Imaging optical system

An imaging optical system (10) having a focal length (f) and a maximum imagewise numerical aperture. The system combines diffractive and refractive optical elements such that aberrations, including axial chromatic aberration, are well-corrected over a large wavelength range from the visible into the infrared. The system comprises, objectwise to imagewise along an optical axis (A), a front lens group (G.sub.F) having at least one refractive optical element, a diffractive optical element (DOE) provided with a diffractive surface (D) having positive diffractive power and a focal length (f.sub.DOE). The system also includes a rear lens group (G.sub.R) having at least one refractive optical element. The system has an amount of spherical aberration at the maximum imagewise numerical aperture, is measured with respect to e-line wavelength light, in the limit as the positive diffractive power of the diffractive surface approaches zero, of RSA. The system also has a maximum amount of axial chromatic aberration of the imaging optical system over a wavelength range of 400-850 nm, as measured with respect to e-line reference wavelength light, of LA. Further, the following conditions, as well as several others, are preferably satisfied: EQU -0.01<RSA/f<0.01 (1) EQU -0.002<LA/f<0.002. (2)

FIELD OF THE INVENTION 
The present invention relates to an imaging optical system capable of 
forming an image of an object arranged at a finite distance, as in a 
scanner optical system, and more particularly relates to the improved 
correction of chromatic aberration of such imaging optical systems over a 
wide wavelength range extending from the visible wavelength region to the 
infrared wavelength region. 
BACKGROUND OF THE INVENTION 
An optical system for a scanner requires the ability to faithfully read the 
information of the original picture or object being scanned. Accordingly, 
it is necessary to correct various aberrations like spherical aberration 
for a single color, as well as to satisfactorily correct axial and 
transverse chromatic aberration. Axial chromatic aberration increases 
proportionate to the square of the imaging magnification in an optical 
system that forms the image of an object arranged at a finite distance, 
such as in an optical system for a scanner. Thus, the correction of 
chromatic aberration is even more critical. 
Generally, it is necessary with an optical system for a scanner to 
faithfully reproduce the original picture or object in the visible 
wavelength region. However, it has also become necessary in recent years 
for such optical systems to be corrected for chromatic aberration over a 
wide wavelength range extending from the visible region to the infrared 
region in the vicinity of 850 nm. 
Accordingly, to make such a system apochromatic (i.e., wherein chromatic 
aberration is corrected over a wide wavelength range), use of anomalous 
dispersion glass of the phosphoric acid series (e.g., phosphosilicate 
glass) may be considered. However, in addition to the high cost of the 
materials for these glasses, there is also the problem of an increase in 
fabrication costs due to poor workability. Also, temperature changes cause 
large changes in the refractive index of finished parts made from such 
glass as compared to ordinary glass, thereby adversely impacting imaging 
performance. Furthermore, since the refractive index of anomalous 
dispersion glass is typically low at around 1.5, the number of lenses 
needed to image with sufficient brightness increases, further increasing 
cost. 
On the other hand, the application of a diffractive optical element (DOE) 
to an imaging optical system has been proposed for the purpose of 
correcting chromatic aberration. Due to the deflection of the light by 
diffraction, the wavelength-dispersion characteristics of a diffractive 
optical element differs from that of an ordinary refractive optical 
element. Accordingly, combining a diffractive optical element with a 
refractive optical element has received attention as a new 
aberration-correcting means. 
The article entitled "The Phase Fresnel Lens," in the Journal of the 
Optical Society of America, Vol. 51, No. 1, 1961 ("the JOSA reference") 
discusses a Fresnel lens wherein the phase differential between light 
passing through adjacent phase rings is 2.pi. with respect to a specified 
wavelength. The JOSA reference proposes that this phase Fresnel lens is 
effective as an aberration-correcting means, and discloses an optical 
system that corrects, for example, spherical aberration by arranging a 
phase Fresnel lens at the pupil position of the imaging optical system. 
The Figures show a Schmidt lens and a triplet lens as Working Examples. In 
addition, the JOSA reference mentions that, taking the wavelength 
characteristics of the phase Fresnel lens into consideration, it is 
effective in the correction of the secondary spectrum of a doublet lens. 
Also, actual design values of a collimator lens are provided. Furthermore, 
the JOSA reference suggests that axial chromatic aberration can be 
corrected over a broad wavelength range by arranging a diffractive optical 
element at the pupil position of the optical system. 
Japanese Patent Application Kokai No. Hei 2-1109 discloses an imaging 
optical system having high resolving power and that corrects spherical 
aberration and chromatic aberration using a particular type of diffractive 
optical element called a binary optical element (BOE). In a binary optical 
element, a step-shaped surface is formed on a light transmitting member 
using a lithography process. This step-shape surface is capable of 
partially varying the optical path length. The above-cited Japanese Patent 
Application discloses also an imaging optical system having a high 
resolving power and which comprises EL refractive lens element and a 
transmissive grating element on which is formed a plurality of concentric 
circular rings. The grating element is arranged at the aperture of the 
optical system. 
However, the invention disclosed in Japanese Patent Application Kokai No. 
Hei 2-1109 relates to a reduction projection lens for a stepper that uses 
a KrF excimer laser as the light source. The main objective of the 
invention is to principally correct spherical aberration to obtain a 
high-resolution imaging system. The spectral width of the excimer laser is 
approximately 0.08 nm and chromatic aberration is corrected only over this 
narrow range. In other words, the disclosed invention uses a limited 
amount of glass material to correct chromatic aberration in an extremely 
limited wavelength range of the ultraviolet region, rather than in a 
wavelength range extending from the visible region to the infrared region. 
