 .times. 10.sup.-6 
A.sub.10 = 7.71240 .times. 10.sup.-8 
______________________________________ 
Surface interval 
Wide angle Intermediate 
Telephoto 
______________________________________ 
d.sub.5 
0.55 10.99 17.95 
d.sub.11 
20.35 9.91 2.95 
d.sub.15 
4.62 1.55 10.70 
d.sub.20 
7.76 10.83 1.68 
______________________________________ 
Refractive index distribution coefficient 
Radial GRIN rod lens 
n.sub.0 i 
n.sub.1 i 
______________________________________ 
d 1.56602 -0.27475 .times. 10.sup.-2 
F 1.57968 -0.30700 .times. 10.sup.-2 
C 1.56051 -0.26256 .times. 10.sup.-2 
______________________________________ 
Concerning the second group 
.nu.dG = 6.2 
fG/fH = -11.7 .nu.dH/.nu.dG = 8.3 
Consequently -fG/fH = 1.4 .nu.dH/.nu.dG 
______________________________________ 
In the lens system of the present invention, as can be seen in the above 
examples, when a radial GRIN rod lens and a lens of homogeneous refractive 
index are combined, ON-axis chromatic aberration and chromatic aberration 
of magnification which are caused in a lens system of homogeneous 
refractive index can be very excellently compensated by a relatively thin 
radial GRIN rod lens. 
Especially, in the case where the lens of homogeneous refractive index is a 
single lens or a doublet, ON-axis chromatic aberration and chromatic 
aberration of magnification deteriorate the quality of images, so that the 
lens system of the present invention is very effective. In the case where 
the lens system of the present invention is applied to a zoom lens, the 
lens size can be reduced and difficulties caused in the manufacturing 
process can be avoided. 
In the above examples, the present invention is applied only to a simple 
picture-taking lens and a zoom lens, however, it should be noted that the 
present invention can be applied to an optical system such as an objective 
lens of a microscope and an laser optical system to which more severe 
chromatic aberration compensation is required. Further, the present 
invention can be applied to various optical systems in which chromatic 
aberration can not be satisfactorily compensated by the prior art. 
Whereas the radial GRIN rod lens is formed into a parallel plane plate, it 
can be supplied at low cost.