U.S. patent application number 09/840867 was filed with the patent office on 2001-11-22 for optical system comprising a diffractive optical element, and method of designing the same.
Invention is credited to Ishii, Tetsuya.
Application Number | 20010043397 09/840867 |
Document ID | / |
Family ID | 15026079 |
Filed Date | 2001-11-22 |
United States Patent
Application |
20010043397 |
Kind Code |
A1 |
Ishii, Tetsuya |
November 22, 2001 |
Optical system comprising a diffractive optical element, and method
of designing the same
Abstract
The invention relates to a method of easily and quantitatively
determining the diffraction efficiency of a relief type diffraction
optical element, and provides a method of designing an optical
system comprising a diffractive optical element using diffraction
efficiency as at least one estimation index. The diffractive
optical element is a transmission blaze grating constructed of a
relief having a sawtooth shape in section. The transmission blaze
grating having a side wall substantially vertical to a grating
surface. The estimation index is given by a predetermined
mathematical expression including at least a phase difference due
to a relief discontinuity and shadow effects thereof. The
mathematical expression includes as the shadow effects of said
discontinuity at least a light attenuation effect dependent on an
angle-of-incidence and a light attenuation effect independent on
the angle-of-incidence in a separate manner.
Inventors: |
Ishii, Tetsuya; (Tokyo,
JP) |
Correspondence
Address: |
Intellectual Property Group
Pillsbury Winthrop LLP
East Tower, Ninth Floor
1100 New York Avenue, N.W.,
Washington
DC
20005-3918
US
|
Family ID: |
15026079 |
Appl. No.: |
09/840867 |
Filed: |
April 25, 2001 |
Related U.S. Patent Documents
|
|
|
|
|
|
Application
Number |
Filing Date |
Patent Number |
|
|
09840867 |
Apr 25, 2001 |
|
|
|
09291180 |
Apr 14, 1999 |
|
|
|
6246522 |
|
|
|
|
Current U.S.
Class: |
359/571 ;
359/565 |
Current CPC
Class: |
G02B 5/1866
20130101 |
Class at
Publication: |
359/571 ;
359/565 |
International
Class: |
G02B 027/44; G02B
005/18 |
Foreign Application Data
Date |
Code |
Application Number |
May 13, 1998 |
JP |
10-130105 |
Claims
What we claim is:
1. A method of designing an optical system comprising a diffractive
optical element using diffraction efficiency as at least one
estimation index, characterized in that: said diffractive optical
element is a transmission blaze grating constructed of a relief
having a sawtooth shape in section, said transmission blaze grating
having a side wall substantially vertical to a grating surface
thereof, and said estimation index is given by a predetermined
mathematical expression including at least a phase difference due
to a relief discontinuity and shadow effects thereof, said
mathematical expression including as the shadow effects of said
discontinuity at least a light attenuation effect dependent on an
angle-of-incidence and a light attenuation effect independent on
said angle-of-incidence in a separate manner.
2. An optical system comprising at least one diffractive optical
element on an optical axis thereof, characterized in that: said
diffractive optical element is a transmission blaze grating
constructed of a relief having a sawtooth shape in section, said
transmission blaze grating having a side wall substantially
vertical to a grating surface thereof, and said optical system
comprises said optical axis and at least one entrance surface
vertical to a grating groove, wherein an angle .theta..sub.en of an
incident light ray with a normal to said grating groove on said
entrance surface and a construction of said blaze grating in a
region of said grating surface on which said incident light is
incident satisfies the following relationship with respect all
light rays on said entrance surface upon incidence on said
diffractive optical element:1.gtoreq.PSE.times.SE>0.8where
PSE=.vertline.sin .pi.(m-.phi.)/{.pi.(m-.phi.)}.vertline..sup.2
.phi.=h/.lambda..multidot.(- n.sub.ex cos
.theta..sub.ex.sup.(m)-n.sub.en cos .theta..sub.en) n.sub.ex sin
.theta..sub.ex.sup.(m)=n.sub.en sin .theta..sub.en+m.lambda./p
SE={1-{square root}(SE1.sup.2+SE2.sup.2)}.sup.2
SE1=k.sub.1.multidot.(h/p- ) tan .theta..sub.ave
SE2=k.sub.2.multidot..lambda./p
.theta..sub.ave=(.theta..sub.en+.theta..sub.ex.sup.(m))/2
k.sub.1=0.85
k.sub.2=0.3.multidot..vertline.h/.lambda..vertline..sup.0.35
wherein n.sub.en is a refractive index of an entrance side medium,
n.sub.ex is a refractive index of an exit side medium, h is a
grating groove depth, p is a grating pitch, m is an order of
diffraction at which diffraction efficiency is maximized,
.theta..sub.ex.sup.(m) is an angle of an m-order diffracted light
ray with the normal to the grating surface, and .lambda. is a
wavelength of light used.
3. The optical system according to claim 2, which satisfies the
following relationship:1.gtoreq.PSE.times.SE>0.9
4. The optical system according to claim 2 or 3, wherein a sign of
an angle .theta..sub.en Of incidence of all chief rays on said
entrance surface upon incidence on said diffractive optical element
with respect to the normal to said grating surface coincides with a
sign of said order of diffraction m at which said diffraction
efficiency is maximized.
5. The optical system according to claim 4, which satisfies the
following
relationship:5>.vertline.n.sub.ex-n.sub.en.vertline.>0.5
6. The optical system according to any one of claims 2 to 5, which
satisfies the following relationship:.vertline.m.vertline.=1
7. The optical system according to any one of claims 2 to 6, which
is an image formation optical system for forming an image of an
object.
