The present invention based on applications of Fresnel Diffraction of light from phase step in optical metrology.
It is more than a decade that the Fresnel diffraction from phase steps of one and two dimensions has been studied comprehensively ((Optical diffractometry; Tavassoly et al. 2009) and (Fresnel diffraction from 1D and 2D phase steps in reflection and transmission mode; Amiri et al. 2007)). It is shown that the phase difference (PD) or the corresponding optical path difference (OPD) between the waves diffracted from the two sides of a step edge can be varied by changing the step height and the angle of incidence of light. The latter changes OPD very finely. This has led to many applications including: precise measurements of displacement (Nanometer displacement measurement using Fresnel diffraction; Khorshad et al. 2012), film thickness (Application of Fresnel diffraction from a phase step to the measurement of film thickness; Tavassoly et al. 2009), plate thickness (Applications of Fresnel diffraction from the edge of a transparent plate in transmission; Tavassoly et al. 2012), refractive indices of solids and liquids (High precision refractometry based on Fresnel diffraction from phase plates; Tavassoly et al. 2012) and (Optical refractometry based on Fresnel diffraction from a phase wedge; Tavassoly et al. 2010). Thus, the Fresnel diffraction from phase steps is emerging as a powerful metrological technique with remarkable advantages.
The main part of the disclosed invention is a 1D step with variable height that has optically flat mirrors on either sides of the step. The PD between the lights diffracted from the two sides of the step edge is varied by changing the light incident angle and the step height.
The criterion of the measurement is the change in the average visibility of the three central fringes that varies between zero and one for a PD change of π. Thus, on one hand the PD is changed finely by varying the light incident angle and on the other hand the visibility of the fringes is very sensitive to phase change. Thus, combination of these properties makes the device a very accurate measuring tool.
In addition, the repetition of exactly similar diffraction patterns as PD changes and existing a universal curve for the visibility versus PD, improves the accuracy of the measurements considerably. Thus, the device permits to measure wavelength, plate thickness, surface topography coherence length, spatial coherence width more accurate than those are measured by conventional interferometers.
Since, there isn't any optical element between the step and the device detector; measurements are performed in a very wide range of wavelengths.
The device is indispensable for specifying the line shapes of broad spectral lines. Because, one can acquire a large volume of data by choosing suitable step height and varying the PD by changing light incident angle.
Since the active parts of the device are two rather small mirrors that are closely located; the device can be fabricated in a very compact form with very low mechanical noise. This allows using the device in rather rough circumstances.
Replacing the mirrors by two plates with different reflectances and illuminating them with P and S polarized lights, the recorded intensity distributions for different angles of incidence, lead to reliable determination of the optical constants of the plates. Thus, the device can act as an ellipsometer.
Fabricating flat mirrors with fixed steps, but rather small heights, say 10, 50 and 100 micrometers allows to perform Fourier spectroscopy on light source with broad spectral lines like LED and optical filters like interference filters. It is however obvious that the numbers indicated above are simply examples of preferred embodiments and other numbers, say less than 10 and more than 100 micrometers can be used in other embodiments as needed.
a, displays general experimental setup for different measurements.
b, displays the orientation of light beam relative to the device. Light incident angle (θ) is changed by rotating the goniometer around its vertical axis.
In this invention, we introduce a phase step diffractometer for application in optical metrology.
Mirrors (2, 4) are mounted on two separate stands (6, 8) with a very narrow gap between the mirrors' sharp edges for the step of zero height. In order to change the step height (12), one of the mirrors; first mirror; (2) is displaced in the direction perpendicular to its surface by a provided micrometer (10). However, the precise step height (12) is determined by the experiment. The planes of mirrors (2, 4) are aligned by manipulating the screws (14, 16 and 18), (that are equipped with restoring springs which are not shown in
General setup for performing different measurements is shown in
In general, the mirrors (2, 4) are not parallel and depending on their mutual orientations in the neighborhood of the aligned position, one of the Fresnel diffraction patterns shown in
When light beam (34) strikes the step device (1) at angle of incidence θ, the phase difference (PD) φ and optical path difference (OPD) Δ of the diffracted lights from two sides of step edge is:
Where h and λ stand for the height of step device (1) and light wavelength. (see for example Optical diffractometry; Tavassoly et al. 2009 and Fresnel diffraction from 1D and 2D phase steps in reflection and transmission mode; Amiri et al. 2007).
