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Multiple-beam fringes of equal chromatic order interferometry is a powerful technique to extract optical properties over a continuous range of spectrum. In this paper we develop a theory for the spectral shape of the fringes of equal chromatic order (FECO) that are formed across a double-clad fiber. The modified single term Sellmeier dispersion formula is used to construct the refractive index dispersion curves for the liquid, claddings, and core. Expressions for the paraxial optical path length for several rays passing through the fiber and the liquid are developed. The condition of bright fringe is applied to get an analytical expression for the spectral shape of the FECO across a double-clad fiber with an elliptical/circular inner cladding. A potential application of this theory is to determine the dispersion of Kerr constant of the claddings and the core of the fiber. To illustrate the theory and its application, simulation examples are provided and discussed.

Multiple-beam fringes of equal chromatic order (FECO) are formed when white light source illuminates a silvered air wedge. Each fringe corresponds to a specific optical path difference. However, since white light is used, these fringes are overlapped. To disperse the fringes, an auxiliary dispersion system is required. FECO interferometry is a powerful and sensitive tool in the field of metrology.

FECO. Bennett and Eastman used multiple-beam FECO interferometry to determine accurately thin film thickness and surface roughness [

A great advantage of the FECO interferometry is that a single interferogram gives the required information to deduce the fiber optical parameters across the visible spectrum. Barakat et al. derived a mathematical expression for the shape of Fizeau fringes and FECO crossing an optical fiber with single circular cladding [

A double-clad fiber (DCF) is an optical fiber that has a core and two cladding layers [

Fringes of equal chromatic order are produced when a liquid wedge interferometer is illuminated with a parallel beam of white light and is crossed with a spectrograph. The layout of the optical setup for producing the FECO across a DCF is shown in

where t is the interferometric wedge thickness.

The step index DCF is inserted in a liquid wedge interferometer such that the fiber axis is perpendicular to the edge of the wedge. The axes are chosen such that the x-axis is parallel to the edge of the wedge, the y-axis is along the wedge thickness, and the fiber axis is along the z-axis.

where

where A, B, and C are constants. Equation (3) along with Equation (2) maybe used to estimate the refractive index dispersion for the liquid, inner cladding, outer cladding, and the core.

Since the difference in the refractive indices of the liquid, outer cladding, inner cladding, and the core is small, refraction can be neglected in the calculation of the optical path length for rays passing through the DCF. The optical path length for ray AB that is passing through

But:

Substituting Equation (5) in Equation (4) gives

The optical path length for ray CD that is passing through the liquid

But

Substituting Equation (8) in Equation (7) gives

The optical path length for ray EF that is passing through

But

Substituting Equation (11) in Equation (10) gives

(12)

The conditions of bright fringe with order m across

Transforming the origin to

In the previous derivations, it is assumed that there are no fabrication defects in manufacturing the DCF. Equations (16)-(18) give the wavelength shift of the FECO across a DCF (with elliptical inner cladding) relative to the wavelength in the liquid region. Wavelength shift is attributed to the variation of optical thickness across the DCF. The contribution of the outer cladding, inner cladding, and the core to the wavelength fringe shift are half

ellipses. The semi-axes of the half ellipses of the outer cladding, inner cladding, and core are

As an example, consider a step index DCF with dispersion curves as shown in

cular outer cladding with a displaced core

To illustrate the spectral fringe shape across a DCF, consider a W DCF with claddings and core refractive index dispersion curves that are shown in

The first potential application of this theory is to measure experimentally the linear and nonlinear refractive index dispersion curves across double-clad fibers. This may be achieved by rewriting Equations (16)-(18) into

In order to determine the linear refractive index dispersion curve experimentally, a reference wavelength is superimposed on the fringe pattern and a calibration equation is used to get the values of

and the values of

to be dependent on the wavelength as well as the irradiance E. This maybe represented for the outer cladding by the following equation

where

So far, we assumed that there are no defects on the refractive index profile during the fabrication process. The second potential application is to measure fabrication defects. During the fabrication process, a refractive index dip in the core’s refractive index may show up. The core’s refractive index dip has a radius of

where r is the radial distance and

where

In this paper, the theory for the formation of the fringes of equal chromatic order across a double-clad fiber is presented. The double-clad fiber is perpendicular to the edge of the wedge. Each fringe as it passes through the fiber makes a wavelength shift with respect to the wavelength of the liquid fringe. Expressions for the wavelength fringe shift as a function of position along the DCF are developed. The developed expressions are applicable to DCFs that have elliptical or circular inner cladding with shifted or centered core. Furthermore, the

theory is extended to include the refractive index dip in DCF that has a centered core. To illustrate the theory, simulation examples are presented. The simulation examples show that the amount of wavelength fringe shift depends on the geometrical and the spectral optical properties of the fiber under study. Using a high power laser causes each order to be shifted. In other words, FECO interferometry is a very powerful technique to extract information about the linear and nonlinear dispersions of optical properties over a wide spectrum from a single interferogram. In addition, it may be used to measure the fabrication errors such as the refractive index dip. Future work includes the application of the theory to experimentally measure the linear and nonlinear refractive index dispersions of the optical parameters of DCF.

Mona F.Omar,Rania H. AbdEl-Maksoud, (2016) The Formation of Fringes of Equal Chromatic Order across Double Clad Fiber in the Presence of High Power Lasers. Optics and Photonics Journal,06,29-38. doi: 10.4236/opj.2016.62005