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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 13 — Jun. 25, 2007
  • pp: 8346–8359
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Heterodyne two beam Gaussian microscope interferometer

J. Mauricio Flores, Moisés Cywiak, Manuel Servín, and Lorenzo Juárez P.  »View Author Affiliations


Optics Express, Vol. 15, Issue 13, pp. 8346-8359 (2007)
http://dx.doi.org/10.1364/OE.15.008346


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Abstract

We present a novel microscope interferometric technique based on the heterodinization of two Gaussian beams for measuring roughness of optical surfaces in microscopic areas. One of the beams is used as a probe beam, focussed and reflected by the surface under test. The second beam interferes with the first beam and introduces a time varying modulating signal. The modulating light beam is obtained from the first diffraction order of a Bragg cell. The two beams are superimposed and added coherently at the sensitive plane of a photodetector that integrates the overall intensity of the beams. We show analytically that it is possible to find appropriate working conditions in which the system has a linear response. Under these conditions, the size of the probe beam at the plane of detection as well as the amplitude of the time varying signal at the output of the photodetector, are both proportional to the local vertical height of the surface under test. As a narrow bandwidth amplifier is used to detect the time varying signal the system exhibits a high signal to noise ratio. We also include experimental results of the measurement of the topography of a sample consisting in a blazed-reflecting grating.

© 2007 Optical Society of America

1. Introduction

Roughness measurement of surfaces is a valuable tool in several optical applications, in particular in the electronic industry. Due to its importance, different methods have evolved for surface profiling. A review of the different techniques in use can be found in Ref. [1–9

1. J. M. Bennett, “Comparison of techniques for measuring the roughness of optical surfaces,” Opt. Eng. 24, 380–387 (1985).

]. For the inspection of high quality optical surfaces, interferometric techniques are preferred due to their very high vertical sensitivity [10–16

10. G. E. Sommargren, “Optical heterodyne profilometry,” Appl. Opt. 20, 335–343 (1981). [CrossRef]

]. Additionally, scanning systems are preferred for better lateral resolution in the inspection of microscopic areas [17–19

17. B. S. Lee and T. C. Strand, “Profilometry with a coherence scanning microscope,” Appl. Opt. 29, 3784–3788 (1990). [CrossRef] [PubMed]

]. Interferometric scanning techniques can be combined with heterodyning or homodyning techniques in an effort to improve the lateral and/or the vertical resolution [20–21

20. J. Murakowski, M. Cywiak, B. Rosner, and D. van der Weide, “Far field optical imaging with subwavelength resolution,” Opt. Commun. 185, 295–303 (2000). [CrossRef]

].

The optical system proposed here is a scanning microscope interferometric heterodyne system intended to inspect microscopic areas with high vertical resolution (λ/100) and moderate lateral resolution (≈ λ), with high signal to noise ratio. It uses coherent light from a laser as the illuminating source, and it is based on the heterodyning of two beams in the form of a scanning interferometer. One of the beams, after being reflected by the surface under test, is directed to a photodiode where it is coherently superimposed with the first diffracted order of a Bragg-cell. To obtain a line profile, the object under test is then scanned.

The analytical description is based on the propagation of Gaussian beams by means of the scalar Fresnel diffraction integral. By using this integral, we show that the proposed system can work in appropriate conditions in which the size of the probe beam at the plane of detection is proportional to the local vertical height of the surface under test. At the same time, the amplitude of the time varying signal at the output of the photodetector, is also proportional to the same local vertical height.

To better describe the system, the presentation is divided in 7 sections including this one. In Sec. 2 we describe the experimental setup and analytical description. In Sec. 3 we present the calculation of the power detected by the photodiode. In Sec. 4 we discuss the selection of an operating point. In Sec. 5 we present experimental evidence to support our proposal. In Sec. 6 we present some experimental considerations. Finally, in Sec. 7 we give our conclusions.

2. Experimental setup and analytical description

Figure 1 depicts the experimental setup. A He-Ne laser with a Gaussian intensity profile is used as the coherent illuminating source. The illuminating beam is transmitted through a Bragg-cell that consists on an acousto-optical medium of tellurium dioxide (TeO2) excited at 80 MHz. Only the diffracted orders zero and one are used.

