Optics and Photonics Journal, 2013, 3, 351-359
Published Online November 2013 (http://www.scirp.org/journal/opj)
http://dx.doi.org/10.4236/opj.2013.37055
Open Access OPJ
Aplanatic and Telescopic Lens with a Radial
Gradient of Refraction Index
Vladimir Ivanovich Tarkhanov
Joint stock Company “InfoTeCS”, Moscow, Russia
Received September 12, 2013; revised October 9, 2013; accepted October 29, 2013
Copyright © 2013 Vladimir Ivanovich Tarkhanov. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
It is shown that aplanatic lens with a radial gradient of refraction index is simultaneously a telescopic lens, notably not
only for an axial beam, but also for an off axis parallel beam. Consideration is carried out by an algebraic way on the
basis of regularities of ray paths. It is also shown that aplanatic and telescopic properties of the lens are independent of
the refracting surface shapes. Various versions of lens performance are shown below.
Keywords: Aplanatic Lens; Telescopic Lens; Radial Gradient; Refraction Index
1. Introduction
Aplanatic lenses with an axial and radial gradient of the
refraction index are known by now. In these lenses,
spherical aberration for a point on the optical axis is
strictly eliminated.
Earlier possibility of creation of an aplanatic lens with
an axial gradient of the refractive index was shown [1].
The homocentric beam leaving a point of 1
M
on an
optical axis, reaches to the first surface of a lens, refracts
on it and then propagates parallel to an optical axis in the
lens medium. Then, refracting on the second surface of a
lens, the rays of the beam forms a homocentric dispers-
ing beam again (Figure 1).
Virtual continuations of the ray forms the virtual image
at the axial point 2
M
 on the optical axis. Various gra-
dients of the refractive index and the related refracting
surfaces can be used for formation of a lens. The calcula-
tions for lens parameters were provided with spherical
and parabolic surfaces of revolution.
An aplanatic gradient lens [2] limited by the first and
the second refracting surfaces of revolution with thickness
by the axis , which is multiple of double nominal
focal length, made of a material with a radial distribution
of the refraction index determined from the equa-
tion
dz

ny


   
00
sech 2expexpny naynayay 
, (1)
where 0 is the refraction index value on the axis; is
the constant; having generatrix
n a

1
y
z of the 1st convex
surface defined by the equation:






 
22 2
112 1
F
yzny zsny z 

, (2)
where
F
s
is the front distance; and having generatrix


2
y
z of the 2nd concave surface defined by the equa-
tion:
 


   
2
22
2121 F
yzny zdsnyzd




, (3)
where
F
s


is the rare distance; hence,
F
F
s
s

.
The shape of both refracting surfaces in the known lens
is equal. Generatrixes of Equation (2) and Equation (3)
may be conditionally called hyperbolas of the higher or-
der.
It is known that the pitch (periodicity length) for
the refractive index (RI) distribution is determined by the
Еquation (1) and also called the hypersecans one equals
[3]
L
2πLa

,
as a consequence, the half of the periodicity length is
2πLa
The nominal focal length of the lens with hypersecans
RI distribution is
04π2
f
La


,
double nominal focal length will be equal to half the pe-
riodicity length
0
22π
F
fL a




V. I. TARKHANOV
352
Figure 1. The known aplanatic lens with axial gradient.
When selecting certain thickness of the lens with the RI
distribution of type Equation (1) and flat 1st and 2nd sur-
faces normal to the optical axis, focusing, diverging or
telescopic lens can be obtained [3]. Hence, it is quite ob-
vious that the telescopic lens is obtained in the case of
selecting the thickness of the lens equal to twice the
nominal focal length. As a result, the input parallel beam
of rays, which is also parallel to the optical axis, becomes
strictly parallel when leaving the lens.
The ray path equation in the medium with RI distribu-
tion Equation (1) is also known. For instance, if the initial
height of a point, from which the ray is initiated, equals
C, the initial coordinate is C, and the initial direction
coefficient (the initial ray tangent) 1
b z
is

