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We present an effective method of testing the surface shape of quasi-parallel plates which requires only two phase-shifted three-beam interferograms. We derive a general formula for difference of two three-beam interferograms as a function of the phase shift value. The phase shift does not have to be precisely determined and uniform in the image domain. We show and compare results of extracting the fringe set and corresponding phase distribution related to the plate front surface shape using the two dimensional continuous wavelet transform, Hilbert-Huang transform and Fourier transform methods. Simulated and experimental data is used to verify the algorithm performance and robustness.
© 2016 Optical Society of America
1. Introduction
It is well known that surface shape testing of quasi-parallel plates conducted in conventional interferometric setups poses a problem. The beam reflected from the plate rear surface interferes with two other beams: the reference one and the one reflected from the plate front surface. Three-beam interference leads to a complex fringe pattern composed of three high contrast fringe sets. From the point of view of the plate front surface evaluation two of them are parasitic and therefore undesired. Widely used methods of fringe pattern analysis such as temporal phase shifting or the Fourier transform method are not capable of retrieving a single fringe set from the three-beam interferogram intensity distribution.
Several solutions to the problem of back surface reflection which take advantage of various types of light sources and different interferometer configurations were proposed. Wavelength-scanning interferometry [1–5, and references therein] uses special tunable light source and restricts dimensions of the element under test and its distance from the reference flat. It uses m-frame phase shifting algorithms for phase extraction, where m starts from 13 and may reach a few dozen. This is the cost of identification and suppression of the back surface reflection. Short coherence interferometry manipulating temporal [6] or spatial [7] coherence introduces a dedicated source of light which is an interferometer itself. In the first case the source generates two orthogonally polarized beams with precisely defined and controlled optical path difference. A phase mask along with a quarter-wave plate located just before the detector allows for observation of interference of beams reflected from the selected surfaces and processing the image with the use of the spatial phase shifting method. Only one camera frame is registered. In the second case the interferometric source generates so called Ring of Fire, a sinusoidal pattern with an intensity distribution similar to a Fresnel zone plate. It enables registration of two-beam interferograms which may be phase-shifted to apply standard five-frame algorithm. In the phase-shifting Fizeau setup [8] it is necessary to conduct measurements in three interferometric modes corresponding to different configurations of the interferometer cavity. The method requires tilting the object under test and using additional reference flat. Data from three measurements is combined but in order to obtain information in the full field of view it is also necessary to aid the phase shifting method with dedicated iterative algorithm based on the Fourier domain filtration.
It is seen that suppression of the back surface reflection is possible, but it requires either considerable complexity of the interferometric setup or complicated data analysis and multistage measurement procedure. The simplest solution, therefore, is based on covering rear plate surface with an immersion fluid. It is still a common choice in optical workshops despite the plate contamination and the risk of damage.
Our goal is to provide a method of surface shape testing of quasi-parallel plates which does not require any modification of Fizeau or Twyman-Green setups. A few attempts based on analyzing a single three-beam interferogram have already been made [9–11]. In this paper we significantly advance our previous approaches. We assume that the interferometer is equipped with a piezoelectric phase shifter, although even this demand could be relaxed under conditions we will present later in this paper. Our innovative solution is to record two phase-shifted three-beam interferograms with a phase shift in a wide range of permitted values. This two-shot only data acquisition and automatic processing of their subtraction pattern is sufficient to obtain information on topography of the plate surface. We believe our method may be particularly useful for research conducted in far-IR, UV and X-ray wavelength regions where tunable or a variety of light sources are not available.
2. Theoretical grounds
The three-beam interference intensity pattern recorded in testing a quasi-parallel optical plate can be expressed as [12–14]:
where Af, Ab, Ar are the amplitudes and θf, θb, θr are the phases of the three beams, respectively. Subscripts f, b, r refer to the front, back and reference beam, respectively. For notation brevity the spatial (x, y) dependence of all terms has been omitted. Example of the three-beam interference intensity pattern is presented in Fig. 1(a). The phase of a fringe set related to front surface shape is given by: where k is a wave number and zf, zr are front and reference surface shapes including their tilts. Actually zr is the tilt only, when assuming a perfectly flat reference. Since we want to obtain Zf, that is front surface shape without tilt, Fig. 1(b), the computed phase distribution θfr has its linear term removed and then scaled to height units. Scaling quantities given in radians to height units, assuming the measurement was conducted in a Twyman-Green interferometer, is carried out as follows: where θf [rad] is detrended θfr and λ is wavelength used.