8. An optical system, characterized by comprising a transmission
blaze grating constructed of a relief having a sawtooth shape in
section and satisfying the following
relationships:0.8<h/h.sub.0.vertline.<16&l-
t;.vertline.p/h.sub.0.vertline.where
h.sub.0=.lambda./(n.sub.ex-n.sub.en), .lambda. is a wavelength of
light used, n.sub.en is a refractive index of an entrance side
medium, n.sub.ex is a refractive index of an exit side medium, h is
a grating groove depth, and p is a grating pitch.
9. The optical system according to claim 8, wherein a sign of an
angle .theta..sub.en of incidence of all chief rays on an entrance
surface upon incidence on said transmission blaze grating with
respect to a normal to a grating surface coincides with a sign of
an order of diffraction m at which diffraction efficiency is
maximized.
Description
BACKGROUND OF THE INVENTION
[0001] The present invention relates generally to a relief type
diffraction optical element of sawtooth shape in section, an
optical system comprising a relief type diffraction optical element
and a method of designing the optical system, and more particularly
to a relief type diffraction optical element having a lens
action.
[0002] A lens (diffractive lens) bending light by diffraction has a
unique yet useful feature different from the feature of a
refractive lens. For instance, it is possible to achieve an
achromatic optical system by making use of a diffractive lens
having dispersions (inverse and anomalous dispersions) different
from those of a refractive lens, and reduce the size and weight of
an optical system by making use of the feature of the diffractive
lens that has substantially no thickness. Such a diffractive lens
can be fabricated in the form of a relief type diffractive element
by use of recent micromachining technologies. Recently,
applications with an actually mounted diffractive lens have been
put to practical use. Thus, diffractive lenses have already been at
the commercial stage.
[0003] However, a diffractive optical element represented by a
diffractive lens has a unique property called diffraction
efficiency that is not found a refractive element. In some cases,
this property is a deterrent to the application of a diffractive
lens to a phototaking optical element. One example of that property
is the angle-of-incidence dependence of diffraction efficiency.
Even though a diffractive optical element is constructed in such a
manner that sufficient diffraction efficiency is obtainable at a
specific angle of incidence, the diffraction efficiency often drops
with a change in the angle of incidence--a phenomenon called the
angle-of-incidence dependence of diffraction efficiency. In
principle, this problem is inherent to every diffractive optical
element. Given an actual diffractive optical element, this problem
arises necessarily to one degree or another.
[0004] A lowering of diffraction efficiency with respect to major
orders of light is tantamount to an increase in other orders of
light (unnecessary light). Such unnecessary light takes the form of
ghosts or flares, causing a deterioration in the performance of an
optical system. Even when a diffractive lens is used in a
photodetection optical system such as a pickup lens, it is
therefore desired to achieve at least 60% diffraction efficiency.
Especially when the diffractive lens is used in an image formation
optical system to form images, at least 80% diffraction efficiency
is needed so as to keep a minimum of image quality. To achieve
high-definition image quality, at least 90% diffraction efficiency,
or at least 95% diffraction efficiency in some cases, is
needed.
[0005] In some cases, grooves in a relief type diffractive optical
element, for instance, are made flat by use of a transparent
optical material. The diffractive optical element of such
construction is characterized by being resistant to extraneous
contamination because the fine structure of a relief pattern is
protected, and so enables such optical design as to expose a relief
pattern surface substantially to the outside. This technique is
very effective to put a diffractive optical element to practical
use. However, this construction may possibly make the
angle-of-incidence dependence of diffraction efficiency worse than
that of an ordinary relief grating. Therefore, when this
construction is applied to an actual optical system, it is very
important to pay quantitative attention to the angle-of-incidence
dependence of diffraction efficiency.
[0006] To apply a diffractive optical element to an optical system
for practical purposes, it is then inevitable to pay quantitative
attention to the angle-of-incidence dependence of diffraction
efficiency. In such an ordinary method of designing a diffractive
lens as represented by an ultrahigh index method (Sweatt Model)
wherein a diffractive optical element is substituted by a virtual
refractive lens, however, whether the diffractive optical element
is a relief grating or a volume hologram is not designated at the
design stage. The angle-of-incidence dependence of diffraction
efficiency varies for each embodiment of the diffractive optical
element. In other words, such a design method cannot achieve design
with the angle-of-incidence dependence of diffraction efficiency in
mind. Even though an optical system comprising a diffractive
optical element is constructed according to such design, the
optical properties as designed would often be unexpectable.
[0007] As one means for providing quantitative analysis of the
angle-of-incidence dependence of diffraction efficiency in a relief
type diffractive optical element, electric field analysis based on
Maxwell's equations is generally available. Electric field analysis
based on Maxwell's equations may make it possible to have a
quantitative understanding of the angle-of-incidence dependence of
diffraction efficiency. When this method is used, however, it is
not easy to obtain the results of calculation, because a solution
of a differential equation should be found for each specific state
(the wavelength of light used, grating pitch, groove depth,
refractive index, and so on). To obtain a solution of the
differential equation, approximately P.times.P matrix calculations
are needed at a pitch P standardized with wavelength. At a pitch
level (usually 20 or more pitches standardized with wavelength)
often used for a diffractive lens in an image formation optical
system, it is very difficult to obtain the results of calculation.
Thus, it is impractical to perform such analysis parallel with lens
design.
