Generalized confocal imaging systems for
free-space optical interconnections
Qing Cao, Matthias Gruber, and Ju ¨rgen Jahns
A generalized confocal imaging system, which is composed of two confocal lenses and one field lens, is
proposed for free-space optical interconnections.Unlike in a conventional 4-f system, both the object
distance and the image distance can be almost arbitrarily chosen.
tant for practical setups in which the object distance and the image distance cannot be designed to be the
same. As a concrete example, we have designed and experimentally tested a planar-integrated micro-
optical imaging system.The result is in good agreement with the theoretical prediction.
the conventional 4-f imaging system and the light-pipe imaging system, the system proposed here can
also be used as one important part of a hybrid imaging setup.
200.4650, 350.3950, 110.1650, 050.1940, 080.2730.
This advantage is especially impor-
© 2004 Optical Society of America
The conventional 4-f system is widely used for imag-
ing in free-space optical interconnections because it
offers telecentricity and good imaging performance
for extended fields.It is also an important part of a
hybrid 4-f imaging system.1,2
one characteristic feature of this setup is that object
distance d1, image distance d2, and focal length f are
the same. However, design restrictions in practical
setups, for example, for planar-integrated free-space
optical interconnections,3,4often do not allow one to
choose the parameters d1, d2, and f in this way.
overcome this disadvantage and to provide additional
design freedom, we propose here a generalized con-
focal imaging system, shown in Fig. 2, for free-space
It is well known that propagation through paraxial
optical systems can be described by matrix optics.5,6
In matrix optics, a specific transfer is denoted by a
specific ABCD matrix, which has the form
Generalized Confocal Imaging System
As shown in Fig. 1,
where A–D are the four matrix elements.
to matrix optics,5,6total transfer matrix M from the
input plane to the output plane in Fig. 2 can easily be
determined to be
2f ? d1? d2?f2
It is well known that the intensity distribution at
the output plane is the image of that at the input
plane when matrix B is equal to 0.
?1?, one can get the following imaging condition for
the setup of Fig. 2:
Then, from Eq.
d1? d2? f2?F ? 2f. (2)
Also according to matrix optics,5,6when the imaging
condition B ? 0 is satisfied, the value of matrix ele-
ment A indicates the magnification.
one can see that the magnification is ?1, where the
negative sign indicates a coordinate reversal.
find that the conventional 4-f setup corresponds to
the special case of F ? ? and d1? d2? f.
find that the light-pipe imaging system7–9corre-
sponds to another special case of d1? d2? 0 and F ?
f?2.Because the two f lenses still have a common
focal plane and added field lens F is located just at
From Eq. ?1?
The authors are with Optische Nachrichtentechnik, FernUni-
versita ¨t Hagen, Universita ¨tsstrasse 27?PRG, 58084 Hagen, Ger-
many.Q. Cao’s e-mail address is firstname.lastname@example.org.
Received 3 November 2003; revised manuscript received 9
March 2004; accepted 22 March 2004.
© 2004 Optical Society of America
3306APPLIED OPTICS ? Vol. 43, No. 16 ? 1 June 2004
this plane, we call this imaging system a generalized
confocal imaging system.
and d2values, one can always find a suitable F value
to make the equality of Eq. ?2? hold.
significantly improves design freedom and flexibility.
The solution for F can be given explicitly by
For predetermined f, d1,
F ? f2??2f ? d1? d2?. (3)
When 2f ? d1? d2, the field lens is a positive lens
because F ? 0. When 2f ? d1? d2, the field lens is
a negative lens because F ? 0.
the field lens is not needed because F ? ?.
worth mentioning that the relaxed design freedom
still has limits that are imposed by the lens technol-
ogy used. Limitations exist, for example, for the
range of focal lengths that can be implemented, the
numerical aperture, and the efficiencies.10
When 2f ? d1? d2,
As one concrete application of the generalized confo-
cal imaging system mentioned above, we designed
and experimentally tested a planar-integrated micro-
optical imaging system.
folded system images the input signal at the input
plane into the corresponding position at the output
plane, through a zigzag path.
the transverse and the vertical directions, respec-
tively, of the cross plane shown in Fig. 3.
direction is that which is perpendicular to the cross
plane.We denote by OS the optical substrate used
M the input plane, the output plane, a grating that
Experimental Test As an Example
As shown in Fig. 3, this
We denote by x and z
deflects the light, a confocal lens, the field lens, and a
reflective mirror, respectively.
rors are made from highly reflective metal films.