, then the ray
path equation becomes as follows [4]


 

1
1arshshcoshsin
R
CCC
yz
abaz zcabaz z
a
 
C

(4)
As a consequence, considering that

2
arsh ln1xxx
and

2
arsh 11xx

one can write down that
 
1
shcosh sin
CCCC
x
abaz zcabaz z   
,


2
ln 1
R
y
zxxa
,
and the derivative is obtained from the expression


 
 
1
2
1
h cossh sin
sh coshsin1
R
CCCC
CCCC
yz
cabazzabaz z
abaz zcabazz



 


 
 
(5)
Note that if the length between the coordinates of the
ray exits point from the lens
Е
z and the ray entrance
point of the lens
D
z
E
D
dz z
equals to the periodicity length
2πLa

,
then according to Equation (4) the height of ray exit point
from the lens
E
y is equal to the height of ray entrance
point to the lens
D
y, or
E
D
yy (6)
As a result, the RI value at the ray entrance point to the
lens will be equal to the RI value at the ray exit point
from the lens


E
D
ny ny (7)
In this case, according to Equation (5), the ray tangent
2

occurring in the ray exit point from the lens before
the refraction will equal to the initial tangent 1

21

  (8)
and the derivative will correspond to the initial one


R
DRE
y
zyz

 (9)
If the length between coordinates of the ray exit
point from the lens
d
E
z and coordinates of the ray en-
trance point to the lens
D
z equals to the half periodicity
length,
2πLa

,
then according to Equation (4) the height of the ray exit
point from the lens
E
y will be equal to the height of the
ray entrance point to the lens
D
y with the sign reversed
E
D
yy (10)
The RI value at the ray entrance point to the lens will
be equal to that at the ray exit point from the lens




E
D
ny ny
The ray tangent 2

occurring at the ray exit point
from the lens before the refraction will be equal to the
initial tangent 1

with the sign reversed
21

  (11)
The value of derivative will be equal in the absolute
magnitude to the initial one taken with the sign reversed


R
DRE
y
zyz
  (12)
According to the design, in the known aplanatic lens,
every ray of the homocentric radiation beam exiting from
a point on the optical axis, after the refraction on the 1st
surface becomes parallel to the optical axis. Then spread-
ing inside the lens, in a gradient medium by a curvilinear
symmetrical path, each ray reaches the 2nd refractive sur-
face. In the cross-point with the 2nd refractive surface the
ray is also parallel to the optical axis.
After the refraction on the 2nd surface, each ray obtains
the initial direction which allows re-forming a homocen-
Open Access OPJ

V. I. TARKHANOV 353
tric diverging beam with the center located on the optical
axis, at a distance of
F
F
s
s

 from the 2nd surface cen-
ter. Hence,
F
s

 is the rear distance (Figure 2).
Depending on the selected thickness of the lens,
which is multiple to the double nominal focal length, the
refraction on the 2nd surface may happen both above and
below the axis (if we conditionally assume the initial re-
fraction on the 1st surface occurring above the optical
axis). Thus, however causes no effect on the lens ability
to form a homocentric diverging beam at the exit (Figure
3).
d
The known lens allows formation of a homocentric di-
verging radiation beam at the exit with the only help of
two refractive surfaces of revolution of the same shape,
which generatrix is rather complicated, Equation (2) and
Equation (3), that makes its manufacture rather complex.
Thus, the fact that the other surfaces, including one
simpler shape, cannot be used as the refractive surfaces is
the disadvantage of the known lens restricting possibili-
ties of its manufacture.
2. Analytical Treatment
Disadvantages of the known lens can be avoided, if we
consider a remarkable, previously unknown property of
the primary lens, which has a thickness multiple to the
periodicity length, to preserve initial direction of the ray
at the exit point, which appears on the 1st surface, inde-
pendently of a chosen shape of the 1st and the 2nd sur-
faces, as well as front distance value and location of the
radiation source, respectively.
The lens having a thickness multiple to half the perio-
dicity length possesses a similar property of preserving
the absolute value of the direction coefficient at the exit
point for the entrance ray appearing on the 1st surface,
Figure 2. The known aplanatic lens with thickness . L
Figure 3. The known aplanatic lens with thickness 2L.
independently of the shapes selected for the 1st and 2nd
surfaces, as well as of the front distance and location of
the source of radiation, respectively. However, in this
case the direction coefficient sign at the exit point for the
refracted ray reverses.
Let us show the effect of these properties, firstly for the
lenses of the minimal thickness with the thickness equal
to the periodicity length and half periodicity length L
2L.
For the lenses of a greater thickness multiple to half the
periodicity length, these properties also take effect.
A lens of thickness , by the optical axis at a distance
equal to the periodicity length is made from a mate-
rial with the radial RI distribution
d
L


