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White light interferometry is a well-developed and very old technique for optical measurements. The thesis describes the design of a vertical scan interferometer system to study the surface profile of surfaces down to nanometers. The desired properties of the system are its simplicity, portability and compact size, making it suitable for use in general labs and for educational purposes. By acquiring a sequence of images of the deformed fringe pattern, the change in the surface topology can be observed, giving greater understanding of the surface profile.
The principle behind the system is coherence peak sensing where the resulting fringe pattern of the object gets changed in accordance with its surface topology. To accomplish this individual component for the interferometer were studied and a prototype was built in the lab. A series of experiments were performed which validate the working of the system. The results obtained were promising except for the presence of excessive amount of speckle noise in the output. The results of the validation which are produced in the report give the accuracy of the system. The output from the prototype interferometer is processed by MATLAB to decode the surface profile of the object under measurement. The design of the prototype is also discussed. Possible application of this device for sensing the surface profile of a cylindrical object is also put forward.
Even-though the white light interferometer is more common, making them simple and cost effective will be more advantageous for the whole research community.
1 Introduction 1
Advantages of using a white light source 13
Interference Objectives 14
List of Figures
White light interferometry is a widely used technique for measurement of surface profile of objects over large areas. Systems which use this technique can measure areas which are equal to the field of view of the instrument. Interferometric optical profilers are widely used to study surface profiles due to the high measurement accuracy, non-contact, rapid data acquisition and analysis.
Even though the conventional interferometers are extensively used they are built for ruggedness and have many features and different techniques of measurement integrated which are not effectively used by all. Moreover they have features for automated adjustment which can be done manually to some extent.
In this project we try to replicate the white light interferometer by constructing a proto type in the lab which uses white interferometry measurement technique. With measurement results from the constructed interferometer the feasibility for practical usage is discussed.
The report also presents the use of the technology on surfaces and analyzes the accuracy of the prototype compared with the commercial interferometer (MicroXAM).
Aim of this project is to construct a coherence peak white light interferometer and define its accuracy. To achieve this goal this project includes the design, construction, experiments and validation of the interferometer system.
Light Interferometry system
Interferometry is the technique of analyzing the properties of two or more waves by studying the pattern of interference resulting from their superposition. The instrument that uses this concept is known as an interferometer.
Superposition of waves
The principle of superposition combines separate waves so that the result of their combination has a meaningful information about the original state of the waves. The resulting interference pattern represents the difference in phase between the two waves of same frequency. The waves which are in phase will result in constructive interference while waves which are out of phase will result in a destructive interference. Common interferometers use light as diagnostic element to examine the surface profile.
[ Pal, Bishnu P. (1992). Fundamentals of Fibre Optics in Telecommunication and Sensor Systems. New Delhi: New Age International. p. 663 (read section 3). ISBN 8122404693.]
Normally a single beam of coherent light will be divided into two identical beams by a partial mirror or a grating. Each of these beams will travel a different path, until they are recombined before arriving at the detector. The optical path difference between the beams creates a corresponding phase difference between them. It is this phase difference between the initially identical waves that creates the interference. The figure below show the interference of waves with zero phase difference and waves which are completely out of phase.
If a single beam has been divided into two then the phase difference is diagnostic of the changes in phase along the paths. This can be a change in the path length itself or a change in the refractive index along the path. The resulting interferogram might look like the one below.
To obtain a fringe pattern the two sources need to be in phase or should have a constant relative phase between them. This property of waves always marching together is called coherence and it is a fundamental requirement for the sources producing interference. The coherence of the interfering beams determines the fringe contrast and sharpness. Common sources produce light that is a mix of photon wavetrains exhibiting different frequency.
A fixed phase relationship between the waves at different locations or at different times defines coherence.
At each point of the light in space there is a net field that oscillates nicely resembling a sinusoidal wave before it randomly changes phase and this measure is known as temporal coherence. The average time interval during which the light wave oscillates in a predictable way we have already designated as the coherence time of the radiation. The longer the coherence time, the greater the temporal coherence of the source. The corresponding spatial extent over which the lightwave oscillates in a regular, predictable way is the coherence length (Ï„c).
