142 Insights into the behavior of certain optical systems glea ned from Feynman s approach to quantum electrodynamics Masud Mansuripur

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Insights into the behavior of certain optical systems gleaned from
Feynman’s approach to quantum electrodynamics
Masud Mansuripur
James C. Wyant College of Optical Sciences, The University of Arizona, Tucson
[Published in the Proceedings of SPIE 12197, Plasmonics: Design, Materials, Fabrication,
Characterization, and Applications XX, 1219703 (3 October 2022); doi: 10.1117/12.2632902]
Abstract. Richard Feynman’s method of path integrals is based on the fundamental assumption that a
system starting at a point and arriving at a point takes all possible paths from to , with each
path contributing its own (complex) probability amplitude. The sum of the amplitudes over all these
paths then yields the overall probability amplitude that the system starting at would end up at . We
apply Feynman’s method to several optical systems of practical interest and discuss the nuances of the
method as well as instances where the predicted outcomes agree or disagree with those of classical
optical theory. Examples include the properties of beam-splitters, passage of single photons through
Mach-Zehnder and Sagnac interferometers, electric and magnetic dipole scattering, reciprocity, time-
reversal symmetry, the optical theorem, the Ewald-Oseen extinction theorem, far field diffraction, and
the two-photon interference phenomenon known as the Hong-Ou-Mandel effect.
1. Introduction. Many problems in quantum optics have their classical counterparts.1,2 For
example, Thomas Young’s famous double-slit experiment, when carried out with single photons,
continues to exhibit the iconic interference fringes that were so crucial in understanding the wave
nature of light against the Newtonian corpuscular ideas. Similarly, the passage of a single photon
through a Mach-Zehnder interferometer is governed by principles that have an uncanny
resemblance to those that guide the behavior of classical coherent light. And Glauber’s coherent
state mimics the characteristic features of classical coherent light in many consequential ways.3,4 In
fact, there are numerous instances in classical optics where the observed behavior of light can be
described in the language of quantum optics (or quantum electrodynamics) in ways that deepen our
understanding of light and help bridge the gap created by its mysterious wave-particle duality.
It is a goal of the present paper to showcase a few situations where well-known classical optical
phenomena submit to elementary descriptions in terms of single photons, following a methodology
pioneered by Richard Feynman.5 In a nutshell, Feynman asserts that the photon (a Bose particle)
should be assumed to take all the allowed paths through a system, each path having its own
probability amplitude — a complex number. When the various paths taken by the photon are
physically indistinguishable, one must add all the probability amplitudes that lead from a specific
initial condition to a specific final condition along different paths, in order to find the overall
probability amplitude of the corresponding event. The probability of occurrence of the event is then
the squared absolute value of the probability amplitude thus computed.
There exist situations, of course, where the predictions of quantum optics, while agreeing with
experimental findings, contradict those of the classical theory. Here, once again, Feynman’s method
proves its usefulness when one tries to understand (or at least clearly explain) the nature of the
observed phenomena. A rather trivial example is provided by a single-photon wavepacket arriving
at a 50 50
beam-splitter. According to the classical theory, the splitter should divide the incident
optical energy between its two exit ports, with detectors placed in these ports each receiving one-
half of the incident energy. In contrast, the quantum theory of light assigns equal probabilities to
each of the two distinct photodetection events, guaranteeing that, in every instance the experiment is
carried out, the incident photon (in its entirety) will be picked up by only one of the two detectors.5
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Even more interesting perhaps is the case of two identical single-photon wavepackets arriving
simultaneously at the two entrance ports of a 50 50
beam-splitter. While the classical theory
maintains that detectors placed in the two exit ports will each pick up a single photon, the correct
prediction is provided by the quantum theory, which asserts that both photons will arrive together at
one detector or the other (with equal probability), but are never divided between the two detectors.6
These and other examples that reveal the profound differences between the predictions of the
classical and quantum theories of light are discussed in the second half of the paper.