The lenses are reflective multilevel diffractive lenses
with four phase levels.They are designed in accor-
dance with their relationship to elliptical Fresnel
zone plate lenses after the related focal lengths have
been determined.They are fabricated by means of
two-mask binary photolithography and reactive-ion
etching upon the optical substrate.
highly reflective metal films deposited upon surfaces
to cover the diffractive lenses.
lens, the x-directional size is 1.8 mm and the
y-directional size is 4.55 mm.
part of an f lens is shown in Fig. 4.
Fig. 3 the input plane might, for example, consist of
an array of vertical-cavity surface-emitting laser
sources or modulators bonded onto the substrate di-
rectly or onto an intermediate substrate, such that
distance D1to the substrate will be short.
experiment, thickness h of the optical substrate, dis-
tance D1between the input plane and the top plane of
the substrate, distance D2between the output plane
and the bottom surface of the substrate, and deflec-
tion angle ? are chosen such that h ? 9 mm, D1? 1
The reflective mir-
For each diffractive
A photo of the central
For the layout of
Fig. 1.Schematic view of the conventional 4-f system.
Schematic view of the generalized confocal imaging sys-
Schematic view of a planar-integrated micro-optical im-
Fig. 4. Photo of the central part of an f lens.
1 June 2004 ? Vol. 43, No. 16 ? APPLIED OPTICS3307
mm, D2? 0.688 mm, and ? ? 5.711°.
substrate is chosen to be fused silica, whose refractive
index is n ? 1.453 at the vacuum wavelength ? ? 850
nm.Simply by using the value of the refractive in-
dex and of relation ? ? ???n sin????, we determine
both periods ? of the input grating and the output
grating to be 5.879 ?m.According to the structure
of Fig. 3, focal length f of the confocal lenses, object
distance d1, and image distance d2are determined to
be f ? 18 mm, d1? 2f ? nD1? 19.453 mm, and d2?
f ? nD2? 10 mm, respectively.
d1? 2f ? nD1and d2? f ? nD2rather than d1? 2f ?
D1and d2? f ? D2because the fused silica substrate
has refractive index 1.453, which is different from
refractive index 1 of free space.
D2in free space are equivalent to the lengths nD1and
nD2 in the fused-silica substrates, respectively.
Substituting the values of f, d1, and d2into Eq. ?3?,
one can get F ? 49.488 mm.
tions of f, F, d1, and d2, we did not take into account
the oblique propagation property of the light.
is to say, we still assume that the planar-integrated
optical system is a normal paraxial optical system in
which all the components have a common optical
axis; strictly speaking, such is not the case.
ever, the differences and the connections between a
planar-integrated optical system and its correspond-
ing unfolded optical system are investigated in detail
in Refs. 11 and 12.It is shown that11,12the lenses in
a planar-integrated optical system are not circular
lenses but elliptical lenses.
the y-directional focal lengths are slightly different.
According to the analyses of Refs. 11 and 12, the x-
and the y-directional focal lengths of the f lens and
the F lens are determined to be fx? f?cos3??? ?
18.271 mm, fy ? f?cos??? ? 18.090 mm, Fx ?
F?cos3??? ? 50.233 mm, and Fy? F?cos??? ? 49.735
mm, respectively.It is worth mentioning that, for
each of these lenses, the difference between the x and
the y directions is small, because the difference be-
tween the x- and the y-directional focal lengths is
small.One can easily verify this fact by taking a
look at Fig. 4.
