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ThreeBeam InterferenceGenerally speaking, the principle of threebeam interference is based on the interference of three reflective surfaces. Figure 1.5 shows one of the typical structures of the threebeam interference based on graded index multimode fiber (GIMMF) [13]. It is fabricated by cascading an air gap and a short section of GIMMF to a singlemode fiber (SMF). The air gap can be formed by fusion splicing the SMF with a chemically etched micronotch on the GIMMF end. The light propagation follows the sinusoidal path as the index profile is parabolic. The raytransfermatrix (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 freespace 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_{s} 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 = (n_{s} — n_{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_{1}( r) = y/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 GIMMF core, which can be given by where n_{1} is the maximum index at r = 0, a is the radius of the GIMMF core, and g is a factor that determines the index profile of the core. The ABCD matrices corresponding to E_{IIr} and E_{m} are given by M_{m} = M_{12}M_{01} and M_{IIr} = M_{2}M_{12}M_{01}, where 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 GIMMF 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{_{I}E_{I}^{,}I(r), where RII = Т— (1 — Aj) R_{n}. T_{I} is the transmittance of surface I and A_{I} is the propagation losses in the air gap. Ф_{п}, the additional phase in E_{n}, is given by Ф_{п} = 2n_{0}kL_{0}. The ABCD matrix corresponding 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_{I} is the propagation losses in the air gap. Ф_{ш}, the additional phase in E_{m}, is given by Ф_{ш} = 2(n_{0}L_{0} + n_{1}L). The ABCD matrices corresponding to E_{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 GIMMF 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 threebeam interference is given by
By using the effective reflectance of the threebeam interferometer, the above RTM theory can be simplified. For the case of n_{0} < n_{1}, the normalized intensity of the threebeam interference can be expressed as
It is well known that the fringe contrast of the twobeam 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 threebeam interference can be deduced by Equation 1.10 and is given by
Unlike the twobeam interference, the constraint condition for the threebeam 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 threebeam interference to obtain high performance than the conventional twobeam interference. Figure 1.6 shows the reflective spectra of the SMF end (black), the air gap (light gray), and the threebeam interferometer introduced above with GIMMF length of 515 pm (gray) [14]. Figure 1.6 Reflective spectra of two and threebeam interferences. 
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