Fig. 1. Experimental setup. (x 0, y 0) are the coordinates of a plane at the output of the Bragg-cell. (x, y) are the coordinates of the focal plane I of lens L1. (x 1, y 1) are the coordinates of the focal plane II. (x 2, y 2) are the coordinates of the focal plane II when the beam reaches again this plane after being reflected from the object under test. (x F, y F) are the coordinates of the focal plane I when the beam reaches this plane again in its way towards the photodedector. Finally, (ξ, η) are the coordinates at the plane of detection. M1 and M2 are mirrors, BS1 and BS2 are 50-50 beam splitters, and L1 is the focusing lens. For the description, the distance between the focal plane II and the object plane has been exaggerated.

For simplicity we will assume that the waist of the laser Gaussian beam is located at the Bragg cell, which is placed at a coordinate plane (x 0, y 0). The photodetector is located at a coordinate plane (ξ,η).

The order-zero propagates towards the focal plane I of lens L1 located at a coordinate plane (x, y). Lens L1 is a 100x microscope objective commercially available with a focal length f of approximately 2 mm. This lens is suitable for capturing as much reflected light as possible from the surface under test in a simple manner. In the analytical treatment below, for simplicity and without loss of generality, the back and front focal lengths will be considered equal to value f.

After being reflected by the object, the probe beam is modulated in its phase by the local surface irregularities and propagates in the direction of lens L1, which transmits the beam towards the focal plane I of lens L1. From this plane, the beam propagates towards the photodetector plane located at a distance z 2 measured from the focal plane I.

The diffracted order one at the output of the Bragg-cell is directed to the photodetector trough the path given by mirrors M1, M2 and by beam splitter BS2. The overall length traveled by this beam is referred as z 3. The temporal frequency of this beam corresponds to the sum of the temporal frequency of the illuminating source and the frequency of excitation of the acousto-optical cell; this beam will be referred as the modulating beam. As depicted in Fig. 1, the two beams are superimposed and coherently added at the plane of a photodetector, whose sensitive area is large enough to integrate the overall intensity of the beams. The signal at the output of the photodetector is amplified and sent to a lock-in amplifier for A. C. detection and recording.

A simple qualitative picture of the system can be obtained by noticing that, at the plane of the photodetector, two Gaussian beams with different semi-widths and different radii of curvature are coherently superimposed. At this plane, the photodetector integrates the overall intensity of the beams. One of the beams (the modulating beam), has a fixed semi-width and a fixed radius of curvature and angular temporal frequency ωls. The second beam (the probe beam) exhibits a variable semi-width and variable radius of curvature; these variations are due to the local vertical height variations of the sample under test. This beam has an angular temporal frequency ωl. Figure 2 depicts the absolute amplitude distribution of both Gaussians at the plane of detection for beams with a power equal to one.

Fig. 2. Absolute amplitude distribution of the Gaussian beams at the plane of detection.

For clarity, we divide the calculations of the overall propagation of the zero-order-beam (probe beam) in five subsections:

2.1. Propagation of the beam from the output of the Bragg-cell to the focal plane I of lens L1.

2.2. Fourier transform performed by lens L1 (This corresponds to the propagation from focal plane I towards focal plane II).

2.3. Propagation from the focal plane II of lens L1 towards the object plane and back to this plane.

2.4. Fourier transform of the reflected beam performed by lens L1. (This corresponds to the propagation from focal plane II to focal plane I of lens L1).

2.5. Propagation of the beam from focal plane I of lens L1 to the photodetector plane.

The propagation of the modulating beam from the output of the Bragg-cell to the photodetector plane is easily performed in only one step and described at the end of this section. In what follows, we present our mathematical analysis.

2.1. Propagation of the beam from the output of the Bragg-cell, (x0, y0), to the focal plane I of lens L1 (x, y).

The amplitude distribution of the Gaussian beam at the plane of the Bragg cell is represented by,

Ψ(x0,y0)=(2P0πr02)12exp[x02+y02r02],
(1)

where P 0 is the beam power and r 0 is the beam radius of the Gaussian beam.

To calculate the amplitude distribution at the focal plane I of lens L1, located at a plane (x, y, z = z), we use the Fresnel diffraction integral [22

22. W. J. Goodman, Introduction to Fourier Optics, Second ed., Mc Graw-Hill, New York, 2000. Chap. 4, 5.

] as,

Ψ(x,y)=exp(iωlt)1iλzΨ(x0,y0)exp{iπλz[(xx0)2+(yy0)2]}dx0dy0,
(2)

ω l is the temporal angular frequency of the laser light, λ is the beam wavelength and i=1.

