Journal of Modern Physics, 2013, 4, 911-922 Published Online July 2013 (
Macroscopic Violation of Duality Generated
on a Laser Beam
Daniel Mirell1, Stuart Mirell2
1Department of Chemistry, University of California at Irvine, Irvine, USA
2Department of Radiological Sciences, University of California at Los Angeles, Los Angeles, USA
Received April 20, 2013; revised May 25, 2013; accepted June 23, 2013
Copyright © 2013 Daniel Mirell, Stuart Mirell. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
It is shown theoretically and experimentally that passage of a laser beam through particular conventional Ronchi grat-
ings prepares the beam in an altered state that violates quantum duality. The violation is characterized by a readily
measured net transfer of energy between the prepared beam and an unmodified beam from a similar, independent laser.
Notably, the experiment is conducted with the beams at macroscopic power levels where measurability of the dual-
ity-violating transfer is vastly improved over that of the discrete photon regime. These results are consistent with other
recently reported experiments that have challenged the validity of the duality-related principle of complementarity.
Keywords: Quantum Duality; Local Realism; Probabilistic Interpretation
1. Introduction
Wave-particle duality is a central principle of the standard
probabilistic interpretation of quantum mechanics. We
report the results of an experiment violating that principle.
The experiment demonstrates a net power transfer be-
tween a specially prepared state of a continuous-wave (cw)
laser beam ΦG and an ordinary beam ΦR at the same
wavelength but generated by an independent laser where
both beams are at macroscopic power levels.
The net transfer is established by first intersecting ΦG
and ΦR over a common coupling path and measuring beam
power on a sample region of the ΦG beam spot separated
from ΦR at the terminus of that path. ΦR is then blocked
from the coupling path and beam power of ΦG in the same
sample region is re-measured to provide a baseline power
for the calculation of the net power transfer.
ΦG is prepared in three different states, “depleted”,
“enriched”, and “ordinary”, in order to provide for three
distinct experimental trials. The total beam power of ΦG is
substantially identical for these states. The results of these
trials show that when ΦG is prepared in a depleted state,
coupling to ΦR results in a net transfer of power to ΦG.
Alternatively, when ΦG is prepared in an enriched state, a
net transfer of power out of ΦG occurs when coupled to ΦR.
Lastly, ΦG is prepared in an ordinary state which is shown
to yield no net transfer upon coupling with ΦR.
From the perspective of the probabilistic interpretation,
independently generated ΦG and ΦR will still demonstrate
interference along an intersection region of those beams,
despite Dirac’s dictum that a photon can interfere only
with itself [1], based upon a theoretical analysis by
Mandel [2]. However, the probabilistic interpretation
provides no mechanism by which the observed net trans-
fer would occur when ΦG is in a presumptive altered state
of depletion or enrichment. Specifically, we show here
that the source of the duality violation originates with the
preparation of ΦG in these presumptive altered states.
As an alternative, a theoretical construct derived di-
rectly from principles consistent with local realism phy-
sically represents these altered states and quantitatively
predicts the respective transfers reported here.
2. Background
The probabilistic interpretation of quantum mechanics is
distinguishable from local realism by the self-interaction
of discrete photons incident on mechanisms such as a
beam splitter or a double slit. For local realism, the pro-
babilistic property of duality in the discrete photon re-
gime is violated by the wave packet that emerges from
one of the two outputs of such mechanisms as an “empty”
wave. That empty wave is forbidden by duality which
requires an inseparability of a photon’s wave-like prop-
erty, probability, and its particle-like property, the energy
quantum. Duality effectively imposes a fixed proportion-
opyright © 2013 SciRes. JMP
ality between those two properties. In local realism, the
treatment of wave-like and particle-like properties as
manifestations of two separate and real entities has its
origins in de Broglie’s initial reality-based pilot wave re-
presentation of quanta phenomena [3].
Empty waves, if they exist, should in principle yield
testable consequences for a variety of phenomena com-
monly associated with the wave-like properties of pho-
tons. Important advances in the study of these phenom-
ena have been made by a number of investigators in-
cluding Croca, Garuccio, Lepore, Moreira, Selleri, and
others e.g. [4-7]. Popper, in seeking a rational hypothesis
for empty waves, has raised compelling philosophical
arguments that question the validity of the prevailing
probabilistic interpretation [8].
Testability for empty waves in the context of double
slit experiments is related to determining which slit is
traversed by an energy-bearing photon without destroy-
ing accompanying interference arising from both slits, a
result consistent with an empty wave traversing the other
slit but in conflict with Bohr’s duality-related principle of
complementarity [9]. Recently, two notable investiga-
tions [10] and [11] employing different methodologies
have experimentally realized slit traversal determination
with verified accompanying interference.
The methodology of one investigation has its origins
in an analysis by Aharonov et al. [12] showing that a
“weak” measurement of a system can provide some de-
gree of information about the system without signifi-
cantly altering its subsequent interactions thereby pro-
viding an effective investigative tool [13-16]. Wiseman’s
assessment that such weak measurements could be used
to establish average trajectories [17] is experimentally
demonstrated in the recent investigation by Kocsis et al.
where averaged slit-specific trajectories are measured
from a double slit accompanied by interference in the far
field of the slits [10].
In a different approach, Rabinowitz proposes a novel
double slit experiment using correlated photon pairs in
which the passage of discrete photons through a particu-
lar slit can be definitively established together with an
observation of interference in the far field beyond the
slits [18,19]. A recent investigation by Menzel et al. re-
ports the results of an analogous correlated photon pair
experiment in which the slit traversed by a discrete pho-
ton is determined and far-field interference arising from
both slits is verified [11].
The investigations referenced above in this section are
inherently associated with the microscopic regime of
discrete quantum entities. In this regard it is significant
that local realism imposes no constraints on duality vio-
lation in the discrete quantum regime as well as in the cw
regime where a parameter such as beam power can be at
a macroscopic level.
