 # 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) . 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 A0 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 E0(r) and corresponding complex parameter is q0 = insp&2S /l. ns 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 = (ns n0)2/(ns + n0)2, and n0 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 E1( r) = -y/R"E0 (r). The reflectance of the etched micronotch (surface II) is represented by where and ЮПг 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) - n0]2/[n(r) + n0]2, where n(r) is the refractive index profile of the GI-MMF core, which can be given by where n1 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 EIIr and Em are given by Mm = M12M01 and MIIr = M2M12M01, 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 p1 is the radius of curvature of the etched micronotch on the GI-MMF end and L0 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 EII(r) = R{IEI,I(r), where RII = Т— (1 — Aj) Rn. TI is the transmittance of surface I and AI is the propagation losses in the air gap. Фп, the additional phase in En, is given by Фп = 2n0kL0. The ABCD matrix corresponding

to En is presented as Mn = M10M2rM2M12M01 with M2V = M12 and

Г1 о ]

Min = ~ / . The effective reflectance of surface II can be

0 no /ns

expressed as where 2as is the mode field diameter of the SMF.

Analogizing to the reflection Rn, the reflectance of surface III is given by The electrical amplitude of the light beam reflected by surface 111 is Em{r) = ylR{nЕ{ц(г) with Ащ = 7]2Tn(1 - An)Am. 7} is the transmittance of surface I and AI is the propagation losses in the air gap. Фш, the additional phase in Em, is given by Фш = 2(n0L0 + n1L). The ABCD matrices corresponding to EIffi, EIIIr, and EIII are presented as Mmi = M34M23M12M01> MIIIr = M44,M34M23M12M01> and

MIII = My0’M2YM32 M33/M23M12M01, 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, RnIff is smaller than RIII, 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 n0 < n1, 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) . Figure 1.6 Reflective spectra of two- and three-beam interferences. 