In The Classical Electromagnetic The
ENGINEERING
The Uncertainty Principle in Classical Optics
MASUD MANSURIPUR
expand as they propagate along Z
In the classical electromagnetic the-
ory the wave-vector k = (2 / )
and, although their centers drift
underlies the Fourier space of propa-
apart, there is the distinct possibility
gating (or radiative) fields. The k-
that they will never be completely
vector combines into a single entity
separated. Roughly speaking, we ex-
the wavelength and the unit vector
pect the beams to remain more or
that signifies the beam’s propaga-
less collimated between z = 0 and z =
tion direction. The Fourier trans-
D2/ , the Rayleigh range2 for a beam
form relation between the three-di-
of diameter D and wavelength . If
mensional space of everyday experi-
at the Rayleigh range the distance
ence and the space of the wave-vec-
between the beam centers is greater
tors (the so-called k-space) gives rise
than D, the beams should be separa-
to relationships between the two do-
ble; otherwise their drifting apart will
mains analogous to Heisenberg’s un-
go hand in hand with their expan-
Figure 1. Two beams of the same wavelength ,
certainty relations.
propagating in slightly different directions, pass
sion, and the beams remain entangled as
Considering that in quantum theory
through an aperture of diameter D.The angle be-
they propagate beyond the Rayleigh range.
the electromagnetic k-vector is propor-
tween the two k-vectors is
, giving rise to k ≈
x
The necessary condition for separability is
tional to the photon’s momentum1
(2 / )
.The beams separate from each other at
thus (D2/ )
> D, or equivalently,
(p = -h k , where -h = h /2 , h being the
the observation plane located a distance z from
Planck constant), one should not be sur-
the aperture, provided the uncertainty relation
D kx > 2 .
(1)
prised to find relationships between di-
D k ≥
x
2
is satisfied.
mensions of a beam in the XYZ-space and
The lower bound 2 on the product of D
its momentum spread in the k-space. Such
and kx appearing in Ineq. (1) is not ex-
relationships impose fundamental limits
act, but depends on the definition of beam
on the ability of measurement systems to
diameter D and the adopted criterion for
determine the various properties of elec-
separability, which are typically imprecise.
tromagnetic fields.
For all practical purposes, the number ap-
In this article we address two problems
pearing on the right-hand side of Ineq. (1)
that have widespread applications in opti-
should be on the order of unity, say,
cal metrology, spectroscopy, telecommu-
greater than 1 but less than 10.
nications, etc., and discuss the constraints
Invoking the quantum nature of light,
imposed by the uncertainty principle on
if the aperture diameter D is interpreted as
these problems. The first topic of discus-
a measure of the uncertainty x about the
sion is the separation of two overlapping
photon position along X, while kx is re-
beams of identical wavelength having
lated (through the relation p = -hk) to the
slightly different propagation directions.
linear momentum uncertainty px along
This will be followed by an analysis of the
the same axis, then Ineq. (1) is equivalent
limits of separating co-propagating beams
to Heisenberg’s uncertainty relation
having slightly different wavelengths.
x px > h.
Figure 2 shows the intensity and phase
Angular separation
profiles of two plane waves as well as those
and the limit of resolvability
of their superposition at the aperture de-
picted in Fig. 1 (diameter D = 500 ). The
Figure 1 shows an aperture of diameter D,
phase distributions in Figs. 2(b) and 2(d)
which transmits two plane waves of the
indicate that one of the beams is slightly
same wavelength propagating in slightly
tilted towards the upper right corner of
different directions. Denoting the angular
Figure 2. Plots of intensity (left) and phase (right)
the XY-plane, while the other is tilted by
separation between the beams by
, we
at the entrance aperture of the system of Fig. 1.
an equal amount towards the lower left
find that the projections of the two k-vec-
Two uniform beams, one propagating with a slight
corner. The angular separation between
tors along the X-axis differ by k ≈
x
tilt toward the upper right, another with a slight
these beams is
= 0.23° = 0.004 radians.