Fig. 1 Simulated intensity distribution (256×256 pix) of Ifbr (a), front surface shape map Zf [nm] (b), ΔI(π) (c), SNR ratio of ΔI(α) intensity distribution (d).
If we have two interferograms with mutual phase shift α equal to π, denoted by Ifbr and Iπ, where Iπ is given by:
we can easily derive a formula for their difference: It is a sum of two fringe sets which may be separated (in order to retrieve the phase maps, either θfr or θbr) by one of the algorithms mentioned in this paper. Intensity distribution of very similar form obtained in a different way was investigated in previous works [9–11]. In this paper we generalize Eq. (5) to a form which takes into account an arbitrary phase step α between the subtracted frames. In order to achieve this we first made three initial generalizations concerning sets of three-beam interferograms with a constant mutual phase shift Δα. The initial generalizations take the form: where i, j are the frame numbers, j > i. Using trigonometric identities it can be proved that:- If we have four three-beam interferograms I1, I2, I3, I4 with a constant mutual phase shift Δα = π/2 then: where The assumption given by Eq. (10) holds for the air-glass interface characterized by intensity reflection and transmission coefficients R = 0.04 and T = 0.96, respectively.
So, in this point we may conclude that:
- If we have four three-beam interferograms I1, I2, I3, I4 with a constant mutual phase shift Δα = π/3 then:
We obtain:
- If we have four three-beam interferograms I1, I2, I3, I4 with a constant mutual phase shift Δα = π/4 then:
We get:
Equations 11,15 and 19 may be generalized to a final form, which takes into account an arbitrary phase step α between subtracted frames:
The intensity distribution ΔI(α) consists of the sum of two fringe sets with amplitude modified by factor 2 sin(α/2) and phase with a phase shift α′ = α/2 − π/2. Subtracting two frames removes the slowly varying background and the fringe set generated by interference of the two object beams, see Fig. 1(c). We know that α′ is a constant function (or more realistically speaking a linear one), so at the stage of scaling it will be removed from the obtained phase map. Therefore, it is not necessary to know the precise value of α to estimate the front surface shape Zf. Moreover, a tilt-shift error [15] of the PZT shifter does not influence the results because detrending algorithms remove all the linear terms. These facts allow for using other techniques of linear movement of the optical elements, less precise than the one using the piezoelectric transducer.We expect that only for α close to 2mπ a signal to noise ratio SNR of ΔI(α) will be very low and it will disenable phase retrieval from ΔI(α), see Fig. 1(d) (SNR is calculated using built-in Matlab function snr). Exact permitted range of α will be determined in the next section.
Having the intensity distribution ΔI(α) it is necessary to identify the fringe family connected with the front surface reflection. To achieve this we have to suppress the back surface reflection at least partly or for a while. It is enough to put a drop of water or blow at it because but we do not need permanent full-field suppression.
When the fringe family related to the front surface reflection is identified we may extract it from the intensity distribution ΔI(α). For fringe separation we used three algorithms: two dimensional continuous wavelet transform (CWT) [16], Hilbert-Huang transform (HHT) [17–19], and well-known Fourier transform (FT) method [20].
3. Simulation and experimental results
First, we present results of phase retrieval from the intensity distribution ΔI(π) obtained from simulated and experimental phase-shifted three-beam interferograms, see Fig. 1 and Fig. 2, respectively. The simulated and experimental interferograms are 8-bit. The dense fringe set in the intensity distribution ΔI(π) shown in Fig. 1(c) and Fig. 2(c) corresponds to the front surface shape and its phase is to be determined. The sparse fringe family codes back surface shape, but is also dependent on front surface shape and refractive index distribution. Under the assumption of homogeneous refractive index distribution and with knowledge of front surface shape, back surface shape can also be computed from the intensity distribution ΔI(π).
In Fig. 1(c) and Fig. 2(c) fringe components cross almost perpendicular to each other. To separate them with the use of the CWT, HHT or FT algorithms we must ensure either difference in the spatial frequency or in orientation (or both). Therefore, the right angle between fringe sets is not necessary but may be helpful in the separation process. Such (or similar) alignment of mutual fringe orientation is usually possible when testing quasi-parallel plates.
The experimental data was recorded in the Twyman-Green setup equipped with a PZT phase shifter, Fig. 3, when testing a plate of thickness 3 mm and refractive index n = 1.526. The reference measurement was conducted using the temporal phase shifting method after covering the rear plate surface with an appropriate immersion fluid.