Probability in local realism is a relative entity associ-
ated with a real wave structure and retains its original
restricted interpretation in Born’s rule [20] as the likeli-
hood or “propensity” [8] of a particle-like energy quan-
tum progressing onto a particular probability channel in a
complete set of out-going probability channels. A re-
scaling of the total probability distributed to the complete
set does not alter the likelihood of the particle-like entity
entering onto that particular channel. We continue to use
the term “probability” here bearing in mind that its usage
is potentially misleading since that term suggests equi-
valence to a mathematical absolute probability. That equi-
valence is manifested in the probabilistic interpretation as
the principle of duality.
In the macroscopic cw regime the treatment of prob-
ability as a relative quantity implies that for any number
of resident energy quanta on some arbitrary beam seg-
ment there is not an imposed, fixed proportionate value
of inclusive probability. Consequently, a mechanism
predicted by local realism to generate even a modest (du-
ality-violating) disproportion on a cw beam of macro-
scopic power contributes to conclusive experimental test-
ability. This advantage arises because the net “excess” or
“deficit” of particle-like energy quanta residing on the
probability’s wave-like structure would provide a readily
measurable macroscopic power increment.
Clearly, a beam splitter does not provide local realism
with the requisite mechanism to achieve duality violation
in the macroscopic cw regime. In the transition from the
discrete to the macroscopic regimes as beam power is
increased, statistical distribution of the numerous inci-
dent energy quanta onto the beam splitter’s two output
channels restores a proportionality of probability and
energy quanta on both of those channels that is not dis-
tinguishable from that of duality. However, there are a
number of mechanisms consistent with local realism, e.g.
[21], that generate duality violation at macroscopic pow-
er levels. We report here on one of those mechanisms, its
theoretical basis, and on the experimental demonstration
of duality violation using that mechanism.
3. Mechanism for Duality Violation
The basic mechanism for duality violation presented here
can most readily be appreciated by first considering a
simple gedanken experiment before proceeding to the
particular realization of that experiment reported here. In
the gedanken experiment we use a Ronchi transmission
grating (ruling) and a laser beam at normal incidence. A
Ronchi grating consists of a linear array of equal-width
opaque and transmissive bands resulting in a grating pe-
riod 2pw
where w is the “slit” width represented by
the transmissive bands. In the gedanken experiment, we
hold the laser beam wavelength constant at some λ and
consider the consequences of progressively decreasing w
Copyright © 2013 SciRes. JMP
from large values of w
down to small values of w ~
λ while maintaining .
The irradiated slits produce the Figure 1 “output” set
of wavelets that do not intersect in the grating’s near
field. The total output probability emerging on these
wavelets is a constant Po independent of w since the
transmissive factor for a Ronchi is 0.5 wp. In arbi-
trary units, we can set Po = 1. As the individual wavelets
expand in the grating’s far field, they generate smoothly-
varying single-slit envelopes that intersect and interfere
thereby producing the “resultant” set of highly direc-
tional diffraction orders j shown in Figure 2 for the
Fraunhofer approximation. The envelope intensity is
proportional to 22 2
sin csin
and θ is the physical azimuthal angle.
The depicted envelope in Figure 2 is equivalent to an
individual slit envelope scaled up by the total number of
irradiated slits to correctly show the essential equivalence
of the output integral and the integral over the resultant
peaks. For Ronchi gratings, the side lobes are bisected by
the symmetrically disposed odd ±j orders for |j| and the
integral over a particular side lobe is equal to the integral
of the odd jth resultant peak bisecting that side lobe. Fig-
ure 2 is parameterized in terms of the continuous vari-
Figure 1. Emergent output diffraction wavelets from irra-
diated individual slits fully determine collective output pro-
bability (and energy) prior to the formation of resultant
orders by interference as those wavelets intersect.
Figure 2. Single slit output diffraction intensity envelope and
associated resultant diffraction intensity peaks for a Ronchi
grating of those slits. The single slit envelope is scaled up by
number of irradiated slits N = 20 to correctly depict total
envelope output intensity relative to that of the resultant
peaks. The depicted total envelope output and the resultant
peaks are both scaled up by ×15 in the detail graph of the j =
2 to 6 range.
able 2πj
as well as α since j conveniently as-
sumes integer values at the side lobe bisector points and
null points.
Before proceeding further, we introduce a convention
useful in the present theoretical analysis. Beam probabil-
ity P is evaluated by volumetric integration of intensity
over a beam segment of some arbitrary selected length.
Correspondingly, this defines an inclusive energy E on
that beam segment. We use the convention of considering
beam segments purely as a convenience in order to pro-
ceed with the theoretical analysis using the parameters of
probability and energy rather than the associated flux
densities of those parameters. Nevertheless, the results of
the analysis are fully applicable to the experimentally
relevant parameter of the integrated energy flux density,
i.e. power, which is proportionate to that inclusive energy
From a straight-forward classical calculation [22], the
resultant probability of the diffraction orders
sin d
rt j
is given by the integral over the resultant order intensities,
expressed here as a Riemann sum, divided by the integral
of the output envelope intensity. The output envelope
truncates at πw
where the azimuthal .
Truncation on the j continuum is given by
is shown as a function of the con-
venient parameter jt, but that functional dependence is
fundamentally the linearly related parameter
is subscripted by F to indicate that the quantity is valid in
the Fraunhofer approximation. Because of symmetry,
dependence on jt is equivalent to that for –jt. From these
expressions, we have a useful relationship for the Ronchi
slit width in terms of jt,
wj (2)
The Riemann sum truncates at the ±nth final resultant
orders inclusive within ±αt. Those ±nth orders are respec-
tively in the neighborhood of and are said to be
near the grating “threshold”.
In particular, the total output probability Po was shown
The basis for the derivation of Equation (1) [22] can
be succinctly summarized here. An integral over resul-
tant order intensities (or its Riemann sum equivalent)
would normally by itself constitute the requisite resultant
probability unlike the form of Equation (1). However, in
the present case the integral over resultant orders as well
as the output envelope integral are both computed over α
and not the physical azimuthal angle θ causing both inte-
grals to exhibit an artifactual decrease as the variables αt,
jt, and w in our gedanken experiment mutually decrease.
Copyright © 2013 SciRes. JMP
above to physically remain constant independent of w.
Accordingly, the divergence from constancy of the out-
put integral evaluated over α yields the requisite nor-
malization factor for the corresponding resultant order
integral over α to provide a probability Pr for which the
α-related artifactual decrease is selectively eliminated.
