Interferometric instruments with dispersion introduced in the reference arm have previously been created, as the controlled dispersion can be used to generate a signal that contains a clearly identifiable point, the location of which relates to the position of the scattering surface in the measurement arm. In the following, we illustrate that the linear approximations that have been used previously can lead to significant errors, and that second-order terms need to be included in order to correct this. These corrections are vital if these instruments are to be used for metrological applications.
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Dispersed reference interferometry (DRI) is a category of spectral interferometry in which dispersion is purposely introduced into one arm of an interferometer. This methodology differs from the majority of broadband interferometric methods, such as white light interferometry, where dispersion must be minimized by design or compensated for in order to attain high measurement accuracy . The analysis of spectral interferograms produced by dispersion in interferometric systems is useful for such applications as the characterization of dispersive fiber , thickness measurement of transparent samples , and chirped mirror characterization . In these instances, the dispersion is produced by the element under test and the analysis focuses on determining the dispersion parameters from the interferograms. In DRI, the primary aim is to measure optical path length and thus displacement or surface topography; as such, the dispersion is applied through a suitable mechanism located in the reference arm of the instrument, for instance, by a pair of diffraction gratings. Axial (height) resolution approaching a nanometer has been reported in surface topography measurement applications using DRI .
The dispersion introduced in the reference arm leads to the generation of a characteristic chirped spectral interferogram, with clear changes to the signal as the location of the object varies. Previous work surrounding the analysis of this signal [6,7] has used a linear approximation to describe an aspect of the elements in the reference arm, either the variation in refractive index of a dispersive material or the variation in path length that light of different wavelength takes. In this paper, we look at the effect that the linear approximation has, showing that the change in the signal as the object being measured moves is significantly different from the result obtained when the full expressions are used. This is because, even though the refractive index change or path length change may be close to linear, the short wavelength of the light means these small changes are significant in describing the output of the signal. This means if the signal is analyzed using the results from the linear approximation to calculate the location of the measurement object, then significant errors can arise. As such, we present the exact solutions to the equations, and it is these that need to be used if high-accuracy measurements are to be obtained over large measurement ranges. While we limit our attention to DRI in this work, the ideas presented here are applicable to any system where dispersion is introduced in a similar manner and can provide a basis for determining when linear approximations are suitable for modeling the system in setups such as that in Ref. .
The paper is laid out as follows: In Section 2, we examine the instrument introduced in Ref. , which takes the form of a Michelson interferometer with a dispersive element introduced into the reference arm. We examine the validity of the linear approximation used when the dispersive element takes the form of a glass plate and show that second-order terms play a significant role in capturing the behavior of the system. In Section 3, we look at the case of an instrument where the dispersion has been introduced geometrically via a pair of diffraction gratings in the reference arm [5,7]. Again, we demonstrate that the linear approximation is insufficient to accurately model the response of the system and derive more accurate expressions. In Section 4, we present experimental results from an apparatus of the form analyzed in Section 3 to demonstrate the linearity improvement resulting from analysis using the newly derived expressions.
2. DISPERSION VIA A GLASS PLATE
The first experimental setup to be considered was introduced by Pavlicek et al.  and is shown in Fig. 1. It takes the form of a Michelson interferometer with a dispersive element (glass plate) in the reference arm. A broadband collimated light source is incident on the beam splitter before one portion passes along the measurement arm, through a lens, before being reflected from the surface whose position is being measured. The returning light is collected and combined with the light that has passed down the reference arm, containing the dispersive element, as well as a reference mirror and lens. The resulting interfering light is passed into a spectrometer.
The analysis of the system behavior was carried out using a linear approximation applied to the refractive index of the glass plate that takes the form , where
The measured intensity at each wavenumber was given as
A. Linear Approximation6].
B. Removing the Linear Approximation3) reveals that is dependent upon the product of and . As is of the order of , and there is no equivalently large term in the coefficient of in the first term, it is very likely that despite the relative sizes of and , the second term will strongly affect the behavior of . As is equal to , it can be seen that the first term in Eq. (8) is equal to Eq. (6) at the point , though it will only be equal to it elsewhere if the first derivative of the refractive index is constant.
