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TEACHING PHYSICS How does a Mach–Zehnder interferometer work? K P Zetie, S F Adams and R M Tocknell Physics Department, Westminster School, London SW1 3PB, UK The Mach–Zehnder interferometer is a particularly simple device for demonstrating interference by division of amplitude. A light beam is first split into two parts by a beamsplitter and then recombined by a second beamsplitter. Depending on the relative phase acquired by the beam along the two paths the second beamsplitter will reflect the beam with efficiency between 0 and 100%. The operation of a Mach–Zehnder interferometer is often used as an example in quantum mechanics because it shows a clear pathchoice problem. However, it is not at all obvious at first glance that it works as claimed, until reflection phase shifts are considered in detail. The question of how a Mach–Zehnder interferometer works arose out of reading through Alevel Research and Analysis essays. Because our students only study divisionofwavefront interferometers and not division of amplitude they had simply accepted the standard explanation given on websites such as On that site, the basic interferometer is shown as in figure 1 and it is explained that interference between the two paths ensures that the photon always strikes detector A. If one of the paths is lengthened then the interference can be altered to ensure that all photons strike detector B. Whilst this seems eminently plausible (indeed, so plausible that we all accepted it without too much worry) it is grossly misleading. First of all, consider the phase of the photon on following each of the two paths, the lower and upper. Initially we shall assume that there is no phase shift on reflection or transmission. 46 Phys. Educ. 35(1) January 2000 The phase on reaching the second beamsplitter is simply the path length divided by the wavelength, multiplied by 2π . On recombination at the beamsplitter, if the two paths are of equal length, then the phases are equal. So which path shows constructive interference, the path towards A or B? The answer is unresolved. In fact, the entire situation is symmetrical with respect to the two detectors and should one path allow constructive interference, so will the other. Similarly if one path suffers destructive interference, so does the other. This violates conservation of energy. Phase shifts on reflection Clearly there is a false assumption and the obvious place to look is the phase shift on reflection. A stan...
TEACHING PHYSICS
How does a Mach–Zehnder
interferometer work?
K P Zetie, S F Adams andRMTocknell
Physics Department, Westminster School, London SW1 3PB, UK
The Mach–Zehnder interferometer is a
particularly simple device for
demonstrating interference by division of
amplitude. A light beam is ﬁrst split into
two parts by a beamsplitter and then
recombined by a second beamsplitter.
Depending on the relative phase acquired
by the beam along the two paths the
second beamsplitter will reﬂect the beam
with efﬁciency between 0 and 100%.
The operation of a Mach–Zehnder
interferometer is often used as an
example in quantum mechanics because
it shows a clear pathchoice problem.
However, it is not at all obvious at ﬁrst
glance that it works as claimed, until
reﬂection phase shifts are considered in
detail.
The question of how a Mach–Zehnder inter
ferometer works arose out of reading through A
level Research and Analysis essays. Because
our students only study divisionofwavefront
interferometers and not division of amplitude they
had simply accepted the standard explanation
given on websites such as www.qubit.org. On that
site, the basic interferometer is shown as in ﬁgure 1
and it is explained that interference between the
two paths ensures that the photon always strikes
detector A. If one of the paths is lengthened then
the interference can be altered to ensure that all
photons strike detector B.
Whilst this seems eminently plausible (indeed,
so plausible that we all accepted it without too
much worry) it is grossly misleading. First of all,
consider the phase of the photon on following each
of the two paths, the lower and upper. Initially
we shall assume that there is no phase shift on
reﬂection or transmission.
The phase on reaching the second beam
splitter is simply the path length divided by the
wavelength, multiplied by 2π . On recombination
at the beamsplitter, if the two paths are of
equal length, then the phases are equal. So
which path shows constructive interference, the
path towards A or B? The answer is unresolved.
In fact, the entire situation is symmetrical with
respect to the two detectors and should one
path allow constructive interference, so will the
other. Similarly if one path suffers destructive
interference, so does the other. This violates
conservation of energy.
Phase shifts on reﬂection
Clearly there is a false assumption and the obvious
place to look is the phase shift on reﬂection.
A standard piece of physics lore is that on
transmission a wave picks up no phase shift,
but on reﬂection it picks up a phase shift of π.
So now let’s investigate the problem with that
in mind. We shall break the problem into two
parts: ﬁrst the path from the source to the second
beamsplitter, and then the ﬁnal stretch from the
second beamsplitter to the detectors A and B.
On the lower path, the beam undergoes one
transmission and one reﬂection before the second
beamsplitter—a total phase shift of π. On the
upper path there are two reﬂections—a total phase
shift of 2π. Now if we continue on to detector A,
the lower path makes one more reﬂection and the
upper path one transmission. So now each path
has a phase shift of 2π and they will interfere
constructively. All well and good? Until we
look at the path to detector B. Now the lower
path makes one more transmission, picking up a
total phase shift of π. The upper path makes a
further reﬂection, so its total phase shift is 3π. The
46 Phys. Educ. 35(1) January 2000
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