Modeling of the fringe shift in multiple beam interference for glass fibers

A quadratic model is suggested to describe the fringe shift occurred due to the phase variations of uncladded glass fiber introduced between the two plates of the liquid wedge interferometer. The fringe shift of the phase object is represented in the harmonic term which appears in the denominator of the Airy distribution formula of Fabry-Perot’s interferometer. A computer program is written to plot the computed fringe shifts of the described model. An experiment is conducted using liquid wedge interferometer where the fiber of a nearly quadratic thickness variation is introduced between the two plates of the interferometer. The obtained fringe shift shows a good agreement with the proposed quadratic model. Also, it is compared with the previous theoretical shift based on ray optics of semi-circular shape.      

Multiple-beam interference fringes applied to investigate the index profile of the optical fiber dip

A numerical-based phase shift method is presented to study optical fibers having a dip in the refractive index profile at the center of the core. Mathematical expressions for the shape of multiple-beam Fizeau fringes crossing a graded-index optical fiber (GRIN) comprising a central graded-index dip or GRIN fiber with a constant refractive index dip, immersed in a silvered liquid wedge, are derived. The index profile parameters of the central dip are determined by comparing the theoretical fringe shape across the fiber cross-section and the experimental fringe shape obtained from micro-interferograms of the wedge interferometer.     

马赫-曾德尔型相移矢量剪切干涉仪

为了实现对激光波面的等光程测试,设计了一个新颖的相移矢量剪切干涉仪。该干涉仪以马赫曾德尔结构为基础,为等光程干涉仪。通过在相干的两路光束中分别插入楔角方向正交放置的楔板实现矢量剪切;其中一块楔板分成两块(平板部分和楔板部分),楔板部分沿平板部分的表面移动,通过改变光程差来实现相移,2π相移所需的移动距离为几毫米的量级,该相移方法在相移精度控制上比较简单。文中还对楔板引入的横向剪切误差和相移误差进行了分析,最后给出了实验所得的一组相移干涉图。     

Rapid extraction of the phase shift of the cold-atom interferometer via phase demodulation

Generally, the phase of the cold-atom interferometer is extracted from the atomic interference fringe, which can be obtained by scanning the chirp rate of the Raman lasers at a given interrogation time T. If mapping the phase shift for each T with a series of measurements, the extraction time is limited by the protocol of each T measurement, and therefore increases dramatically when doing fine mapping with a small step of T. Here we present a new method for rapid extraction of the phase shift via phase demodulation. By using this method, the systematic shifts can be mapped though the whole interference area. This method enables quick diagnostics of the potential cause of the phase shift in specific time. We demonstrate experimentally that this method is effective for the evaluation of the systematic errors of the cold atomic gravimeter. The systematic phase error induced by the quadratic Zeeman effect in the free-falling region is extracted by this method. The measured results correspond well with the theoretic prediction and also agree with the results obtained by the fringe fitting method for each T.     

Group index dispersion of holey fibres measured by a white-light spectral interferometric technique

We present a new white-light interferometric technique to measure the group index of holey fibres over a wide wavelength range. The technique utilizes an unbalanced Mach–Zehnder interferometer with a fibre under test of known length placed in one of the interferometer arms and the other arm with adjustable path length. In a first step, the differential group index of the fibre is measured over a wide wavelength range. In a second step, the fibre is replaced by the reference sample of known thickness and group dispersion to determine precisely the group index of the fibre at one specific wavelength. The group index as a function of wavelength is measured for two different holey fibres, one made of pure silica glass and the other made of SK222 glass. For both fibres, the wavelength dependence of the group index of the outer cladding and modes supported by the fibre is measured.     

Phase shifting speckle shearing polarization interferometer using a birefringent wedge

A digital speckle pattern shearing interferometer is presented which utilizes a single birefringent wedge. Two orthogonally polarized images of an object measured are sheared by the wedge, and the bias phase difference between two polarized images is shifted by translating the wedge. The phase shifting technique applied to speckle pattern shearing interferometry makes it possible to analyze a slope and surface strain numerically. The fringe formation in the present interferometer is analyzed with statistical theory. Experimental results are also presented in two cases: (1) measurement of a slope and normal displacement with a single illumination; (2) simultaneous and separate measurement of a slope and surface strain with two symmetrical illuminations.     

