White light interferometry

The white-light interferometry ( WLI ) is a contactless optical measuring method, which utilizes the interference broadband light ( white light) and thus 3D profile measurements of structures having dimensions of between a few centimeters and a few micrometers permitted.

Signal generation

By varying the path length difference Az between the two arms of the interferometer, we obtain in the interferometer output signal, as shown in the picture on the right as curve ( 3). The intensity of the signal decreases significantly when the amount of? Z is substantially greater than the coherence length. The exact form of the signal is dependent on the average wavelength and the spectrum as well as the coherence length of the light source used.

Construction

The measured object is placed in one arm of the white light interferometer. The light source is collected by a condenser lens, is coupled into the beam path and separated by a beam splitter into reference and measurement beam. A beam is reflected from the reference mirror and the other beam is reflected or scattered from the surface of the measurement object. The returning beam is passed from the beam splitter to the CCD sensor and form depending on the position of the object for each individual pixel, an interference signal. The Korrelogrammbreite corresponds, as explained below, the coherence length of the light, and therefore depends on the spectral width of the light source.

Operation

A rough surface of the object has a speckle pattern results in that interferes with the light from the reference plane in the detector plane. Each individual has a random speckle phase. The phase remains approximately constant within a speckle. Therefore appears to the camera pixel, an interference when the different optical path lengths of the two arms by less than half the coherence length of the light source. Each pixel of the camera sensor scans a typical white light correlogram (interference signal) when the length of the reference or of the measuring arm is changed with a positioning unit. The interference signal of a pixel has a maximum modulation, when the optical path length of the light incident on the pixel, for the reference and measurement beam is exactly the same. Therefore corresponds to the z- value of the point on the surface which is imaged on that pixel, the Z value of the positioning unit, when the modulation of the correlogram is at its maximum. One can derive a matrix of height values ​​of the surface of the object by using the Z- values ​​of the positioning unit can be determined for each individual camera pixels, in which the modulation is at a maximum. The height uncertainty depends primarily on the roughness of the measured surface. For smooth surfaces, the measurement accuracy is limited by the accuracy of the positioning. The lateral positions of the altitude values ​​will depend on the corresponding object point which is imaged on the array of camera pixels. The x-coordinates described together with the corresponding y- coordinates, the geometric shape of the measured object.

Areas of application

The fact that only in the Reconciled object and reference arm interference occur, can be exploited with appropriate equipment to measure distances ( white light interferometry ). As examples, here are the topographical coherence radar and the volumetric optical coherence tomography to call. Due to dispersion, the synchronicity of the individual wavelengths or frequencies is poor or distorted - the optical path length is a function of frequency. Since the white light interferometer is sensitive on the WLI can also be used for dispersion measurement. Since the spectrum is linked via the Fourier series with its autocorrelation, one can with an interferogram, which is taken over the entire coherence length, also perform spectroscopic measurements. One of the white-light interferometry addition to a procedure is the nonlinear autocorrelation, are measured at the waveforms of optical pulses.

White light interferometer in microscopes

To make microscopic structures visible, the interferometer must be combined with the optical construction of a microscope. The structure is similar to a standard optical microscope. The only differences are an interferometric objective lens and a precise positioning ( a piezo - electric actuator ) to process the interference objective vertical. When the microscope objective images, the measurement object at infinity, the optical magnification of the image does not depend on the CCD chip on the distance between the tube lens and objective lens. The interference objective is the most important part of a Interferometermikroskops. There are several types of lenses. With a Mirau objective of the reference beam is reflected by a beam splitter back in the direction of the front lens back. On the front lens is a tiny mirror is the same size as the illuminated surface on the object being measured. At high magnifications, the mirror is therefore so small that its shadowing can be neglected. By moving the interference objective, the length of the measuring arm is changed. The interference signal of a pixel has a maximum modulation, when the optical path length of the light incident on the pixel, for the reference and measurement beam is exactly the same. Therefore corresponds to the z- value of the point on the surface which is imaged on that pixel, the Z value of the positioning unit, when the modulation of the correlogram is at its maximum.

Relationship between spectral width and coherence length

As described above, defines the z- value of the positioning unit, wherein the modulation of the interference signal for a particular pixel is a maximum height value for that pixel. Therefore, quality and shape of the correlogram have a great influence on the resolution and accuracy of the system. The main parameters of the light source are their wavelength and their coherence length. The coherence length defines the Korrelogrammbreite. The coherence length, in turn, is related to the spectral width of the light source. Therefore Korrelogrammbreite depends on the spectral width of the light source. The picture shows the spectral density function for a Gaussian spectrum is presented, which for example is a good approximation for an LED. It can be seen that the corresponding intensity modulation is only in the area around the position z0 of where the reference and measurement beam have the same length and overlap coherently. The z- range of the positioning unit, in which the envelope of the intensity modulation is more than 1 / e of the maximum value, determines the Korrelogrammbreite. The Korrelogrammbreite corresponds to the coherence length as the difference in optical path length is twice the length difference between the reference and measuring arm of the interferometer. The ratio between Korrelogrammbreite coherence length and spectral width is calculated in the following for the example of a Gaussian spectrum.

Coherence length and spectral width of a Gaussian spectrum

The normalized spectral density function with Equation 1:

Defining the effective 1/e-Bandbreite and the mean frequency. According to the general Wiener- Khinchine theorem, the auto-correlation function of the light field is given by the Fourier transform of the spectral density, see Equation 2,

Which is measured by interference of the light fields of the reference and the measurement beam. Taking into account the fact that the intensities are the same in both interferometer arms, the result for the intensity of which can be observed on the screen, in equation 3:

Context specified. Here is I0 = Iobj Iref with Iobj and Iref as respective intensities from the sensor or at the reference arm. The center frequency can be formulated on the basis of the central wavelength, and the effective bandwidth with reference to the coherence length. From equations 2 and 3 follows the intensity of the screen in equation 4:

Represented context. It must be noted that, where c is the speed of light. Consequently, equation 4 describes the correlogram, as shown in the picture. It can be seen that the intensity distribution is formed by a Gaussian envelope and a periodic modulation with a period. For each pixel of the correlation curve is sampled at a certain pitch of the z displacement. In addition, phase shifts lead to the reflective surface of the object, inaccuracies in the positioning, distribution differences between the arms of a real interferometer, reflections from surfaces other than the object surface and noise in the camera sensor to a deformed correlogram. Therefore, a real correlogram of the result from Equation 4 may be different, but the result illustrates the strong dependence of the correlogram of the two parameters wavelength and coherence length of the light source.

The calculation of the envelope maximum

The envelope is described by the exponential term in Equation 4, see equation 5.

The software calculates the envelope of the Korrelogrammdaten. The principle of the envelope curve is to remove the cosine term in Equation 4. Using a Hilbert transform of the cosine term is converted into a sine term. The envelope is obtained by summing the magnitude of the cosine and sine modulated correlation curves, see equation 6.

For the calculation of the envelope maximum two slightly different algorithms. The first algorithm to evaluate the envelope of the correlogram. The z value is derived from the maximum of the envelope. The second algorithm also evaluates the phase. Using automation interfaces (eg macros), each of the two algorithms. The uncertainty of the calculation of the envelope maximum depends on: the coherence length, the sampling step size of the correlogram, deviations from the z- values ​​against predefined values ​​(eg due to vibrations ), the contrast and the roughness of the surface. The best results are obtained with a short coherence length, a small sampling step, good vibration isolation, high contrast, and a smooth surface.