(2 / )
. In geometrical optics, rays
tilt toward the lower left, enter a D = 500 aper-
The combined beam’s intensity distribu-
propagate along straight lines and, there-
ture.The angular separation of the beams is
=
0.23º. The individual beams are shown in the top
tion in Fig. 2(e) reveals the angular separa-
fore, the two beams must separate from
(a, b) and the middle (c, d) rows; their superposi-
tion of the two superimposed beams
each other after a certain propagation dis-
tion appears at the bottom (e, f ).
through a tell-tale fringe pattern.
tance. In wave optics, however, the beams
44 Optics & Photonics News I January 2002
ENGINEERING
When the composite beam
use a high-NA lens, thus enhancing
(whose intensity and phase distribu-
the polarization effects. The short-
tions are shown in Figs. 2(e, f)) is
est focal length f is obtained when
propagated along the Z-axis, one ob-
the NA of the lens is close to unity,
tains at various distances from the
that is, f ≈ 1_2D. Figure 5 shows com-
aperture the intensity patterns dis-
puted plots of intensity distribution
played in Fig. 3.3 It is seen in these
at the focal plane of an NA = 0.99,
pictures that the two constituent
f = 250
lens, when the incident
beams continue to overlap at first,
beam is the two-beam superposi-
giving rise to interesting interference
tion depicted in Fig. 2.3 The three
patterns. After a sufficient propaga-
columns of Fig. 5 represent three
tion distance, however, the beams
different polarization states. In (a)
separate and go their own ways. The
both incident beams are linearly po-
assumed value of D
k
larized along X, which explains the
x in this ex-
ample is 4 , which satisfies Ineq. (1).
elongation of the spots in this par-
ticular direction. In (b) the two
Separating two beams
beams are linearly polarized at 45°
by means of a lens
to the X- and Y-axes, i.e., the direc-
tion along which the spots are sepa-
In the preceding section it was
rated from each other. The plot in
demonstrated that separating two
(c) corresponds to the case when
beams of a certain angular distance
both beams are circularly polarized.
requires a minimum beam diam-
Frames (d)-(f) are the logarithmic
eter D in accordance with Ineq. (1).
versions of those in (a)-(c), showing
It may be asked whether a similar
their detailed structure by empha-
limitation exists on the propagation
sizing the weaker regions. Since the
distance z before the individual
assumed values of D = 500 and
beams can be resolved. Apparently
= 0.004 rad satisfy the uncertainty
no physical law limits the required
relation in Ineq. (1), the focused
distance z, although practical con-
spots are seen to be resolved irre-
siderations seem to impose certain
spective of their polarization state.
constraints. In free space, the re-
quired propagation distance is typi-
Angular discrimination by
cally less than or equal to the Rayleigh
Figure 3.Two overlapping plane waves depicted in
Fig. 2 propagate along the Z-axis. The various in-
means of a Fabry-Pérot etalon
range, D2/ , but one can substantially re-
duce this distance by employing a lens, as
tensity patterns in frames (a) to (o) are obtained at
Another device that can, in principle, ac-
shown in Fig. 4. Here two overlapping
z/(103 ) = 1, 2, 3, 10, 20, 30, 40, 50, 60, 70, 80, 90,
complish the separation of two beams via
beams of diameter D and angular separa-
100, 125, and 150, respectively. Initially the beams
angular discrimination is a Fabry-Pérot
strongly interfere with each other, but as propaga-
tion
are resolved after going through
etalon,4,5 such as that shown in Fig. 6. This
tion proceeds, they separate and exhibit their indi-
an aberration-free lens. In the focal plane
particular etalon is tuned to transmit a
vidual identities.
of the lens the center-to-center spacing of
plane wave of
= 633 nm at the incidence
the focused spots is f
, which must be
angle of
= 45°. Figure 7 shows the
greater than the Airy disk4 radius of ~
etalon’s computed reflection and transmis-
0.6 /NA = 1.2 f /D. Note that the resolv-
sion coefficients, rs = |rs|exp (i rs) and ts =
ability criterion is independent of f and
|ts| exp(i ts), versus for an s-polarized
NA, requiring only that D (
/ ) > 1.2,
plane-wave of
= 633 nm. It turns out
which is a statement of the uncertainty
that the shapes of the transfer functions
principle in the present context. The re-
|rs( )| and |ts( )| are not quite suitable
quired propagation distance f in this ex-
for complete separation of two finite-
ample can be much less than that needed
diameter beams of differing propagation
in the case of free-space propagation of
directions.
