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A three-beam scanning optical interferometric microscopic technique applied to roughness characterization of optical flats is described. The technique is based on the heterodinization of three coherent optical beams. One of the beams, the probe beam, is focused on the surface under test. A second beam is obtained after being reflected by a reference surface. Finally, the last beam consists of one of the first orders of diffraction that emerges of a Bragg-cell. The three beams are coherently added at the sensitive surface of a photodetector that integrates the overall intensity of the beams. We show analytically that, the electrical signal at the output of the photodetector, is a time-varying signal whose amplitude is proportional to the surface local vertical height. We characterize experimentally the frequency response of the system by measuring the profile of three different gratings. We show measurements of the roughness of an optical flat processed by means of the frequency response of the system.
©2007 Optical Society of America
1. Introduction
Surfaces of high optical quality are highly demanded by industry for several applications, especially in the electronics industry [1, 2], as the performance of the electronic devices depends of the quality of the surface, making it necessary to have accurate instruments for roughness measurements. Several techniques have been developed for roughness measurements of high optical quality surfaces, and can be roughly classified as non-contact or optical and stylus or mechanical.
Non-contact techniques have advantages over the mechanical techniques in the sense that the sample has not to be intimately near to the mechanical probes, lowering the chances of being contaminated or damaged [3–8]. Various methods of optical techniques have evolved as for example, interferometric, confocal, inetrferometric-microscopy techniques, etc. [9–13]. Additionally, when it is necessary to inspect accurately microscopic areas scanning techniques are preferred, due, in principle, to its higher lateral resolution. Additionally, scanning techniques can be combined with heterodyning or homodyning to improve even more the lateral resolution [14–16]. It is however advisable, when high accuracy is required, to use different techniques in performing the same measurement for comparison purposes [17].
The work described here can be considered as a follow-up of the “three-beam Gaussian heterodyne interferometer” presented in Ref. [18]. The purpose of the new report is to show how the spatial frequency response of the profilometer can be calibrated, resulting in a more accurate height profile. Then, in using this response to process the raw data obtained from the instrument to obtain a profile that best matches the actual profile of a high quality optical flat. This is usually a difficult task with other optical techniques. For example, for obtaining profiles of high quality surfaces in micrometric areas, microscopic interferometers of different configurations are used Refs. [19, 20]. Due to the complexity of these optical systems, it is in general difficult to obtain a simple analytically expression of the spatial frequency response, making it hard to process the raw data obtained from these instruments to make a good fitting with the actual profiles under inspection. In contrast, the “three Gaussian beam interferometer,” exhibits a simple spatial frequency response that consists of Gaussian function centered at the origin. This allows processing the readout in an easy manner to obtain a result that resembles better the actual profile. An additional advantage to consider in the proposed interferometer is the use of a carrier temporal frequency that is used to filter out spurious signals. When this carrier is not used, spurious signals may mislead the actual height measurements. These spurious measurements arise when the non-modulated reflected beam (base-band) is treated as the actual-real height, as is the case of other instruments reported up to date.
The “three Gausian beam interferometer” consists of a focussed laser beam which is used as a local probe to illuminate the surface under test and the reflected beam is combined with two additional reference beams for heterodyning. Usually, in interferometry two beams are combined; as we show bellow, the use of a third beam allows the system to present a temporal carrier whose amplitude is modulated by the local vertical height of the sample improving the vertical sensitivity of the system. However, as indicated above, for accurate profiling the frequency response of the system should be taken in to account. In order to obtain the frequency response and calibration of the system, we use three different diffraction gratings with periods of 300, 600 and 1200 lines/mm, thus. The system response is then used for characterizing the profile of a high quality optical flat.
Our presentation is divided in the remaining six sections: In section 2, we describe the experimental setup. In section 3, we present the analytical formulation of our method. In section 3.1 we describe the experimental results and compare them with the one obtained by using an atomic force microscopy, in section 4 we describe the roughness measurement of an optical flat. Finally, we give our conclusions in section 5.