By substituting the amplitude distribution given in Eq. (1) into Eq. (2) and evaluating the integrals, Eq. (2) can be written as,

Ψ(x,y)=r02λzr022P0πr02exp(iωlt)exp[(π2r02iπλzλ2z2+π2r04)(x2+y2)],
(3)

where z is the distance between planes (x 0, y 0) and (x, y). Eq. (3) represents the amplitude distribution of the beam at the focal plane I of the lens L1.

2.2. Fourier transform performed by lens L1. (This corresponds to the propagation from focal plane I, (x, y) , towards focal plane II, (x1, y1).

The Fourier-transform performed by lens L1 is given by [22

22. W. J. Goodman, Introduction to Fourier Optics, Second ed., Mc Graw-Hill, New York, 2000. Chap. 4, 5.

],

Ψ(x1,y1)=1λfΨ(x,y)exp[-i2πλf(xx1yy1)]dxdy.
(4)

Applying the Fourier-transform given by Eq. (4) to the amplitude distribution given by Eq. (3), we obtain the transmitted amplitude distribution at the focal plane II of lens L1 located at (x 1, y 1, z = f) as

Ψ(x1,y1)=r02λf2P0πr02exp(iωlt)exp[(ππr02+iλzλ2f2)(x12+y12)].
(5)

2.3. Propagation from the focal plane II, (x1, y1), of lens L1 towards the object plane and back to this plane, (x2, y2).

The amplitude distribution given by Eq. (5) is again propagated using the Fresnel diffraction integral towards the object plane. To do this, we note that the probe beam propagates a definite distance to the reflecting surface, and that it travels back the same distance again to the focal plane II of lens L1. By defining zp this overall optical path length of propagation, the reflected amplitude distribution, precisely at the focal plane II of L1, located at a coordinated plane (x 2, y 2, z = zp), can be written as

Ψ(x2,y2)=1zpΨ(x1,y1)exp{iπλzp[(x2x1)2+(y2y1)2]}dx1dy1.
(6)

By substituting the amplitude distribution given in Eq. (5) into Eq. (6), and evaluating the integrals, Eq. (6) can be written as

Ψ(x2,y2)=πr02fπr02zp+(zzpf2)2P0πr02exp(iωlt)×exp{ππλf2r02i[π2r04zp+λ2z(zzpf2)]λ[π2r04zp2+λ2(zzpf2)2](x22+y22)}
(7)

Equation (7) is now used to find the amplitude distribution of the beam at the focal plane II of lens L1.

2.4. Fourier transform of the beam performed by lens L1. (This corresponds to the propagation from focal plane II, (x2, y2), to focal plane I of lens L1, (xF, yF)).

By calculating the Fourier-transform given by Eq. (4) when the reflected beam is transmitted by lens L1 in its path towards BS1, we obtain the amplitude distribution at the plane (xF, yF, z = f) as

Ψ(xF,yF)=A2P0πr02exp(iωlt)exp[(1r2+iπλR)(xF2+yF2)],
(8)

where A represents a complex amplitude, r the semi-width, and R the radius of curvature of the probe beam at this plane. Their respective values are given by,

A=r02πr02zp+(zzpf2)×π2r04zp2+λ2(zzpf2)2πλf2r02i[π2r04zp+λ2z(zzpf2)],
(8.1)
r=π2λ2f4r04+[π2r04zp+λ2z(zzpf2)]2[π2r04zp2+λ2(zzpf2)2]π2r02,
(8.2)

and,

R=f2×π2λ2f4r04+[π2r04zp+λ2z(zzpf2)]2[π2r04zp2+λ2(zzpf2)2][π2r04zp+λ2z(zzpf2)]
(8.3)

It will be noticed that the semi-width and the radius of curvature of the probe beam at the focal plane, according to Eq. (8.2) and (8.3), are respectively functions of the parameters z, zp, f.

2.5. Propagation of the beam from the focal plane I of L1, (xF, yF), to the photodetector plane (ξ,η).

The amplitude of the distribution given by Eq. (8) is finally propagated towards the photodetector plane using the Fresnel diffraction integral again. As depicted in Fig. 2, z 2 is the distance of propagation. By using the Fresnel diffraction integral, the amplitude distribution at the photodetector plane located in a coordinated plane (ξ, η, z = z) can be written as

Ψp(ξ,η)=B2P0πr02exp(iωlt)exp[(1rp2iπλRp)(ξ2+η2)],
(9)

where the amplitude B, the semi-width rp and the radius of curvature Rp of the probe beam at this plane are given by,

B=A×r2Rz2+r2(z2R)
(9.1)
rp=(Rλz2)2+π2r4(z2R)2π2r2R2
(9.2)

and

Rp=(Rλz2)2+π2r4(z2R)2()2z2+π2r4(z2R)2.
(9.3)

The set of equations (9), give the final expression of the amplitude distribution at the photodetector plane of the overall process that conveys the probe beam. As r,R are functions of zp as given by Eq. (8), rp, Rp given by Eqs. (9) are also functions of zp.