ny

, which corre-
sponds to (1) and is known for the given wavelength, and
is limited by the refracting 1st and 2nd surfaces of the
same shape with generatrixes

1
y
z and


2
y
z and
disposed in a homogeneous medium (assume homogene-
ous medium RI
1
1n

(air)).
In the case under consideration, the same shape of the
refracting surfaces of the lens thickness, which is
equal to the periodicity length , means that the 2nd
surface represents the 1st surface after parallel translation
along the optical axis by a distance .
d
L
z d
As a result, a distance along the axis between any
separate initial point on the 1st surface, which has a cer-
tain height above the axis , and a point on the 2nd sur-
face having the same height above the axis and lo-
cated on a straight line parallel to the axis with the
initial point of the 1st surface will be the same and equal
to the thickness
z
z
z
z
dL

.
In the simplest case, the refracting 1st and 2nd surfaces
of the same shape may be the surfaces of revolution, but
in more general case, this is optional.
The 1st surface vertex is located in the origin of the co-
ordinate system. The axis represents the optical axis
of the lens. Assuming that the lens is axially symmetrical,
consideration is made in the meridional plane.
z
The incident beam for the 1st surface originating from
the point 1
M
of the optical axis has a direction coeffi-
cient
1К1
u


and is refracted in the point A having
coordinates 1C, 1C. Let us denote: the angle of inci-
dence of a ray on the 1st surface as 1
z y

, the angle of re-
fraction on the 1st surface as 1

, normal to the 1st sur-
face in a point A as , the angle of incidence of a ray
on the 2nd surface as 2
1
N

, the angle of refraction on the
2nd surface as 2


, and normal to the 2nd surface in a
point B as .
2
Let us denote: direction coefficient of the entrance ray
refracting on the 1st surface in point A as 1T, direction
coefficient of the ray refracted on the 1st surface in a
point A as 1
N
u
E
u, direction coefficient of the ray refracting
on the 2nd surface in a point B as , and direction co-
2T
u
Open Access OPJ

V. I. TARKHANOV
354
efficient of the ray refracted on the 2nd surface in a point
B as 2
E
u.
Direction coefficient 1
N
u of the normal 1 to the 1st
surface in a point A can be expressed as follows:
N

11
1
N
uy
  z, (13)
where

1
y
z
 is the 1st derivative of

1
y
z.
Direction coefficient 2
N
u of the normal 2 to the
2nd surface in a point B can be expressed as follows:
N


22
1
N
uyz

 , (14)
where

2
y
z
 is the 1st derivative of

2
y
z.
Let us consider the case of the 1st convex and the 2nd
concave surfaces for a lens with the thickness multiple to
( Figure 4). L
The refraction scheme in Figure 3 relates to cases
when after the refraction the direction coefficient 1
E
u of
the exit ray refracted on the 1st surface is negative.
For this case, a general scheme of the ray path in the
lens is shown in Figure 5.
For the refraction schemes shown in Figure 3, the an-
gle of incidence 1

in the point A on the 1st surface can
be determined from the following expression:
 
111 1
tan 1
TN TN
uu uu   1 , (15)
and the angle of refraction 1


is obtained from the ex-
pression
 
111 1
tan 1
EN EN
uu uu 
 1
1
, (16)
Moreover, according to Snell's refraction law

11 1
sin sin
C
nny


 ,
where is the RI in the refraction point A.