Turning now to the beginning of our gedanken ex-
riment in the limit of large w, which corresponds to
“coarse” gratings, the significant central region of the
output envelope and all of the correspondingly signifi-
cant resultant diffraction orders are tightly clustered in
the forward direction about 0˚ and, equivalently, about α
= 0 consistent with the Fraunhofer approximation. The
more distal envelope and higher orders extending out to
90 are vanishingly small where both “truncate” at
large |j|. In this limit the Equation (1)
Pj is
very nearly a constant value of unity. This con for
large w merely expresses the high degree to which prob-
ability is conserved for a coarse grating in the transition
that occurs as the expanding wavelets interfere and gen-
erate in their place the highly directional resultant orders.
However, as w
for “fine” gratings, only the en-
v maxim
he resu
elope’s central um and significant proximal side
lobes are included over the –90˚ to +90˚ full azimuthal
span as truncation occurs at some small |j|. Correspond-
ingly, that same truncation also leaves only a very few
orders, including those with the most significant intensi-
ties, widely dispersed over –90˚ to +90˚. This departure
from the Fraunhofer approximation necessitates that we
amend the Equation (1)
Pj to properly represent
the relative intensities of tltant orders as a func-
tion of inclination from 0˚ by incorporating the classi-
cally well-known “obliquity” correction appropriate for
the presently considered Ronchi grating diffraction sys-
tem. The relative actual intensities of orders nearer
threshold are reduced below that predicted by the
function. A physically accurate obliquity cor-
n the present experiment is readily available em-
pirically from a direct measurement of the beam powers
of the resultant jth orders emerging from an appropriate
fine grating and comparing those powers to the 2
rection i
function. Our primary interest in this investig
confined to gratings for which jt ranges from ±4 down to
±2 respectively over the first side lobes. For a Ronchi
grating the probabilities (integrated intensities) of those
output side lobes are expressed in turn by the probabili-
ties associated with the bifurcating resultant ±3rd orders.
Accordingly, we measure the powers of the resultant
orders with a Ronchi grating that places the ±3rd orders
near threshold. From these measurements, we obtain an
obliquity correction of f = 0.56 for those ±3rd orders near
threshold and a correction negligibly differing from unity
for the lower orders within the diffraction envelope’s
central maximum. Similarly, that obliquity correction f is
also applicable to the output integral over the first side
ation is
obliquity-amended expression for the Equation (1)
resultant probability
0.5π0.5sin sin
sin dsin d
sin dsin d
2.539 0.071
2.532sin d
fS c
is valid when the truncation j resides in the first side lobe
. Because of symmetry about 0 on the jt con-
ative j terms have been excluded in the Rie-
mann sum. Since the negative αt limits in the envelope
integration have also been excluded, Equation (3) still
represents the total resultant probability for all positive
and negative orders within .
tinuum, neg
S3 is a step function, zero
when the j = 3 order is excl and unity when the j = 3
order is included in the integrated output envelope.
Pj exhibits a discontinuous perturbation about unity
t. This perturbation is also present for the other odd
j, but these perturbations rapidly diminish with
sing jt as depicted in Figure 3. In this figure, the
obliquity correction is applied to the second side lobes as
well as the first side lobes in order to depict the diminution
of the perturbation at jt = 5 relative to that at jt = 3. The
perturbations are not true mathematical discontinuities but
approach that status as the number of irradiated slits be-
comes large.
The origin of t
at j = 3
hese perturbations is understood from
Equation (3). Interference of the Figure 1 wavelets con-
verts the output probability into resultant probabilities
represented by the highly directional diffraction orders.
As w λ, truncation leaves only a very few orders prop-
agating over 90˚ to +90˚. Consider for example the be-
Figure 3. Total resultant probability Pr(jt) inclusive of all
diffraction order probabilities within the truncation limits
±jt. For large jt, the total resultant probability Pr(jt)F = 1 is
Copyright © 2013 SciRes. JMP
havior of the Equation (3) probability beginning at jt = 4,
i.e. P(4
r). Truncation at jt = 4 includes the envelope’s
central maximum and the two entire adjacent side lobes.
The only non-zero propagating orders are the 0th, ±1st, and
41.0015 1
shows that probability is sub-
stantially conserved in the output resultant transition at
the jt = 4 truncation value. As we progress from
 (where the notation 3+ designates marginal
inclusion of the ±3rd order peaks), the intensities of the
propagating orders remain constant relative to each other
as seen from the Riemann sum in the numerator of Equa-
tion (3). However, concurrently the normalization factor
provided by the output envelope integral over α progres-
sively decreases as truncation reduces the first side lobes.
In this process as 43
 , output wavelet interfer-
ence generates a progressively increasing resultant prob-
Pj to about 1.3% above unity.
Most significantly for our gedanken experiment, in the
incremenuncation point reduction j
tal tr33
t the
±3rd orders are abruptly excluded from the Riemann sum.
The resultant probability
Pj exhibits a nearly dis-
continuous fall of about 2.8% ending at 1.5% below unity
as the normalizing integral of the output envelope is vir-
tually unaltered in that incremental 33
 reduc-
tion. (Figure 2 necessarily shows resultant orders formed
from only 20 irradiated slits to clearly depict the envelope
on the same vertical scale. For an experimentally realistic
N ~500 irradiated slits, the base widths of the ±3rd order
peaks are a mere ~0.2% of the respective side lobe widths
emphasizing the incremental nature of the 33
transition that removes a significant pair of resultant
probability channels with a vanishingly small change in
the normalizing integral.)
Over the subsequent transition 32
, interfer-
ence of the individual output envelopes continues to de-
crease the normalization integral in Equation (3).
increases asymptotically to
P and at jt =
2 probability is again substantially conserved in the in-
putoutput transition.
These excursions of resultant probability
Pj are
still, by themselves, unremarkable. Classically, th
inherent violation associated with the interference of an
initial wave set producing a final wave set where the re-
spective integrated wave intensities of each (identified in
local realism as relative probabilities) may differ as a
result of net destructive or constructive interference.
Correspondingly for local realism, relative probability is
not necessarily a conserved quantity.