2. Experimental setup
Figure 1 depicts the experimental setup that corresponds to a three-beam interferometer. A 15-mW He-Ne laser with a wavelength of 632.8 nm with a Gaussian intensity profile is used as the coherent illuminating source. The illumination beam goes through a Bragg cell that consists of an acousto-optical medium of tellurium dioxide (TeO2) excited at 80 MHz. In the present technique, only the Bragg cell orders 0 and 1 are used. The order zero is directed to a beam splitter BS1 where it is divided into two beams. The first beam is focused on the surface under test by lens L1. This beam is called the probe beam. The reflected beam is transmitted again by lens L1 and directed by BS1 to the photodetector. The second beam is directed to a high quality mirror, M3, and the reflected beam is directed to the photodetector in a similar way as the probe beam; this beam is called the reference beam. The first diffracted order beam at the output of the Bragg cell is guided towards the photodetector by mirrors, M1, M2 and beam splitter BS2. The temporal frequency (color) of this beam corresponds to the sum of the frequencies of the light and of the frequency of excitation of the acousto-optical cell. For brevity, we will refer to this beam as the modulating beam. The three beams are superimposed and coherently added at the plane of the photodetector. The electrical signal at the output of the photodiode is amplified and sent to a lock-in amplifier which is tuned to the frequency of excitation, thus, discriminating all other signals different from the reference frequency.
The lock-in amplifier communicates with a personal computer by means of a GPIB channel. The lock-in is programmed to sample 3000 points when scanning a line of approximately 12 μm. Each line scan takes approximately 10 seconds. The focusing lens L1 consisted of a high quality microscope objective with a working distance of 6 mm and a focal length of 2 mm.
In the experimental setup, two phase-shifters (1 and 2), introducing φP and φM rad, for the probe and modulating beams respectively, are used to make the output signal a function of the local roughness under measurement as shown in the next section. Additionally, two attenuators are used to make equal the intensity of the beams at the plane of detection, this condition is not necessary but it simplifies the analytical formulation described in the next section.
Fig. 1. Experimental set up. M and BS stand for a mirror and a beam splitter respectively. Phase-shifter 1 introduces a phase shift of value ϕP in the probe beam with respect to the reference beam. Phase shifter 2 introduces a phase shift of value ϕM in the modulating beam with respect to the reference beam. The attenuators are introduced to make equal the three intensities at the plane of detection.
3. Analytical description
For simplicity of the analytical description we will consider plane waves. A more formal analysis of the system calculated in terms of Gaussian beams can be found in [18].
The reference beam, can be expressed as
where i = √-1 , I 0 is the intensity of the beam, k = 2π/λ. λ and ω1 are the wavelength and the temporal angular frequency of the light. Variable zR represents the overall path traveled by this beam.
The probe beam, at the plane of detection, after being reflected from the surface under test, can be expressed as
where h(x) represents the one-dimensional amplitude vertical height distribution of the surface under test. Similarly, the variable zp represents the path traveled by this beam and zP represents the phase shift introduced by phase shifter 1. In writing Eq. (2) we are considering that the surface under test has constant reflectivity in the area under inspection.
The third beam, the modulating beam, can be expressed as
As this beam comes from the first diffraction order that emerges form the Bragg cell, its angular temporal frequency is the sum of ωl and ωS; the angular frequency of the laser beam, and the excitation frequency respectively. The variable zM represents the path traveled by this beam and ϕM represents the phase shift introduced by phase shifter 2.
By superimposing coherently the three beams, we can write the total light amplitude distribution at the plane of the detector as,
Next, considering that
zp = zR + ΔzP, zM = zR + ΔzM, and that the phase shifters are adjusted such that and , (m,n being integers), then, by A. using Eqs. (1–3), Eq. (4) can be written as,
The overall distances zR,zP, zM are not critical, and should be approximately equal, in order to obtain similar spot sizes for the three beams under detection at the plane of the photodiode. The size of the probe beam at the plane of detection is the only one that suffers changes as a function of the local vertical height of the surface under test. Thus, the surface under test is placed approximately in the back focal plane of lens L1, and it is adjusted, before performing the scanning, to attain approximately the same spot size at the plane of detection.