C1
ny


Then, with regard to the fact that , the refraction
angle is expressed as
11n

11C
sinsin ny 
1
(17)
Figure 4. Schemes of refraction on the 1st convex and the
2nd concave surfaces for a lens with the thickness multiple
to . L
Figure 5. The ray path scheme for the lens with the thick-
ness multiple to , having the 1st convex and the 2nd con-
cave surfaces.
L
Since the ray refracted in the point A on the 1st surface,
which has the initial direction 1
E
u, spreads after the re-
fraction by a curvilinear path with a period equal to the
periodicity length , then in the point B, after complet-
ing the full period:
L
1) according to the condition Equation (6), the height
of the refraction point B will be equal to the height
of the refraction point A;
2C
y
C
y1
2) according to the condition Equation (8), direction
coefficient 22T
u


of the ray incident to the 2nd sur-
face will be equal to 1E
u1


and, respectively,
2T
uu1E ; (18)
3) according to the condition Equation (8) and ratios
Equation (9), Equation (13) and Equation (14), and since
shapes of the 1st and 2nd surfaces are the same, direction
coefficient 2
N
u of the normal 2 to the 2nd surface in
the point B will be equal to the direction coefficient 1
N
N
u
of the normal to the 1st surface in the point A, and
thus
1
N
21
N
N
uu (19)
The angle of incidence 2

in the point B will be equal
to

 
222 2
tan 1
TN TN
uu uu  2
(20)
Taking into account Equation (18) and Equation (19)
and substituting them into Equation (20), we get:

 
2111
tan 1
EN EN
uu uu   1 (21)
Comparing Equation (16) and Equation (21), we get:
12




On this basis, we can write down that
2
sinsin1



 (22)
For the point B on the 2nd surface, according to Snell's
refraction law,


C22 12
sin sinny n


 ,
where


C2
ny is the RI on the refraction point B.
Considering that 11n

, the angle of refraction will be
as follows:


22
sin sin
C
ny 2



Taking into consideration relations Equation (17) and
Equation (22), we get:


221
sin sin
CC
ny ny 
1
C
Since 21C
yy

in the point B, as mentioned above,
and according to Equation (7), respectively, the expres-
sion




2C
ny ny1C is true, and we get:
21
sin sin




Then, as a consequence,
Open Access OPJ

V. I. TARKHANOV 355
21



and
21
tan tan


 (23)
The refraction angle 2

 in the point
ex

B may also be
pressed as follows:

2
tan uu 

22 22
1
EN EN
uu 
If we use ratio Equation (19) in this expression, we get:



221 21
tan 1
EN EN
uu uu 
  (24)
Based on Equation (23), and equatin
Eq
g expressions
uation (15) and Equation (24) and making simple
transformations, we get:
21
E
T
uu
Thus, direction of the exit ray refracted in the point B will
coincide with direction of the entrance ray in the point A.
Let us consider the case of the 1st convex and the 2nd
concave surfaces for the lens with the thickness multiple
to 2L (Figure 6).
The refraction scheme shown in Figure 6 also relates
to the case, when after refraction the direction coefficient
1
E
u of the ray refracted on the 1st surface is negative.
For this case, a general scheme of the ray path in the
lens is shown in Figure 7.
For the refraction scheme shown in Figure 6, the angle
of incidence 1

in the point A on the 1st surface can be
determined from the following expression:
 