The origin of duality violation relates to the associated
energy output of the grating. Each in
ere is no
dividual slit irradi-
ated by an ordinary incident beam (Figure 1) produces an
output probability sampling that is accompanied by a
proportionate energy sampling. Summation over all irra-
diated slits yields collective output quanta with energy Eo
proportionate to the collective output probability Po which
was earlier assigned a unit value. Because of the propor-
tionate samplings of these two output quantities, in di-
mensionless units Eo may also be set to unity giving
The equality of the wave-like
ticle-like energy is effectively a
probability and the par-
statement that the output
the grating in the near field is still in agreement with
duality. This concurrence with duality can be formalized
by defining an “occupation” value as the ratio of resi-
dent energy quanta on a wave of some probability. For the
grating output, the occupation value is
 (5)
The critical observation to be m
distribution of the output quanta represented by Eo that
n the
ade here concerns the
itially reside on the emergent near-field wavelets. As the
wavelets expand and intersect, the resident energy quanta
distribute without loss onto the resultant
Pj far-field
probability channels (the diffraction orders) in proportion
to the respective relative probabilities oindividual
channels. The resultant energy
and is fully conserved for all val
of jt. The resultant occupation va
ues of w and, equivalently,
  
rt r
 
is simply the inverse of the resultant proba
in Figure 4 and, by symmetry, is equally applicable to any
bility as shown
jj is re-identified as the theoretically predicted
G th on that graph. Because of non-conservation of
there are regions with
j where the
resultants are defined as “enriched” and regions with
where the resultants are “depleted”
defined as
Figure 4. Theoretical occupation value curve G th, equiva-
lent to the Equation (7) r(jt). The three experimentally
measured G ex’s are plotted for comparison to the theo-
retically predicted G th. The no-grating NG control ex-
periment value NG ex is plotted to the right relative to the
vertical ax
Copyright © 2013 SciRes. JMP
in reference to the respective disproportionalities of resi-
dent energy quanta relative to probability.
Resultant beams with = 1 are defined as “ordinary”, a
designation that also applies to the incident beam and the
grating output. In the realization of our gedanken ex-
periment, we use particular Ronchi gratings with w values
that are predicted to respectively generate enriched and
depleted resultant beams. Any individual jth order of the
resultant orders has the same occupation value as that of
the entire set, i.e. rj r
, because occupation values
are intrinsic variables formed from quotients of the ex-
trinsic variables of energy and probability. A particular jth
order beam with a probability Prj acquires a rjr
of the total energy or
EE as it leaves the near-field of
the grating.
Before proceeding with the performed experiment, we
note that the discontinncountered in a transition such
uity e
t presumptively duality-
probabilistic interpretation.
ular grating where a prepared beam ΦG may spe-
that from the depleted region to the enriched region
when 33
 is characterized by the energy quanta
that had been on the ±3rd orders being redistributed onto
the remaining propagating orders. This abrupt redistribu-
tion of energy onto remaining orders as an order passes
threshold is superficially analogous to a “Rayleigh grating
anomaly”. Historically, the incident beam used to study
grating anomalies is multi-wavelength where the beam
energy (or power) is at most a slowly varying function of
wavelength. Rayleigh anomalies relate to those photons at
some particular wavelength λt that have an order “at
threshold” i.e. at the grating plane. In Rayleigh’s analysis
[23], the photons on that threshold order are coherently
scattered off of deep grating grooves (relative to λt) and
are redistributed onto the remaining propagating orders of
the λt photons. As a result, those remaining orders of the λt
photons exhibit an abrupt increase in energy relative to the
energy on the corresponding orders of marginally shorter
wavelength λnt photons that have an unscattered order
near threshold (denoted by “nt”) but not at threshold.
Significantly, the scattered λt photons carry not only the
energy quanta but also the associated wave packet onto
the remaining propagating orders. Consequently, the
Rayleigh anomaly is consistent with probabilistic duality
(as well as with local realism). In this context, we note that
the Ronchi gratings used in this experiment have thin
opaque bands relative to wavelength (see Figure 1 where
t” refers to thickness), and would not be expected to
provide the photon scattering mechanism associated with
gratings actually demonstrating the Rayleigh anomaly.
Clearly, a methodology to directly measure the duality
state of the Ronchi grating propagating orders must be
utilized in order to establish whether those orders are
ordinary, as would be expected for Rayleigh anomalies, or
are in fact “duality-modulated” ( deviations from unity)
as predicted by local realism.
The experiment presented in the next section to provide
that direct test for duality modulation utilizes a transient
coupling between a resultan
olating beam from the grating (a propagating order) and
an independent ordinary beam [22]. The coupling setup is
analogous to the intersection of two independent beams
used in numerous investigations to experimentally assess
duality violation by determining the presence or absence
of interference between the intersected beams as a test of
the probabilistic interpretation. An excellent review of
these investigations is given by Paul [24]. An observa-
tion of interference would seemingly violate Dirac’s dic-
tum that a photon in the probabilistic interpretation can
interfere only with itself [1].
The outcomes of these numerous investigations are
conclusive demonstrations that interference does occur in
apparent contradiction to the
owever, in a theoretical analysis of this phenomenon,
Mandel makes the critical argument that for any given
photon measured in the interference we do not know on
which beam that photon had initially resided [2]. Because
of that lack of knowledge, each photon is treated in
Mandel’s analysis as interfering with itself. Consequently,
interference in the intersection of independent beams is
widely regarded as consistent with the probabilistic in-
terpretation and as not providing a test of that interpreta-
In a variant of those independent beam investigations,
we prepare one of the beams by transmission through a
fically violate probabilistic duality, i.e. the beam is in a
depleted or enriched state from the perspective of local
realism. Spatially transient coupling of that prepared beam
with an independent ordinary “restoration” beam ΦR by
mutual interference over a coupling path should then yield
a net equilibrating transfer of energy quanta for local
realism but not for the probabilistic interpretation. That
net energy transfer relative to ΦG is experimentally readily
measurable in the cw regime at macroscopic powers by
detecting the beam power on a sampling of ΦG that is
substantially separate from ΦR at the end of the coupling
path with and without ΦR present on the coupling path.