Japanese Patent Application Kokai No. Hei 8-43767 discloses a photographic 
optical system for the purpose of correcting the chromatic aberration of a 
photographic telephoto lens. To correct spot profile (convergence of the 
ray bundle) as well as distortion and field curvature over the entire 
image plane, this photographic optical system arranges a diffractive 
optical element objectwise of a conventional all-refractive telephoto-type 
optical system. This arrangement corrects characteristic chromatic 
aberration of the telephoto lens. 
Although the invention disclosed in Japanese Patent Application Kokai No. 
Hei 8-43767 has the objective to correct chromatic aberration over a wide 
wavelength range when shooting at close range, the field-angle 2.omega. 
covered is less than 15.degree.. In addition, the imaging optical system 
disclosed therein arranges all diffractive optical elements most 
objectwise in the optical system. If an attempt is made to cover a wider 
field-angle with this configuration, the correction of transverse 
chromatic aberration becomes problematic even if axial chromatic 
aberration is corrected. As such, the image quality at the periphery of 
the field can, no longer be ensured. 
SUMMARY OF THE INVENTION 
The present invention relates to an imaging optical system capable of 
forming an image of an object arranged at a finite distance, as in a 
scanner optical system, and more particularly relates to the improved 
correction of chromatic aberration of such imaging optical systems over a 
wide wavelength range extending from the visible wavelength region to the 
infrared wavelength region. 
The present invention has the goal of obtaining an imaging optical system 
for finite distances having a high image quality. Aberrations like 
chromatic aberration are satisfactorily corrected over a wide wavelength 
range extending from the visible region to the infrared region in the 
vicinity of 850 nm without making much use of anomalous dispersion glass. 
Moreover, the system has a field-angle of 20.degree. or greater. 
Accordingly, a first aspect of the invention is an imaging optical system 
having a focal length f and a maximum imagewise numerical aperture. The 
system (comprises, objectwise to imagewise along an optical axis, a front 
lens group having at least one refractive optical element, a diffractive 
optical element provided with a diffractive surface having positive 
diffractive power and a focal length F.sub.DOE, and a rear lens group 
having at least one refractive optical element. A quantity RSA is the 
amount of spherical aberration at the maximum imagewise numerical 
aperture, as measured with respect to e-line wavelength light, in the 
limit as the positive diffractive power approaches zero. Also, LA is I 
maximum amount of axial chromatic aberration of the imaging optical system 
over a wavelength range of 400-850 nm, as measured with respect to e-line 
reference wavelength light. Further, the following conditions are 
preferably satisfied: 
EQU -0.01&lt;RSA/f&lt;0.01 (1) 
EQU -0.002&lt;LA/f&lt;0.002. (2) 
A second aspect of the invention is an imaging optical system as described 
above, further satisfying the following conditions: 
EQU 0&lt;LA.sub.Rs /f&lt;0.1 (3) 
EQU -0.1&lt;LA.sub.Ds /f&lt;0, (4) 
wherein LA.sub.Rs is an amount of s-line axial chromatic aberration of the 
imaging optical system, as measured with respect to an e-line reference 
wavelength light, in the limit as the diffractive power of the diffractive 
surface approaches zero, LA.sub.Ds.tbd.LA.sub.s -LA.sub.Rs, and LA.sub.s 
is an amount of s-line axial chromatic aberration of the imaging optical 
system with respect to the e-line reference. 
A third aspect of the invention is an imaging optical system according as 
described above, satisfying the following condition: 
EQU 10&lt;f.sub.DOE /f&lt;100. (5)

DETAILED DESCRIPTION OF THE INVENTION 
The present invention relates to an imaging optical system capable of 
forming an image of an object arranged at a finite distance, as in a 
scanner optical system, and more particularly relates to the improved 
correction of chromatic aberration of such imaging optical systems over a 
wide wavelength range extending from the visible wavelength region to the 
infrared wavelength region. 
Due to the deflection of light by diffraction, the wavelength-dispersion 
characteristics of a diffractive optical element differ from that of an 
ordinary refractive optical element. Accordingly, as is generally known, a 
diffractive optical element can be used as a new aberration-correcting 
means. That is to say, a chromatic aberration-corrected optical system can 
be obtained by combining diffractive refractive optical elements. Also, as 
mentioned above, for an optical system imaging at finite object-distances 
(e.g., scanners), the correction of chromatic aberration is even more 
critical than for an optical system for photography used at infinity. This 
is because axial chromatic aberration increases proportionate to the 
square of the imaging magnification. 
With reference to FIG. 1 and imaging optical system 10, the present 
invention is an imaging optical system comprising, in order along optical 
axis A from an object plane (not shown) to an image plane 14 (i.e., 
objectwise to imagewise), a front lens group G.sub.F having at least one 
refractive optical element, a diffractive optical element DOE provided 
with a diffractive surface D having a positive diffractive power, and a 
rear lens group G.sub.R having at least one refractive optical element. 