Fig. 1. It must be emphasized that the un-
Computed plots of intensity distribu-
certainty principle does not impose any
tion in Fig. 8 confirm that the etalon of
constraints on z, the requirement for re-
Fig. 6 can only partially separate two
solvability being only a restriction on the
beams of diameter D = 2 104 and angu-
product of D and
.
lar separation
= 0.115° = 0.002 rad,
Figure 4.Two identical beams of diameter D and
An interesting feature of separating
even though the value of D (
/ ) = 40 in
angular separation
may be isolated after going
two beams by means of a lens is the result-
through an aberration-free lens. In the focal plane,
this case amply satisfies Ineq. (1). Figure
ing dependence of the focused spots on
the distance between the focused spots is f
,
8(a) shows the incident pattern of intensi-
the state of polarization. To reduce the re-
which must be greater than the Airy disk radius of
ty distribution of the superposed beams
quired propagation distance z, one may
1.2 f /D if the individual spots are to be resolved.
upon arriving at the etalon. One of these
January 2002 I Optics & Photonics News 45
ENGINEERING
beams propagates along the
this case the beam diameter
direction that makes a 45° an-
D turns out to be irrelevant,
gle with the etalon’s surface
but the available propagation
normal, while the other devi-
distance z is critical for iso-
ates from this direction by
lating the individual beams.
= 0.115°. The reflected inten-
A straightforward method
sity profile depicted in Fig.
of separating two beams of
8(b) contains mostly the latter
differing wavelengths is
beam, plus a small fraction of
shown in Fig. 9. This Mach-
the former. This is due to the
Zehnder interferometer4
imperfect transfer function of
splits each input beam into
the etalon, which cannot fully
two equal halves, provides a
transmit the angular spec-
separate path for each half,
trum of the 45° beam, nor can
then recombines the halves
it fully reflect the spectrum of
into a single beam at the out-
the 45.115° beam. Either
put. For one of the wave-
beam’s angular spectrum has
lengths, say,
1, the path-
a width of ~ /D ≈ 0.003°,
length difference z between
which would readily pass
the two arms of the device
Figure 5. Total electric field intensity distribution
through a narrow rectangular transfer
( |E |2 = |E
may be an integer-multiple of
x |2 + |E y |2 + |E z |2 ) at the focal plane of a
1, in which
function, but is partially blocked by the
0.99NA lens. (Rainbow colors: red = maximum,
case the corresponding half-beams inter-
sharply peaked transfer functions of the
blue = minimum).The beam at the entrance pupil is
fere constructively and emerge from one
etalon (see Fig. 7(a)). The same arguments
the superposition of two D = 500 beams of angu-
exit channel of the interferometer. For the
apply to the transmitted intensity distrib-
lar separation
= 0.23°, as shown in Fig. 2. In (a)
other wavelength, 2, the path-length dif-
ution shown in Fig. 8(c) which, although
the assumed polarization state of both incident
ference may be a half-integer-multiple of
primarily composed of the 45° incident
beams is linear along the X-axis. In (b) the two
2, in which case interference will be de-
beams are linearly polarized at 45° to the X-axis,
beam, still contains a fraction of the
structive and the beam will emerge from a
i.e., along the direction of separation of the spots. In
45.115° beam.3
different exit channel of the device. There-
(c) one of the beams is right-circularly polarized,
To summarize the results of this and
fore, separability condition for this inter-
while the other is left-circularly polarized. Frames
the preceding sections, there are several
(d)-(f) in the bottom row are the logarithmic ver-
ferometer is z/ 1 - z/ 2 = 1_2 , or
ways of separating two overlapping beams
sions of frames (a)-(c) in the top row. Like an over-
z k
exposed photographic plate, a logarithmic plot re-
≈
of the same wavelength and differing
z
2
z
/ 2 = .