4. Experimental Configuration and Methods
We begin with a description of the experimental con
figuration shown in Figure 5. A HeNe laser generates a
horizontally linearly polarized beam Φ of several milli-
watts at 633 nm. Beam Φ traverses a variable attenuator
AttG and an optical beam chopper wheel Ch with a 0.5
duty cycle generating square wave pulses at 40 Hz. Φ is at
normal incidence on a grating G. The grating is one of
three Ronchi gratings (rulings) with respective slit widths
w = 833 nm, 1000 nm, and 1250 nm formed from opaque
bands of 150 nm-thick reflective chromium deposited on a
Copyright © 2013 SciRes. JMP
Figure 6. Detail of Figure 5 coupling path, not to scale,
showing the substantial separation of ΦR from ΦG at the
detector det by convergence of the former onto a disk mas
t of duality violation with this Figure 5
onfiguration imposes some essential general criteria on
us parameters in the region
At BS the diameter
aque disk mask beam
general identifiers of the respective beams, but
Figure 5. Experimental apparatus configuration showing a
potentially duality-modulated beam ΦG after passage through
a particular grating G. ΦG is equilibrated with an ordinary
are 600, 500, and 400. The slit widths uniquely
haracterize each grating through Equation (2) which
it widths. Accordingly, the in-
dividual gratings are also uniquely chara
Equation (8) j-continuum truncation point
beam ΦR along a coupling path extending from beam splitter
BS. Coupling occurs with beam blocker BR shifted to trans-
mit ΦR.
glass substrate. The respective grating frequencies in
gives the w-dependent j-continuum truncation points in
terms of the slits’ widths where λ = 633 nm in this invest-
tigation. These points are
2.63,3.16,and 3.95
jw (8)
for the three respective sl
cterized by the
s. In our nota-
tion a Ronchi grating of some arbitrary slit width w is
denoted as
Gj or simply G. Conversely, a grating
identified with a numerical
-equivalent truncation
point identifies a particular grating with an implicitly
expressed slih as in G(2.63), G(3.16), and G(3.95)
for the three selected gratings use of jt continues our
convention in which variables are most instructively
identified by the critical j-continuum truncation value.
The motivations for selecting these three gratings are
their duality properties based on Equation (7). G(2.63),
G(3.16), and G(3.95) are respectively predicted to pr
t widt
. This
ic) TM (S) polari-
ce enriched, depleted, and ordinary resultants. Data are
also acquired with no grating present. These trials, which
are designated by NG, should produce ordinary resultants
and serve as control experiments.
The particular grating under study is mounted with the
grating bands on the exit face and vertically oriented
thereby providing (transverse magnet
tion with respect to G in the usual classical configura-
tion for observing grating anomalies. The 0th order dif-
fraction beam identified as ΦG is incident on a 50:50 me-
tallic plate beam splitter BS with a 3 mm thick glass sub-
strate and the metallic deposition on the exit face. An
independent HeNe laser generates a horizontally linearly
polarized beam ΦR initially several milliwatts in power.
ΦR traverses a variable attenuator AttR, a retractable beam
blocker BR and enters a beam expander, L1 (f = +100 mm)
and L2 (f = +200 mm), before forming a beam spot con-
centric with that of ΦG on the beam splitter BS as shown in
the Figure 6 detail view. This concentricity is a critical
beam stop Bm.
alignment for the apparatus.
the set of beam and apparat
signated as the “coupling path” that extends from BS to
the final optical components. We include those general
criteria below, augmented by specific examples of pa-
rameter values given in parentheses that are taken from an
experimentally utilized parameter set. In that set, specified
beam widths are Gaussian diameters.
The equivalency of the ΦG and ΦR polarization angles
should be verified on the coupling path itself and cor-
rected by rotation of ΦR if necessary.
ΦR (1.8 mm) is expanded slightly beyond that of ΦG
(1.7 mm) as a result of the L1 and L2 relative spacing. The
beam components exiting BS utilized here are the trans-
mitted component of ΦG and the reflected component of
ΦR. The orientation of BS is adjusted to concentrically
align the ΦR beam spot to the ΦG beam spot at the terminus
of the coupling path. This critical, second beam spot
alignment coaxially aligns the ΦR and ΦG beams over the
coupling path length (~2000 mm).
An assembly of a beam blocker Bm, an iris diaphragm Ir,
and a photodiode detector Det is located at the coupling
path terminus. Bm consists of an op
op (diameter 1.7 mm) mounted on a glass substrate as
detailed in Figure 6. At Bm, natural divergence from the
source laser has further increased the diameter of ΦG (3.8
mm) to a value significantly larger than that of the Bm disk
mask. Conversely, the relative spacing of L1 and L2 is
critically adjusted to converge the ΦR diameter (1.0 mm)
to a value significantly less than that of the Bm beam mask.
The iris (3.3 mm dia.) is set to exclude the peripheral
portion of ΦG from the detector Det. Beam directors on the
coupling path (not shown in Figures 5 and 6) are used to
provide concentric alignment of ΦG and ΦR with Bm, Ir,
and Det.
Data are acquired with one of the three Ronchi gratings
in the Figure 5 position of G. We continue the use of ΦG
and ΦR as
e are reminded that these wave functions in local realism
are exclusive of the energy quanta residing on those
Copyright © 2013 SciRes. JMP
Consistent with our prior notation, the complete ex-
pression for occupancy of that ΦG beam would formally
be identified as
j since jt designates the unique
ifies its
g is tha
the wave struc-
e end of the
coupling path. We use the added
ables such as to denote values a
is criterion is satisfactorily ap-
ncation point on the j-continuum for any Ronchi grat-
GGj, r denotes that ΦG is a resultant diffraction
beam and 0 specorder. These identifiers are al-
ready understood in the present context and the compact
expr is used here in place of
j. Similarly,
an R = 1 designates the occupation value of the initially
ordinary restoration beam ΦR.
The basic premise of beam couplint a duality
modulated beam equilibrates with an ordinary beam by a
net transfer of energy quanta that leaves
res of both beams unchanged and ideally converges the
occupation values toward a common value. For the ob-
jective of achieving complete equilibration,
Gc Rc
 (9)
as the two beams ΦG and ΦR approach th
subscript “c” on vari-
t the end of the coupling
path where equilibration of ΦG and ΦR has potentially
altered those values (in contrast to the respective values of
those variables without ΦG and ΦR simultaneously present
on the coupling path).