The above equation shows that the faster a wave decorrelates (and hence the smaller Ï„c is) the larger the range of frequencies Î”f the wave contains. The figure below shows the two wave of slightly different frequencies having a temporal coherence for time Ï„c.
Simply, the temporal coherence is a manifestation of the spectral purity of the light source. By contrast, if the light were ideally monochromatic, the wave would be a perfect sinusoid with an infinite coherence length. [Eugene Hecht, Optics, Fourth edition, ISBN 0-321-18878-0]
Temporal coherence is a strong correlation between the disturbances at one location but at different times. In white light interferometry the temporal coherence of the white light defines the extent to which the fringes are visible in the field of view of the focused surface.
For point source of light the disturbances at the points on its wavefront is completely correlated and so these laterally separated points are in-phase and stay in-phase. So if we consider a point source which changes the frequency moment to moment, even then the waves exhibit complete spatial coherence.
By contrast, suppose the source is broad, that is, composed of many widely spaced point sources, the disturbance at laterally spaced point will be completely uncorrelated depending on the size of the source.
Spatial coherence therefore means a strong correlation in phase between the waves at different locations across the beam profile.
Given below are examples to better understand the types of coherence.
A laser beam with spatial and temporal coherence
A laser beam with spatial coherence, but poor temporal coherence
A laser beam with poor spatial coherence, but good temporal coherence
Theory of Optical Interference
When a light from a source is split into two beams, then the inherent variation in the two beams are generally correlated, and the beams are said to be completely or partially coherent depending on the existing correlation. In light beams from two independent sources, the phase functions are usually uncorrelated and such beams are called incoherent beams. When coherent waves superpose, they produce visible interference effects because their amplitudes can combine, whereas for the incoherent waves, their intensities combine. Interference produced by incoherent waves varies too rapidly in time to be practically observed.
When two mutually coherent beams pass through a point, we can observe the phenomena of interference between the wavefront. The medium at that point is subjected to the total effect of the superposition of the two vibrations, and under certain conditions, this superposition results in stationary waves, known as interference fringes.
Consider the superposition of two monochromatic plane waves U1 and U2 of the same frequency and with different complex amplitudes. The result is a monochromatic wave of the same frequency and the complex amplitude is the sum of the individual amplitudes, i.e.
Expressing the plane waves in terms of their intensities, we get
Thus we have,
where the asterisk denotes complex conjugation.
The above equation is known as the interference equation, and the term
is known as the interference term. At different points in space, the resultant irradiance can be greater, less than or equal to I1 + I2, depending on the value of the interference term, i.e. depending on Ï†. Irradiance maxima occur for Ï† = 2Ï€m and minima occur for Ï† = (2m+1)Ï€. The dark and light zones that would be seen on a screen placed in the region of interference are known as interference fringes.
An interferometer is, in the broadest sense, a device that generates interference fringes. Interferometers can basically be classified into two types: wavefront splitting interferometers and amplitude splitting interferometers.
Wavefront splitting interferometers recombine two different parts of a wavefront to produce fringes. The earliest experimental arrangement for demonstrating the interference of light was Young's experiment, which employed a double slit to obtain two sources of light. The principle of double slit experiment is illustrated below.
Young's Double Slit Experiment
The light from a monochromatic source S falls on the two pinholes S1 and S2, which are close together and equidistant from S. The pinholes act as secondary monochromatic sources, which are in phase when SS1=SS2, and the beams from these sources are superposed in the region beyond the pinholes. An interference pattern can be observed on the screen.
Amplitude-splitting interferometers, on the other hand, divide a wavefront into two beams (splitting the amplitude), which propagate through separate paths and are then recombined. Typically, beam splitters are used for splitting and recombining wavefront amplitudes.
One of the most important interferometers based on amplitude-splitting technique is the Michelson's interferometer. Its basic arrangement is shown below.
Basic Principle of Michelson and Twyman Interferometers
It consists of a partially silvered mirror B, which acts as a beam splitter, dividing the incident beam coming from S into two beams of equal intensity, one reflected and the other transmitted. These beams strike the reference mirror M1 and the test surface M2 at normal incidence and are reflected back and meet at the beam splitter and combine to create interference patterns, which can be seen at the point of observation O.
