## Abstract

In interferometers where a wave is divided into two beams that propagate along separate branches before being recombined, the closed circuit formed by the two branches must be threaded by wave dislocation lines. For a large class of interferometers, it is shown that the (signed) dislocation number, defined in a suitable asymptotic sense, jumps by +1 as the phase difference between the beams increases by 2*π*. The argument is based on the single-valuedness of the wave function in the branches and leaking between them. In some cases, the jumps occur when the phase difference is an odd multiple of *π*. The same result holds for the Aharonov–Bohm wave function, where the waves passing above and below a flux line experience different phase shifts; in this case, where the wave is not concentrated onto branches, the threading dislocations coincide with the flux line.

## 1. Introduction

In simple interferometers, a beam of light or matter waves, here assumed to be scalar or linearly polarized, is split into two at the entrance port. Each beam accumulates a different phase shift while propagating, essentially one dimensionally, along a different branch of the interferometer. The beams are recombined at the exit port, where the phase difference *Γ* is detected, for example by the shift of an interference pattern beyond the exit port. My purpose here is to point out a curious consequence of the fact that the wave cannot be completely confined to the two branches, so there will always be some leakage between them, especially near the entrance and exit ports, and a wave *Ψ*, albeit of very low intensity, must exist throughout the three-dimensional space in which the interferometer is situated.

The wave *Ψ*—a complex scalar field—must be single-valued in the space between the branches. This will be true whatever the nature of the intervening medium (refracting, reflecting, absorbing, active or inactive), or the mechanism of propagation (e.g. diffraction, scattering, tunnelling), or the wave equation (linear or nonlinear). It follows that the total phase change around any circuit formed by the branches of the interferometer must be an integer multiple *S* of 2*π*. *S* is the total strength of wavefront dislocation lines (Nye & Berry 1974; Nye 1999) threading the interferometer. On a dislocation line (also called a nodal line or a wave vortex), the phase arg*Ψ* is singular. The threading dislocations will pierce any surface spanning the branches (figure 1), at points *P* around which arg*Ψ* increases by 2*πn*, in a sense defined by a normal to the surface; usually *n*=±1. *S* is the sum of the (signed) strengths *n* over all *P*. In general, the phase difference *Γ* measured by the interferometer will not be an integer multiple of 2*π*, so *S*≠*Γ*/2*π*. Therefore, it is interesting to ask: how does *S* depend on *Γ*?

The question is imprecise unless we specify more closely what is meant by the ‘circuit formed by the branches’. In order for the beams in an interferometer not to spread by diffraction, they must be many wavelengths wide; hence, the branches cannot be regarded simply as lines without thickness. What follows will be an asymptotic theory, in which the ratio *ϵ* of the width of the beams to their total length is small. Thus, the overlap between the beams, necessary for *Ψ* to be single-valued, is confined to the regions *ϵ* close to the entrance and exit ports, and the circuit can be any closed curve (as in figure 1) that does not stray outside the beams.

As parameters describing the interferometer vary, *S* can change only by a dislocation entering or leaving the circuit while passing through one of the branches of the interferometer. After giving a general theory (§2), I will show that under reasonable assumptions, *S* jumps by +1 as *Γ* increases by 2*π*, i.e.(1.1)

In some cases (§§3 and 5), *Γ* is the only relevant parameter determining *S*, and a model (§3) for a wide class of interferometers shows that the jumps in *S* occur as *Γ* passes half-integers. Mach–Zehnder interferometers conform to equation (1.1), but the details are slightly different: as a separate treatment (§4) shows, the jumps occur not at half-integers but at values of *Γ* that depend on the phases in the individual branches, as well as the reflection and transmission coefficients of the beam splitters.

This phenomenon is not restricted to conventional interferometers. For example, the Aharonov–Bohm effect (AB; Aharonov & Bohm 1959), where a localized magnetic flux line scatters a beam of electrons incident on it, can be regarded as an interferometer inducing a phase difference between waves passing on either side of the flux line (Ehrenberg & Siday 1949; Feynman *et al*. 1965). In terms of the quantum flux , the phase difference 2*πΓ*=2*πα*. In the AB situation, the interfering waves are distributed throughout space, rather than being concentrated in branches. Therefore, we can choose as the ‘interferometer circuit’ any loop enclosing the flux line. Nevertheless (§5), the result (1.1) holds and, reinterpreting an old result (Berry *et al*. 1980), all the dislocation strengths are concentrated onto the flux line.