ΦR should ideally serve as an infinite source for a de-
pleted ΦG or an infinite sink for an enriched ΦG in the
equilibration process. Th
oximated when the inequality
PP (10)
is satisfied by a ratio of ~100:1 leaving the final equili-
brated ΦG and ΦR both as or
Equation (9) equality to a un
ing G, a net transfer of energy ΔE w
and ΦR that changes the initial grat
ng path terminus is approached. A positive-
signed ΔE corresponds to an energy
transfer from ΦR where ΦG had ini
ling path terminus for our choice of arbitrary
units. A sampling of this EGc is acquired by the detector.
Ideally, the wave-like prob
dinary and extending the
it-valued ordinary value,
Gc Rc
 (11)
If ΦG is depleted or enriched as it emerges from a grat
ill occur between ΦG
ing-emergent energy
of ΦG to
Gc G
EE E (12)
as the coupli
gained by ΦG in a
tially been depleted.
Similarly, if ΦG had initially been enriched, ΔE is nega-
tively signed. Alternatively, if ΦG emerging from G is
initially ordinary, no net transfer occurs and ΔE is zero.
The coupling equilibration of ΦG to an ordinary state (if
it is not already in an ordinary state) provides the impor-
tant result
PE (13)
at the coup
ability PR is entirely con-
fined to the mask of Bm as PRm = PR and the residual
probability of PR in the detector’s annular sampling region
PRa = 0, but this extreme criterion is impractical for the
Figure 5 apparatus. However, this criterion can be satis-
factorily approximated by
PP (14)
The criteria given by the inequalities of Equations (10)
and (14) are expressed in terms of probabilities which are
not directly amenable to measu
rement. Nevertheless, for
e modest deviations of from unity on the order of one
percent realized in the present experiment, the two ine-
qualities are equivalently representable in terms of the
corresponding energies ER, EG, ERm, and ERa. These ener-
gies, in turn, are proportional to the experimentally
measurable corresponding beam powers PWRR (~600
μW), PWRG (~6 μW), PWRRm (~600 μW), and PWRRa
(~2.5 μW) which can be substituted for the probabilities in
the two inequalities for the purposes of setting up the
apparatus. Correspondingly, the ΦG pulse height is
measured from the power PWRGa (~2.5 μW) incident on
the detector. (With the chopper wheel in rotation, the
detector amplifier registers an average power of PWRGa/2
for the pulsed ΦG). The beam power objectives on the
coupling path are achieved by adjustment of the variable
attenuators AttG and AttR.
For data acquisition in the experiment with a particular
GGj in place, the output voltage of the detector
amplifier provides a proportionate instantaneous measure
of the energy incident on the detector in the annular sam-
g rgin. With the chopper wheel in rotation, the de-
tector amplifier signal received by a digital oscilloscope
produces a square wave of a height proportional to the
pulsed ΦG energy.
A trapezoidal-like deviation from a true square wave is
caused by partial eclipsing of the ΦG beam by the vanes of
the chopper wheel. That deviation, which represents in-
pline o
mplete detection sampling of ΦG, is reduced to an in-
significant level relative to the complete sampling when
the ratio of the Gaussian diameter of ΦG to the arc span
between adjacent vanes is very small (~1:100 in the pre-
sent apparatus). This reduction is most readily achieved
when using a chopper wheel with a minimal number of
With the oscilloscope set to dc coupling of the signal
input, the ΦG square wave rides on the baseline bias level
produced by any steady state flux of photons in the an-
nular sampling region. When ΦR is blocked from the
coupling path, that flux consists only of background pho-
tons. When ΦR is unblocked, that steady state flux addi-
tionally includes ΦR photons residually in the annular
sampling region and accordingly significantly elevates the
baseline bias level. Conversely, with the oscilloscope set
Copyright © 2013 SciRes. JMP
to ac coupling of the signal input, the ΦG square wave
appears as alternating positive and negative half-height
square pulses symmetrically distributed about the hori-
zontal zero voltage midline independent of a blocked or
unblocked ΦR. For either mode of signal coupling, in-
cremental changes in the full pulse height are readily
measured as ΦR is blocked and unblocked.
With ΦR unblocked, i.e. coupled to ΦG, the pulse height
of the square wave, measured as a differential voltage
between the upper and lower levels,
GvGc G
 (15)
gives the ΦG post-coupled energy EGc and, very impor-
tantly, the probability PG as well to within a multiplicative
constant κ.
After ΔVGc is acquired, ΦR is blocked
constant κ.
The measurement precisionGc G
improved by using ac coupling rather than dc
Consequently, dc coupling
from the coupling
path by BR. The detector then samples the same annular
region of ΦG but now the pulse height measurement
 (16)
provides the beam’s grating-emergent energy EG, un-
modified by coupling, to within the same multiplic
of ΔV and of ΔV is op-
timized by using real-time waveform averaging. More-
over, the measurement precision of ΔVGc relative to ΔVG is
upling of the input signal. With ac coupling, as ΦR is
alternately blocked and unblocked, the respective upper
and lower levels of the square wave each shift by only
~1% or less relative to the full pulse height of the square
wave. The confinement of the respective upper and lower
levels to these narrow ranges of the vertical measurement
scale effectively eliminates any artifactual effects of
range-related non-linearity.
Conversely, for dc coupling as ΦR is unblocked the
square wave shifts upward on the vertical measurement
scale by the bias voltage produced when a power PWRRa is
deposited on the detector.
cessitates an extremely linear response by the oscillo-
scope in order to distinguish an actual transfer-related
incremental change of ΔVGc relative to ΔVG from an arti-
factual change arising from a non-linear response over the
full range being used. (For example, with PWRRa ~
PWRGa, unblocking ΦR shifts the waveform on the vertical
measurement scale by a bias that is ~100% of the square
wave pulse height and the full utilized range in assessing
ΔVGc relative to ΔVG is about two orders of magnitude
greater for dc coupling than for ac coupling.)