(2)
propagation directions. Some of these
veals weak regions of an intensity distribution.
Figure 10 shows computed detector signals
methods may be more effective than oth-
S1, S2 of the system of Fig. 9 versus the in-
ers, but none could violate the uncertainty
put wavelength in the vicinity of
= 633
relation given by Ineq. (1). Moreover,
nm.3 For the particular path-length differ-
Ineq. (1) remains valid even if the beams
ence chosen in this example ( z = 1.266
are observed within a transparent medi-
mm), it is observed that, in compliance
um of refractive index n > 1. For instance,
with Eq. (2), a pair of beams having
=
in Fig. 1 if the region to the right of the
0.158 nm can be readily separated from
aperture happens to be filled with such a
each other.
medium, the angular separation
of the
An alternative form of the uncertainty
beams shrinks by a factor n upon entering
relation may be obtained in this case by in-
the medium, but the length of the k-vector
voking the quantum mechanical relation
increases by the same factor, thus preserv-
between the magnitude k of the wave-vec-
ing the magnitude of
k
Figure 6. Fabry-Pérot etalon designed for opera-
x. Similarly, in
tor and the photon energy E = h , namely,
Fig. 4 if the index of the medium on the
tion at
= 633 nm,
= 45°. Dielectric mirrors
k = 2 / = 2
/c = E/-hc. For two beams of
right-hand-side of the lens happens to be
each contain six pairs of high/low-index layers
(n
wavelengths and +
, co-propagating
n, the focused spot diameters will be n
1 = 2.0, d1 = 84.6 nm; n2 = 1.5, d2 = 119.6 nm).
Both mirror substrates are glass (nsub = 1.5), and
in the Z direction, kz = E/-hc. Also z =
times smaller, but their center-to-center
the medium separating the mirrors is air (dair =
c t, where c is the speed of light and t is
spacing will also be reduced by the same
55.95 m).The incidence angle on the etalon is in
the time needed for light to travel a dis-
factor, resulting once again in the preser-
the vicinity of = 45°; within the substrate, how-
tance z in free space. The product z kz
vation of Ineq. (1).
ever, the angle of incidence on the stack is close to
is thus proportional to E t, with -h being
= 28.1255° (sin = nsub sin ). The etalon can
the proportionality constant. One may
Co-propagating beams
separate two beams of identical arriving through
thus reinterpret Eq. (2) as a statement of
of differing wavelengths
an aperture of diameter D, but differing in propa-
the time-versus-energy uncertainty. When
gation direction, namely,
A problem of general interest in spec-
1 = 45°,
2 = 45° +
.
the observations are made in a transparent
One beam is reflected by the etalon while the oth-
troscopy is that of separating two beams
er is transmitted. Only s-polarized light is consid-
medium of refractive index n > 1, the in-
of slightly different wavelengths, 1 and
ered here, although p-polarized beams exhibit sim-
crease of the k-vector by a factor of n dic-
2, propagating in the same direction. In
ilar behavior.
tates a corresponding decrease in z. This
46 Optics & Photonics News I Month 2001
ENGINEERING
Figure 9. The Mach-Zehnder interferometer can
be used to separate two beams of differing wave-
lengths, 1 and 2.The beams have identical diame-
ters and arrive in the same direction. The two
beams are split equally at the first 50/50 splitter,
travel the two arms of the device, and are recom-
bined at the second 50/50 splitter. The lengths of
the two arms of the interferometer differ by z.