## 2. General theory

Referring to figure 2 and with the chosen circuit, let *s* be a parameter measuring the distance along the branches a and b of the interferometer, with entrance port *s*=0 and exit port *s*=1, and let the wave functions in the branches be *Ψ*_{a}(*s*) and *Ψ*_{b}(*s*), where *Ψ*_{a}(0)=*Ψ*_{b}(0) and *Ψ*_{a}(1)=*Ψ*_{b}(1), reflecting single-valuedness. Then, the dislocation strength around the circuit from *s*=0 to *s*=1 on branch b and from *s*=1 back to *s*=0 on branch a is(2.1)where, here and henceforth, subscripts denote differentiation. We will use this formula for numerical calculations.

The only mechanism by which *S* can change as parameters vary is by *Ψ* becoming zero as a dislocation crosses either branch. Only one parameter needs to vary, and the parameter we are interested in (which as we will see is sometimes the only relevant one) is the phase difference *Γ*. Let a dislocation cross one of the branches at *s*=*s*_{c}, *Γ*=*Γ*_{c}. We seek the sign of the change Δ*S* as *Γ* increases through *Γ*_{c}. Δ*S*=+1(−1) corresponds to an *n*=+1 dislocation entering (leaving) the interferometer, or an *n*=−1 leaving (entering). We will sometimes incorporate the sign of *n* and refer to Δ*S*=+1(−1) simply as a dislocation entering/leaving.

Consider the case where the crossing is on the b branch, on which *s* increases. In the space *s*, *Γ*, the change Δ*S* is 1/2*π* times the difference between the phase change accumulated passing just above *s*_{c} (i.e. *Γ*>*Γ*_{c}) and that just below *s*_{c} (i.e. *Γ*<*Γ*_{c}). Thus, Δ*S* is minus the dislocation strength, defined as 1/2*π* times the phase change in a positive circuit of the dislocation. If the dislocation crosses the branch a, on which *s* decreases, the sign is reversed. Thus, using a well-known formula for dislocation strength (Berry 1998, 2004),(2.2)We will use this formula in later sections.

For the unperturbed waves in the branches, ignoring the leakage between them, we write (figure 2)(2.3)

The picture envisaged here is that a single locally plane wave propagates in each branch; the amplitudes are *c*_{a} and *c*_{b}, and the evolving phases are *γ*_{a}(*s*) and *γ*_{b}(*s*). These must satisfy (figure 3*a*)(2.4)where *Γ*_{a} and *Γ*_{b} are the phases accumulated along the branches, so the phase difference is(2.5)

In numerical computations, we will use the simple model(2.6)

The expressions (2.3) approximate the waves everywhere, except near the entrance and exit ports. Near *s*=0 and *s*=1, the full waves *Ψ*_{a}(*s*) and *Ψ*_{b}(*s*) differ from *Ψ*_{a0}(*s*) and *Ψ*_{b0}(*s*) because of the leakage, which is necessary for the wave in the space containing the interferometer to be single-valued.

We model the leakage with a function *f*(*s*, *ϵ*) (figure 3*b*), in which *ϵ* represents the ratio of the extent of the leakage region to the total length of the branches. We shall consider *ϵ*≪1, corresponding to an interferometer with well-separated branches, for which *f*≪1 except near *s*=0 and *s*=1, where *f*=1. In the numerical computations below, I will use(2.7)

## 3. Simple interferometer model

In this section, we regard the leakage as a direct contamination of the wave in each branch from that in the other. This simple model applies, for example, to the Young and Rayleigh interferometers (Born & Wolf 2005), as well as single-crystal neutron (Rauch & Werner 2000) and X-ray (Hart 1975) interferometers. Thus, the total wave in each branch is given by the following modification of the unperturbed expression (2.3):(3.1)*Ψ*_{a} and *Ψ*_{b} agree at *s*=0 and *s*=1, as they must. Equation (3.1) can be simplified by defining the amplitude ratio and evolving phase difference(3.2)giving(3.3)

Dislocations are determined by the vanishing of the factors in square brackets.

Thus, the dislocation strength (2.1) becomes(3.4)

Figure 4 shows a numerical example using the models (2.6) and (2.7). Evidently, *S* increases by unity as *Γ* increases by 2*π*, as claimed in equation (1.1). Moreover, the jumps occur as *Γ* increases through *Γ*_{c}=(2*n*+1)*π*.