The imaging optical system of the present invention preferably satisfies a 
number of preferred design conditions. The first two preferred conditions 
are: 
EQU -0.01&lt;RSA/f&lt;0.01 (1) 
EQU -0.002&lt;LA/f&lt;0.002, (2) 
wherein RSA is the spherical aberration at the e-line wavelength (546.1 nm) 
at the maximum numerical aperture of a system wherein a plane (i.e., 
non-diffractive) surface is substituted for the diffractive surface of the 
diffractive optical element. In other words, RSA is the amount of e-line 
spherical aberration at the maximum numerical aperture in the limit as the 
amount of diffractive power of diffractive surface D approaches zero. 
Also, LA is a maximum amount of axial chromatic aberration of the imaging 
optical system for 400-850 nm wavelength light, measured relative to the 
e-line wavelength. The overall focal length of the imaging optical system 
is f. The reasons why these conditions are preferably satisfied are 
elucidated below. 
With reference now to FIG. 2, the curve 22 (solid line) describes the 
typical axial chromatic aberration characteristics (LA) of an imaging 
optical system according to the present invention. For purposes of 
comparison, the chromatic aberration characteristics of an optical system 
constructed with only refractive optical elements having a specification 
identical thereto are described by curve 23 (dashed line). The identical 
specification means a system wherein a planar surface is substituted for 
diffractive surface D. For example, where a diffractive surface is formed 
on one side of a plane parallel plate, the identical all-fractive system 
comprises front group G.sub.F, a plane parallel plate, and rear group 
G.sub.R. 
It can be clearly seen from FIG. 2 and curves 22 and 23 that, by combining 
a diffractive optical element with an ordinary refractive optical element, 
axial chromatic aberration is satisfactorily corrected from the visible 
region, extending from the vicinity of 400 nm to 700 run to as far as the 
infrared region in the vicinity of 850 nm. 
Thus, the present invention first comprises front group G.sub.F and rear 
group G.sub.R so that, in a system wherein a planar surface is substituted 
for diffractive surface D, spherical aberration, coma, astigmatism, 
curvature of field and distortion are generally corrected. Moreover, 
diffractive surface D ensures that axial chromatic aberration and 
transverse chromatic aberration are sufficiently corrected in the final 
imaging optical system. 
Thus, the imaging optical system wherein a plane surface is substituted for 
diffractive surface D must be constructed so that spherical aberration RSA 
satisfies condition (1). If the total sum of the spherical aberration due 
to front group G.sub.F, the element wherein the planar surface is 
substituted for diffractive surface D, and rear group G.sub.R is such that 
RSA/f falls below the lower limit in condition (1) or exceeds the upper 
limit in condition (1), the burden of aberration correction placed on the 
diffractive surface to satisfactorily correct spherical aberration becomes 
excessive. In addition, spherochromatism becomes conspicuous. 
An image satisfactorily corrected for chromatic aberration can be obtained 
by correcting the spherical aberration of the imaging optical system 
wherein a planar surface is substituted for the diffractive surface to the 
range indicated in condition (1), and by constructing diffractive surface 
D so that the axial chromatic aberration of the entire system satisfies 
condition (2). 
With reference now to FIG. 6 and optical imaging system 30, it is 
preferable that the present invention have a construction wherein front 
group G.sub.F has at least one positive lens (e.g., lens L31) and, 
imagewise of that positive lens, at least one negative lens (e.g., lens 
L33) having an imagewise concave surface. Further, it is preferable that 
rear group G.sub.R have at least one positive lens (e.g., lens L36), and 
objectwise of that positive lens, has at least one negative lens (e.g., 
lens L34) having an objectwise concave surface. 
In an imaging optical system used at finite distances, as in an imaging 
optical system for a scanner, the entire system must be made compact by 
shortening the distance from the object plane to the image plane (i.e., 
image plane 14) wherein a light-receiving element (e.g, a detector or 
image pick-up device) is arranged. Consequently, the field-angle 2.omega. 
covered by the imaging optical system must be widened. 
To ensure satisfactory imaging across a wide field-angle 2.omega. of at 
least 20.degree., the various aberrations of a single color (wavelength) 
must be satisfactorily corrected by a refractive optical element that 
assumes the main burden of refractive power. Accordingly, a lens type is 
known wherein the Petzval sum can be corrected and a flat image plane 
ensured by providing, objectwise of the aperture stop, a negative lens 
having an imagewise concave surface, and by providing, imagewise of the 
aperture stop, a negative lens having an objectwise concave surface. For 
example, it is effective to combine this existing lens type with a 
diffractive optical element. 
Next, it is preferable the present invention satisfy the following 
conditions: 
EQU 0&lt;LA.sub.RS /f&lt;0.1 (3) 
EQU -0.1&lt;LA.sub.DS /f&lt;0, (4) 
wherein LA.sub.Rs is the amount axial chromatic aberration of the s-line 
(852.1 nm), with the e-line as the reference, wherein diffractive surface 
D is substituted with a planar surface. Further, LA.sub.DS .tbd.LA.sub.S 
-LA.sub.Rs, wherein LA.sub.S is the amount of axial chromatic aberration 
of the s-line, with the e-line as the reference. 
Now, let LA.sub.R be defined as the amount of axial chromatic aberration of 
each wavelength, with the e-line as the reference, wherein a planar 
surface is substituted for diffractive surface D. Then LA.sub.D 
.tbd.LA-LA.sub.R, where LA is the amount of axial chromatic aberration of 
each wavelength of the imaging optical system, with the e-line as the 
reference. Here, LA.sub.D is the difference between axial chromatic 
aberration LA of the imaging optical system having a diffractive surface 
and axial chromatic aberration LA.sub.R of the system wherein a planar 
surface is substituted for the diffractive surface. Therefore, LA.sub.D is 
the amount of axial chromatic aberration of diffractive surface D. The 
values LA.sub.R, and LA.sub.Ds in condition (3) and condition (4) are 
values at the s-line of axial chromatic aberrations LA.sub.R and LA.sub.D, 
respectively. 