[0008] As another means for estimating the diffraction efficiency
of a relief type diffractive element, a method of applying a thin
diffraction grating model (thin model) to Scalar diffraction
formula is available. This method has a feature of making it
possible to easily obtain the results of calculation, because the
diffraction efficiency of the relief type diffractive element is
described by a formulated (so-called closed form of) expression.
However, the method based on such a thin model is little applied to
the estimation of the angle-of-incidence dependence of diffraction
efficiency that is one specific object of the invention, because no
care is generally taken of the influence of oblique incidence. In a
model obtained by extension of the thin model (E. K. Popov et al.,
Opt. Commun. 80, (1991) 307), care is taken of the oblique
incidence feature of the relief grating. This model, too, is not
sufficient in terms of quantitative accuracy with which diffraction
efficiency is quantitatively estimated as contemplated in the
invention. Especially when grooves in the relief type diffractive
optical element are made flat with a transparent optical material,
the results obtained in this method are in no quantitative
coincidence with the results of stringent calculation based on
electric field analysis at all.
[0009] As explained above, any method of making quantitative
estimation of the diffraction efficiency of a relief type
diffractive optical element including its angle-of-incidence
dependence in an easy manner has not been established in the prior
art. For this reason, little care is taken of the
angle-of-incidence dependence of diffraction efficiency of an
optical system at the design stage; there is a large significant
difference between the optical performance and the design
performance of an actually constructed optical system.
SUMMARY OF THE INVENTION
[0010] In view of such problems associated with the prior art, one
object of the present invention is to provide a method of
quantitatively deriving the diffraction efficiency of a relief type
diffractive optical element including its angle-of-incidence
dependence in a simple manner. Another object of the present
invention is to clarify the relationship between the condition for
constructing a blaze grating important to an optical system and the
angle of incidence of light thereon, thereby providing a
high-performance optical system on the basis of the resulting
effective relationship.
[0011] According to one aspect of the invention, the above objects
are achieved by the provision of a method of designing an optical
system comprising a diffractive optical element using diffraction
efficiency as at least one estimation index, characterized in
that:
[0012] said diffractive optical element is a transmission blaze
grating constructed of a relief having a sawtooth shape in section,
said transmission blaze grating having a side wall substantially
vertical to a grating surface thereof, and
[0013] said estimation index is given by a predetermined
mathematical expression including at least a phase difference due
to a relief discontinuity and shadow effects thereof, said
mathematical expression including as the shadow effects of said
discontinuity at least a light attenuation effect dependent on an
angle-of-incidence and a light attenuation effect independent on
said angle-of-incidence in a separate manner.
[0014] According to another aspect of the invention, there is
provided an0 optical system comprising at least one diffractive
optical element on an optical axis thereof, characterized in
that:
[0015] said diffractive optical element is a transmission blaze
grating constructed of a relief having a sawtooth shape in section,
said transmission blaze grating having a side wall substantially
vertical to a grating surface thereof, and
[0016] said optical system comprises said optical axis and at least
one entrance surface vertical to a grating groove, wherein an angle
.theta..sub.en of an incident light ray with a normal to said
grating groove on said entrance surface and a construction of said
blaze grating in a region of said grating surface on which said
incident light is incident satisfies the following relationship
with respect all light rays on said entrance surface upon incidence
on said diffractive optical element:
1.gtoreq.PSE.times.SE>0.8
[0017] where
[0018] PSE=.vertline.sin
.pi.(m-.phi.)/{.pi.(m-.phi.)}.vertline..sup.2
[0019] .phi.=h/.lambda..multidot.(n.sub.ex cos
.theta..sub.ex.sup.(m)-n.su- b.en cos .theta..sub.en)
[0020] n.sub.ex sin .theta..sub.ex.sup.(m)=n.sub.en sin
.theta..sub.en+m.lambda./p
[0021] SE={1-{square root}(SE1.sup.2+SE2.sup.2)}.sup.2
[0022] SE1=k.sub.1.multidot.(h/p) tan .theta..sub.ave
[0023] SE2=k.sub.2.multidot..lambda./P
[0024]
.theta..sub.ave=(.theta..sub.en+.theta..sub.ex.sup.(m))/2
[0025] k.sub.1=0.85
[0026]
k.sub.2=0.3.multidot..vertline.h/.lambda..vertline..sup.0.35
[0027] wherein n.sub.en is a refractive index of an entrance side
medium, n.sub.ex is a refractive index of an exit side medium, h is
a grating groove depth, p is a grating pitch, m is an order of
diffraction at which diffraction efficiency is maximized,
.theta..sub.ex.sup.(m) is an angle of an m-order diffracted light
ray with the normal to the grating surface, and .lambda. is a
wavelength of light used.
[0028] According to yet another aspect of the invention, there is
provided an optical system, characterized by comprising a
transmission blaze grating constructed of a relief having a
sawtooth shape in section and satisfying the following
relationships:
0.8<.vertline.h/h.sub.0.vertline.<1
6<.vertline.p/h.sub.0.vertline.
[0029] where h.sub.0=.lambda./(n.sub.ex-n.sub.en), .lambda. is a
wavelength of light used, n.sub.en is a refractive index of an
entrance side medium, n.sub.ex is a refractive index of an exit
side medium, h is a grating groove depth, and p is a grating
pitch.
[0030] In the present invention, the estimation index is given by a
predetermined mathematical expression including at least a phase
difference due to relief discontinuities and shadow effects
thereof. The mathematical expression includes as the shadow effects
of the discontinuities at least a light attenuation effect
dependent on an angle-of-incidence and a light attenuation effect
independent on the angle-of-incidence in a separate manner. It is
thus possible to provide a quantitatively accurate calculation of
the diffraction efficiency of a transmission blaze grating. By
using this in combination with ray tracing, it is possible to
improve the accuracy of optical design for achieving a diffractive
optical element in the form of a transmission blaze grating and the
efficiency of development thereof and, hence, develop a diffractive
optical element including optical system having ever higher
performance than achieved in the prior art.