To adjust the phases between the beams, and validate Eq. (5), the following procedure is easily accomplished.
To adjust a phase shift of π radians between the probe beam and the reference beam, the modulating beam is momentarily covered. Meanwhile, phase shifter 1 is adjusted to record minimal intensity at the photodiode. To adjust a phase shift of π/2 radians between the reference beam and the modulating beam, the probe beam is momentarily covered and phase shifter 2 is adjusted to observe the desired shift phase in the lock-in readout phase.
Thus, by means of Eq. (5), the intensity on the plane of the photodetector can be written as,
Taking into account the area (Ad) and the responsivity (ρ) of the photodiode, the voltage signal at the output of the photodiode can be written as
It can be noticed that the voltage signal at the output of the photodiode, described by Eq. (7), consists of a DC voltage, the first two summands inside the curled brackets, and an AC, the last summand. The AC term represents a temporal signal whose amplitude is given by a sinusoidal function whose argument is proportional to the local vertical height of the surface under test. It is important to notice that for very small vertical amplitudes the sinusoidal function can be replaced by its argument; thus, the introduction of the modulating beam makes the system highly sensitive due to the fact that the output signal is proportional to the local height. For improving the accuracy of the measurements in this report, this approximation will not be used.
The signal given by Eq. (7) is amplified and sent to a lock-in amplifier for detection and processing. Let A be the gain of the amplifier, thus the amplitude of the voltage signal recorded by the lock-in, is given by,
where P 0 = AdI 0 and x 0 is local point where the measurement takes place.
At this point, it is important to emphasize the result expressed by Eq. (8), It represents the amplitude of a temporal carrier of a well-established frequency. This amplitude is a sine function of the local vertical height under test. When the vertical height becomes smaller, in the nanometric scale, the sine function can be replaced by its argument. Thus the amplitude of the output signal becomes proportional to the vertical height under measurement. Additionally, as a lock-in amplifier is used, if the local vertical height under measurement results very small, as is well known, the lock-in amplifier is still capable in recovering a signal one thousand of times smaller than the noise. Thus, the system becomes highly sensitive to the local vertical topography. In the experimental section, this topic is further discussed.In the next section the frequency response of the system is characterized
3.1 Experimental determination of the frequency system response
As it is well known, in general, an optical system responds in a different manner to objects with different spatial frequencies. The response of this interferometer has been analytically treated in Ref. [18] resulting in a Gaussian frequency response given by,
where (u,v) represents the two-dimensional corresponding spatial frequencies and, r 0 the semi-width of the probe beam at a plane located on the surface under test. In our experiment, r 0 was approximately 0.41 μm. The frequency response represented by Eq. (9) is a consequence of the Gaussian intensity profile of the probe beam.
In order to verify experimentally the frequency response of the system, we used three, commercially available, reflecting gratings. The diffraction gratings are high quality holographic gratings of 300, 600 and 1200 lines/mm, and were measured by means of an atomic force microscope for calibration purposes. Several measurements were taken for each of the gratings inside a small zone of interest by both methods. The AFM, giving the results in real height (nm) and the proposed system in volts according to Eq. (8). The calibration technique is as follows.
First, for each grating an average height distribution over the small zone of interest was estimated. The three averages (one for each grating), obtained with the AFM, are shown in Fig. 2(a). For each of these three average distributions, an rms vertical height value was obtained for each grating.
Second, for each grating an average height distribution was obtained with the proposed technique however, this time, the resulting distribution heights were obtained in volts as expressed by Eq. (8). The three distributions, which correspond to the averages obtained with the AFM, are shown normalized in Fig. 2(b). From the three resulting averages an rms value, in Volts, was obtained for each grating.