Fig. 3 The Twyman-Green interferometer experimental setup: He-Ne laser λ = 632.8 nm, SFS - spatial filter system, MO - microscope objective, PN - pinhole, C - collimation objective, BSW - beam splitting wedge, RF - reference flat (flatness λ/10 PV), PZT-S -PZT phase shifter, PZT-C - PZT controller, QPP - quasi-parallel plate, IS - imaging system, AP - aperture, Det - CMOS camera UI-1240LE-M-GL.
We computed the phase map corresponding to the front surface shape of a quasi-parallel plate, θfr, using the CWT, HHT, and FT methods. The scaled front surface shape maps, Zf, and phase retrieval error distributions, Δθf, are shown in Fig. 4 (simulations) and Fig. 5 (experiment). Δθf is computed as a difference between the reference and obtained phase map detrended and scaled to nanometers.
Fig. 4 The scaled front surface shape maps, Zf [nm] (first row), and phase retrieval error distributions, Δθf [nm] (second row) obtained from simulated data with the use of CWT (a); HHT (b); FT (c).
Fig. 5 The scaled front surface shape maps, Zf [nm] (first row), and phase retrieval error distributions, Δθf [nm] (second row) obtained from experimental data with the use of CWT (a); HHT (b); FT (c).
We determined values of RMS (root mean square) and PV (peak-to-valley) errors in the whole image domain and excluding 20-pixel wide boundary regions (the latter expressed as RMSc, PVc) in order to eliminate the influence of the border effects on final results. Requirements for optical components often relate only to their central part, so in many applications RMSc and PVc are more important. RMS is computed as sqrt(var(Δθf)) using Matlab functions. All the error values are given in Table 1.
Table 1. Plane fit RMS and PV errors for front surface reconstruction [nm]. RMSc and PVc are computed excluding 20-pixel wide boundary regions. The lowest value of a given parameter is marked bold.
For simulated data RMS and RMSc reach value of several hundredths nanometers: RMS (HHT) = 0.47 and RMSc (CWT) = 0.14 in the Twyman-Green interferometer. It is worth to note that both HHT and FT algorithms may be tuned in order to achieve better results in the central part of the image (up to RMSc (HHT) = 0.10 nm) at the cost of worse results at the edges. In all the error distributions in Fig. 4 and Fig. 5 the biggest errors are located at the edges of the images, what is typical for all the methods taking advantage of Fourier transform (CWT and HHT belong to this group).
For experimental data the error parameters reach values a few times higher. The temporal phase shifting method used to compute the reference phase map is also subject to errors. The error distributions presented in Fig. 5, especially the one in Fig. 5(c) show residual fringes which usually accompany the temporal phase shifting method. Also, index-matching treatment is not perfect.
In order to determine the allowed range of α for which we still get good quality phase extraction results, we repeated the phase computation from simulated intensity distribution ΔI(α) for α ∈ (0, 2π) and monitored the calculated values of RMS and PV errors. Figure 6 presents the outcome of analysis using the CWT method. For the HHT and FT algorithms we obtain very similar results.
Fig. 6 Phase extraction errors (plane fit RMS and PV) from the intensity distribution ΔI(α) (the algorithm used: CWT) as a function of the phase step value α.
It can be noted that for α ∈ (π/3, 5π/3) RMS and RMSc errors are almost unchanged. Strictly speaking mean value of RMS in the range α ∈ (π/3, 5π/3) is equal to 1.0007 RMS(π) and mean value of RMSc in the range α ∈ (π/3, 5π/3) is equal to 1.0979 RMSc(π).
Values of RMS and RMSc increase for α ∈ (2mπ − π/3, 2mπ + π/3) but completely unacceptable results are obtained in a very narrow interval: α ∈ (2mπ − π/32, 2mπ + π/32). So, the permitted range of α can be claimed to be quite wide, it reaches 4π/3 around the central value of α = π.
4. Conclusion
In conclusion, we presented a method for measuring the front surface shape of a quasi-parallel plate which may be conducted in the presence of a beam reflected from the plate rear surface. Only two phase-shifted three-beam interferograms are required and the value of the phase shift may lie in a wide range α ∈ (π/3, 5π/3) rad. Subtracting two registered frames enables access to an interesting synthetic intensity distribution which can be treated as the sum of two separate fringe patterns to be extracted and phase-demodulated with the use of one of the algorithms: CWT, HHT or FT. The method is robust to translational and tilt-shift errors of a phase shifter. It can be implemented in standard interferometers (Fizeau and Twyman-Green) without any modifications of the opto-mechanical setup.
Funding
Narodowe Centrum Nauki (NCN), Poland (2014/15/B/ST7/04650 and 2014/15/N/ST7/04881) and, in part, by statutory funds.
References and links
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