Multi-directional lateral shearing interferometer based on rotatable prism pairs

The common camera lens usually includes the spherical glass/plastic lens and aspheric glass/plastic lens. However, spherical/aspheric shape measurement is still a key problem in the process of optical lens fabrication. At present, the in-process measurement of spherical/aspheric shape is conducted mainly by the probe-contacting method. But after a long time, its probe could be scratched severely and cause some big errors. Laser shearing interferometry is a good substitute to some degree. Nevertheless, it is not convenient for general shearing interferometer to carry out the in-process measurement because it is only suitable for certain kind of spherical/aspheric with respect to aperture or asperity. Here a new lateral shearing interferometer is proposed to solve the described problems. It is based on two Jamin plates and rotatable prism pairs which are used not only for shearing displacement and direction, but also for fringe period and tilt degree, in order to meet requirements of various spherical/aspheric shapes or asperities. The new interferometer features a simple optical structure and two symmetric light paths, which makes its system with minimal error. The relation between shearing displacement, fringe period and prism angle of rotation is given in this paper. And the error source is primarily from the manufacture errors of prisms and plates. The final experiments show that one can achieve good-quality fringe patterns according to the requirement of measurement, concerning the shearing direction, shearing displacement, fringe period, tilt degree, etc.     

Interferometric determination of refraction and dispersion of human blood-serum, saliva, sweat and urine

Multiple-beam interference fringes of equal chromatic order are produced in air and liquid sample interferometric gaps. The two gaps are of the same thickness and simultaneously enclosed in a wedge interferometer. A single shot interferogram containing fringes in the two gaps is sufficient to deduce the needed experimental data. Locations of the fringe maxima, in the two gaps, are introduced in a non-numerical procedure for determining the gap thickness and the liquid-phase refractive indices across the visible spectrum. The method has been used for measuring the phase refractive indices of human blood-serum, saliva, sweat, urine and water liquids. A third-order polynomial dispersion relation is applied for fitting the measured phase indices. Group refractive indices have been derived and fitted to the same dispersion formula.     

Slightly dispersive white-light spectral interferometry to measure distances and displacements

A new spectral-domain interferometric technique of measuring distances and displacements is realized when the effect of low dispersion in a Michelson interferometer, which comprises two coated plates of a beam splitter and a compensator, is known and the spectral interference fringes are resolved over a wide wavelength range. First, processing the recorded spectral interferograms by an adequate method, the unmodulated spectrum, the spectral fringe visibility function and the unwrapped phase function are obtained. Then, knowing the dispersion relation for the fused-silica plates, the ambiguity of the unwrapped phase function is removed and the thickness of fused silica and the nonlinear phase function due to the effect of the coatings are determined by using a new procedure. It is based on the linear dependence of the overall optical path difference between interferometer beams on the refractive index of fused silica. Once the thickness and the nonlinear phase function are known, the positions of the interferometer mirror are determined precisely by a least-squares fitting of the theoretical spectral interferograms to the recorded ones.     

Strain and vibration analysis by fibre based speckle shearing interferometry

An optical fibre based, speckle shearing interferometer is described. The instrument uses a highly birefringent optical fibre to illuminate a test object with equal intensities of light guided by the orthogonal polarisation eigenstates of the fibre. A Wollaston prism is used to obtain two sheared images with adjustable shear. Optical phase changes between the sheared images are readily achieved, without mechanical movement of components, by straining the optical fibre. Object strain determination, by fringe analysis with phase stepping techniques, is readily achieved. Vibration analysis by heterodyning is also reported.     

Interferometric determination of refraction and dispersion of a birefringent material: numerical procedure

Air, liquid and solid sample interferometric gaps of the same thickness and simultaneously enclosed in a wedge interferometer are used to produce fringes of equal chromatic order. A mica sample of dimensions 2×5 mm² and an immersion liquid of the same refractive index are used. A single shot interferogram containing fringes in the three gaps is sufficient to deduce the needed experimental data. Locations of fringe maxima are introduced in a numerical procedure to retrieve the sample and liquid refractive indices across the visible spectrum. The numerical procedure is based on a simple dispersion function of wavelength and wavenumber. A modified two-term Sellmeier dispersion formula has been used for fitting the experimental data and deducing the needed dispersion parameters.     

An interferometric method for refractive index profiling of planar gradient index waveguides

In this paper, we present an interferometric method to determine the refractive index profile in graded-index planar waveguides. To begin, part of the soda-lime glass substrate surface was coated with aluminum and then was immersed in molten AgNO₃ to fabricate a planar waveguide on the uncoated part of the surface. After the ion exchange, the coating was removed. Then, the sample was polished obliquely along the boundary between the ion-exchanged and non-ion-exchanged regions, to form a wedge. Thereafter, it was placed inside a Mach–Zehnder interferometer. The fringe pattern was analyzed using the well-known Fourier method and the refractive index profile was determined. The sample preparation, data analysis and experimental results are presented.     

Ring fibre laser with interferometer based in long period grating for sensing applications

This work presents two distinct configurations based on phase-wavelength conversion using a ring fibre laser with two different long period gratings interferometer topologies. The sensors are interrogated by analysing the wavelength change of the emission laser, which is directly dependent on the interferometer phase change. The first configuration integrates a Mach-Zehnder interferometer, which is based on a pair of long period gratings and is used as sensing head for bending radius and longitudinal strain measurement. The second configuration, comprehends a Michelson interferometer, which is based on a single LPG and a fibre end mirror and is used as a liquid level sensor or as an optical refractometer.     

Determination of GR-IN optical fibre parameters from transverse interferograms considering the refraction of the incident ray by the fibre

The core–cladding refractive index difference Δn and the index gradient profile parameter of the graded index (GR-IN) optical fibre has been determined. The curve representing the real path, in the core region, due to refraction of the beam and the change in the exit wavefront are considered. Multiple-beam Fizeau fringes and two-beam interference Pluta polarizing microscope, are used for this investigation. A new method, based on a derived mathematical expression, is used with a prepared computer program to estimate the fringe shift inside GR-IN of the fibre core. The estimated and experimental values of the fringe shift, along the core radius, are used to obtain the refractive index profile of the optical fibre. A comparison between the new method and a previous model considering the refraction has been shown. Microinterferograms are given for illustrations.     

Electronic speckle-pattern interferometer using holographic optical elements for vibration measurements

We report a simple, compact electronic speckle-pattern interferometer (ESPI) incorporating holographic optical elements (HOEs) for the study of out-of-plane vibration. Reflection and transmission HOEs provide reference and object beams in the interferometer. The alignment difficulties with conventional ESPI systems are minimized using HOEs. The time-average ESPI subtraction method is used to generate the fringe pattern and remove background speckle noise by introducing a phase shift between consecutive images. The amplitude and phase maps are obtained using path-difference modulation.     

Observation of Gouy phase anomaly with an interferometer

The intensity variation of bright and dark fringes in Young's double slit experiment shows that a light beam propagating through the focus of a lens experiences a phase shift (called the Gouy phase shift [Gouy CR. Acad Sci Paris 1890;110:1251]) with respect to its plane wave counterpart. The additional phase change of π introduced on focusing the light beam and then further propagating it in one arm of the interferometer changes a bright fringe into a dark fringe and vice-versa. We have, thus, made direct visualization of the Gouy phase shift by a simple experiment.     

Unambiguous measurement of surface profile using a Sagnac interferometer with phase feedback

The fringe profiles from a two beam interferometer with phase feedback are almost unambiguous over a range of 2π or more. This allows phase and therefore surface profile to be determined directly from output intensity of an interferometer. In this paper we describe how feedback can be applied to a polarisation Sagnac common path interferometer using a parallel aligned nematic liquid crystal spatial light modulator. Initial experiments have shown that the device is capable of measuring profiles to accuracies greater than one tenth of a wavelength.     

Simultaneous measurement of the phase and group indices and the thickness of transparent plates by low-coherence interferometry

We propose and demonstrate a novel technique for simultaneous measurement of the phase index, n(p) , the group index, n(g) , and the thickness, t , of transparent plates by use of a low-coherence interferometer. The output light from a superluminescent diode is focused upon the front plane of a transparent plate that is used as the sample. The sample stage is subsequently moved until the light is focused upon the rear plane of the plate. Measurement of the stage movement distance and the corresponding optical path difference allows us to determine both n(p) and n(g) . By placing the sample between two glass plates, we measured n(p) , n(g) , and t simultaneously, with an error of 0.3% or less, for nearly 1-mm-thick transparent plates, including glass and electro-optic crystals.     

Fringe-variable interferometry used for the detection of domain structure in LiTaO3

We design a fringe-variable Jamin interferometer to detect the reversed domain of a ferroelectric crystal in real time. In contrast to the Mach-Zehnder interferometer, this setup is compact and tunable in fringe frequency. In experiments, we use it to detect the partly reversed domains of Stoichiometric LiTaO₃ (SLT). Selecting the proper fringe cycle, we can estimate the average phase shift between the original and reversed domains through eyeballing. Furthermore, in the interferogram processing, we use the FFT methods to reconstruct the phase according to the original fringe and deformed fringe, and obtain the phase variance at the domain wall. The results show that the average phase variance at the domain wall is in good agreement with the theoretical value.     

基于迈克尔逊干涉的傅里叶变换散斑形貌测量技术

提出了电子散斑干涉载频调制测量物体形貌的方法。采用典型的迈克尔逊干涉光路,将物体偏转一微小角度（等效为物面与参考面间形成空气楔）产生等厚干涉,可在物体的表面引入包含物体高度信息的载波干涉条纹。用CCD采集该载波条纹图,利用傅里叶变换法可解调出物体高度的位相信息,从而实现物体的形貌测量。介绍了电子散斑干涉载频调制测量物体形貌的原理,并进行了实物测量,给出了实验结果。由于该方法采用散斑干涉方法测量物体形貌,所以具有灵敏度高的优点。