The vital significance of the ΔVGc and ΔVG pulse height
measurements is that their ratio
Ge i
Gc G
which is the experimentally dete
rmined occupation value
from a sequential pair of measurements ΔVGc and ΔVG,
each of which is derived from an ave
cycles. The subscript “ex” has been added to clearly
raging over 128 pulse
identify this quantity as experimentally determined. The
subscript “i” denotes G ex i as a single trial value for the
particular installed grating G.
A typical incremental change in ΔVGc relative to ΔVG
for two sets of these averaged pulses may be on the order
of ±1% (for G(3.16) or G(2.63), respectively). For beams
of macroscopic power, these incremental changes are
representative of enormous numbers of photons (~1011)
added to or subtracted from ΦG in the annular sampling
region as ΦR is coupled to ΦG. These changes are directly
observable on the oscilloscope waveform as ΦR is blocked
and unblocked. The oscilloscope also generates digital
values for ΔVGc and for ΔVG from which each single trial
value G ex i is calculated for the particular installed grat-
ing G. Ten single trial values G ex i with that particular G
are acquired and averaged to give a final reported value
G ex for that set of individual trials. The process is re-
peated for the other two gratings to give the three tri-
al-averaged values
 
2.63 3.16
An additional set of ten trials is acquired with no grating
(NG) present which provides a control experiment value
NG ex. The resultant beam ΦNG is ordinary and should
on coup trials, exhibit no net transfer hese NG a
upling. In t
th should be observable in real time from a
fixed-value attenuation filter is substituted for the gratings
at position G in the apparatus. For any given setting of the
variable attenuator AttG, that fixed-value filter transmits
approximately the same power to the coupling path as do
the gratings. The four sets of trials comprise a complete
All of the single trials in the complete set, including the
NG trials, are acquired with the same beam power PWRG
on the coupling path. This is facilitated by minor adjust-
ment of attenuator Att prior to acquiring each of the four
sets to maintain some selected beam power PWRGa at the
detector. The objective of this procedure is to exclude any
artifactual power-related influences on the measurements
contributing to the determinations of the three G ex and
the NG ex.
As a practical matter, the apparatus is most readily ini-
tially aligned with either grating G(2.63) or G(3.16) in-
stalled. Optimum coaxial alignment of ΦG and ΦR on the
coupling pa
~1% change in pulse height as ΦR is alternately blocked
and unblocked. It is important to note that deficiencies in
fully achieving the various beam parameter criteria and
accurate concentric alignment of ΦG and ΦR on the cou-
pling path result in an incomplete equilibration transfer
and a resultant experimental underestimate of the actual
magnitude of the duality modulation [22].
Copyright © 2013 SciRes. JMP
5. Experimental Results
The experimentally determined Gex values specific to the
three gratings,
 
2.63 3.16
 and
, are
plotted on the Figure 4 theoretically predicted G th
uation (7) a(equivalent to
j from Eq
an empirical obliquity correction
nd inclusive of
e no-grating control
ine/mm grating G(installed
with a du
1 and the t
). Th
experiment value NG ex is also depicted for comparison.
With the 600 l2.63) in the
apparatus, the experimentally determined occupation
2.63 9 0.0024
 which, alternatively
expressed as a “duality modulation” (deviation of from
unity), is +0.59% ± 0.24%. This result and the results
2.63 1. 0043
low are given with ±SE standard error for n = 10 trials.
The corresponding calculated theoretical value at jt = 2.63
is ality modulation of
+0.43%. The significance of the experimentally deter-
is most appropriately assessed relative
to the no-grating control experiment value
0.9988 0.0018
  . Since the theoretical prediction
heoretical prediction of the
no-grating control experiment 1
, the operant
hypothesis is at the true mean of
unity i.e. the true mean of
. Accordingly, the trials
to the ind
re statistically evaluated relative
endent control experiment trials for
e calcul
a one-tailed t test to determine nfidence level.
With the given experimental results, thated con-
fidence level p = 0.015 isy supportive of the hy-
pothesis that tean of
2.63 1
e p c
ue mhe tr
Similarly, with the 500 line/mm G(3.16) grin-
stalled, the value
3.16 0.9923 0.0019
  and the ex-
perimentally measured duality modulation is
0.77 0.19%.0.9912
  giving a predicted dua-
lity modulation of 0.88%. As
sessing this result relative
ean of
ation used
3.95) re
to beatistically distinguishable
this hypothesis is strongly rejected and we conclude that
s th
for the previous
NGex, the theoretical prediction
3.16 1
 yields an
operant hypothesi
is less
than unity. Applying the same statistical evalu
at the true m
grating, the calculated confidence level p
= 0.011 is highly supportive of the present hypothesis that
the true mean of
3.16 1
Finally, the 400 line/mm grating G(sults in
3.95 0.99970.0023
 with a duality modulation of
0.03% ± 0.23%. Since jt = 3.95 is in the neighborhood of
the jt = 4 diffraction null, there is no basis a priori for the
true mean of G
3.95 ex
m unity. This contention is supported by the theoreti-
3.95 0.9985
which closely ap-
proximates NG th = 1. Accordingly, a two-tailed t test is
appropriate for comparing the set of
trials and
the set of NG The operant hypothesis then states
that the true mean of
is significantly different
from that of NG exa calculated p = 0.76,
the true means of
cally predicted
ex trials.
. However, wit
and NG ex are not statistically
One additional statistical evaluation of interest can be
performed with the abolts. We have a theoretical
basis for
 
2.63 3.16GthGth
ve resu
 . Consequently, we can pro-
pose a hypothesis that the true mean of
that of
y a one-tailed t test to find the
relevant p confid
and appl
ence level. With a resultant p = 0.00016,
the present hypothesis is supported at an even higher
confidence level than that for either
relativlthough this hypothesis does
not provide a conclusion with respect tolute, i.e.
that the n of
2.63 1
e to NG ex. A
o an abs
true mea
or that the true mean of
3.16 1
, the confirmation of this hypothesis neces-
sitates that at least one of the true means of
is not unity in violation of duality.