We fabricated and experimentally tested this
planar-integrated micro-optical imaging system.
shown at the left in Fig. 5, we employed a 4-f system
to relay the light source of a laser diode with a wave-
length of 850 nm into a suitable position at an input
plane that is ?1 mm ?i.e., distance D1in Fig. 3? in
front of the left-hand surface ?i.e., the top surface in
Fig. 3? of the planar optical system.
light source was used as the input object.
lengths of the two relay lenses are both 40 mm.
There are two principal reasons for the use of the
relayed light source rather than the direct light
source.One is to prevent damage of the elements
that may result from contact during the adjustment
because the distance from the object to the left-hand
surface of the optical substrate is very short ?only 1
mm?. The other is that the holder of the optical
substrate used in the experiment does not permit a
short distance only, between the direct light source
By the way, we use
The lengths D1and
In the above calcula-
That is to say, the x- and
and the left-hand surface of the planar-optical-
system. As shown at the right of Fig. 5, we used a
large-working-distance micro-objective to image a
suitable plane onto the CCD receiver plane with suit-
able magnification ?this is actually a microscope? and
used a TV for observation.
working-distance micro-objective we can observe not
only the right-hand surface ?and its neighborhood?
but also the left-hand surface ?and its neighborhood?
of the planar optical system by adjusting the position
of the micro-objective. The experimental result is in
good agreement with the design.
position of the micro-objective we could clearly ob-
serve both the input object of the laser spot at the
input plane and the corresponding output image at
the output plane. Photos of the input object and the
may not be sure that there is no distortion, but we are
sure that the potential distortion is small.
tical applications, the receiving targets normally
have sizes of dozens of micrometers by dozens of mi-
crometers. From Fig. 6 one can see that these sizes
are much larger than the spot size of the image.
Therefore, for practical applications, the potential
small distortion can be ignored.
there is a coordinate reversal between the object and
By using this large-
By adjusting the
As we stated above,
integrated micro-optical imaging system.
Obj, and BS the laser diode, the optical substrate, the micro-
objective, and the beam splitter, respectively.
Experimental arrangement for testing the planar-
We denote by LD, OS,
Photos of the input object and the corresponding output
3308 APPLIED OPTICS ? Vol. 43, No. 16 ? 1 June 2004
the image itself of a pointlike object.
coordinate reversal can be easily shown by the loca-
tions of the pointlike object and of the corresponding
pointlike image. For example, if the pointlike object
shifts a certain distance in the positive ?or the nega-
tive? y direction, then the pointlike image will shift a
certain distance but in the opposite direction.
did indeed observe this kind of behavior in the exper-
iment, which accordingly demonstrated the property
of coordinate reversal. In summary, the experiment
has directly demonstrated the good performance of
the generalized confocal imaging system.
This property cannot easily be shown by
Based on matrix optics,5,6we have proposed a gener-
alized confocal imaging system for free-space optical
interconnections. This imaging system is composed
of two confocal lenses and a field lens.
conventional 4-f imaging system and the light-pipe
imaging system7–9are special cases of this general-
ized confocal imaging system.
length F of the field lens, one can almost arbitrarily
choose object distance d1and image distance d2.
This advantage significantly improves design free-
dom and flexibility. At the same time, this general-
ized imaging system still keeps the advantage of
telecentricity of the conventional 4-f system.
tested a planar-integrated micro-optical imaging sys-
tem.The result is in good agreement with the the-
oretical prediction. Similarly to the conventional 4-f
system,7–9this proposed system can also be used as
one important part of the hybrid imaging setup.1,2
By adjusting focal
The authors thank their colleagues for support and
discussions. This study was financially supported
by the Deutsche Forschungsgemeinschaft.
thors are indebted to the reviewers for their com-
ments and suggestions for improving the paper.
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