Finally the amplitude distribution of the modulating beam at the plane of detection has to be calculated in order to add this beam to the probe beam at this plane. For this, we notice that this beam propagates from the Bragg cell plane, (x 0, y 0), to the photodetector plane, (ξ, η), a distance z 3. Thus,

Ψm(ξ,η)=C2P0πr02exp[i(ωl+ωs)t]exp[(1rm2iπλRm)(ξ2+η2)].
(10)

where ωs is the excitation frequency applied to the acousto-optical cell, C is a complex amplitude, r m the semi-width and Rm the radius of curvature given as,

C=r02λz3r02
(10.1)
rm=λ2z32+π2r04π2r02
(10.2)

and

Rm=(λ2z32+π2r04)λ2z3.
(10.3)

Equations (10) give the amplitude distribution of the modulating beam at the plane of detection. The radius of curvature of this beam and its semi-width are constant, as the propagation distance z 3 remains constant.

3. Calculation of the power detected by the photodiode

For this calculation we will use the amplitude distributions given by Eqs. (9) and (10), which correspond to the probe and modulating beams respectively. As both beams are coherently superimposed at the plane of the photodiode, the amplitude distribution at this plane is given by,

ΨT(ξ,η)=Ψp(ξ,η)+Ψm(ξ,η),
(11)

and the corresponding intensity distribution by,

I(ξ,η)=ΨT(ξ,η)ΨT*(ξ,η),
(12)

where (*) is a complex conjugate. By substituting Eqs. (9) and (10) into Eq. (11), and using Eq. (12), gives for the intensity distribution at this plane,

Iξη=DB2exp[2rp2(ξ2+η2)]+DC2exp[2rm2(ξ2+η2)]+BC*Dexp[i(ωst)]exp[(α+)][ξ2+η2]+B*CDexp[i(ωst)]exp[(α)][ξ2+η2]
(13)

where ∣ ∣ is the magnitude of Eqs. (9.1) and (10.1) respectively, and

D=2P0πr02,
(13.1)
α=rm2+rp2rm2rp2,
(13.2)
β=πλRmRp(RpRm),
(13.3)

Equation (13) gives the overall intensity collected by the photodetector. It will be noticed that the intensity at the detection plane is a function of the radius of curvature and the semi-width of both beams, and is modulated by the frequency ωs.

Finally, the total collected power at the plane of detection (ξ, η) is calculated as,

P(zp)=Iξηdξdη.
(14)

Thus,

P=DB2πrp22+DC2πrm22+BC*Dπ(α)α2+β2exp(iωst)+B*CDπ(α+)α2+β2exp(iωst).
(15)

The overall power collected by the photodiode is given by Eq. (15), where the two first terms represent a DC level and the last two summands represent an AC component. The lock-in amplifier is tuned to receive only the AC component at the frequency ωs.

4. Selection of the operating point

We define the defocusing distance zp, as the operating point at which the object under test is to be placed. At the operating point, the amplitude of the A. C. component of the collected power is relatively high and sensitive to small variations of zp, according to the setup shown in Fig. 1, when all the other parameters of the set-up are considered fixed.

The first step to properly select the operating point is to calculate the semi-widths of the two beams at the plane of detection. Figure 3 shows a plot of the semi-width variations at the plane (ξ, η) for both, the modulating and the probe beam, as a function of zp according to Eqs. (9) and (10). The dotted straight line represents the semi-width of the modulating beam. The semi-width of the probe beam changes as zp varies. When zp = 0, the reflective object is placed precisely at the focal plane of lens L1 (Fig. 1). Negative values of zp indicate that the object is closer to the lens, while positive values of zp indicate that the object is moving away. The smallest semi-width of the probe beam does not coincide with the value zp = 0 due to the geometry of the set-up.

Fig. 3. Semi-widths of the probe (continuos trace) and modulating (dotted line) beams at the detection plane, (ξ, η), as functions of the defocusing distance zp.

The second step in the selection of the operating point consists in noticing that the collected power given by Eq. (15) can be written as,

P=PDC+PACcos(ωst+φ).
(16)

According to Eq. (16) the collected power is represented by a DC term plus a temporal sinusoidal carrier modulated in amplitude.