Before proceeding with the experimental description, a discussion on the results depicted in Figs. 2(a) and 2(b) is given. A direct punctual comparison between methods is not simple, as it is not possible to attain exactly the same line scan with both methods. Additionally, the optic system is limited in its lateral resolution because of diffraction; as usual, one can consider λ as a limit. Further, the AFM responds to different physical properties as compared with the optical system. An ample discussion on this topic can be found in Ref. [21]. Figure 2(b) reveals that, even a grating with spatial frequency such as the 1200 l/mm (period = 0.83 μm), can easily be detected with the proposed method. Thus a lateral resolution near to λ is attained. This is attributed to the high vertical sensitivity of the system that in turn improves the lateral resolution. Thus, for a reasonable comparison of the methods, the average rms values of a set of measurements are used.
Third, from the above rms values obtained, the ratios of the rms output voltages to the rms corresponding heights were calculated for each spatial frequency.
A frequency response curve for the above ratios was fitted. A normalized plot is shown in Fig. 3. It will be noticed that the frequency response experimentally obtained is in good agreement with the theoretical response given by Eq. (9).
We point out that the curve of the response was adjusted by using only three points. This is done only for illustrative purposes and more points can be added if desired. For clarity a symmetric spatial frequency axis is shown.
Fig. 3. Plot of the normalized frequency response of the system and the corresponding noise distribution.
Figure 3 also shows the noise distribution in a scale that corresponds to the normalized frequency response, and was obtained as follows. The average profile obtained for each of the gratings was subtracted form each of the measurements. Then, by placing each result of each subtraction one after another, a set was formed. This gives a representative noise of the set of samples. Finally, the Fourier transform of the set was obtained and plotted.
In Fig. 3 the vertical dotted lines indicate the useful area inside the filter that can be used to recover the input profile to be measured by the system.
Finally the limited inverse frequency response of the system is shown in Fig. 4. It was limited by the intersection of the noise with the curve of the frequency response.
In order to calculate a figure of merit of the signal to noise ratio of the system, we used the well-known equation,
where E{f 2(t)} is the mean or expected value of the function f 2(t), likewise for E{n 2(t)}. Sf(w) and Sn(w) define the power spectrums of f(t) and n(t) respectively. The signal to noise ratio obtained, was approximately 24 dB. In the next section, we proceed in measuring the topography of a high quality optical flat.
4. Roughness measurement of an optical flat
The sample to be characterized consisted of a circular λ/4 high quality commercial optical flat with a radius of approximately 1.25 cm. The experimental distribution profile of a line scan of approximately 12 μm obtained with the proposed system, as a normalized output signal, is shown in Fig. 5.
Fig. 5. Normalized output signal obtained from a line scan of an optical flat obtained by the system.
The distribution profile shown in Fig. 5, was then processed in order to be compensated in frequency by using the inverse frequency response shown in Fig. 4. The profile of the optical flat after processing is shown in Fig. 6.
Fig. 6. Profile calculated by using the inverse frequency response of the system for the optical flat.
According to the above discussion, it is expected that, after being properly processing the output signal, the topography shown in Fig. 6 represent better the actual topography of the surface under test.
Before concluding, we notice that it is not possible to attain a comparison of the resulting profile with the AFM because, as indicated above, it is not possible to attain precisely the same line scan with both methods, and due to the randomness of the roughness.
5. Conclusions
We have described a microscope interferometer technique for measuring the profile of optical surfaces in micrometric areas. The technique is based on focusing a Gaussian laser beam on the surface under test. This beam, after begin reflected from the surface under test, it is coherently superimposed at the sensitive area of a photodiode with another two reference beams. We have shown analytically that the signal at the output of the photodetector consists of a time varying sinusoidal signal whose amplitude is a sinusoidal function whose argument is proportional to the local vertical height. We have characterized the spatial frequency response of the system by using three calibrated gratings and we have applied the curve of the frequency response to compensate the measurements. It can be expected that the profile obtained after the processing resembles better the actual profile.
Acknowledgments
The authors thank to Dr. Sergio Calixto for the measurements of the reflective gratings in the atomic force microscope. The authors thank financial support of CONACYT.
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