In the above figure, the interferometer is set up with a point source, which would illuminate only a small part of the field of view. Hence this configuration is used only after further modification. If an extended source (a source with uniform surface brightness) is used in Michelson's mirror arrangement, the form is as shown in figure below. In this arrangement utilizing an extended source, the rays, which reach the point of observation, leave the mirrors at various angles.
Michelson interferometer with an extended source
A Michelson's interferometer modified to work in collimated light is a Twyman-Green interferometer. In the Twyman-Green interferometer, a point source is placed at the focus of a lens so that a plane wave front traverses the mirror system. On reaching the second lens, the wave front is again made spherical and converges on the observation screen. The Twyman-Green interferometer is shown below. In its simplest form, it can be used to test plane mirrors, plane-parallel windows, or prisms. The figure below illustrates its use in testing of a prism.
The Twyman-Green interferometer, as used to test a prism
White Light Interferometry
For an interferometer to be a true white light achromatic interferometer two conditions need to be satisfied.
First, the position of the zero order interference fringes must be independent of wavelength. Second, the spacing of the interference fringes must be independent of wavelength. That is, the position of all interference fringes, independent of order number, is independent of wavelength. Generally, in a white light interferometer only the first condition is satisfied and we do not have a truly achromatic interferometer.
The basic technique involves splitting an optical beam from the same source into two separate beams - one of the beams is passed through, or reflected from, the object to be measured whilst the other beam (the reference) follows a known and constant optical path. The same basic principle can be used in a microscope arrangement as shown below.
Schematic diagram of an interference microscope.
Here a light source provides a beam which is passed through a filter and reflected by the upper beam splitter (acting essentially as a mirror at this stage) down to the objective lens. The lower beam splitter in the objective lens creates and combines the light beams reflected from the sample surface and the reference.
This creates an interference pattern called an interferogram which is magnified by the microscope optics and finally imaged by the CCD camera. This static fringe image would show differences in distance apart of the reference and sample - essentially revealing local 'bow and warp' of the sample. However if the objective lens is moved vertically the path length between sample and beam splitter changes and creates a series of moving interference fringes which will be detected by the camera. The aim is to establish the point at which maximum constructive interference occurs (i.e. at which the image is brightest). The figure shown below gives the irradiance at a single sample point as the sample is translated through focus varying the Optical Path Distance (OPD).
Once this is achieved, provided the vertical movement of the lens can be accurately tracked, it is possible to create a 3D map of the sample surface by measuring the position of the lens required to produce the brightest image at each point on the CCD array. This would normally be carried out using a monochromatic light source - however there could be several different positions at which a maximum in the signal would occur. By using multiple wavelengths (and white light is the ultimate case) it is possible to set the system up so that there is only one point at which this maximum occurs. The only limit on the achievable height resolution is set by how well the measurement algorithm can define the maximum brightness (and thus the surface position) as the objective lens is scanned vertically using a piezo-electric drive. Each pixel of the CCD array effectively acts as an individual interferometer and thus builds up a very accurate map of the surface. [1. M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, "An Application of Interference Microscopy to
Integrated Circuit Inspection and Metrology," Proc. SPIE, 775, 233-247, 1987. 2. G. S. Kino and S. Chim, "Mirau Correlation Microscope," Appl. Opt. 29, pp. 3775-3783, 1990. 3. T. Dresel, G. Hausler, and H. Venzke, "Three-dimensional sensing of rough surfaces by coherence radar," Appl. Opt. 31(7):919-3925, 1992. 4. P. J. Caber, "An Interferometric Profiler for Rough Surfaces," Appl. Opt. 32 (19), pp. 3438-3441,
1993. 5. James C. Wyant, "Computerized interferometric measurement of surface microstructure," Proc. SPIE 2576, pp.122-130, 1995.]