The typical diffraction patterns and the corresponding intensity profiles of lights diffracted from a step with different phases in one period are shown in
Where IL, IR stands for maximum intensities of the bright fringes on the left and the right sides of the central dark fringe with minimum intensity IC.
For the mirrors with the same reflectance, the visibility as function of OPD varies between 0 and 1 and the maximum visibility occurs for PD equals to an odd number of π,
Zero angle of incidence: Precise measurement of the step height (12) is necessary for accurate measurements of different quantities. For the precise measurement of the step height (12), specification of the zero angle of incidence is vital. To find the latter we illuminated the device's minors (2, 4) perpendicularly with a parallel beam of light (34) and record the corresponding reading on the goniometer (30). Then, we turn the step device (1), say to the right very slowly and record the corresponding readings on the goniometer (30) for several visibility maxima as precisely as possible. Then, we repeat the same procedure for the other direction. Finally, we determine the best zero angle of incidence by adjusting the readings on both sides. Then we measure the step height (12) by the following methods.
After setting the step device (1) at the state of zero angle of incidence, we turn the goniometer stage to a maximum visibility state at small incident angle θ0. At this angle, we have:
Where m0 is the order of the interference (
Subtracting Eq. 4 from Eq. 3 leads to:
that provides the step height.
Dividing Eq. 3 by Eq. 4 and solving the equation for m0 we get
Uncertainty in m0 is given by:
By choosing θ0 as small as possible and increasing θ1, the second term in Eq. 6 tends to 1 and tan θ0 can be ignored. For the case Δm0<0.5 the interference order is determined exactly at incident angle θ0. The rough estimation of m0 can be obtained by the first method (see S R Hosseini and M T Tavassoly, The application of a phase step diffractometer in wavemetry, J. Opt. 2015).
For example, using a goniometer of 10 arc second precision for a step height about 1.5 mm, and λ=632.8 nm, θ0=5° for θ1=57° uncertainty, Δm0 can be reduced below 0.5 and interference order is determined exactly. The step height is obtained by substituting m0 and θ0 in Eq. 3.
When the step heights are short, say less than few micrometers, counting the repetition of diffraction patterns does not provide very accurate height. In these cases, we change the light incident angle to get visibility in the 0-0.7 interval. Then by plotting the visibility versus the incident angle cosine in the latter interval, we get a straight line similar to those represented in
Before describing the applications of the device, the main features of the phase step is reviewed. a) There is a state of OPD=0 for incident angle of θ=π/2 which provides a natural origin for various measurements. b) In this device the plot of OPD versus the cosine of light incident angle is linear and passes through the origin. c) The OPD varies very finely by changing the light incident angle. d) The measurement criterion is the change in fringe visibility, not fringe shift as in interferometry; this leads to more reliable and precise measurements. e) The effective area of the phase step can be as small as few square millimeters; thus, the device can be very compact with very low mechanical noise. f) Since there is no optical component between the step and the detector, a very wide range of wavelengths can be covered. g) For a given step-screen distance, practically, the shapes and the sizes of diffraction fringes are fixed; independent of the step height and light incident angle, therefore, by choosing suitable step-screen distance the size of diffraction pattern can be adjusted to the size of the monitor. h) The technique can be applied easily and does not require complex and high quality optics. These features are utilized in different measurements.
The unknown wavelength of a monochromatic light can be determined by using the general setup of
In the first method we illuminate the step device (1) with a given height that is measured by the methods described in preceding section and count the repetitions of diffraction patterns in an incident angle interval θ0- θ1 and use Eq. 5 for wavelength determination.