The organization of the paper is as follows. Section 2 contains a detailed analysis of the lossless
beam-splitter, which plays an important role in numerous optical systems. Here, we derive the
fundamental relations between the Fresnel reflection and transmission coefficients of beam-splitters
that will be needed in some of the subsequent sections. In Sec.3, we examine the passage of a single
photon through a Mach-Zehnder interferometer and derive the conditions under which the photon
could emerge from one or the other of its exit channels. A similar analysis of a single photon going
through a Sagnac interferometer is the subject of Sec.4. Section 5 is devoted to the problem of
electromagnetic (EM) scattering from a small spherical particle, where we relate the susceptibility
of the host material to the polarizability of the particle in the presence of an external EM field. The
scattering amplitude of an electric dipole in various directions (relative to that of the incident
photon) is subsequently discussed in Sec.6, with the polarization state of the scattered photon
properly taken into account. Here, we also demonstrate the underlying principle of reciprocity in
EM systems using the inherent symmetries of single-photon scattering. The results of Sec.6 are then
generalized in Sec.7 to cover EM scattering from magnetic dipoles. Section 8 is a brief description
of diffraction from a blazed grating as an elementary example of the principle of reciprocity.
Another example is provided in Sec.9 in the context of optical transmission through slabs and
multilayer stacks involving multiple internal reflections. Section 10 describes a thought experiment
that involves the scattering of a single photon from a pair of identical particles, then raises a
question as to whether quantum interference is still viable when a certain amount of angular
momentum is transferred from the incident photon to its scatterer.
In Sec.11, we examine the reflection and transmission coefficients of an EM plane-wave
arriving at a thin dielectric sheet, then use the results to provide an elementary demonstration of the
principle of time-reversal symmetry in classical optics. The results of Sec.11 are also used in Sec.12
to elucidate the fundamental argument behind the Ewald-Oseen extinction theorem. Section 13 is
devoted to yet another important theorem of classical electrodynamics; here, we use the notion of
single-photon scattering to straightforwardly prove the so-called optical theorem, revealing the
intimate relation between the scattering cross-section of a material body and its forward scattering
amplitude.
The classical problems of scalar and vector diffraction are visited in Sec.14 from the viewpoint
of single-photon scattering. We derive the standard formulas of far field diffraction from an object,
then invoke the properties of beam-splitters to argue that the results should apply not only to single
photons but also to any coherent state of the incident beam. Section 15 is a concise description of
the quintessential quantum-optical phenomenon known as the Hong-Ou-Mandel effect. This section
provides the motivation for, and a segue to, our subsequent discussions of multi-photon states.
We return to the problem of beam-splitter in Sec.16, this time examining two wavepackets, one
in the number state |, the other in |, that simultaneously arrive at the entrance ports 1 and 2 of
a lossless beam-splitter. Here, we explain how the splitter distributes these + photons between
its two output ports, thus arriving at the probability of detecting photons in port 3 and the
remaining + photons in port 4.
3/42
Up to this point, we have relied primarily on the principle of superposition and the notions of
distinguishability and indistinguishability of events to reach conclusions that either reaffirm the
well-known results of classical electrodynamics or provide plausibility arguments for decidedly
non-classical phenomena. To proceed further, we must employ some of the more formal tools and
techniques of quantum electrodynamics. This we do in the remainder of the paper, starting with
Sec.17, where the orthonormal eigenmodes of the EM field in free space are formally defined. The
annihilation and creation operators, and , are then introduced in Sec.18 and used to describe
Glauber’s coherent state | and its properties. Next, we introduce in Sec.19 the EM field operators
pertaining to the electric field (,), magnetic field (,), vector potential (,), Poynting
vector (,), and the overall energy of EM modes. This is followed by an examination of certain
physical characteristics of the number states, the coherent state, and the (mixed) thermal state.