The processing of the interferogram using modern computerized technique has helped to increase the accuracy and the dynamic range of surface profile measurements. In white light interferometry the point of highest fringe contrast, the coherence peak, is the most distinctive feature of the broadband interferogram. Most optical profilers based on white light process the interference data in order to determine the position of the coherence peak [ J. C. Wyant, Laser Focus World (September), 131-135 (1993) ]. These profilers differ mostly in the way they detect and process the fringe-contrast envelope. Balasubramanian makes the reference mirror oscillate with a PZT to calculate the fringe contrast, and then repeats the calculation for a succession of points in space in order to trace out the contrast envelope [ N. Balasubramanian, U.S. Patent #4,340,306 (Jul. 20, 1982) ]. Caber has developed more flexible and accurate formulas for calculating the fringe contrast, and also with more powerful methods of estimating the fringe contrast peak with sparse data using curve fitting [P. J. Caber, Appl. Opt. 32(19), 3438-3441 (1993) ]. Kino and Chim transform their data to the spatial-frequency domain, eliminate the negative frequencies, center the positive frequency packet around zero and then transform back to the original data domain to reveal a carrier-suppressed envelope for further processing [Gordon S. Kino and Stanley S.C. Chim, Appl. Opt. 29(26), 3775-3783 (1990)].
Advantages of using a white light source
Lasers are generally used as the light source for the interferometer system because it is easy to obtain interference fringes due to their long coherence length regardless of the path difference between the two interfering beams. However, there are several advantages of using the white light as the source in interferometric optical profilers.
The first advantage is that the noise due to spurious interference fringes is avoided because the coherence length of white light is very short and interference can be obtained only when the path length are within a few microns or less. [Advances in interferometric optical profilers]. Thus, even though the spurious reflections still exist in the white light interferometer they don't produce fringes which can add to the noise.
For an optical profiler, it is very important that the sample is in focus or else the measurements will be incorrect. It is very difficult to determine the focus on a smooth surface due to the absence of structures. A major advantage here when using white light source is the presence of interference fringes which precisely defines focus. The maximum contrast fringes are obtained only when the path lengths are exactly matched. Hence moving through the sample looking for maximum contrast fringes the correct focus can be obtained. [Cohen, D.K., E.R. Cochran, and J.D. Ayres (1989). Development of an automatic focusing mechanism for an interference microscope. Proc. SPIE, 1164].
The third advantage of using a white light is that it can be filtered to produce different wavelengths for measurement on the same sample. Here, the multiple wavelength techniques can be used to measure steps or surfaces having steep slopes [Creath. K. (1987). Step height measurement using two-wavelength phase-shifting interferometry. Appl. Opt. 26, 2810-2816]. The surface height change allowed in the measurement using a single wavelength is limited to one-quarter wavelength. By performing the measurement at two wavelengths, Î»1 and Î»2 and subtracting the two measurements, the limitation in the height between two adjacent detector points is now one-quarter of Î»eq, where
The measurement is essentially tested at a synthesized equivalent wavelength, Î»eq. While this technique increases the dynamic range of measurement, it decreases the precision by the ratio of Î»eq/ Î». The precision can be regained by using the equivalent wavelength results to correct the 2Ï€ ambiguities of the single wavelength data. Thus a large dynamic range is obtained maintaining the precision of single wavelength data. [Advances in interferometric optical profilers]
An important component in an interferometer optical profiler is the microscope interference objective. Due to wide range of optical profile measurements, no single type of interferometer objective can be used.
The Michelson interferometer is used for low magnifications such as 1.5X, 2.5X and 5X. Advantage of this interferometer is that only a single objective is used and hence first order aberrations do not contribute to errors in measurements. Only long working distance objectives can be used because the beam splitter must be placed between the objective and sample.
The Mirau interferometer is used for medium magnifications such as 10X, 20X and 40X. It also has the advantage that only a single objective is needed. Even though some optics needs to be placed between the objective and the sample, not much space is used when compared with Michelson. The disadvantage of the Mirau is that a central obstruction is present in the system but this is not a problem for medium magnifications since the size of the obstruction is equal to the field of view of the sample.
The Linnik interferometer is used for high magnifications such as 100X and 200X. The disadvantage is that two matched objectives are needed. However, since no optics is needed between the objective and the sample, large numerical aperture and short working distance measurement can be performed. Due to large numerical aperture the optical resolution of the system is very high. The optical resolution is given by 0.61Î»/(2NA), where NA is the numerical aperture.
The Mirau interferometric objective is most widely used because only one objective is required for measurement and it offers medium magnification which is right for most applications. Another advantage of this objective is that adjusting the distance for the reference surface is not necessary since it is firmly fixed inside the objective enclosure and hence can be used easily.