The a
of G th F
gratings G(2
uce enri
and depl.95) is pre-
rrent experiment is to test for statisti-
bove reported experimental results of Gex pro-
vide for the determination of the three points plotted on
the theoretical graphigure 4. The particular
.63) and G(3.16) are used because they are
theoretically predicted to respectively prodched
eted beams at λ = 633 nm while G(3
cted to produce an ordinary beam at that wavelength.
Moreover, in the interests of facilitating replication of the
present experiment, all three gratings are commercially
readily available.
A detailed verification of the Figure 4 theoretical curve
by the acquisition of a large number of experimental data
points is certainly of interest. However, that detailed
verification is clearly not possible with three data points
nor is that the present intent of the authors. The primary
objective of the cu
lly significant violations of duality when particular
resultant beams ΦG are coupled with a beam ΦR. Spe-
cifically, G(2.63) and G(3.16) are predicted to prepare the
resultant outgoing beams in a state not representable by
duality. In the interest of eliminating potential systematic
sources of experimental error, the respective resultant
beams for all three gratings and for the NG control are
maintained at the same power level for the complete set of
trials. With this constraint of equivalent power, duality
necessarily imposes a fundamental physical equivalence
of the four resultant beams. Duality also forbids a net
transfer for these or any other beams upon coupling with
an independently generated beam. Nevertheless, when the
four resultant beams are respectively coupled with ΦR,
statistically significant net transfers consistently occur for
with positive and negative net
transfers, respectively.
In addition to the trials reported here, many hundreds of
preliminary trials have been conducted. Various beam
powers for ΦG and ΦR have been employed for these trials
while maintaining the general coupling criteria. Several
different gratings of each of the three types were used.
Copyright © 2013 SciRes. JMP
Throughse prel
ed here. An independent apparatus
ly through
rough entanglement of correlated quan-
e duality violation presented here is
tates of those
pendent laser beam, the
sfers either +0.59% or 0.77% of its
endent beam for two of the gratings,
er phenomenon is shown to be theoretically
out theiminary trials the results were con-
sistent with those report
s also been assembled that similarly demonstrates G ex
comparable to those of the present apparatus.
6. Discussion
From a theoretical perspective there remains an important
consideration relating to the demonstration of duality
violation as a validation of local realism. The probabilistic
interpretation is distinguished from local realism as a
consequence of non-locality manifested not on
duality but also th
tum entities [25]. Th
consistent with local realism. However, that leaves the
dilemma of entanglement, seemingly conclusively con-
firmed through Bell’s theorem [26] by reported experi-
mental results [27,28], in support of the probabilistic
interpretation despite compelling arguments to the con-
trary [29,30]. Any viable physical representation must
necessarily demonstrate a self-consistent basis supporting
or refuting both duality and entanglement.
In this regard, one of the present authors derived a lo-
cally real representation of quantum mechanical states
from fundamental principles that gives agreement with
performed experiments for correlated photons and for
correlated particles [31]. That representation, demon-
strating locality for correlated entities, is based on non-
conservation of probability for individual s
tities in accord with the present locally real representa-
tion demonstrating duality violation through non-conser-
vation of probability in the diffraction process. Non-
conservation of probability is of particular relevance in the
present context. Clauser and Horne deduce that the locally
real representations that can be excluded by Bell’s theo-
rem are constrained by an implicit supplementary as-
sumption of “no-enhancement” (for which insertion of a
polarizer does not increase detection) [32]. That exclusion
does not apply to the locally real representation in [31]
where non-conservation of probability inherently pro-
vides for enhancement. The resultant locally real repre-
sentation is fully consistent with the underlying mathe-
matical formalism of quantum mechanics. That formalism
is “completed” [25] in the sense of maintaining local
realism for quantum phenomena by admitting the degree
of freedom to treat the relevant wave functions as relative
entities of the field separable from resident particle-like
entities. With this degree of freedom, specific examples of
probability non-conservation emerge naturally.
Experimentally, the configuration reported here, con-
sisting of a single beam incident on a conventional Ronchi
grating, provides for a modest duality modulation. Nev-
ertheless, even this modest duality modulation translates
to a readily measurable increment of beam power in the
macroscopic cw regime. Moreover, the associated un-
derlying locally real basis for this configuration
r a particularly straightforward and compelling under-
standing of a duality-violating phenomenon. There are,
however, other configurations, e.g. [21], that exceed the
present modest duality modulation. Ultimately, there is no
inherent limitation on duality modulation in local realism.
The utility of a duality modulation with very high en-
richment or depletion at macroscopic powers can be ap-
preciated from the transient coupling used in the present
configuration. Transient coupling shows that interaction
of a duality-modulated beam with an ordinary beam re-
sults in a significant net transfer of energy from one of the
beams to the other in the process. For example, a highly
riched beam can be used to directly amplify a weak
ordinary “signal” beam. Transient coupling equilibrates
the two beams causing an enrichment of the signal beam
that enhances conventional detectability of the signal’s
wave modulations. Similarly, empty wave beams (or at
least highly depleted beams) of macroscopic wave inten-
sity would also be of utility in various applications such as
probes of material samples where energy deposition into
those samples by a probe beam must be minimized or
eliminated. The interaction of an essentially empty probe
beam with a sample would be made observable by equi-
librating the post-interaction probe beam with an ordinary
beam. That equilibration renders the wave of the post-
interaction probe beam measurable by conventional de-
tectors as a macroscopic power.
7. Conclusions
The experiment reported here demonstrates violations of
quantum duality with an apparatus reduced to the simplest
of elements. A laser beam is prepared by passage through
one of three particular Ronchi gratings. When the pre-
pared beam is coupled to an inde
prepared beam tran
power to the indep
respectively, whereas the power transfer associated with
the prepared beam from the third grating is 0%. Quan-
tum duality requires that the power transfer must be 0%
for all three.
Notably, the experiment is conducted with beams of
macroscopic power. The resultant duality-violating power
transfers, representing extraordinarily large numbers of
~1011 photons, are readily measurable by a conventional
detector. The transfer phenomenon is robust and highly
That transf
presentable from basic principles and the phenomenon
is consistent with recent reports [10] and [11] of other
performed experiments that also demonstrate violations of
Copyright © 2013 SciRes. JMP
Copyright © 2013 SciRes. JMP
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