111 1
tan 1
TN T
uu uu   1N , (25)
and the angle of refraction 1


is obtained fr

om the ex-
pression

111 1
tan 1
EN EN
uu uu 
 1
(26)
Figure 6. Schemes of refraction on the 1st convex and the
2nd concave surfaces for a lens with the thickness multiple
to 2L.
Figure 7. The ray path scheme for the lens with the thick
ness multiple to- 2L, having the 1st convex and the 2nd
concave surfaces.
1
Moreover, according to Snell's refraction law

11 1
sin sin
C
nny




Then, with regard to the fact that 11n, the refraction
angle is expressed as

11
sin sinC ny 

For the lens with t
1
he periodicity length 2L, the ray
refracted in the point A on the 1st surface, having the ini-
tial direction 1
E
u, spreads by periodical curvilinear path
af
ht
int t, according to the condition Equa-
tio
ter the refraction and in the point B, after passing the
half period:
1) the heig2C y of the refraction point B will be
equal, by the absolute value, to the height 1C
y of the
refraction po A, bu
n (10), will be negative;
2) direction coefficient 22T u

 of the ray ident to
the 2nd surface will be equal to 11E u
inc

 by the absolute
value but, according to the condition Equation (11), will
be negative and, thus,
21TE
uu

; (27)
3) direction coefficie2
N
u nt of the normal to the
1st surface in the point B will, b
equal to the direction coefficient
2
N
y the absolute value, be
1
N
u of the normal N
1
but, according the con-
dition Equation (12), will be negative:
21
to the 1st surface in the point Ao t
N
N
uu

 (28)
The angle of incidence 2

in the point B will be equal
to

 
22222
tan 1
NT TN
uu uu    (29)
Taking into account Equation (27) and Equation (28)
and substituting them into Equation (29), we get:

 
211 11
tan 1
EN EN
uu uu    (30)
Comparing Equation (26) and Equation (30), we get:
12




On this basis, we can write down that
21
sin sin




For the point B on the 2nd surface, according to Snell's
refraction law,


2212
sin sin
C
ny n


 ,
where


2C
ny is the RI on the refraction point B.
Subseq, with respect to the fact that uently
1
1n

, the
refraction angle will be expressed as follows:


222
sin sin
C
ny

 
Taking into account ratios Equation (18) an

d Equation
(19), we get:
Open Access OPJ

V. I. TARKHANOV
356
 
221
sin sin
CC
ny ny 

1
As mentioned above, since in the point B 21CC yy

,
but at that

n

21CC
yn y

, we get:
21
sin sin



Then, as a consequence,
12



and
21
tan tan

  (31)
The refraction angle 2

 in the point B may also be
ssed as follows: expre
 
222 22EN
uu tan 1
NE
uu 

(

If we use ratio Equation28) in this expression, we get:

22121
tan 
1
EN EN
uu uu  (32)
Based on Equation (31), and equating expressions
Eq
transformations, we get:
uation (25) and Equation (32) and making simple
21
E
T
As a result, the direction coefficient 2
uu
E
u of the exit
ray refracted in the point B will, by the absolute value, be
equal to the direction coefficient 1T u
e
of the entrance ray
in the point A, but will be negativ
th
pendently of
se
.
Let us show execution of aplanatic properties of the
lens on the example of the above-considered case of re-
fraction.
It is commonly known that the two conjugated points in
e space of objects and images are called aplanatic, if
spherical aberration is absent in the image and the sine
condition (or the Abbe sine law) is fulfilled.
The above proved property of the considered lenses
having a thickness multiple to half the periodicity length,
which is a preservation of direction of the initial ray inci-
dent to the 1st surface at the exit point inde
lected shape of the 1st and 2nd surfaces, provides the
absence of the spherical aberration for a point on the op-
tical axis at a finite distance from the lens.
Let us show now that for a pair of the conjugated points
1
M
and 2
M
 on the optical axis the Abbe sine law is
fulfilled.
In the general case, the Abbe sine law looks as follows:
sin sinnn





or
sin sinnn


 ,
where is the RI of the medium, in which the object is
ted;
is the refractive index of the meum, in which the
image is formed;
n
loca
n di

iting from the axial object point;


is the angle between the optical axis and the ray
exiting from the optical system and passing through the
axial image point;

is linear magnification of the optical system.
For considered versions of the suggested lens, the ratio
will become as follows:
21
sin sin