To correct axial chromatic aberration over a wide wavelength range, as 
discussed earlier, the achromatism characteristics of the refractive 
optical element(s) and the wavelength-dispersion characteristics of the 
diffractive optical element must be considered. Also, the axial chromatic 
aberration LA.sub.R of the system wherein a planar surface is substituted 
for diffractive surface D, and the axial chromatic aberration LA.sub.D 
including the diffractive surface, must be set based on an appropriate 
ratio. 
With achromatism due to just refractive optical elements wherein a 
low-dispersion glass is used for the positive lens, a high-dispersion 
glass is used for the concave lens. If these lenses are cemented together, 
the axial chromatic aberration increases sharply in the infrared region, 
even if sufficiently corrected for practical purposes in the visible 
region. This is shown by curve 23 of FIG. 2. On the other hand, the 
diffractive action of a diffractive optical element is linear with respect 
to the wavelength and strengthens as the wavelength lengthens, in contrast 
to a refractive optical element. The use of this difference in dispersion 
characteristics is effective in the correction of the focal point position 
particularly in the infrared region. However, if chromatic aberration is 
corrected by combining a diffractive optical element and a refractive 
optical element, negative chromatic aberration remains in the infrared 
region if correction by the diffractive optical element is too strong. 
Therefore, axial chromatic aberration from the visible region to the 
infrared region in the vicinity of 850 nm can be satisfactorily corrected 
in the final imaging optical system by setting the correction of axial 
chromatic aberration LA.sub.R and the correction of axial chromatic 
aberration LA.sub.D such that condition (3) and condition (4) are 
satisfied. 
The satisfactory correction of axial chromatic aberration by appropriately 
combining a diffractive optical element and a refractive optical element 
will now be explained with reference to FIGS. 3A-3C. With reference first 
to FIG. 3B, it can be se,en in curves a, b and c that axial chromatic 
aberration LA.sub.D of the diffractive surface is linear with respect to 
the wavelength, and only its gradient can be adjusted. Accordingly, to 
satisfactorily correct axial chromatic aberration LA of the final imaging 
optical system, it is necessary to first correct axial chromatic 
aberration LA.sub.R so that it is as linear as possible with respect to 
the wavelength, as shown in curve b of FIG. 3A. 
If the value of LA.sub.Rs at the s-line of LA.sub.R is reduced more than 
the state indicated by curve b of FIG. 3A, the wavelength characteristics 
of LA.sub.R lose their nearly linear form and become downwardly convex, as 
indicated by curve c of FIG. 3A. Accordingly, even if LA.sub.D is shaped 
so that the LA.sub.R wavelength characteristics are optimally corrected 
(as indicated by curve c of FIG. 3B), the final LA wavelength 
characteristics inevitably become convex in the downward direction, as 
indicated by curve c of FIG. 3C. Accordingly, satisfactory correction 
cannot be achieved. 
In particular, if LA.sub.Rs is reduced and LA.sub.Rs /f falls below the 
lower limit in condition (3), LA.sub.R on the short wavelength side 
becomes overcorrected in the positive direction more than the e-line and, 
for example, the g-line in the visible region, as indicated by curve c in 
FIG. 3A. If an attempt is made at this time to correct LA on the short 
wavelength side by adding with LA.sub.D, the increase in LA on the long 
wavelength side, due to the wavelength linearity of the diffractive 
optical element, becomes conspicuous. As a result, LA in the infrared 
region becomes overcorrected, as indicated by curve c in FIG. 3C. 
Conversely, if LA.sub.Rs increases more than the state as indicated by 
curve b of FIG. 3A, the wavelength characteristics of LA.sub.R lose their 
nearly linear shape and become upwardly convex, as shown in a of FIG. 
3(A). Accordingly, even if LA.sub.D is shaped so that the wavelength 
characteristics of LA.sub.R are optimally corrected, as indicated by curve 
a of FIG. 3B, the final wavelength characteristics of LA inevitably become 
upwardly convex, as indicated by curve a of FIG. 3C. Accordingly, 
satisfactory correction cannot be achieved. 
In particular, if LA.sub.Rs is increased and LA.sub.Rs /f exceeds the upper 
limit in condition (3), LA.sub.R is overcorrected in the infrared region, 
as shown in FIG. 3A. Also, if the, C-line, for example, of the long 
wavelength is overcorrected more than the e-line in the visible region, 
LA.sub.R is undercorrected in the negative direction with respect to light 
of short wavelength like the g-line. Thus, chromatic aberration in the 
form of LA.sub.R unfortunately remains. 
If an attempt is made to correct LA on the short wavelength side in the 
positive direction by adding LA.sub.D, LA becomes undercorrected at the 
s-line, as indicated by curve a in FIG. 3C, due to the reduction in LA on 
the long wavelength side, particularly the significant reduction in LA in 
the infrared region, simultaneous with the increase in LA on the short 
wavelength side. 