C. Example for Typical Glass
To examine if the second-order terms are significant for a typical case, we consider a glass plate made of BK7, representing its refractive index by a two-pole Sellmeier equation  of the form,9].
The calculation of and from Eq. (9) is straightforward but yields unwieldy expressions, which are not included for brevity. However, for BK7 it is found that the magnitudes of the two terms in Eq. (8) are of the same order of magnitude; thus the linear approximation is not sufficient for this material. As the thickness of the dispersive element, , just scales both results by the same amount, it has been set to 1.The solid line in Fig. 2(a) shows the value of needed for the minima in to occur at the given wavenumber, while the dotted dashed line shows the plot of as given by Eq. (5), with . Thus the error that would be introduced by assuming varies linearly with can be seen in part (b), which is a plot of the difference between the two lines, where it can be seen that, for this case the error in over this wavenumber range can be up to 500 μm per meter of . It can also be seen that the growth in the error is increasing rapidly away from the point .
3. DISPERSION USING GRATINGS
An alternative method by which dispersion has been introduced into the reference arm of an interferometer uses diffraction gratings and is illustrated in Fig. 3 [5,7]. Here is the object whose axial position is being monitored, and is a reference mirror. Light passing into the reference arm is normally incident on a transmission grating , after which light from the first diffracted order meets a second identical grating, , parallel to the first. The recollimated light reflects from the reference mirror before once again passing through the grating pair. As the angle at which the light is diffracted is dependent upon the wavelength, the length of the path that the light takes between the gratings varies with wavelength, and thus the desired dispersion is induced geometrically.
The light from the two arms is recombined, leading to interference between the two beams. The light is incident on a spectrometer, allowing the resulting interference to be analyzed by wavenumber, and generating a spectral interferogram of the form shown in the inset of Fig. 3. As the position of the object changes, the location of the extrema in the broad central fringe moves along the wavenumber axis [5,7]. The distance the light travels between the gratings, , is5,7], where and , while the wavelength, , is in the range 818 nm . The phase difference between light propagating down each arm of the interferometer, , is then found to be
In prior work on this system [5,7], the linear approximation was made. It can be seen that , where is the wavenumber about which the linear approximation is made. Applying the approximation to Eq. (11) gives
Returning to the exact form of and differentiating gives18) is equal to the solution to Eq. (15) at ; however, the contribution from the second derivative term has again been lost, and because it is multiplied by a large-valued wavenumber, it is potentially nonnegligible. Indeed, with the parameters for the system listed above the difference between and is highly significant, as is illustrated in Fig. 4. The solid line in this figure shows the value of needed for the minima in to occur at each value of according to the exact equations, while the dashed line gives the value of when .
The dashed line has a constant gradient, suggesting that the minima in would move along the wavenumber axis linearly as the scattering object moves; however, looking at the solution to the exact equation, it can be seen that the minima in actually moves along the wavenumber axis in the opposite direction, in a manner that is not quite linear with the change in position of the scattering object.
It is interesting to note that while the location of the extremum in given by the full expression and the linear approximation differs significantly, as shown in Fig. 4, the form, or appearance of the signal predicted in both cases, is very similar. This is illustrated in Fig. 5, which shows plots of and the signals these would lead to calculated for . Figure 5(a) shows the value of for each wavenumber calculated from the exact equations, with part (b) showing a plot of . Parts (c) and (d) show the results calculated when the linear approximation is used.
While the difference between the value of and its linear approximation is very small compared to the magnitude of , such a small difference is sufficient to change the phase of the light significantly, converting the minima in in the exact case to a maxima; however, the shape of the curve is similar despite its inversion. The inversion does mean that when a linear phase ramp is added, as is the case when the object being examined is moved, changing , the minima moves in opposite directions along the wavenumber axis for the two cases.