 (33)
where 1

is the angle between the optical axis and the
ray exiting from the axial (object) point 1
M
;
2


is the angle between the optical axis and the ray
exiting from the lens and passing through the axial image
point 2
M

(the virtual image).
Fo
is shown in Fi gure
8,
Let us consider the lenses with the 1st convex and the
2nd concave surfaces.
r the lens L thickness, with the 1st convex and the
2nd concave surfaces, the refraction scheme on the con-
vex 1st surface for the case 1E
u0
and for the 2nd surface—in Figure 9.
For the lens 2L thickness, with the 1st convex and
the
8-10, the ratio
Eq
e 2nd concavsurfaces, the refraction scheme on the
convex 1st surface for the case 10
E
u is shown in
Figure 8, and for the 2nd surface—in Figure 10.
For all cases considered in Figures
uation (33) will become as follows:
Figure 8. The refraction scheme on the 1st convex surface
of the lens.
Figure 9. The refraction scheme on the 2nd concave surface
for the lens with the thickness multiple to L.
is the angle between the optical axis and the ray ex-
Open Access OPJ

V. I. TARKHANOV 357
Figure 10. The refraction scheme on the 2nd concave surface
for the lens with the thickness multiple to 2L.
 

2
2
2
22
1
2
2
11
C
CCF
C
CCF
y
yzds
y
yzs






Since generatrixes of the 1st and 2nd surfaces have the
same shapes, then the deflections of the surfaces C

for
the same height are also equal, respectively:
Moreover, according to the initial data, the front dis-
12CC C
zzd
tance and the rear distance are equal, too:
FF
s
ss


As a result, we can write down that


2
2
2
C
y
ys 
As mentioned above, for the lenses thickness:
2
1
2
2
1
CC
C
CC
y
ys
 
 
L
12CC
yy,
and for the lenses of the thickness 2L
12CC
As a result, linear magnification of
yy
the lenses will be
constant and, for the lenses with the thickness multiple to
, will be equal to: L
1

,
and for the lenses with the thickness multiple to 2L:
1


Then in the lens under consideration, for a pair of the
conjugated points 1
M
and 2
M
 on the axis, spherical
d fny ray exiting
from an axial object point, the Abbe sine law will be ful-
filled. Subsequently, points
aberration will also be absent, anor a
1
M
and 2
M

