Imaging Multiple Planes Simultaneously with a Diffraction Grating

Imaging Multiple Planes Simultaneously
with a Diffraction Grating
Carlos O. Font, Iván Guzman, Orlando Marrero,
Joel Vega de Jesús, Zugel P. Vidarte Quintero,
Advisor: Mark Chang∗
∗ Department of Physics,University of Puerto Rico, Mayagüez Campus,Mayagüez,
Puerto Rico 00681
E-mail: mark@feynman.uprm.edu
This article describes a simple technique for
the simultaneous imaging of multiple focal
planes within an object field onto a single
image plane. The method makes use of a
binary diffraction grating in which the lines
are distorted in such a way as to introduce a
different level of defocus with each diffraction
order. In this article we discuss the
grating design and the experimental work
done to validate the theory.
1. Introduction
Many fields of study require the ability
to image several layers within an object
field at the same time. Some examples
of these are in vivo microscopy, studies
of optical propagation, wavefront sensing,
ocular imaging and three dimensional
optical data storage. In order to
image these layers, we evidently have to
split a single beam. A diffraction grating
is one type of beamsplitter. A system
that is based on a single optical train
wherein there is found a grating has the
advantages of preserving the resolution
of the input lens in each of the images
and gives an accurate registration of the
series of images. That is to say, the image
locations are precisely known. Ignoring
the problem of chromatic dispersion,
which in many applications is not
an issue, a quasi–monochromatic object
imaged through a diffraction grating in
the far–field would produceon a common
image plane a series of images of the object.
We will describe the steps required to
modify the standard grating structure in
order to use it as a beamsplitter, while at
the same time adapting it to form simultaneous
images of multiple object planes
on a single camera (or image plane). The
application differs from most normal usage
of diffractive optical elements in that
it makes simultaneous use of the zeroth,
first and potentially higher diffraction
orders. The descriptions given here and
the experimental results shown are based
on the use of amplitude gratings. Of
course, amplitude gratings have the disadvantages
of low optical efficiency and
reduced brightness in the higher diffraction
orders, but these disadvantages can
be overcome through the use of phase
gratings of similar design.
2. Defocus Gratings
A binary diffraction grating is one that
consists of alternating regularly spaced
strips of different transmissivity, reflectivity
or optical thickness. Let the y axis
be defined so that it is parallel to the
grating stripes, and the x axis be perpendicular
to said stripes. If the grating
geometry is distorted locally by a
displacement of the strips along the x
axis, a phase shift is introduced into the
wavefront scattered from the displaced
region. Relative to the zeroth order, the

local phase shift φ m (x, y) is dependent
on the amount of the local grating shift
through
φ m (x, y) = 2πm∆ x(x, y)
d
[1]
where d is the grating period, m is the
diffraction order into which the wavefront
is scattered and ∆ x is the displacement
of the grating strips relative to
their undistorted positions.
All non zeroth orders therefore undergo
a phase shift that is linearly dependent
on the grating displacement. This phase
shift is equal in magnitude but opposite
in sign to the positive and negative orders
of the same coefficient. This principle
is sometimes known as the detour
phase effect and has been used to encode
diffractive optics elements.
What is of interest is the ability to generate
different levels of defocus in the wavefronts
that are diffracted into each order.
This may be done by applying phase delays
that modify the wavefront curvature.
How might the required phase delays
be calculated? In general, regular
diffractive elements (gratings) produce
regular sets of images along the length
of the optical axis of the system. This
can be taken to mean that, for the zeroth
diffracted order say, the image of a point
object is a line that extends along the
optical axis. The simplest phase profile
of a wavefront that generates this type of
image is spherical. So consider the phase
profile of a diffractive element that forms
a spherical wavefront,
φ(r) = 2π λ
(f − √ f 2 − r 2 )
[2]
where f is the effective focal length of
the element and r is a general radial coordinate.
This is justified in Figure (1).
Equation (2) may be expanded to give
( r
2
2f + r4
8f 3 +
)
r6
16f 5 + . . .
φ(r) = 2π λ
[3]
The above expansion is easily simplified
if the paraxial case is considered. Hence
we only need the first term of Equation
(3). Higher order terms are valid, but
will not be considered in detail. In the
paraxial case defocus is described by a
phase shift which varies as the square of
the distance from the optical axis, relative
to the in–focus system. A simple
way to encode this defocus into a grating
is to first consider a standard line grating
in a circular pupil. See Figure (1). By
considering a grating with the quadratic
phase shift encoded into it as a series of
rows, where each row is a normal grating
translated in the x direction, it is possible
to find a relation for the local phase
shift in terms of the offset ∆ x .
This defocus offset can be written as
( )
∆ x W20
= ± r 2
d
λ
( )
W20 (x 2 + y 2 )
= ±
λ R 2 [4]
Here r 2 = (x 2 + y 2 )/R 2 where R, the radius
of the full aperture, is the normalisation
constant. The amount of defocus
is contained in the standard coefficient of
defocus W 20 , the extra path length introduced
at the edge of the aperture, which
in this case is shown for the wavefront
diffracted into the ±1 orders. The phase
change φ m of the wavefronts diffracted
into each order is then given by
φ m (x, y) = m 2π W 20
λ R 2 (x2 + y 2 ) [5]
What form do the grating lines take,
given this phase change? The grating
lines’ equation is
n = x d 0
+ W 20(x 2 + y 2 )
λR 2 [6]
where d 0 is the grating period at the centre
of the aperture and the integer values
of n define the loci of the grating lines.
When n = 0, the equation describes the
grating line that passes through the centre
of the aperture mask. The first term
x/d 0 represents the undistorted grating,
while the second term represents the
quadratic distortion. It is easy to show
the Equation (6) describes circles centered
at
x n = −
λR2
[7]
2W 20 d 0
with radii ρ n
(
nλR 2 ( ) λR
2 2
) 1/2
ρ n = +
[8]
W 20 2d 0 W 20