[0031] Still other objects and advantages of the invention will in
part be obvious and will in part be apparent from the
specification.
[0032] The invention accordingly comprises the features of
construction, combinations of elements, and arrangement of parts
which will be exemplified in the construction hereinafter set
forth, and the scope of the invention will be indicated in the
claims.
BRIEF DESCRIPTION OF THE DRAWINGS
[0033] FIG. 1 is a schematic for defining the relationship between
the configuration of a blaze grating used herein and a incident
light ray.
[0034] FIG. 2 is a schematic for illustrating the phase shift
action of a relief grating.
[0035] FIG. 3 is a schematic for illustrating the shadow effects of
a relief discontinuity in the relief grating.
[0036] FIGS. 4(a), 4(b), 4(c) and 4(d) are graph representations
for illustrating the results of calculation by the approximate
expression of the invention and the results of stringent
calculation on an electric field analysis basis when a relief is
formed on the surface of an optical material.
[0037] FIGS. 5(a), 5(b), 5(c) and 5(d) are graph representations
similar to FIGS. 4(a), 4(b), 4(c) and 4(d) when a relief is formed
across the boundary of two optical materials.
[0038] FIGS. 6(a), 6(b), 6(c) and 6(d) are graph representations
similar to FIGS. 4(a), 4(b), 4(c) and 4(d) when groove depth is set
such that the transmission secondary diffraction efficiency is
maximized.
[0039] FIGS. 7(a), 7(b), 7(c) and 7(d) are graph representations
for illustrating the relationship between the groove depth of the
blaze grating and the angle-of-incidence dependence of diffraction
efficiency.
[0040] FIGS. 8(a), 8(b), 8(c) and 8(d) are representations in which
contours lines at which an estimation index EC for first-order
diffracted light is 80%, and 90% are drawn using the angle of
incidence and the groove depth as parameters.
[0041] FIGS. 9(a), 9(b), 9(c) and 9(d) are representations similar
to FIGS. 8(a), 8(b), 8(c) and 8(d).
[0042] FIG. 10 is a conceptual rendering of the phototaking optical
system of the invention.
[0043] FIG. 11 is a conceptual rendering of the optical system of
the invention wherein a light source is incorporated.
[0044] FIG. 12 is a sectional schematic of a blaze grating formed
on the surface of the boundary of two different optical
materials.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0045] To accomplish the above objects, the inventors come up with
a method of applying an extended thin model to Scalar diffraction
formula, i.e., a method of improving the above attempt made by E.
K. Popov et al., thereby making calculation (approximate
calculation) on an extended thin model basis coincide
quantitatively with stringent numerical calculation on an electric
field analysis basis. How to design a relief type diffractive
optical element to achieve the first object of the invention will
now be explained in detail.
[0046] The relationship between the configuration of a blaze
grating used herein and an incident light ray is defined in FIG. 1.
In FIG. 1, an x-y plane is assumed to lie on an entrance surface
vertical to grating grooves and including an optical axis, an
x-axis to be contiguous to a reference (grating) surface with a
relief formed thereon, and a y-axis to be vertical to the x-axis at
its point of contact. Here let the sign of each axis be positive in
the direction of an arrow in FIG. 1. A thick line stands for a
boundary between a medium 1 having a refractive index n.sub.en and
a medium 2 having a refractive index n.sub.ex, and defines a
sectional shape of the blaze grating. As shown, the blaze grating
has a sawtooth sectional shape having a side wall blaze vertical to
the x-axis, i.e., vertical to the grating surface. The pitch is
given by p, and the groove depth is given by h (tan
.theta..sub.b=h/p where .theta..sub.b is a blaze angle). It is here
to be noted that when h is negative, light rays incident on the
blaze grating are separated into plural orders of diffractive
light. At this time, the relationship between the angle of
incidence .theta..sub.en of the light ray on the x-y plane, i.e.,
the entrance surface and an m-order angle of diffraction
.theta..sub.ex.sup.(m) is defined by
n.sub.ex sin .theta..sub.ex.sup.(m)=n.sub.en sin
.theta..sub.en+m.lambda./- p (1)
[0047] Here let the signs of the angle of incidence and the angle
of diffraction be positive in the counterclockwise direction, and
the sign of the order of diffraction be positive in the direction
in which the angle of diffraction increases. Unless otherwise
stated, the angle of incidence or the angle of diffraction is
assumed to be determined on the x-y plane (entrance surface).
[0048] Among factors for determining the diffraction efficiency of
such a blaze grating as shown in FIG. 1, the following factors are
considered to be significant:
[0049] (1) effect of phase shift action by relief (PSE),
[0050] (2) efficiency drop due to the shadow effects of a relief
discontinuity (SE), and
[0051] (3) drop of energy transmittance by surface reflection (FT).
From these factors, it is possible to obtain an expression giving
transmission m-order diffraction efficiency, as described
below.
.eta..sub.m=FT.times.PSE.times.SE (2)
[0052] The inventors have built up a model for each factor, and
made a quantitative comparison of the results of approximate
calculation obtained from the expression with the results of
stringent numerical calculation, and found that the models of the
following constructions are preferable because of their simplicity
and quantitatively precise coincidence. The construction of the
model for each factor will now be explained.