direction coefficient of the
y locat
iple to
will form a
pa
see entrance ray at the exit point (or
preserve the absolute value of
equce,
beam of rays (with the center mae both on the
op
iding the pr
thicult
ir of aplanatic points, and the lens in this case may also
be called aplanatic.
The indicated property of the suggested lenses to pre-
rve direction of th
ray with reversing the coefficient sign), as a consen
leads to the fact that the entrance homocentric diverging
tha
ov
kness m
t
tical axis and outside of it) will preserve homocentricity
at the exit of the lens, and virtual extensions of the rays at
the exit point will concur forming an virtual image of the
object point.
Conditions for the lenses proved property
can be formulated:
1) as the lens is
F
, the lens
thickness projection on the optical axis in any plane con-
taining the optical axis, for any two points of the 1st and
2nd surfaces having the same absolute height relative to
the optical axis but located on different sides of it, equals
to
F
; or
2) as the lens is thickness multiple to 2
F
, the lens
thickness projection on the optical axis in any plane con-
taining the optical axis, for any two points of the 1st and
2nd surfaces having the same height relative to the optical
axis and located on the sa
eq
me side of the optical axis,
uals to 2
F
.
Thus, the lens having refractive surfaces, which meet
th
lend the homocentricit at the en-
tra
the lens (with respect to restrictions applied to
th
),
beam will re-
m
e above conditions, will possess the proved properties.
It is notable that the 1st consequence of this property will
be the absence of the necessary fixed location of an object
point on the optical axis. This means that the object point
may be located on the optical axis at any distance from
thens, ay of the beam
xit
nce of the lens will not be disturbed.
The 2nd consequence of this property will be the ab-
sence of the necessary fixed location of an object point
directly on the optical axis. This means that the object
point may be located outside the optical axis at any dis-
tance from
e refraction conditions on the 1st surface associated
with full internal reflection and finite diameter of the lens
and the homocentricity of the beam at the exit point of the
lens will not be disturbed.
The third consequence of this property will be simulta-
neous telescopic properties of the lens, since the entrance
parallel beam of rays at the exit point of the lens will pre-
serve its direction, and all rays of the e
ain parallel, hence, the entrance parallel beam is not
necessarily parallel to the optical axis of the suggested
lens.
As a result, the suggested lenses will possess aplanatic
and telescopic properties simultaneously, which has not
been known before.
Open Access OPJ

V. I. TARKHANOV
358
Thus, for the known lens [2], the presence of aplanatic
properties was proved before, but the simultaneous tele-
scopic properties were not known yet.
t and 2nd surfaces, in particular, for the 1st concave
an
sted lenses with a
th
For another known lens with the refracting 1st and 2nd
flat surfaces, which are normal to the optical axis [3], the
telescopic properties were indicated, hence, exclusively
for the exit ray parallel to the entrance ray. However, the
aplanatic properties were not known simultaneously.
Consideration of other cases and refraction versions on
the 1s
d the 2nd convex surfaces, for flat 1st and 2nd surfaces
etc., also allows proving the indicated property of the
lenses.
Numerical computation performed proves the above
indicated property of the lenses.
The indicated property of the sugge
ickness multiple to L and 2L can be used for
forming gradient lenses possessing both aplanatic and
telescopic properties simultaneously, as well as various
refractive surfaces.
It is natural to use surfaces of revolution with various
si
ith inflection points.
tive surfaces may be un-
sy
o inclined planes with different in-
cl
the optical axis of the lens is
co ay be reduced to one of
th
rix sections with different curvature
si
rmal to the
op
mpler generatrixes than type Equation (2) generatrix - a
straight line, a circle, etc. In this case, refractive surfaces
will be symmetrical relative to the optical axis. One may
also note that the refractive surfaces may have genera-
trixes w
However, this is optional.
In the general case, the refrac
mmetrical relative to the optical axis. There is a possi-
bility to use the refractive surfaces as the inclined planes,
for example, as well as combined refractive surfaces as a
combination of an inclined plane and a plane normal to
the optical axis, tw
ined angles, etc.
Note that if two inclined planes, 1st and 2nd, are used,
aplanatic and telescopic properties of the lens can be
proved using the above-mentioned approach, if each ray
path of the homocentric or parallel beam entrance in the
plane containing this ray and
nsidered. Then consideration m
e above-considered cases (not shown here).
It is possible to prove aplanatic and telescopic proper-
ties of a lens that has the refractive surfaces of revolution
with generatrixes having inflection points using the
above-mentioned approach, if we consider a path of each
ray of the homocentric or parallel beam entrance sepa-
rately for generat
gns. As a consequence, consideration for every section
can be reduced to one of the above-considered cases (not
shown here, either).
3. Possible Versions of the Lens Performance
Obviously, the simplest version of the lens performance
will be a lens with flat 1st and 2nd surfaces no
tical axis.
Other versions of suggested lens performance with the
thickness multiple to L and 2L are also possible:
with spherical refractive surfaces (Figures 11, 12);
with conic refractive surfaces (Figure 13);
with flat refractive surfaces (Figure 14);
with flat inclined refractive surfaces (Figure 15);
with combined refractive surfaces having flat and in-
clined flat surfaces (Figure 16).
In the context of considered versions of the suggested
lens, the known lens [2] is a particular case, for which
1) surfaces of revolution are selected as the refractive
ones;
Fith scheme for
th e 1st
co
gure 11. Parallel and homocentric beam pa
e lens with the thickness multiple to L, having th
nvex and the 2nd concave spherical surfaces.
Figure 12. Parallel and homocentric beam path scheme for
the lens with the thickness multiple to 2L, having the 1st
concave and the 2nd convex spherical surfaces.
Figure 13. Parallel and homocentric beam path scheme for
the lens with the thickness multiple to 2L, having the 1st
convex and the 2nd concave conic surfaces.
Figure 14. Parallel and homocentric beam path scheme for
the lens with the thickness multiple to 2L, having the 1st
and the 2nd flat surfaces normal to the optical axis.
Open Access OPJ