Figure 6 shows the size of errors that would be introduced if it is assumed that the turning point moves linearly with changing , as has been done in previous work [5,7]. This is obtained by taking a tangent to the against curve at the point , with all other parameters corresponding to the experimental setup described above. It can be seen that an error of about 8 μm can be seen at either end of the measurement range, which covers a distance of 704 μm. It can be seen from Fig. 6 that the error grows at an increasing rate the farther from the point the minima in lies. Thus, while the errors shown here represent only about 1% of the percent of the range measured, it is significant if high-accuracy measurements are desired, or will become more significant if the wavelength range is increased.
4. EXPERIMENTAL COMPARISON
In this section, we present experimental results that were obtained using an instrument of the form illustrated in Fig. 3, with and . The measurement mirror, , was mounted on a precision closed-loop piezo stage (PI Hera 625.1CL), moved axially in 5 μm steps throughout a 290 μm range. The location of the stage, as reported by an integrated capacitance sensor, was compared to the interferogram generated by the apparatus. The aim was to determine the linearity of the minima of the curve in wavenumber with changing . The magnitude of the interference signal generated depends predominantly on source spectrum (a 55 nm bandwidth centered on 855 nm), and this variation needs to be accounted for before the signal is analyzed.
The envelope on the interference signal is measured, and the signal is rescaled with lower and upper envelopes being set to 0 and 1, respectively. The resultant signal takes the form illustrated in Fig. 7(a). The minima in is determined using the method described in Ref.  where the signal, after subtracting the mean, is autoconvolved, producing a result such as is illustrated in Fig. 7 (b), where the peak in the signal corresponds to the location of the minima in .
The value of at which the minima of is found was recorded for each stage position. The solid line in Fig. 8 shows the nonlinearity of the measured minima, calculated as the deviation from a best fit straight line, i.e., the result that would be expected if the minima moved along the axis linearly with . The dashed line is the result predicted by the equations describing the system. In order to generate it, the value of at which the minima lies for the starting position of the stage was found, and the corresponding value of is calculated using Eq. (17). The values that correspond to the minima as is incremented in 5 μm intervals were then calculated. As was done for the measured values, the deviation from a least squares best-fit line is plotted. The differences between the measured and calculated plots can be accounted for by a combination of several factors. First, as can be seen in Fig. 7, the upper and lower envelopes to the signal do not perfectly match the limits of the fringes at this position. This may be due to imperfect collimation of the light, temporal coherence effects leading to a loss of modulation on the signal, high-order dispersion due to, for instance, small misalignments, e.g., rotations of gratings. This highlights a need to determine the true dispersion in the system through calibration in order to reach the highest possible measurement accuracy.
Taking the gradient of the best-fit straight line to the measured data gives a value of , meaning an error in of about , as shown in Fig. 8 would correspond to an error of about 2.78 μm over this range. As the range of increases, this error grows rapidly, so it is essential to take the nonlinearity of the position of the minima in into account.
5. DISCUSSION AND CONCLUSIONS
In this paper, we have examined the use of linear approximations in the analysis of spectral interferograms generated by dispersed reference interferometers. The use of such approximations can lead to significant measurement error when the dispersion is introduced via paired diffraction gratings [5,7], glass plates , or high-dispersion optical fiber . While in the examples presented here, the difference between the results for the linear approximation and the exact solution is far greater for the case where the dispersion is introduced via diffraction gratings, the magnitude of the errors when dispersion is introduced via a glass plate will be dependent on the exact material characteristics.
The source of the error is that the signals that are generated depend on the second derivative of the variable that is replaced by a linear approximation. The second derivative of the approximation is zero, and although the true second derivative is small compared to the first, in expressions describing the inteferograms generated, it is multiplied by the wavenumber, which is large in magnitude. This results in a significant term containing the second derivative, which should not be ignored. Experimental results are included to support our analysis.
We conclude that to make high-precision measurements at submicron scales using instruments based on DRI, the more exact expressions presented here should be used instead of those based on a linear approximation.
Engineering and Physical Sciences Research Council (EPSRC) (EP/P006930/1); Renishaw; Royal Academy of Engineering.
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