Condition (3) quantitatively stipulates, by the LA.sub.Rs value in the 
s-line the wavelength, characteristics of LA.sub.R needed to 
satisfactorily correct the final LA wavelength characteristics. In 
contrast, condition (4) quantitatively stipulates, by the LA.sub.Ds value 
in the s-line, the wavelength characteristics of LA.sub.D needed to 
satisfactorily correct the final LA wavelength characteristics. 
If the power of the diffractive surface is increased and the value of 
LA.sub.Ds is such that LA.sub.Ds /f falls below the lower limit value in 
condition (4), a large negative LA unfortunately remains in the infrared 
region if combined with a refractive optical element. This is the opposite 
of the case if only an ordinary refractive optical element were used. 
Conversely, if the value of LA.sub.Ds is such that LA.sub.Ds /f exceeds the 
upper limit in condition (4), the correction of LA in the infrared region 
is inadequate if the value of LA.sub.Ds is 0, the same as the case of just 
a refractive optical element. Furthermore, if the value in condition (4) 
becomes positive, positive axial chromatic aberration in the infrared 
region is added more than the case of only a refractive optical element. 
This invites a significant deterioration in imaging performance. 
As described above, axial chromatic aberration can be satisfactorily 
corrected over a range extending from the visible region to the infrared 
region in the vicinity of 850 nm by combining, under appropriate 
conditions, axial chromatic aberration LA.sub.R in the infrared s-line of 
the system and axial chromatic aberration LA.sub.D. 
Next, it is preferable the imaging optical system of the present invention 
satisfy the following condition: 
EQU 10&lt;f.sub.DOE /f&lt;100 (5) 
wherein f.sub.DOE is the focal length of diffractive optical element DOE. 
If the diffractive power of the diffractive optical element is weak to the 
point where f.sub.DOE /f exceeds the upper limit in condition (5), the 
difference between the present invention and existing optical systems 
based on only a refractive optical element becomes insignificant with 
respect to the correction of the secondary spectrum. Also, the correction 
of chromatic aberration cannot be achieved over a wide wavelength range. 
Conversely, if the diffractive power of the diffractive optical element is 
strong to the point that f.sub.DOE /f falls below the lower limit in 
condition (5), axial chromatic aberration becomes undercorrected in the 
infrared region. 
With reference again to FIG. 1, it is also preferable the imaging optical 
system of the present invention satisfy the following condition: 
EQU -0.1&lt;h/y&lt;0.1 (6) 
wherein h is the maximum incident height of a principal ray 18 impinging on 
diffractive surface D, and y is maximum image height. The reasoning for 
including this condition is as follows. 
To ensure satisfactory performance over the entire image plane, it is 
essential to correct monochromatic and chromatic aberration as well as to 
correct axial chromatic aberration. In addition, it is essential to 
satisfactorily correct transverse chromatic aberration in the periphery of 
image plane 14. 
Consider the case where a single diffractive optical element is combined 
with a plurality of refractive optical elements. As mentioned earlier, the 
dispersion characteristics with respect to wavelength differ greatly 
between diffractive and refractive optical elements. On the other hand, 
transverse chromatic aberration is significantly affected by the height 
from the optical axis of the principal ray. Consequently, if a diffractive 
optical element is arranged at a position greatly removed from the pupil 
of the optical system, then the correction of transverse chromatic 
aberration becomes impossible. 
If the field-angle of the imaging optical system is not very wide, it is 
rot difficult to determine where to arrange the diffractive optical 
element in the optical system. However, to ensure a field-angle exceeding 
20.degree. as in the present invention, it is preferable to arrange 
diffractive optical element DOE such that maximum principal ray height h 
at the position of diffractive surface D is within the range of condition 
(6). 
If h/y falls below the lower limit in condition (6) and an attempt is made 
to correct negative transverse chromatic aberration of the diffractive 
optical element by another refractive optical element, the third 
wavelength is undercorrected even if the second wavelength is corrected. 
In other words, the secondary spectrum in the transverse chromatic 
aberration cannot be corrected. 
Conversely, if h/y exceeds the upper limit in condition (6) and an attempt 
is made to correct positive transverse chromatic aberration of the 
diffractive optical element by another refractive optical element, the 
third wavelength is overcorrected even if the second wavelength is 
corrected. In other words, the secondary spectrum in the transverse 
chromatic aberration cannot be corrected. 
In a preferred embodiment of the imaging optical system according to the 
present invention, diffractive surface D is a kinoform (i.e., a 
saw-toothed shape ring), such as shown in FIG. 4. Further, it is 
preferable that the minimum radial pitch of the sawtooth ring of the 
kinoform be in the range of 1.times.10.sup.-3 f-9.times.10.sup.-3 f, and 
that the height H of the sawtooth ring be 0.5-1.5 .mu.m. 
In addition, in another preferred embodiment of the imaging optical system 
according to the present invention, it is preferable that diffractive 
surface D be binary, wherein the height of the kinoform shape is 
distributed over at least eight levels. Further, it is preferable that the 
minimum radial pitch of the binary sawtooth ring be 1.times.10.sup.-3 
f-9.times.10.sup.-3 f, and that height H of the sawtooth ring be 0.5-1.5 
.mu.m. 
With continuing reference to FIG. 6, to satisfactorily correct spherical 
aberration and ensure brightness in the imaging optical system of the 
present invention, it is preferable to arrange at least one positive lens 
in front group G.sub.F and rear group G.sub.R, respectively. In this case, 
it is preferable the imaging optical system of the present invention 
satisfy the following condition: 
EQU n.sub.p &gt;1.6, (7) 
wherein n.sub.p is the average value of the refractive index of the 
positive lenses in the imaging optical system. 