A 10X Mirau objective is used in our white light interferometer prototype for medium magnification.
White-Light Interferometry Equation
In the white-light interferometry, the detected light intensity I is a function of the reference mirror scanning position z and the height of the reflectance layer of the sample z0,
where, Î»0 is the center wavelength of the light source, and A(z) is the envelope function which is normally a Gaussian.
By substituting the above equation, I(z) is expressed in the simple form
In 3D shape measurements using white-light interferometry, the aim is to obtain the parameter z0 from the detected light intensity I(z).
MicroXAM interferometer from ADE Phase Shift
MicroXAM is a white light scanning interferometer which can perform surface profile measurements using coherence peak scanning and phase measurements techniques. For coherence peak scanning it uses a filtered white light from a halogen lamp. A narrow bandwidth green light is used for phase shift measurements.
First the object to be measured is tested under this interferometer. A software application known as MapView is used to decode the surface profile information from the interferogram. Then the map files that are obtained from MapView software are given into another program called MountainsMap for the generation of the actual 3d surface visualization.
The scanning process takes a maximum of 30 seconds and the device can scan for a range of 100µm using coherence peak scanning.
Prototype Coherence Peak Scanning Interferometer
The construction of the device is made as simple as possible while maintaining high quality components for imaging. The basic differences in this new design is that the components required for coherence peak scanning interferometry is only used and also the piezo motors in the scanning direction is reduced to one. The new design as shown below uses a white LED as the light source. The LED and the beam splitter are mounted together on the optical axis. The light from the source is directed towards the surface by the first beam splitter.
The light is then divided by the beam splitter in the Mirau objective as shown below. One goes to the reference surface and the other beam is focused on the object under examination.
The resulting pattern is captured by the camera. The piezo motor is used to adjust the focus very precisely in nanometers. The piezo motor is mounted on a motorized rotational stage to align the optical axis perpendicular to the object under measurement.
A 10X Nikon Interferometry objective is used for the setup. The objective has a numerical aperture of 0.3 and a focal length of 20mm. The working distance of the objective is 7.4mm and field of view is 0.88mm x 0.66mm.
A monochromatic CCD camera Basler piA2400-12gm is used for capturing the images. The maximum resolution of this camera is 2456x2058 and is interfaced by a Gigabit Ethernet port.
Rotor table controller
The rotational table is interfaced to the computer through a compatible controller BSC101 from Thorlabs. The device can be driven through ActiveX controls embedded in the Virtual Interface of LabView. The rotor table is used for aligning the optical axis perpendicular to the object surface.
Piezo drive system
The most important device for the vertical scanning is the Piezo drive system used for precise movements in the order of nanometers. The total range of the system is 20µm. The system is used in closed loop for accurate movement. Controller BPC 201 from Thorlabs is used for driving the system and also provides interface for control from LabView.
Programming the Scanning System
Labview was chosen for developing a GUI for automatic scanning system. ActiveX controls required for interfacing Labview with the controller were available from Thorlabs which makes the use of Labview more convenient. A screenshot of the programmed control interface is shown below.
Parameters required for the scan can be specified before recording the measurements. The system is designed to capture a frame for very step it takes while scanning. The video is saved as an uncompressed AVI file.
The block diagram for the virtual instrument is shown below.
The brightest fringe is positioned at halfway through the scan length before starting the measurements.
The most important part of the whole interferometer system is the processing capability of the system in order to decode the actual surface profile from the captured interferogram. The process of post processing the interferogram should have the ability to reject noise and resolve phase ambiguities. In a coherence peak scanning system the phase ambiguity does not affect the processing since the maximum intensity for a pixel is obtained only when the path lengths are same.
For post processing Matlab from Mathworks is used. The ability to process images as a 2D matrix makes Matlab the right candidate for programming with ease. The program employs the process of splitting the individual frames of the captured video and recognizing the frame at which the intensity was highest for a given pixel. Since there is a direct relation between the frame number and the relative position of the focus while scanning, each pixel is located in 3D space for reconstruction of the surface profile.
The algorithm for post processing of the interferogram in Matlab is shown below.
Comparison of surface profile measurements: