An alternative method for measuring the index profile of the GRIN lens is proposed based on Fresnel's equations and the
common-path phase-shifting interfermetry. A light beam composing of the right- and the left- circularly polarized
components is obliquely incident on the tested GRIN lens and the reflected light passes through an analyzer. The light is
collected and imaged by an imaging lens to a CMOS camera. Four interferograms under different additional phases are
taken and Carre's phase-shifting method is used to calculate the full-field phase distribution. Next, the estimated data are
substituted into the special equations derived from the Fresnel's equations, and the index profile of the GRIN lens can be
obtained. Its validity has been demonstrated. It has both merits of the common-path interferemetry and the phase-shifting
interferemetry.
The processes to derive the associated phase of an interference signal from the data of a series of recorded frames are performed, and we find that the sampling frequency being lower than the Nyquist sampling rate can also be applied to the full-field heterodyne interferometry. Two optimal sampling conditions for a commonly used CCD camera are proposed based on the relation between the heterodyne frequency and the contrast of the interference signal under the condition that the phase error is set to be 0.05 deg.
A simple method for measuring a step-height sample is presented with the heterodyne central fringe identification
technique and a precision translation stage. This method can accurately point out the zero optical path difference position
such that the optical path lengths of two arms of the interferometer are absolutely equivalent. Thus, the two surfaces of
the step-height sample can be identified sequentially with the translation stage. The displacement of the translation stage
is equal to the step-height of the test sample. The feasibility of the technique is demonstrated. The measurable range is
not limited by the coherence length of the light source. The measurement accuracy depends on the uncertainties of the
heterodyne central fringe identification method and the translation stage. In our setup, we have a 100 mm measurable
range and a 4 nm uncertainty. The wavelength stability of the light source has a minor effect on the measurement.
Based on the Fresnel's equations and the phase-shifting method, an alternative method for measuring the refractive index
distribution of a GRIN lens is presented. A linearly/circularly polarized light in order enters a modified Twyman-Green
interferometer, in which an electro-optical modulator is used as a phase shifter. In the interferometer, the light beam is
divided by a beam-splitter into two beams, a reference beam and a test beam. After they are reflected by a plane mirror
and the tested GRIN lens, respectively, they are combined together and pass through an analyzer. The analyzer extracts
the same polarized components to interfere each other, and the
full-field interference signals produced by the
components of the s- and the p-polarizations can be obtained. The full-field interference signals are taken by a CMOS
camera. The phase differences can be obtained by using the four-step phase-shifting interferometric method. Substituting
these two groups of data into special equations derived from Fresnel equations, and the two-dimensional refractive index
distribution of the GRIN lens can be calculated. Its validity is demonstrated and has some merits such as simple optical
configuration, easy operation and high resolution.
A novel method for full-field absolute phase measurements in the heterodyne interferometer with an electro-optic
modulator is proposed in this paper. Instead of the commonly-used half-wave voltage to drive the electro-optic
modulator, a saw-tooth voltage signal with the amplitude being lower than its half-wave voltage is used. The interference
signals become a group of periodical sinusoidal segments. The initial phase of each sinusoidal segment depends on the
phase difference induced by the test sample. In real measurements, each segment is taken by a fast camera and becomes
discrete digital points. After a series of operations, the starting point of the sampled sinusoidal segment can be
determined accurately. Next, the period of the sampled sinusoidal segments is lengthened and they can be modified to a
continuous sinusoidal wave by using a least-square sine fitting algorithm. The initial phase of the continuous sinusoidal
wave can also be estimated. Subtracting the characteristic phase of the modulator from the initial phase, the absolute
phase measured at the pixel can be obtained without the conventional reference signals. These operations are applied to
other pixels, and the full-field absolute phase measurements can be achieved. The phase retardation of a quarter-wave
plate is measured to show the validity of this method.
Based on the Fresnel's equations and the heterodyne interferometry, an alternative method for measuring the refractive
index distribution of a GRIN lens is presented. A light coming from the heterodyne light source passes through a quarterwave
plate and is incident on the tested GRIN lens. The reflected light passes through an analyzer and an imaging lens;
finally it enters a CMOS camera. The interference signals produced by the components of the s- and the p-polarizations
are recorded and they are sent to a personal computer to be analyzed. In order to measure the absolute phases of the
interference signals accurately, a special condition is chosen. Then, the interference signals become a group of periodic
sinusoidal segments, and each segment has an initial phase ψ with the information of the refractive index. Consequently,
the estimated data of ψ are substituted into the special equations derived from Fresnel's equations, and the refractive
index distribution of the GRIN lens can be obtained. Because of its common-path optical configuration, this method has
both merits of the common-path interferometry and the heterodyne interferometry. In addition, the phase can be
measured without reference signals.
A collimated heterodyne light enters a modified Linnik microscope, and the full-field interference signals are taken by a fast CMOS camera. The sampling intensities recorded at each pixel are fitted to derive a sinusoidal signal, and its phase can be obtained. Next, the 2-D phase unwrapping technique is applied to derive the 2-D phase distribution. Then, Ingelstam's formula is used to calculate the height distribution. Last, the height distribution is filtered with the Gaussian filter, the roughness topography and its average roughness can be obtained and its validity is demonstrated.
KEYWORDS: Heterodyning, Interferometry, Digital cameras, Signal processing, Fourier transforms, Optical engineering, Cameras, Digital signal processing, Visibility, Transform theory
An area scan digital camera is used to record the full-field heterodyne interference signals, and the processes to derive the associated phases from the data of the recorded frames under a convenient condition are described. By calculating the possible errors under several different cases, an optimal condition to get better results is proposed.
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