Techniques for designing a diffractive optical element include a lattice 
model and a high-refractive index model. These techniques are disclosed 
in, for example, in the reference "Introduction to Diffractive Optical 
Elements," Japanese Society of Applied Physics, Optical Society of Japan, 
Optics Design Research Group; Optronics, Inc. In the design stage, both 
techniques handle the diffractive surface as a virtual phase transform 
surface having no real shape. Then, a procedure is executed which 
transforms the phase function .phi. into a real shape in the final design 
stage. The phase function .phi. is expressed by, for example: 
EQU .phi.(r)=C.sub.2 .times.r.sup.2 +C.sub.4 .times.r.sup.4 +C.sub.6 
.times.r.sup.6 +C.sub.8 .times.r.sup.8 +C.sub.10 .times.r.sup.10 
wherein C.sub.2 -C.sub.10 are coefficients and r is the height from the 
optical axis. Every time the optical path differential is an integer 
multiple of wavelength .lambda., the real shape forms a ring on the planar 
surface. 
The pitch p of the lattice of the sawtooth-shaped ring is defined by: 
EQU p=m.lambda./(d.phi.(r)/dr) 
wherein m is the diffraction order, and .lambda. is the reference 
wavelength. 
Pitch p in the above expression is a continuous function. However, the 
width of each ring in FIG. 4 has a discrete pitch p.sub.i. In the Working 
Examples of the present invention explained below, pitch p.sub.i is 
minimized at the outermost periphery. 
To satisfactorily correct axial chromatic aberration over a wide range f-Om 
the visible region to the infrared region, it is preferable that the 
following condition be satisfied: 
EQU 1.times.10.sup.-3 &lt;p.sub.min /f&lt;9.times.10.sup.-3 (8) 
wherein f is the focal length of the entire system, and p.sub.min is the 
minimum pitch in the radial direction of the sawtooth-shaped ring. 
In addition, height H of the sawtooth-shaped ring is defined by: 
EQU H=m.lambda./(n-1) 
wherein n is refractive index of the plate. 
If the design reference wavelength .lambda. is set to 546.1 nm (e-line), 
the diffractive order m is set to m=+1, and quartz, for example, is used 
as the plate upon which diffractive optical element DOE is formed, then: 
EQU H=1.0385 .mu.m. 
If the goal of the present invention takes into consideration the working, 
wavelength, diffractive order M and refractive index n of the plate glass, 
then it is preferable that height H of the sawtooth-shaped ring satisfy: 
EQU 0.5 .mu.m&lt;H&lt;1.5 .mu.m (9) 
Furthermore, with reference to FIG. 5, a binary optical element (BOE), in 
which the cuneiform is step approximated can also be used. The diffractive 
efficiency of a stepped approximation is 41% at two levels, 81% at four 
levels and 95% at eight levels. Since the approximation error invites a 
deterioration in image quality like flare, it is preferable for the 
purposes of the present invention that the number of binary levels be 
eight or greater. 
When using a binary optical element, it is preferable to satisfy the 
abovementioned conditions (8) and condition (9). However, height H.sub.b 
of the sawtooth-shaped ring of the binary optical element is given by: 
EQU H.sub.b =H(b-1)/b 
wherein, H is the height of the original cuneiform sawtooth-shaped ring;, 
and b is the number of levels. 
WORKING EXAMPLES 
Even-numbered FIGS. 6-16 show Working Examples 1 to 6, respectively, of the 
imaging optical system according to the present invention. With reference 
to FIG. 1 and imaging optical system 10, the imaging optical system of 
each Working Example is provided with, objectwise to imagewise along 
optical axis A, front group G.sub.F having at least one refractive lens, 
diffractive optical element DOE provided with diffractive surface D having 
positive diffractive power, and rear group G.sub.R having at least one 
refractive lens. The preferred use of the imaging optical system of the 
present invention is as an optical system for an optical scanner. 
In each Working Example, front group G.sub.F has at least one positive lens 
and a most imagewise negative lens having an imagewise concave surface. In 
addition, rear group G.sub.R has at least one positive lens and a most 
objectwise negative lens having an objectwise concave surface. The 
aperture stop in each Figure is represented by AS. 
Odd-numbered FIGS. 7-17 are aberration plots for spherical aberration, 
astigmatism, distortion and transverse chromatic aberration for Working 
Examples 1-6, respectively. In each aberration plot, g is the g-line 
(435.8 mu), e is the e-line (546.1 nm), C is the C-line (656.3 nm) and s 
is the s-line (852.1 nm). In the spherical aberration plots (odd-numbered 
FIGS. 7A-17A), the broken line indicates offense against the sine 
condition. In the astigmatism plots (odd-numbered FIGS. 7B-17B), the 
broken line indicates the meridional image plane and the solid line 
indicates the sagittal image plane. 
Tables 1-6 below set forth the design specifications, including 
coefficients of phase functions, for Working Examples 1 to 6, 
respectively. In the Tables, NA represents the imagewise numerical 
aperture, .beta. represents the imaging magnification, S represents number 
of each optical surface from the object side, r represents the radius of 
curvature of each optical surface, d represents the axial distance from 
each optical surface to the next optical surface (or image plane), n.sub.e 
represents the refractive index with respect to the e-line of the optical 
member (blank indicates air) arranged from each optical surface to the 
next optical surface, V.sub.e represents the Abbe number (e-line) of each 
optical member, and the last column labeled "Element" lists the reference 
symbol of each optical member or optical surface. 