propagation
direction
spherical
wavefront
r
x
standard grating
grating with
offset in x
φ
k
d
∆ x
d
f
φ m
FIG. 1: (Left) Geometrical deduction of φ(r). (Right) Definitions of quantities.
which shows that ρ n ∝ √ n.
If the Equations (3) and (5) are compared,
an equivalent focal length f m per
order may be found. The focal lengths
come about physically because the grating
has focusing power in the nonzero orders
from the quadratic phase function.
We find
f m =
R2
. [9]
2mW 20
R is the grating aperture radius and
mW 20 is the path length difference introduced
at the edge of the aperture in
the mth diffraction order. So a single
grating distorted to the above prescription
acts as a set of lenses of positive,
neutral and negative power.
For most practial purposes it will be useful
to use such a grating together with a
lens, where the lens provides the major
part of the focusing power, with the grating
modifying the focal length. Using
the thin lens approximation, it is easy
to find that a single lens of focal length
f in contact with a defocus grating has
a focal length per diffraction order of
f m =
fR 2
R 2 + 2fmW 20
. [10]
2.1. Imaging a Single Object
Plane
The system of a defocus grating and a
single lens will map a single object plane
onto a series of image planes, each with
a different level of defocus. The images
produced will appear in the different
diffraction orders, as shown in Figure
(2). We can now demonstrate this
ability to image an object plane under
different defocusing conditions in each
diffraction order experimentally. Simple
binary amplitude gratings are fabricated
by laser printing the grating structure
onto letter sized paper. These structures
have been reduced by photocopying; for
better line resolution they will be photographically
reduced onto a 35-mm transparent
slide at a later date.
2.2. Imaging Multiple Object
Planes
The other application is to image simultaneous
three different object planes.
This is the original motivation of this
study. The situation is illustrated in Figure
(2). If a camera is placed at the image
plane, the images formed in the three
central diffraction orders should correspond
to in–focus images of the three object
planes. The zero order is the sum of
the out–of–focus images of objects A and
C and an in–focus image of object B. So
if the degree of defocus is high enough,
then a clear image of object B will result.
The distance from the object or the image
plane in the mth order to the position
of the zero order is given by
δz m = − 2mz2 W 20
R 2 + 2mzW 20
[11]
with z being the distance from the central
object/image plane to the nearest
principal plane of the optical system.
This means that in general the plane
separation between pairs of adjacent orders
is not equal. In the limiting case

0
δ z +1
δz
-1
O
+1
A
+1
O
O
A B C
B
0
O -1
C
-1
z
z
FIG. 2: Single object imaging (left); Multiple object imaging (right).
of 2mzW 20