(1) Effect of the Phase Shift Action by Relief (PSE)
[0053] A ray bundle incident on a relief grating at a certain angle
of incidence .theta..sub.en leaves the relief grating at an angle
of diffraction .theta..sub.ex.sup.(m) of m-order diffracted light
where m is an integer). On the basis of such a ray tracing
assumption as shown in FIG. 2, the phase shift action .phi. of the
relief grating (the amplitude of a phase shift function) is then
given by a difference between two optical paths AB and A'B' shown
in FIG. 2. Hence,
.phi.=h/.lambda.-(n.sub.ex cos .theta..sub.ex.sup.(m)-n.sub.en cos
.theta..sub.en) (3)
[0054] By applying this result to an ordinary Scalar diffraction
formula, the following expression can be obtained as the effect PSE
of the phase shift action of a transmission blaze grating:
PSE=.vertline.sin .pi.(m-.phi.)/{.pi.(m-.phi.)}.vertline..sup.2
(4)
[0055] With this expression, it is possible to obtain an effective
expression of the asymmetry of the angle-of-incidence of
diffraction efficiency because the m-order diffracted light is
assumed to be a ray leaving a relief slope.
(2) Drop of Efficiency Due to the Shadow Effects of a Relief
Discontinuity (SE)
[0056] When light is incident on a blaze grating from an oblique
direction, a part of the incident ray bundle is considered to be
cut off by a relief discontinuity, as shown by a hatched region in
FIG. 3. For such shadow action, independent hypotheses are set up
about the geometrical shadow effect of the side wall of a grating
groove, i.e., an effect SE1 dependend on the angle of incidence,
and a light attenuation effect SE2 that is found in the vicinity of
the boundary of discontinuities and is independent on the angle of
incidence. Here these actions are expressed in the following
forms:
SE1=k.sub.1.multidot.(h/p) tan .theta..sub.ave (5)
SE2=k.sub.2.multidot..lambda./p (6)
[0057] where .theta..sub.ave is an average of the angle of
incidence and the exit angle or
.theta..sub.ave=(.theta..sub.en+.theta..sub.ex.sup.(m))/2 (7)
[0058] and k.sub.1 and k.sub.2 are arbitrary correction
coefficients introduced for quantitatively combining together the
results of calculation. With expressions (5) to (7), the drop of
efficiency by the shadow effect of relief discontinuities is herein
expressed in the following form:
SE={1-{square root}(SE1.sup.2+SE2.sup.2)}.sup.2 (8)
[0059] With the term SE1 that expresses the geometrical shadow
effect, it is possible to obtain an effective expression of the
angle-of-incidence property when the relief groove becomes deep,
and with the term SE2 independent on the angle of incidence, it is
possible to obtain an effective expression of diffraction
efficiency in the vicinity of vertical incidence. With the
arbitrary correction coefficients k.sub.1 and k.sub.2, it is
possible to obtain results with quantitatively high precision.
(3) Drop of Energy Transmittance Due to Surface Reflection (FT)
[0060] To deal with reflection at the surface of a relief grating,
assume that the slope surface of a blaze grating is of sufficient
magnitude. From angles of incident light and zero-order diffractive
light with respect to the normal to the slope surface, the
transmission coefficient C.sub.t of the slope surface of the blaze
grating is given in such a form as described below:
C.sub.t=2n.sub.en cos (.theta.en-.theta..sub.b).div.{n.sub.en cos
(.theta..sub.en-.theta..sub.b)+n.sub.ex cos
(.theta..sub.ex.sup.(0)-.thet- a..sub.b)} (9)
[0061] where .theta..sub.b is a blaze angle. It is here to be
understood that this holds of, for instance, when the electric
field vector of the incident light is parallel with the grating
groove (TE mode). From a comparison of the energy of the incident
light with the energy of the m-order diffractive light on the basis
of this result, an expression with respect to energy transmittance
is given by
FT=(n.sub.ex cos .theta..sub.ex.sup.(m)/n.sub.en cos
.theta..sub.en).multidot..vertline.C.sub.t.vertline..sup.2 (10)
[0062] By the application of each of the above constructions to
equation (2), approximate formula Tim with respect to the
transmission m-order diffraction efficiency can be obtained. It is
here to be noted that the above expression includes arbitrary
correction coefficients k.sub.1 and k.sub.2. For instance, these
coefficients may be
k.sub.1=0.85 (11)
k.sub.2=0.3.multidot..vertline.h/.lambda..vertline..sup.0.35
(12)
[0063] The same shall hold for all the following embodiments.
[0064] It is to be understood that the values of k.sub.1 and
k.sub.2 indicated herein are to balance the precision of
approximate formula .eta..sub.m against practically possible
conditions. In other words, the values of k.sub.1 and k.sub.2 are
not limited to these values, and so may be effectively determined
depending on the conditions. While separate correction coefficients
are determined with respect to SE1 and SE2, it is to be understood
that a correction coefficient common to these may be
determined.
[0065] To enhance the precision of approximate formula (2),
however, it is preferable that correction coefficient k.sub.1 is a
constant while correction coefficient k.sub.2 is a variable
proportional to (h/.lambda.), as can be seen from equations (11)
and (12).