Tables 7A and 7B list the values for the design conditions (1) to (9) for 
Working Examples (WE) 1-3 and 4-6, respectively. 
TABLE 1 
______________________________________ 
DESIGN SPECIFICATIONS 
______________________________________ 
f = 100 
NA = 0.083 
.beta. = 
-1.223 
2 .omega. = 
31.3.degree. 
y = 62 
= 546.1 nm (e-line) 
M = +1 
C.sub.2 = 
-1.73008 .times. 10.sup.-8 
C.sub.4 .about. C.sub.10 = 
0 
______________________________________ 
S r d n.sub.e 
.nu..sub.e 
Element 
______________________________________ 
0 .infin. 128.13 
1 75.435 8.28 1.776210 
49.39 L.sub.31 
2 453.628 3.92 
3 34.675 10.45 1.720550 
47.80 L.sub.32 
4 113.343 2.61 1.677648 
31.93 L.sub.33 
5 25.024 11.33 
6 .infin. 6.53 1.460118 
64.49 DOE 
7 .infin. 11.33 D 
8 -25.024 2.61 1.677648 
31.93 L.sub.34 
9 -113.343 10.45 1.720550 
47.80 L.sub.35 
10 -34.675 3.92 
11 -453.628 8.28 1.776210 
49.39 L.sub.36 
12 -75.435 166.65 
______________________________________ 
TABLE 2 
______________________________________ 
DESIGN SPECIFICATIONS 
______________________________________ 
f = 100 
NA = 0.078 
.beta. = 
-0.55506 
2 .omega. = 
28.3.degree. 
y = 39.2 
= 546.1 nm (e-line) 
M = +1 
C.sub.2 = 
-1.00810 .times. 10.sup.-8 
C.sub.4 .about. C.sub.10 = 
0 
______________________________________ 
S r d n.sub.e 
.nu..sub.e 
Element 
______________________________________ 
0 .infin. 230.87 
1 52.981 2.57 1.554642 
49.85 L.sub.41 
2 34.949 9.41 1.654256 
58.24 L.sub.42 
3 148.781 0.22 
4 26.625 8.63 1.605201 
65.14 L.sub.43 
5 504.415 1.90 1.615937 
44.17 L.sub.44 
6 21.080 9.63 
7 .infin. 2.57 1.460118 
64.49 DOE 
8 .infin. 16.58 D 
9 -18.740 1.90 1.615937 
44.17 L.sub.45 
10 -167.228 8.63 1.605482 
65.14 L.sub.46 
11 -27.804 0.22 
12 -378.334 9.41 1.654256 
58.24 L.sub.47 
13 -39.298 2.57 1.554642 
49.85 L.sub.48 
14 -54.799 100.84 
______________________________________ 
TABLE 3 
______________________________________ 
DESIGN SPECIFICATIONS 
______________________________________ 
f = 100 
NA = 0.111 
.beta. = 
-0.31496 
2 .omega. = 
24.8.degree. 
y = 29 
= 546.1 nm (e-line) 
M = +1 
C.sub.2 = 
-1.88472 .times. 10.sup.-8 
C.sub.4 = 
-7.08689 .times. 10.sup.-12 
C.sub.6 .about. C.sub.10 = 
0 
______________________________________ 
S r d n.sub.e 
.nu..sub.e 
Element 
______________________________________ 
0 .infin. 347.57 
1 58.774 9.98 1.619921 
53.73 L.sub.51 
2 384.448 0.14 
3 38.023 13.60 1.747931 
44.75 L.sub.52 
4 115.850 2.89 1.762584 
31.40 L.sub.53 
5 25.794 11.15 
6 .infin. 2.89 1.532350 
55.92 DOE 
7 .infin. 14.04 D 
8 -23.945 2.17 1.762584 
31.40 L.sub.54 
9 -347.425 10.42 1.747931 
44.75 L.sub.55 
10 -38.524 0.14 
11 -178.460 11.00 1.748009 
49.28 L.sub.56 
12 -53.957 0.28 
13 2687.835 7.23 1.748009 
49.28 L.sub.57 
14 -183.061 79.33 
______________________________________ 
TABLE 4 
______________________________________ 
DESIGN SPECIFICATIONS 
______________________________________ 
f = 100 
NA = 0.087 
.beta. = 
-0.63457 
2 .omega. = 
29.7.degree. 
y = 43.3 
= 546.1 nm (e-line) 
M = +1 
C.sub.2 = 
-1.36243 .times. 10.sup.-8 
C.sub.4 .about. C.sub.10 = 
0 
______________________________________ 
S r d n.sub.e 
.nu..sub.e 
Element 
______________________________________ 
0 .infin. 212.50 
1 51.792 11.35 1.654256 
40.06 L.sub.61 
2 168.629 0.20 
3 26.231 8.67 1.605482 
60.39 L.sub.62 
4 261.056 1.85 1.610741 
40.06 L.sub.63 
5 20.934 8.05 
6 .infin. 2.37 1.460118 
64.49 DOE 
7 .infin. 1.54 D 
8 -18.838 1.85 1.615937 
40.06 L.sub.64 
9 -132.453 8.67 1.605482 
60.39 L.sub.65 
10 -28.544 0.20 
11 -214.596 8.98 1.654256 
40.06 L.sub.66 
12 -47.191 109.30 
______________________________________ 
TABLE 5 
______________________________________ 
DESIGN SPECIFICATIONS 
______________________________________ 
f = 100 
NA = 0.087 
.beta. = 
-1.173 
2 .omega. = 
19.6.degree. 