In the second method for a step device (1) with given height we determine the exact order of the interference in the same manner described in preceding section and use Eq. 3 to derive the wavelength. (see S R Hosseini and M T Tavassoly, The application of a phase step diffractometer in wavemetry, J. Opt. 2015).
In the third method, we illuminate the device (1) by the reference and the unknown wavelengths simultaneously
A step (1) with known height (12), is illuminated by a broadband light (setup of
The origin of decay in visibility in
Fourier spectroscopy of broadband sources (including LED and white light sources) can be performed by the introduced phase step, by recording the intensity on the line in diffraction pattern that corresponds to step edge. The intensity at step edge vs OPD is sinusoidal pattern of interference. Then, intensity at the line corresponding to the step edge vs OPD, say interferogram, can be recorded by detector. The Fourier transform of interferogram provides the spectrum of the source as in conventional Fourier spectroscopy based on using interferometer.
The advantage of this method compared with interferometry is that in the former OPD is varied by changing the incident angle very finely. Thus, by choosing suitable step height the Fourier spectroscopy can be performed reliably.
One more noteworthy point is that the device is indispensable for the measuring of average wavelengths of broad-band spectral lines and the transmittance of filters. For this kind of measurements, one needs rather small step height permitting smooth change of OPD that is provided by the fine change of incidence angle. At the beginning, one increases the angle of incidence and reduces the OPD below the coherence length of LED light. Then, the visibility maxima are counted for the three central fringes by increasing the angle of incidence smoothly.
The phase difference between two visibility maxima is 2π. The first distinguishable visibility maximum is associated by number m. Then, the successive maxima are labeled by m-1, m-2, m-3, . . . . The plot of visibility maxima number (m) versus the cosine of incident angle (cos θ) is a straight line shown in
The device (1) in
D=2h sin q. (9)
(Investigation of correlation properties of light fields by Fresnel diffraction from a step; by Hosseini et al. 2013)). For perpendicular illumination of the step θ=0, D is zero and the visibility of the fringes is maximum.
By increasing the incident angle the distance between interfering waves increases according to Eq. 9.
Fresnel diffraction from a step can be applied to the measurements of opaque and transparent pates thickness. Mounting an opaque plate (80) on an optically flat surface (82), the required step (84) is formed
For the transparent plate, it is just sufficient to install the plate vertically on a goniometer and illuminate the upper edge of plate with a parallel beam of light. In this case a phase step is formed at the boundary of the plate and the surrounding medium because of sharp change of the refractive index. By counting the repetition of transmitted diffraction pattern in an incident angle interval the plate thickness is determined. ((Applications of Fresnel diffraction from the edge of a transparent plate in transmission; by Tavassoly et al. 2012) and (High precision refractometry based on Fresnel diffraction from phase plates; by Tavassoly et al. 2012)).
The device (1) can be used for fine displacement of the objects. For this purpose, we mount the object on the supporting plate (6) of the first movable mirror (2). By moving the first minor (2), the object is displaced and by counting the repetition of diffraction patterns or recording the changes in the visibility of the fringes, very fine displacement of the object is measured.
So far we have used mirrors (2, 4) with the same material. We have already shown that the intensity distribution on the diffraction pattern of light diffracted from a step with different materials on each sides of the step edge, depends on optical constants of the materials, step height and light incident angle. ((Optical diffractometry; by Tavassoly et al. 2009) and (Fresnel diffraction from 1D and 2D phase steps in reflection and transmission mode; by Amiri et al. 2007)).
Illuminating such a step with poly chromatic P and S polarized lights and recording the intensity distributions on the diffraction patterns at several incident angles provides a large volume of data for specification of the optical constants of the mirrors' materials. There is no need to a retarder in this method and the technique is applicable in a very large range of wavelengths.
The described step device (1;
The above identified embodiments describe the invented device in working condition, however it is obvious that other configurations and measurements may be carried out using such device. These embodiments were not intended to limit the functionality and working range of the device, only the description was simply for describing the best mode. It is obvious that the ranges and materials used and the configurations described can be modified for best use in different environments.