[0066] FIGS. 4 and 5 illustrate the results of calculation by the
approximate formula according to the invention together with the
results of stringent numerical calculation on an electric field
analysis basis. Here, the groove depth h is determined such that
the transmission first-order diffraction efficiency reaches a
maximum on the basis of the thin model. In each figure, symbols
.largecircle. and .quadrature. show the results of stringent
calculation of diffraction efficiency DE (O:p=50 [WL] and
.quadrature.:p=20 [WL] where WL is wavelength). Solid and broken
lines show the results of calculation of diffraction efficiency DE
by approximate formula (2) (solid line:p=50 [WL] and broken
line:p=20 [WL] where WL is wavelength. FIG. 4 shows a simple case
where a relief is merely formed on the surface of an optical
material and one of media with the relief interleaved between them
is an outside atmosphere (substantially air with a refractive index
of 1), and FIG. 5 shows a case where a relief is formed across the
boundary of two optical materials (the relief grating grooves are
filled up with another optical material). From the results of these
calculations, it is found that for practically possible
combinations of entrance side media and exit side media, the
results of calculation by the approximate formula of the invention
are in good coincidence with the results of the stringent numerical
calculation.
[0067] FIG. 6 shows the results of calculation when the groove
depth h is determined such that the transmission second-order
diffraction efficiency reaches a maximum. All conditions in FIG. 6,
except that the groove depth h is twice as large, are the same as
those in FIG. 4. As can be seen from FIG. 6, approximate formula
(2) can quantitatively express well the angle-of-incidence
dependence of diffraction efficiency even in the case where the
second-order diffraction efficiency reaches a maximum.
[0068] As explained above, the present invention enables the
diffraction efficiency of a transmission blaze grating to be
quantitatively estimated in a simple manner because approximate
formula (2) provides an accurate expression of the diffraction
efficiency of the transmission blaze grating. Approximate formula
(2) has the construction conforming to the physical action of a
blaze grating, and so makes it possible to obtain the results of
quantitatively accurate calculation of the diffraction efficiency
of every practical blaze grating according to the invention.
Therefore if this is used in combination with ray tracing, it is
then possible to enhance the precision of an optical design for
realizing a diffractive optical element in the form of a
transmission blaze grating and improving the efficiency of
developing the optical design. This in turn makes it possible to
develop an optical system comprising a diffractive optical element
higher in performance than achieved in the prior art.
[0069] Further, the inventors have found an estimation index EC as
a criterion for determining practical conditions for constructing a
transmission blaze grating from the standpoint of the
angle-of-incidence dependence of diffraction efficiency, as
described below:
EC=PSE.times.SE (13)
[0070] This index EC is a subtraction of factor FT given by
equation (1) or the drop of diffraction efficiency due to the
influence of reflection from the expression given by formula (2).
Such an influence of reflection may be compensated for by the
application of an antireflection coating, and so the nature
inherent to a diffractive optical element can be effectively
expressed by estimation index EC given by equation (13). This in
turn enables an optical system comprising a more effective
diffractive optical element to be developed.
[0071] In what follows, a detailed account will be given of the
condition preferable for constructing a blaze grating to accomplish
the second object of the invention while the angle-of-incidence
dependence of diffraction efficiency is taken into
consideration.
[0072] Here an index to the minimum condition to form a
satisfactory image is assumed to be 80% diffraction efficiency.
According to an article by D. A. Buralli et al. (APPL. OPT. 31,
(1992) 4389), it is pointed out that unncessary light occurring
with a drop of diffraction efficiency acts as background light to
lower image contrast until MTF at zero spatial frequency decreases
to a value of the order of diffraction efficiency. This teaches
that the lowering of diffraction efficiency to 80% implies that MTF
at zero spatial frequency drops to 80% and, at the same time, MTF
drops with this value even in a region having a high spatial
frequency. Although depending on applications where an optical
system is used, it is generally believed that the optical system
has not sufficient image formation capabilities on this level. From
this, the 80% diffraction efficiency is regarded as an index to the
formation of a satisfactory image.
[0073] It is here to be noted that the D. A. Buralli et al.
argument is applied to the image formation capabilities of a single
diffractive lens on the premise that the unnecessary light is
uniformly distributed in an image plane. Accordingly, such an
argument is not correct on a general optical system. However, this
argument appears to be well grounded as an index to a full
understanding of MTF degradation in the spatial frequency region of
interest.
[0074] As already explained, the diffraction efficiency of the
transmission blaze grating can be described in terms of estimation
index EC that is given by equation (13) and expresses the nature
inherent to a diffractive optical element, and factor FT or the
drop of energy transmittance by surface reflection. Here, the drop
of energy transmittance can be avoided by use of an antireflection
coating, etc.; the above condition for the formation of a
satisfactory image is achieved by determining the structure of the
blaze grating in such a manner that estimation index EC exceeds
80%.
[0075] In order to form a satisfactory image by an image formation
optical system comprising a transmission blaze grating, therefore,
the optical system should be constructed in such a manner that the
relationship between the angle of incidence of light on at least an
entrance surface and the construction of a blaze grating in a
region corresponding thereto satisfies the following condition (8)
with respect to estimation index EC given by equation (13):
EC>80% (14)
[0076] With this construction, at least 80% diffraction efficiency
can be kept with respect to all ray bundles on at least the
entrance surface, so that the optical system can have satisfactory
image formation capabilities.