y = 37.5 
= 546.1 nm (e-line) 
M = +1 
C.sub.2 = 
-1.75864 .times. 10.sup.-8 
C.sub.4 .about. C.sub.10 = 
0 
______________________________________ 
S r d n.sub.e 
.nu..sub.e 
Element 
______________________________________ 
0 .infin. 146.05 
1 79.708 6.44 1.732340 
54.44 L.sub.71 
2 6978.164 0.26 
3 37.078 13.02 1.758440 
52.09 L.sub.72 
4 197.360 3.15 1.754570 
34.81 L.sub.73 
5 26.658 8.16 
6 .infin. 5.26 1.532350 
55.92 DOE 
7 .infin. 13.29 D 
8 -25.524 4.34 1.754570 
34.81 L.sub.74 
9 -83.849 6.84 1.758440 
52.09 L.sub.75 
10 -34.347 1.31 
11 -368.880 13.94 1.791950 
47.26 L.sub.76 
12 -75.807 45.19 
______________________________________ 
TABLE 6 
______________________________________ 
DESIGN SPECIFICATIONS 
______________________________________ 
f = 100 
NA = 0.083 
.beta. = 
-0.945 
2 .omega. = 
20.3.degree. 
y = 34.8 
= 546.1 nm (e-line) 
M = +1 
C.sub.2 = 
-1.99471 .times. 10.sup.-8 
C.sub.4 .about. C.sub.10 = 
0 
______________________________________ 
S r d n.sub.e 
.nu..sub.e 
Element 
______________________________________ 
0 .infin. 154.08 
1 70.425 5.17 1.732340 
54.44 L.sub.81 
2 1188.665 0.24 
3 36.534 8.89 1.758440 
52.09 L.sub.82 
4 98.820 6.06 1.754570 
34.81 L.sub.83 
5 26.173 11.97 
6 .infin. 3.23 1.532350 
55.92 DOE 
7 .infin. 12.37 D 
8 -27.511 7.43 1.754570 
34.81 L.sub.84 
9 -1199.000 
9.38 1.758440 
52.09 L.sub.85 
10 -41.989 1.21 
11 -237.755 6.46 1.791950 
47.26 L.sub.86 
12 -82.333 1.61 
13 -350.467 6.46 1.791950 
47.26 L.sub.87 
14 -143.453 137.54 
______________________________________ 
TABLE 7A 
______________________________________ 
DESIGN CONDITIONS FOR WORKING EXAMPLES 1-3 
DESIGN CONDITION 
WE1 WE2 WE3 
______________________________________ 
(1) RSA -0.008 -0.002 -0.001 
(2) LA 0.0014 0.0007 0.0005 
(3) LA.sub.R 0.039 0.011 0.012 
(4) LA.sub.D -0.038 -0.010 -0.012 
(5) f.sub.DOE /f 
28.9 49.6 26.5 
(6) h/y 0.017 0.019 0.046 
(7) n.sub.p 1.74838 1.629799 
1.72236 
(8) p.sub.min /f 
5.62 .times. 10.sup.-3 
3.39 .times. 10.sup.-3 
1.93 .times. 10.sup.-3 
(9) H (.mu.m) 1.19 1.19 1.03 
______________________________________ 
TABLE 7B 
______________________________________ 
DESIGN CONDITIONS FOR WORKING EXAMPLES 4-6 
DESIGN CONDITION 
WE4 WE5 WE6 
______________________________________ 
(1) RSA -0.002 -0.009 -0.004 
(2) LA 0.0008 0.0008 -0.0007 
(3) LA.sub.R 0.017 0.039 0.030 
(4) LA.sub.D -0.016 -0.039 -0.037 
(5) f.sub.DOE /f 
36.7 28.4 25.1 
(6) h/y 0.000 0.000 0.000 
(7) n.sub.p 1.629869 1.760293 
1.76624 
(8) p.sub.min /f 
2.41 .times. 10.sup.-3 
3.42 .times. 10.sup.-3 
2.36 .times. 10.sup.-3 
(9) H (.mu.m) 1.19 1.03 1.03 
______________________________________ 
As can be seen from the aberration plots corresponding to the Working 
Examples 1-6, the imaging optical system in each Working Example has a 
wide field-angle and superior imaging performance over a wide wavelength 
range from 400-850 nm. 
As described above, by appropriately combining a refractive optical element 
and a diffractive optical element according to the present invention, an 
imaging optical system for use at finite distances is obtained wherein 
various aberrations are satisfactorily corrected after correcting axial 
chromatic aberration and transverse chromatic aberration over a wide 
wavelength range extending from the visible region to the infrared region 
in the vicinity of 850 nm. Moreover, the imaging optical system has a 
large field-angle of 20.degree. or greater. 
While the present invention has been described in connection with preferred 
embodiments, it will be understood that it is not so limited. On the 
contrary, it is intended to cover all alternatives, modifications and 
equivalents as may be included within the spirit and scope of the 
invention as defined in the appended claims.