# Three-Beam Interference

Generally speaking, the principle of three-beam interference is based on the interference of three reflective surfaces. Figure 1.5 shows one of the typical structures of the three-beam interference based on graded- index multimode fiber (GI-MMF) [13]. It is fabricated by cascading an air gap and a short section of GI-MMF to a singlemode fiber (SMF). The air gap can be formed by fusion splicing the SMF with a chemically etched micronotch on the GI-MMF end. The light propagation follows the sinusoidal path as the index profile is parabolic.

The ray-transfer-matrix (RTM) theory is used to describe the principle. A Gaussian beam can be expressed as

where *A _{0}* remains constant for energy conservation.

*k =*2n/X is the wave number X with the free-space wavelength. Ф =

*nkz*denotes the phase shift of the light beam as it propagates and

*n*is the refractive

index of the medium light propagating. The complex beam parameter, *q,* is given by

where p and Ю are the radius of the curvature and the beam radius of the Gaussian beam, respectively. Suppose *M* is the transfer matrix from input plane to output plane, the transformation of the complex beam parameter is given by

where *A =* M(1,1), *B =* M(1,2), *C =* M(2,1), and *D =* M(2,2), and *q* and *q* are the complex beam parameters at the input and output planes, respectively.

The electrical amplitude of the incident beam at location 0 is E_{0}(r) and corresponding complex parameter is q_{0} = *in _{s}p&^{2}S* /l.

*n*is the reflective index of the SMF core and is the beam radius of the SMF. The reflectance of surface Rj is determined by the Fresnel equation, Rj =

_{s}*(n*n

_{s}—_{0})

^{2}/(n

_{s}+ n

_{0})

^{2}, and n

_{0}is the reflective index of the ambient medium, which is approximately 1 in the air. The electrical amplitude of the reflective beam at surface I is

*E*) = -y/R"

_{1}( r*E*(

_{0}*r*). The reflectance of the etched micronotch (surface II) is represented by

where and Ю_{Пг} are the beam radii of the incident light and the reflected light by surface II, respectively. *R*(*r*) is the reflectance given by the radial distribution, *R(r) =* [n(r) - n_{0}]^{2}/[n(r) + n_{0}]^{2}, where n(r) is the refractive index profile of the GI-MMF core, which can be given by

where *n _{1}* is the maximum index at

*r*= 0,

*a*is the radius of the GI-MMF core, and

*g*is a factor that determines the index profile of the core. The ABCD matrices corresponding to

*E*and

_{IIr}*E*are given by

_{m}*M*and

_{m}= M_{12}M_{01}*M*where

_{IIr}= M_{2}M_{12}M_{01},*Mj*is the matrix describing the transformation of the complex beam parameters between locations

*i*and

*j.*The elementary matrices are given by

where p_{1} is the radius of curvature of the etched micronotch on the GI-MMF end and *L _{0}* is the effective cavity length of the air gap, which is smaller than the distance between the SMF end and the bottom of the etched micronotch. The transformation of the complex beam parameters is determined by Equation 1.13. The electrical amplitude of the light beam reflected by surface II is E

_{II}(r)

*= R{*E

_{I}_{I}

^{,}I(r), where RII =

*Т—*(1 — Aj) R

_{n}.

*T*is the transmittance of surface I and

_{I}*A*is the propagation losses in the air gap. Ф

_{I}_{п}, the additional phase in E

_{n}, is given by Ф

_{п}=

*2n*The ABCD matrix corresponding

_{0}kL_{0}.to E_{n} is presented as M_{n} = M_{10}M_{2r}M_{2}M_{12}M_{01} with *M _{2V}* = M

_{12}and

Г1 о ]

M_{in} = ~ / . The effective reflectance of surface II can be

0 no */n _{s}*

expressed as

where *2a _{s}* is the mode field diameter of the SMF.

Analogizing to the reflection R_{n}, the reflectance of surface III is given by

The electrical amplitude of the light beam reflected by surface ^{111 is} *E _{m}{r*

^{)}

*= ylR{nЕ{ц*

^{(}г^{) with}Ащ = 7]

^{2}Tn

^{(1 -}A

_{n}

^{)}A

_{m}. 7}

^{is the }transmittance of surface I and

*A*is the propagation losses in the air gap. Ф

_{I}_{ш}, the additional phase in E

_{m}, is given by Ф

_{ш}=

*2(n*The ABCD matrices corresponding to E

_{0}L_{0}+ n_{1}L)._{Iffi}, E

_{IIIr}, and E

_{III}are pre

^{sented as}

*M*

_{mi}=^{M}34

^{M}23

^{M}12

^{M}01>

^{M}IIIr =^{M}44

^{,M}34

^{M}23

^{M}12

^{M}01>

^{and}

*M _{III} = My_{0}’M_{2}’_{Y}M_{3}’_{2}’* M

_{33}/M

_{23}M

_{12}M

_{01}, respectively, with

Considering the coupling losses of the light beam reflected from III into the SMF, the effective reflectance of surface III can be expressed as

*R тф* is mainly dependent on the coupling coefficient of the light beam into the SMF as the GI-MMF length changes. In general, *R*_{n}Iff is smaller than *R*_{III}, because of the coupling losses and propagation losses. The reflective signal of the three-beam interference is given by

By using the effective reflectance of the three-beam interferometer, the above RTM theory can be simplified. For the case of *n _{0} < n_{1},* the normalized intensity of the three-beam interference can be expressed as

It is well known that the fringe contrast of the two-beam interference becomes maximum when the reflectance of the surfaces is equal, which is a strict constraint. The corresponding constraint condition on the effective reflectance of the three surfaces for the three-beam interference can be deduced by Equation 1.10 and is given by

Unlike the two-beam interference, the constraint condition for the three-beam interference to obtain the optimal fringe contrast is an inequality; that is, the requirement on the reflectance is a relatively wide range rather than a decided value. This makes it easier for sensing based on three-beam interference to obtain high performance than the conventional two-beam interference. Figure 1.6 shows the reflective spectra of the SMF end (black), the air gap (light gray), and the three-beam interferometer introduced above with GI-MMF length of 515 pm (gray) [14].

**Figure 1.6 **Reflective spectra of two- and three-beam interferences.