[0077] FIG. 10 is a conceptual rendering of an phototaking optical
system constructed as mentioned just above. A phototaking optical
system 4 comprises a refractive lens 1, a diffractive lens 2, an
aperture stop 3 and an image pickup means 5 for picking up an image
of an object, all disposed on an optical axis. By optimizing this
construction, aberrations of the phototaking optical system 4 are
well corrected in a practically satisfactory range. The surface of
diffractive lens 2 is shown on an enlarged scale in a circle on the
right side in FIG. 10. As shown, the diffractive lens 2 is a
transmission blaze grating constructed of an optically transparent
material substrate with a refractive index N and a relief formed on
the surface thereof with a sawtooth shape in section. The relief is
made up of a concentric form of relief pattern so that the
diffractive lens 2 can have a predetermined lens action. According
to the invention, the relief has a groove depth h and a pitch p,
and the lens action of diffractive lens 2 is balanced against the
construction of phototaking optical system 4 in such a manner that
the maximum angle of incidence .theta..sub.max of a light ray on
the relief region satisfies condition (14) with respect to the
refractive index N of the substrate. Thus, unnecessary light
resulting from diffractive lens 2 can be effectively reduced so
that the phototaking optical system can have high image formation
capabilities. The present invention is particularly effective for
an image formation optical system for forming an image of an
object, wherein the influence of unnecessary light becomes a grave
problem.
[0078] When an optical system comprises a photodetector member such
as image pickup means 5 as in this embodiment, the wavelength
.lambda. of the light used may be any wavelength in a wavelength
band to which the photodetector member is sensible.
[0079] It is evident that the higher the diffraction efficiency of
a diffractive optical element, the more improved the image
formation capabilities of an optical system comprising the
diffractive optical element is. To achieve even higher image
formation capabilities, it is thus preferable that the relationship
between the relief structure and the above maximum angle of
incidence is determined in such a manner that estimation index EC
satisfies the following condition 15):
EC>90% (15)
[0080] From detailed review of the results shown in FIGS. 4 and 5,
it is found that when the sign of the angle-of-incidence is
positive in an angle-of-incidence range where generally practical
diffraction efficiency is achieved, the diffraction efficiency is
less dependent on the angle-of-incidence. According to what is
implied by estimation index EC given by condition (13), the case
where the sign of the angle-of-incidence coincides with the sign of
the order of diffraction is generally characterized in that the
angle-of-incidence dependence is reduced. For an optical system
constructed as mentioned above, it is practically effective to
locate the angle-of-incidence in the direction satisfying this
condition (or in the direction in which the sign of the
angle-of-incidence coincides with the sign of the order of
diffraction). In other words, it is desired for an optical system
comprising a blaze grating that the signs of the angles of
incidence of all chief rays on the entrance surface of the blaze
grating be in coincidence with the sign of the m-order of
diffraction at which the maximum diffraction efficiency is
obtained. With this arrangement, the angle-of-incidence of
diffraction efficiency is reduced while high diffraction efficiency
is maintained, so that the optical system can have ever higher
performance.
[0081] It is here to be noted that a phenomenon wherein the
angle-of-incidence dependence of diffraction efficiency becomes
asymmetric, for the most part, is ascribable to the effect of phase
shift action given by equation (4). According to what is implied by
estimation index EC given by equation (13), the effect prevails
when there is large difference in refractive index between the
entrance side and the exit side. For the above arrangement
constructed on the assumption that practical optical materials are
used, it is preferable that the media on the entrance and exit
sides are determined in such a manner that the following condition
(16) is satisfied:
5>.vertline.n.sub.ex-n.sub.en.vertline.>0.5 (16)
[0082] From FIG. 6 it is evident that the angle-of-incidence
dependence of diffraction efficiency becomes worse with an increase
in the order of diffraction m where the maximum diffraction
efficiency is obtained. To make effective use of an diffractive
optical element in a wider range of specifications, it is thus
desired that the order of diffraction m used satisfy the following
condition (17):
.vertline.m.vertline.=1 (17)
[0083] With this arrangement, it is possible to widen the range of
application of a diffractive optical element to an optical system
and, hence, achieve an ever higher-performance optical system,
because it is possible to use the +first-order or -first-order
diffracted light where the angle-of-incidence dependence of
diffraction efficiency is minimized.
[0084] FIG. 11 is a conceptual rendering of the optical system of
the invention wherein a light source is incorporated. In FIG. 11,
an illumination optical system 9 comprises a light source 6, a
condenser lens 7 and a band-pass filter 8, and illuminates the
surface of a sample. Reference numeral 10 stands for an image
formation optical system for forming an image of the sample. The
image formation optical system 10 has a transmission blaze grating
therein, and the construction of the blaze grating and the angle of
incidence of a light ray thereon satisfy the aforesaid
predetermined condition according to the invention. When the
optical system of the invention comprises a light source as in this
embodiment, the wavelength .lambda. of the light used may be any
wavelength in a wavelength band of light emitted from the light
source. When the optical system of the invention comprises a
band-pass filter as in this embodiment, the wavelength .lambda. of
the light used may be any wavelength in a wavelength band of light
which transmits through the band-pass filter.
[0085] For the optical system of the invention as explained above,
it is effective that the wavelength .lambda. of the light used is
substantially a center wavelength in a wavelength band of light
that transmits substantially through the optical system. By
selecting the wavelength .lambda. of the light used, it is thus
possible to obtain a satisfactory image in a wide wavelength region
in a wavelength range of substantially usable light.
[0086] Next, the structure of the blaze grating used to accomplish
the second object of the invention will be explained.