When z/ 1 – z/ 2 = 1⁄2, constructive interfer-
ence at the second beam-splitter for one of the
two wavelengths coincides with destructive inter-
Figure 7. Computed reflection and transmission
ference for the other.The beams are thus separat-
coefficients versus the incidence angle for the
ed at the second splitter, one is captured by detec-
etalon of Fig. 6 at = 633 nm for an s-polarized
tor 1, the other by detector 2. The 45° mirrors
plane wave. The magnitude and phase of the re-
(three in each arm) have a reflectivity of 90%, re-
flection and transmission coefficients are defined
sulting in an overall system transmission of 73%.
through the relations rs = |rs| exp(i rs) and ts =
The 50/50 splitters are identical, each consisting of
|ts| exp(i ts).At = 633 nm the stack is tuned to
a glass substrate coated with a six-layer dielectric
fully transmit at = 45°. A small deviation from
stack as follows:
45° incidence causes a sharp drop in |ts| and a
corresponding rise in |r
(Substrate, nsub = 1.5) / (d1 = 30 nm, n1 = 2.64) /
s|.
(d2 = 140 nm, n2 = 1.76) / (d3 = 50 nm, n3 = 2.64) /
(d4 = 105 nm, n4 = 1.76) / (d5 = 60 nm, n5 = 2.64) /
is consistent with the reduced speed of
(d6 = 100 nm, n6 = 1.76) / Air
Figure 8. Two overlapping beams of uniform am-
light in the medium of index n, which
plitude and circular cross-section ( = 633 nm, D =
Although the above stack works for both p- and
yields the same travel time t for the
2 x 10 4 ) arrive at the etalon of Fig. 6. One beam
s-polarized light, its splitting ratio is much closer to
shorter propagation distance z/n. Need-
travels at = 45° relative to the etalon’s surface
50/50 for s-light than for p-light. In our simulations
less to say, E = h
is independent of n.
normal, the other at = 45.115°. (a) Intensity dis-
the polarization state of the incident beam was
tribution of the superposed beams at the entrance
fixed at s.
aperture. (b) Reflected intensity distribution, con-
Wavelength discrimination
sisting mainly of the second beam plus a small frac-
using a Fabry-Pérot etalon
tion of the first. (c) Transmitted intensity distribu-
finition, kz = 2 / and, therefore, kz =
The etalon of Fig. 6 may also be used to
tion, consisting mostly of the first beam plus a
2
/ 2. Figure 12 shows the above
small fraction of the second.
separate co-propagating beams of slightly
beams arriving at an incidence angle 0° ≤
different wavelengths, say,
and +
.
< 90° on a grating of period P. The N th
Figure 11 shows computed plots of reflec-
diffracted order emerges from the grating
tion and transmission coefficients versus
that of the incident beam. (In the present
at an angle , in accordance with Bragg’s
for a resonator having an air gap d
example, G is 3.0 for s-light and 1.94 for p-
law,4,5
air =
55.95 m. From Eq. (2) at = 633 nm,
light.) The effective z is thus G times the
considering that z = 2d
effective gap width, resulting in a corre-
sin
= sin +N /P,
(3a)
air cos(45°) =
79.125 m, we find
= 2.53 nm, in
sponding increase in the resolution of the
agreement with the peak-to-valley dis-
device.
which yields,
tance in the simulated results of Fig. 11.
The figure, however, indicates the feasibil-
Spectral analysis using
cos
= (N /P)
.
(3b)
ity of resolving beams with a smaller
as
a diffraction grating
well; this is due to the high finesse of the
Consider two co-propagating beams of
Now, the emergent beam diameter is D =
Fabry-Pérot etalon. In other words, multi-
wavelengths and +
, where it is as-
D |cos /cos |. Since the lens is expected to
ple back and forth reflections within the
sumed for convenience that
> 0. These
resolve the two wavelengths, Ineq. (1) re-
etalon’s cavity build up an optical field
beams travel along the Z-axis and pass
quires that |
| ≥ /D , which leads to
whose amplitude is G times stronger than
through an aperture of diameter D. By de-
|cos
| ≥ cos /D, which in turn leads
January 2002 I Optics & Photonics News 47
ENGINEERING
Figure 10. Computed detector signals S
Figure 11. Computed plots of amplitude reflec-
1 and S2
versus the input wavelength in the Mach-Zehn-
tion and transmission coefficients versus for the
der interferometer of Fig. 9. The assumed path-
Fabry-Pérot etalon depicted in Fig. 6. The air gap
length difference between the two arms of the
and the incidence angle are fixed at dair =
device is z = 1.266 mm. In the vicinity of
=
55.95 m and
= 45°. The incident beam is p-
633 nm the adjacent peaks of S
polarized in (a) and s-polarized in (b).