To explain these observations, we must determine the zeros of *Ψ*_{a} and *Ψ*_{b}. From equation (3.3), destructive interference can occur only near the exit port, and there are two possibilities(3.5)

Now,(3.6)hence, zero-crossing requires *Γ*_{c} to be a half-integer multiple of 2*π*, and the crossings occur near the exit port, as claimed. With the particular phase evolution (2.5) and leakage function (2.6), the approach to these asymptotic values is(3.7)

To determine whether the change of dislocation strength Δ*S* is +1 or −1 as *Γ* increases through *Γ*_{c}, we use the formula (2.2). Evaluating the derivatives of *Ψ*_{a} and *Ψ*_{b}, together with the information that near *s*=1 both *γ*(*s*) and *f*(*s*) are increasing functions of *s*, and *γ*(*s*) is also an increasing function of *Γ* and exp{i*γ*(*s*_{c})} is negative real, we find, for both the a and b branch crossings,(3.8)

To determine the absolute value of *S*, as well as its increase as *Γ* passes *Γ*_{c}, we study the case *Γ*=0. Consider *r*<1 (an analogous argument works for *r*>1), then it follows from equation (3.3) that the only possibility of *Ψ* winding round the origin, i.e. of a non-zero *S*, is on the b branch. But such winding does not occur, because *γ*(*s*) is close to zero near the entrance and exit ports *s*=0 and *s*=1, and for *ϵ*≪0 any possible large excursions in *Γ*(*s*) must take place in regions where *f*≪1; it follows that Im *Ψ*≪Re *Ψ*, thus excluding winding. We conclude that for this class of interferometers, the dislocation strength is independent of the amplitude ratio *r* and also independent of *ϵ* when *ϵ*≪1, and is given by(3.9)as figure 4 illustrates.

## 4. Mach–Zehnder interferometer

With this type of interferometer (figure 5), and the similar Jamin and Sirks–Pringsheim interferometers (Born & Wolf 2005), the contamination is different from the case just considered, in which the a and b beams are contaminated with the b and a beams. To understand what happens instead, consider the circuit indicated in figure 5, corresponding to the centre lines of the beams in the branches. At the entrance port, the ‘a’ beam is contaminated by the wave incident on the beam splitter B_{1}, and the ‘b’ beam is uncontaminated; at the exit port, the ‘a’ beam is contaminated by the ‘a’ wave reflected by B_{2} and the ‘b’ wave transmitted by B_{2}, and the ‘b’ beam is contaminated by the ‘a’ wave transmitted by B_{2} and the ‘b’ wave reflected by B_{2}.

We denote the complex reflection and transmission amplitudes of the beam splitters by *ρ* and *τ*. In this section, it is convenient to employ the leakage functions *g*(*s*) and *g*(1−*s*) for the entrance and exit ports individually, as in equation (2.7) (which corresponds to Gaussian beams). Thus, the waves in the branches can be written as(4.1)

In the simplest model, the beam splitters can be regarded as infinitely thin, reversible and non-absorbing, so that the incoming and outgoing waves on the two sides are connected by a unitary matrix. Furthermore, single-valuedness demands that the total wave (incoming+outgoing) must be the same on both sides. These requirements imply the following relations between *ρ* and *τ*:(4.2)

The unique solution, involving a single parameter *θ*, is(4.3)

These expressions are periodic in *θ* with period *π*.

For simplicity, we henceforth consider the ‘democratic’ case |*c*_{a}|^{2}=|*c*_{b}|^{2}=1/2, where *θ*=*π*/4, so(4.4)

The qualitative behaviour of the dislocation strength is similar for other values of *θ*. In addition, we use the model phase and leakage functions (2.6) and (2.7). Then, the waves (4.1) become, after neglecting a common factor,(4.5)

Now, there is a non-trivial dependence on the phase *Γ*_{a} as well as the phase difference *Γ*.

Near the entrance port, the branch b consists of just the transmitted wave, so there is no interference; therefore, *Ψ*_{b} cannot vanish and no dislocations cross this branch. For the a branch, equation (4.5) gives, for *ϵ*≪1 and again neglecting an overall factor,(4.6)

It follows that dislocations can cross the branch a as *Γ*_{a} varies, but these events are independent of *Γ*, as is physically obvious. Moreover, the crossings can occur only when *s*<*ϵ*√(log 2), so the corresponding values of *Γ*_{a} are separated by phase differences of order 1/*ϵ*. We do not consider such crossings further.