[0087] FIG. 7 illustrates the relationship between blaze grating
groove depths h and the angle-of-incidence dependence of
diffraction efficiency. The groove depths h in FIGS. 7(a), 7(b),
7(c) and 7(d) are determined such that h=0.8.multidot.h.sub.0,
h=0.9.multidot.h.sub.0, h=1.1.multidot.h.sub.0, and
h=1.2.multidot.h.sub.0 where h.sub.0 is the optimum value of
transmission first-order diffractive light found on an ordinary
thin model base or
h.sub.0=.lambda./(n.sub.ex-n.sub.en) (18)
[0088] The refractive indices of the entrance and exit sides, too,
are determined such that n.sub.en=1.0 and n.sub.ex=1.5,
respectively. As in FIGS. 4 to 6, symbols .largecircle. and
.quadrature. show the results of stringent calculation (O:p=50 [WL]
where WL is wavelength, and .quadrature.:p=20 [WL]), and solid and
broken lines show the results found by approximate formula (2)
(solid line:p=50 [WL], and broken line:p=20 [WL]).
[0089] As can be seen from FIG. 7, if a slight lowering of
diffraction efficiency in the vicinity of vertical incidence is
within allowable limits, it is then possible to make the blaze
grating groove depth h smaller than the optimum value found on a
thin model basis, thereby improving the angle-of-incidence
dependence of diffraction efficiency. By detailed examination of
what is implied by the aforesaid estimation index EC, the inventors
have found that the angle-of-incidence dependence of diffraction
efficiency is improved by permitting the construction of a blaze
grating to satisfy the following conditions (19) and (20):
0.8<.vertline.h/h.sub.0.vertline.<1 (19)
6<.vertline.p/h.sub.0.vertline. (20)
[0090] By satisfying these conditions, it is thus possible to keep
the diffraction efficiency needed to form satisfactory images in a
wide angle-of-incidence range.
[0091] In FIGS. 8 and 9, contour lines at which m=1, i.e.,
estimation index EC is 80% and 90% with respect to first-order
diffracted light are drawn using the angle of incidence and the
groove depths as parameters. In FIG. 8 the contour lines are drawn
at a pitch p=20 [WL], and in FIG. 9 the contour lines are drawn at
a pitch p=50 [WL]. In FIGS. 8(a), 8(b), 8(c) and 8(d), and FIGS.
9(a), 9(b), 9(c) and 9(d), .vertline.p/h.sub.0 is 5, 6, 7 and 8,
respectively. Here the refractive indices of the entrance and exit
sides are assumed to be n.sub.en=1.3 and n.sub.ex>n.sub.en,
respectively. However, this assumption has little or no influence
on the angle-of-incidence dependence. As can be seen from FIGS. 8
and 9, unless the construction of the blaze grating satisfies
condition (20), the angle-of-incidence dependence of diffraction
efficiency then becomes worse when the sign of the angle of
incidence is positive, and unless condition (19) is satisfied,
there is then a drastic reduction in the angle-of-incidence range
where the diffraction efficiency needed to form satisfactory images
is achievable.
[0092] From FIGS. 8 and 9 it is also found that the
angle-of-incidence dependence of diffraction efficiency of the
above construction is improved when the angle of incidence is
negative, and generally when the angle of incidence and the order
of diffraction m are of opposite sign. For an optical system
comprising the blaze grating constructed as mentioned above, it is
thus preferable that the signs of all chief rays incident on the
entrance surface of the blaze grating are opposite to the sign of
the order of diffraction m at which the diffraction efficiency
reaches a maximum. With this construction, it is possible to keep
the angle-of-incidence of diffraction efficiency low as a whole
and, hence, achieve an ever higher-performance optical system.
[0093] For the constructions of the blaze grating and optical
system as explained above, it is possible to arbitrarily select the
wavelength .lambda. of the light used from the following range:
0.4 .mu.m<.lambda.<10 .mu.m (21)
[0094] Alternatively, the wavelength .lambda. may arbitrarily be
selected from a range of wavelength substantially transparent to
the entire optical system. Especially when satisfactory image
characteristics are obtained in a white image formation optical
system, however, it is effective to select the wavelength k from
the following range:
0.45 .mu.m<.lambda.<0.55 .mu.m (22)
[0095] By selecting the wavelength .lambda. from this range, it is
possible to satisfactorily keep the angle-of-incidence dependence
of diffraction efficiency in a wide range of wavelengths taking
part in the formation of white images.
[0096] It is here to be noted that the blaze grating of the
invention or the blaze grating used with the optical system of the
invention is advantageously formed on the surface of the boundary
of two different optical materials, as shown in FIG. 12 wherein
reference numeral 11 represents the contour of a section of the
blaze grating, 12 a first optical material, and 13 a second optical
material. In general, a blaze grating having such a layer structure
has deep relief grooves and narrow allowable limits to oblique
incidence. With the arrangement of the invention, however, it is
possible to obtain satisfactory images less susceptible to
deterioration due to unnecessary light. It is also possible to
effectively prevent contamination of the surface of a relief
pattern with fingerprints, etc., thereby enabling such optical
design as to expose the relief pattern surface substantially to the
outside. This in turn makes it possible to enlarge the range of
application of a diffractive optical element and, hence, provide an
ever higher-performance optical system.
[0097] According to the optical system comprising the diffractive
optical element of the invention and the method of designing the
optical system, the estimation index is given by a predetermined
mathematical expression including at least a phase difference due
to relief discontinuities and shadow effects thereof. The
mathematical expression includes as the shadow effects of the
discontinuities at least a light attenuation effect dependent on an
angle-of-incidence and a light attenuation effect independent on
the angle-of-incidence in a separate manner. It is thus possible to
provide a quantitatively accurate calculation of the diffraction
efficiency of a transmission blaze grating. By using this in
combination with ray tracing, it is possible to improve the
accuracy of optical design for achieving a diffractive optical
element in the form of a transmission blaze grating and the
efficiency of development thereof and, hence, develop a diffractive
optical element-including optical system having ever higher
performance than achieved in the prior art.
* * * * *