V. I. TARKHANOV
Open Access OPJ
359
Figure 15. Parallel and homocentric beam path scheme for
the lens with the thickness multiple to 2L, having the 1st
and the 2nd flat inclined surfaces.
Figure 16. Parallel and homocentric beam path scheme for
the lens with the thickness multiple to 2L
wit
, having the 1st
and the 2nd combined refractive surfacesh flat inclined
and flat surfaces normal to the optical axis.
d t
s of revolution, special
functions Equation (2) and Equation (3) are selected; for
the entrance ray, after the refraction on the 1st surface,
application of these functions provides direction parallel
to the optical axis.
It may also be noted that the maximum attainable nu-
merical aperture for all considered aplanatic lenses will
qua
eans that then,
hen passing the lens material, maximal height of the ray
on
his angle. According to the design, in
th
lanatic lens.
copic properties of the lens are inde-
cting surface shapes, which simplifies
creation of boroscopes, objectives, condensers,
co
[1]
ience Publishers, New
York, 2007, pp. 23-30.
[3] A. L. Mikaelyyered medium for
are parallel to the optical surface. Then, while passing
through the lens material, the maximum ray height on the
path does not exceed the ray entrance height.
4. Conclusions
As shown, aplanatic lens with a radial gradient of refrac-
tion index is simultaneously a telescopic lens, notably not
only for an axial beams, but also for an off axis parallel
beams.
Maximum reachable numerical aperture of the lens is
principally lower than that of the known ap
Aplanatic and teles
pendent of the refra
production of the lens.
Various suggested versions of the lens performance
may be applied to fiber optics and optical instrument-
making,
uplers for fiber-optic communication lines with sources
of radiation and photodetectors, etc.
REFERENCES
2) a version with the 1rst convex anhe 2nd concave
surfaces is selected;
3) for generatrixes for surface V. I. Tarkhanov, “Calculation of an Aplanatic Lens with
an Axial Gradient of the Refractive Index,” Optico-Me-
chanicheskaya Promyshlennost, No. 5, 1990, pp. 35-37.
[2] V. I. Tarkhanov, “Aplanatic Lens with a Radial Gradient
of Refraction Index,” In: I. Chen, Ed., Lasers, Optics, and
Electro-Optics Research, Nova Sc
obviously be principally lower than that for the known
lens [2], all other conditions being el and the same
lens diameters, because after refraction on the 1st surface,
rays are inclined to the optical axis. This m
an, “Application of the la
foc
Ind
using of waves,” Doklady Akademii Nauk SSSR, Vol.
81, No. 4, 1951, pp. 569-571.
[4] V. G. Ilyin, et al., “Physical Fundamentals for Gradient
ex Optics (textbook),” Leningrad Polytechnic Institute,
Leningrad, 1990.
w
the path will exceed the entrance height by a certain
value depending on t
e known lens, after refraction on the 1st surface all rays