The operator algebra is deployed in Sec.20 to re-examine the characteristic behavior of beam-
splitters previously studied in Sec.16, and to confirm the results obtained in that earlier section. We
rely on the operator algebra once again in Sec.21 to demonstrate that a pair of coherent beams, |
and |, arriving at the input ports of a beam-splitter give rise to another pair of coherent beams,
| and |, that emerge from the corresponding output ports. The passage of thermal light through
a beam-splitter is the subject of Sec.22. Section 23 is a brief foray into the Sudarshan-Glauber P-
representation. The paper closes with a summary and a few concluding remarks in Sec.24.
2. Characteristics of beam-splitters. We consider a beam-splitter constructed from thin-film layers
of homogeneous, isotropic, and transparent materials of differing thicknesses and refractive indices,
as depicted in Fig.1. A monochromatic plane-wave of frequency , -vector =( 
)
, and
linear polarization (either - or -polarized) arrives at oblique incidence at the front-facet of the
splitter. The Fresnel reflection and transmission coefficients at this front-facet are (,) and
(,) for - and -polarized light, respectively.7,8 In what follows, we examine the properties of
=|| and =|| without specifying the or subscript; we expect the reader to
understand that our arguments apply to both - and -polarized light separately and independently
of each other. Let us mention in passing that and represent the complex probability amplitudes
for reflection and transmission of a single photon occupying a wave-packet of frequency , -
vector =( 
)
, and linear polarization (either or ) that arrives at the splitter along the
direction of the unit-vector
and in the number state |1.
Fig.1. The Fresnel reflection and transmission coefficients at the
front facet of the beam-splitter are (,) and (,) for the - and
-polarized incident light, respectively. The lateral symmetry of the
multilayer dielectric stack guarantees that, for both polarization
states, (,) are the same for incidence from the left- and the right-
hand sides on the front-facet. In contrast, unless the splitter has
structural symmetry in the front-to-back direction, the reflection and
transmission coefficients (
,
) and (,) at the back-side of the
splitter might differ from the corresponding ones at the front-side.
The lateral symmetry of the multilayer stack comprising the beam-splitter ensures that the
Fresnel coefficients (,) are the same for light arriving at the front facet from either the left- or the
right-hand side, so long as the incidence angle and the polarization state of the beam remain the
same. However, unless the splitter has structural symmetry in the front-to-back direction, one
PCM 1
PCM 2
Splitter
Incident beam
Channel 1
Channel 2
4/42
cannot say that its Fresnel coefficients (,) for incidence from the back-side are the same as
those from the front-side — again for a fixed incidence angle and a fixed polarization state. Our goal
in the present section is to demonstrate that, for a lossless beam-splitter, ||=||, =, and
= ½(+
)±2
. Our analysis does not provide any information on the relation between
and
, nor does it specify the sign of 2
in the preceding expression that relates to +
.
We begin by recognizing that, in the absence of optical absorption, conservation of photon
number (or conservation of energy in the language of classical optics) imposes the following
constraints on the various reflection and transmission coefficients:
||+||= 1, (2.1)
||+||= 1. (2.2)
Other constraints are imposed by the requirement of time-reversal symmetry (see Sec.11 for a
brief discussion of time-reversal symmetry). The phase-conjugate mirrors PCM1 and PCM2 shown
in Fig.1 return the reflected and transmitted waves to the front and rear facets of the splitter, albeit
with the complex amplitudes of the returning waves conjugated. The returning waves must
reconstruct the incident beam (now conjugated and propagating backward) in channel 1; they must
also cancel each other out in the direction identified in Fig.1 as channel 2. We thus have
+= 1, (2.3)
+= 0. (2.4)
Equation (2.3) in conjunction with Eqs.(2.1) and (2.2) now yields
=, (2.5)
||=1||=1||=||. (2.6)
Finally, upon combining Eq.(2.4) and (2.6), we arrive at
= 
+= 2± = ½(+
)±2
. (2.7)
This concludes our analysis of the reflection and transmission coefficients of lossless splitters
with lateral symmetry such as those constructed in the form of multilayer stacks with isotropic,
homogeneous, and transparent dielectric layers. In those special cases when the stack has structural
symmetry in the front-to-back direction, we will have =
and, consequently, =±2
.