Near the exit port,(4.7)

It is easy to see that a dislocation crosses the exit port *s*=1 when *Γ*_{c}=(2*n*+1/2)*π*, since then both *Ψ*_{a} and *Ψ*_{b} vanish. From equation (2.2) applied to *Ψ*_{a} or *Ψ*_{b}, the change in *S* when *Γ* increases through *Γ*_{c} is easily calculated as(4.8)

When *Γ*_{a}<0, this corresponds to a dislocation leaving the interferometer as *Γ* increases, rather than entering, and so appears to violate equation (1.1). But we will see that this is not the case, because as *Γ* changes by 2*π*, other dislocations can enter and leave the interferometer, in a manner that maintains the validity of equation (1.1).

These additional dislocation events depend on the value of *Γ*_{a}. It might be thought that this dependence is negligible, because equation (4.7) involves *Γ*_{a}(1−*s*) and 1−*s*∼*ϵ*≪1. This would be wrong, because *Γ*_{a} can be so large that *Γ*_{a}(1−*s*) can be large too. For an interferometer whose arms are of length *l* and contain beams of wavenumber *k* and width *w*,(4.9)since the beams must be sufficiently wide to prevent diffraction spreading.

To determine the dislocation crossings *Γ*_{c}, *s*_{c}, it is convenient to define the exit-port-crossing variable(4.10)so that the exit port corresponds to *σ*=0, with *σ*<0 on branch b and *σ*>0 on branch a. Then, the vanishing of Im*Ψ*_{a} and Im*Ψ*_{b} in equation (4.7) can be written in the common form(4.11)with the values of *σ*_{c} determined by substitution into equation (4.7), which gives(4.12)

All solutions must have *σ*<√log 2, i.e. |1−*s*|<*ϵ*√log 2. Application of equation (2.2), followed by some algebra, now gives the sign of the dislocation jump at *Γ*_{c}, *s*_{c} as(4.13)

Figure 6 shows the locus (4.12), with the positive and negative regions (4.13) superimposed. It is clear that the number of crossings (as *Γ* increases by 2*π*) is odd for any value of *Γ*_{a}, and there is always one more dislocation entering than leaving, demonstrating that equation (1.1) holds for Mach–Zehnder interferometers. For small positive values of *ϵΓ*_{a}, there is a single crossing, at the exit port *σ*=0, *Γ*=*π*/2, and for small negative values there are one negative and two positive crossings. As |*Γ*_{a}| increases, additional crossings appear in groups of four, two positive and two negative. The dependence on *Γ*_{a} is weak: in order to change the number of crossings as *Γ* increases by 2*π*, *Γ*_{a} must change by O(2*π*/*ϵ*), which is large.

Figure 7*a*–*c* shows the locations and signs of dislocation-crossing events in the *σ*, *Γ* plane for three cases, and figure 7*d*–*f* shows how the corresponding dislocation strengths depend on *Γ*. In figure 7*b*, for example, three dislocations enter and two leave as *Γ* increases by 2*π*. In every case, the net number entering is +1, as claimed in equation (1.1).

## 5. Aharonov–Bohm interferometer

The single-valued AB wave function in the plane, for a beam (plane wave) of charged quantum particles, incident from negative *x*=*r* cos *ϕ* and diffracted by an isolated magnetic flux line with strength *α* is in units where the wavenumber is unity (Aharonov & Bohm 1959; Olariu & Popescu 1985),(5.1)

The leading-order large-*r* asymptotics, representing the incident plane wave modified by Dirac's magnetic phase factor, is (Ehrenberg & Siday 1949), for 0<|*ϕ*|≤*π*,(5.2)

This is commonly interpreted interferometrically, as a wave propagating ‘above’ the flux line, that is for *ϕ* decreasing from +*π* to zero through positive *ϕ*, and a wave ‘below’ the flux line, with *ϕ* increasing from −*π* to zero through negative *ϕ*. The phase difference between these waves is *Γ*, where *Γ*=2*πα*, and its observation as a shift of interference fringes is the celebrated AB effect (Olariu & Popescu 1985). Although equation (5.2) is continuous across *ϕ*=±*π* (incident direction), the formula possesses an unphysical discontinuity in the forward direction *ϕ*=0, where, as is well known (Olariu & Popescu 1985), the incident and diffracted waves cannot be separated (for the relationship between the single-valued equation (5.1) and the multi-valued equation (5.2); see Berry 1980).