1 and S2 are sepa-
rated by
= 0.158 nm, in agreement with Eq. (2).
Figure 12. Two beams of wavelengths
and
+
, propagating in the same direction Z, arrive
at an aperture of diameter D.The beams propagate
to |N /P |
≥ cos /D. In other words,
a distance z
tance z from the center of the entrance
1 from the center of the aperture to
a grating of period P, shining on the grating at an
aperture to the focal plane of the lens is
D /cos ≥ ( /
) |P/N |.
(4a)
angle . One of the diffracted orders (the N th or-
given by,
der) leaves the grating at an angle , travels a dis-
tance z2 (from the center of the grating to the
From Eq. (3a) it is clear that |N /P | ≤ 2,
z = z1 + z2 + f
(5a)
center of the lens), then enters a lens of focal
namely, |P/N | ≥ 1_
= 1_2(D/cos ) (sin + |sin | + |cos | ).
2
. Inequality (4a) may
length f and numerical aperture NA ≈ 1. Emerging
thus be written as follows:
from the grating, the two wavelengths deviate from
Since sin ≥ 0, and |sin | + |cos | ≥ 1 for
each other by an angle
, thus forming separate
D /cos ≥ 1_ 2
any , Eq. (5a) yields,
focused spots at the focal plane of the lens. From
2
/
.
(4b)
the entrance aperture to the focal plane, the total
propagation distance is z = z
Inequality (4b) places a lower bound for
z ≥ 1_2D/cos .
(5b)
1 +
z2 + f.
resolvability not on the beam diameter D,
but on the illuminated length of the grat-
Combining Ineqs. (4b) and (5b) then
2
ing, D/cos , in the direction perpendicular
yields z ≥ 1_4 /
, that is,
to the grooves.
References
≥ 1_
Next we examine the propagation dis-
z kz 2 .
(6)
1.
R. P. Feynman, R. B. Leighton, and M. Sands,The Feyn-
man Lectures on Physics,Addison-Wesley, Reading,
tance from the center of the entrance
Massachusetts (1964).
aperture to the focal plane of the lens.
Note that the initial beam diameter D in
2.
A. E. Siegman, Lasers, University Science Books, Cali-
this example is not restricted at all, where-
fornia (1986).
With reference to Fig. 12, the shortest pos-
as the propagation distance z is required
3.
The simulations reported in this article were per-
sible distance from the entrance aperture
formed by DIFFRACT™, and MULTILAYER™; both
to the grating center is z
to be greater than a certain minimum,
programs are products of MM Research, Inc.,Tucson,
1 = 1_2D tan . Sim-
1_
2
4
/
, to ensure resolvability of the
Arizona.
ilarly, the shortest possible distance from
4.
M. Born and E.Wolf, Principles of Optics, 6th edi-
the grating to the lens center (ignoring the
wavelengths and +
.
tion, Pergamon Press, Oxford, 1980.
possibility that the lens might block the
5.
M.V. Klein, Optics,Wiley, New York (1970).
incident beam) is z
Acknowledgment
2 = 1_2D |tan
| =
1
_
OPN contributing editor Masud Mansuripur <masud
2D|sin
|/cos . The smallest feasible focal
I am grateful to Ewan M. Wright of the
@u.arizona.edu> is a professor of Optical Sciences at
length for the lens is f = 1_2D , correspond-
Optical Sciences Center for insightful sug-
the University of Arizona in Tucson and president of
MM Research, Inc.
ing to NA = 1. Therefore, the shortest dis-
gestions.
48 Optics & Photonics News I January 2002