An alternative analysis of lossless splitters starts by assuming that two plane-waves with -
field amplitudes and (both -polarized or both -polarized) enter through channels 1 and 2 of
the splitter depicted in Fig.1. With the phase-conjugate mirrors now removed, energy conservation
requires that the sum of the emergent beam intensities be equal to that of the entering beams; that is,
|+|+|+|=||+||. (2.8)
Expanding and rearranging the terms on the left-hand side of the above equation, we arrive at
(||+||)||+(||+||)||+ 2[(+)]=||+||. (2.9)
We now invoke Eqs.(2.1) and (2.2) to eliminate the terms containing || and ||. Since the
remaining term contains , which can have an arbitrary amplitude and phase, we conclude that
the satisfaction of Eq.(2.9) requires that += 0. Consequently,
cancellation in channel 2
reconstruction in channel 1
5/42
=||||=|||| ||(1||)=(1||)|| ||=||,
=
± +
=+±. (2.10)
As before, it is seen that ||=||, which also implies that ||=||. However, unlike the
previous analysis that was informed by time-reversal symmetry, the present method cannot establish
the equality of and . The best one can do in this case is to conclude that +
differs from
+ by . Only when the splitter has front-to-back structural symmetry will we have =,
=, and =±2
.
3. The Mach-Zehnder interferometer.2,7,8 A wavepacket in the single-photon state |1 enters the
Mach-Zehnder interferometer depicted in Fig.2, where, for the sake of simplicity, both 50 50
beam-splitters are assumed to be symmetric, having reflection and transmission coefficients
= 1 2
and = i 2
, irrespective of the direction of incidence. Depending on the path-length
difference between the two arms of the device, either constructive or destructive interference can
take place at the second splitter, in which case the photon consistently emerges from one or the
other exit channel of the interferometer. Denoting the phase acquired upon propagation in the upper
path by and that in the lower path by , one can adjust the phase difference = by
properly positioning the retroreflector.
Fig.2. A wavepacket in the single-photon state |1 arrives at the entrance facet of a Mach-Zehnder
interferometer whose identical beam-splitters are symmetric and have reflection and transmission
coefficients = 1 2
and = i 2
, irrespective of the direction of incidence. Repositioning the
retroreflector allows for changing the optical path-length of the lower arm of the device, hence adjusting
the phase difference = between the two paths that the photon can take in going from splitter
1 to splitter 2. The photon will consistently emerge at Exit 1 (2) if  is an even (odd) multiple of .
Both alternative paths leading to Exit 1 involve one reflection and one transmission at the
splitters, so the corresponding photon amplitudes will be = ½i and = ½i.
At Exit 2, however, the upper path requires reflections at both splitters whereas the lower path
requires two transmissions; therefore, the corresponding amplitudes are = ½ and
=½. If = 2 for some integer , then the photon emerges at Exit 1 with
probability 1. However, if =(2+ 1), the amplitudes add up to zero at Exit 1 and the photon
emerges at Exit 2, where the sum of its corresponding amplitudes equals 1. This is the fundamental
principle of operation of the Mach-Zehnder interferometer.
Beam-splitter 1
Beam-splitter 2
Mirror 1
Mirror 2
Retroreflector
Entrance
Exit 1
Exit 2
摘要：

1/42InsightsintothebehaviorofcertainopticalsystemsgleanedfromFeynman’sapproachtoquantumelectrodynamicsMasudMansuripurJamesC.WyantCollegeofOpticalSciences,TheUniversityofArizona,Tucson[PublishedintheProceedingsofSPIE12197,Plasmonics:Design,Materials,Fabrication,Characterization,andApplicationsXX,1219...

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