As with other interferometers that we have considered, the phase difference implies that *Ψ*_{AB} is dislocated. But now there is no natural circuit, and we can define a strength that, in principle, could depend on radius, corresponding to dislocations piercing the plane at different points, namely, for circular loops with radius *r*,(5.3)

In fact, all the dislocation strength in *Ψ*_{AB} is concentrated on the flux line at *r*=0 and *S*_{AB}, independent of *r*, is given by equation (1.1). For *r*≪1, this was shown directly from equation (5.1) (Berry *et al*. 1980), and a heuristic argument was given for the same result to hold for *r*≫1.

A better argument is based on the following modification (Berry & Shelankov 1999) of equation (4.12), uniformly valid in *f* for large *r*:(5.4)

(This is the leading order of a series (Berry & Shelankov 1999) in powers of 1/*r*, giving an intriguing representation of *Ψ*_{AB} in terms of the Cornu spiral.) As shown in figure 8, equation (5.4) is extremely accurate. Figure 9 shows a sample wavefront pattern with a dislocation of strength −3; the picture was computed from equation (5.4), but is indistinguishable from a computation based on the exact formula (5.1), showing that the uniform asymptotics is accurate for small *r* too.

Computations directly with equation (5.3), using either *Ψ*_{AB} or *Ψ*_{ABapp}, generate graphs identical to figure 4, so we conclude that(5.5)

As with conventional interferometers, *S*_{AB} jumps as *α*=*Γ*/2*π* increases through a half-integer. But the mechanism is different: instead of a dislocation crossing the interferometer circuit, in the AB case there is a nodal line (in the plane) in the forward direction *ϕ*=0 when *α* is a half-integer, enabling each wavefront arg*Ψ*_{AB}=constant+2*nπ* to unzip and reconnect with its neighbour *n*+1 as *α* increases through a half-integer. Figure 8, for *α*=−2.51, shows some wavefronts that have just experienced the reconnection. The unzipping phenomenon was observed in an AB analogue experiment (Berry *et al*. 1980) based on water waves.

## 6. Concluding remarks

In the phenomenon identified here, the circuit formed by the branches of an interferometer is threaded by dislocation lines, whose signed number changes discontinuously as the phase difference between the arms increases continuously, with a net increase of +1 as the phase difference increases by 2*π*. It occurs in a wide class of interferometers, including that corresponding to the AB effect, where there are no well-defined branches. The only requirement seems to be that there is a single-valued wave function throughout the space occupied by the interferometer.

Is the phenomenon universal? A natural conjecture is that it occurs for all interferometers for which a circuit can be defined. (The ‘circuit’ stipulation excludes interferometers of the Michelson and Twyman–Green types (Born & Wolf 2005), where, at least in the idealized form, each beam path is self-reciprocal, so the beams do not form a circuit that could be threaded by the dislocations.)

For phase differences *Γ* not close to the critical values *Γ*_{c} corresponding to the passage through the branches of the interferometer (e.g. (2*n*+1)*π* in §§3 and 5), the dislocations can lie deep inside the circuit, where the wave *Ψ* is very weak, perhaps unobservably so. But when *Γ*≈*Γ*_{c}, it should be possible to detect the dislocations and follow them as they cross a branch close to the exit port; it would be interesting to do so.

The assumption throughout has been that the waves are scalar or linearly polarized. In optical interferometers where the state of polarization is changed in the branches, the vector nature of the light cannot be ignored, and the analysis must incorporate the spatially varying polarization in the region containing the circuit. With polarized light, there are typically no dislocations. Instead, the singularities threading the circuit are C lines, on which the polarization is purely circular, and L lines, on which the polarization is purely linear (Nye & Hajnal 1987; Nye 1999; Berry 2001; Berry & Dennis 2001). The numbers of C and L lines entering and leaving the interferometer during a 2*π* variation of the phase difference measured by the interferometer remain to be explored. Such a study should incorporate any geometric phase associated with polarization changes (Pancharatnam 1956; Berry 1987; Bhandari 1997).

## Acknowledgments

I am grateful to late Professor Geoffrey Opat for a suggestion, many years ago, that inspired the research reported here, and I would like to acknowledge helpful discussions with Professors John Hannay, John Nye and Sandu Popescu, and Dr Mark Dennis and Mrs P. York.

## Footnotes

- Received January 19, 2007.
- Accepted March 6, 2007.

- © 2007 The Royal Society