Since the time varying power collected by the photodetector changes when the object under test moves back and forth from the focal plane I of lens L1, thus the power collected by the photodiode will change for different distances zp. This collected power as a function of the distance zp can be calculated by using Eq. (15), as can be seen in Fig. 4, where a plot of PAC as a function of the distance zp is shown.

According to Eq. (16), low absolute numerical values of PAC , represent temporal signals with low amplitude, resulting in low signal to noise ratios. To obtain higher electrical temporal signals, high numerical absolute values of PAC must be selected. Additionally, to obtain high sensitivity, it is necessary to choose an operating point with a high slope. In our experiment, this can easily be achieved in a zone around zp = 3μ m, as shown in Fig. 4.

Fig. 4. Total collected power as a function of the defocusing distance zp. The operating point is selected around the value zp = 3μm, within the range marked by the little segments on the graph.

The behaviors shown in Figs. 3 and 4 were experimentally verified. As a very small region around the operating point is used (less than 200 nm) in the measurements, the collected power becomes a linear function of the profile under measurement.

5. Experimental results

In the experimental setup, the illuminating beam consisted of a 15 mW He-Ne laser with a wavelength of 632.8 nm and (1/e 2) beam radius of 0.325 mm. The excitation signal applied to the Bragg cell was a sinusoidal signal at 80 MHz. A GPIB channel allowed the lock-in to communicate with a personal computer for data recording and generation of the required electrical signals and synchronization. Linear scans for roughness recording are performed by means of a flexured piezoelectric translator (FPZT). A FPZT was used to obtain a displacement with a tilt as low as 9 μ rad in an overall scan of 100 μm. The probe beam around the operating point, at the object plane, had a semi-width of approximately 0.65 μm; this value can be estimated by using Eq. (7).

The sample under test consisted of a blazed- reflective grating with a pitch of 300 lines/mm. Before performing the measurements, we confirmed that the grating showed constant reflectivity in the zone of interest. The detected power was converted into a voltage signal, using a transimpedance amplifier with a total gain of approximately 24×103 V/W.

The experimental measures were obtained in a line scan of approximately 10 μm. One of the resulting scans is shown in Fig. 5. For comparative purposes, similar values were obtained by means of an atomic force microscope. One of these measurements is shown in Fig. 6.

Fig. 5. Profile obtained with the proposed technique for the sampled grating. The pitch is 300 lines/mm.
Fig. 6. Profile obtained with the atomic force microscope in a near vicinity of the measure shown in Fig. 5 when scanning a similar distance.

Some differences between measures can be observed and are attributed to the fact that it is not possible to measure precisely in the same zone with both techniques and due to the fact that each system responds to different physical characteristics. Additionally, the lateral resolution of the AFM is higher because it uses a very narrow probe as compared with the optical system. An extended discussion on this topic can found in Ref. [23

23. A. Kühle, B. Rosén, and J. Garnaes, “Comparison of roughness measurement with atomic force microscopy and interference microscopy,” Proc. SPIE 5188, 154–161 (2003). [CrossRef]

]. The measurements with the proposed technique were performed several times to assure repeatability. This topic is further discussed in the following section.

6. Experimental considerations

In this section some parameters that affect the experimental results are discussed.

In order to represent with reasonable precision the grating under measurement, 2000 pixels were recorded for each line scan of approximately 10μm.

From the experimental setup shown in Fig. 1, we may see, that it is necessary to have sufficient spatial separation between the zero-order and the modulating beams to allow the placement of the optical components. Thus, the frequency of the signal applied to the Braggcell should be high. However, using high temporal frequencies increase the complexity of the electronic components used; thus, a moderate frequency of 80 MHz was selected. Now, to widely meet with the Nyquist sampling criteria, the lock-in amplifier could easily register the amplitude of the temporal carrier, say, every 2 μs. This would represent a scanning rate of approximately 2.5μm/ms.

Although the modulation frequency can limit the scan rate, mechanical instabilities in the scanning mechanism represent the chief factor on this rate. This is because the mechanical assembly necessary to mount the surface-under test, supported by the FPZT, introduces undesirable oscillations for fast scanning rates. Thus, a moderate scan rate is needed. Experimentally, we found that for our mechanical assembly a scanning rate of the order 1μm/s can safely be selected. This mechanical scanning rate is so low that it does not affect the lateral resolution.

We plan, in a future work, to improve the scanning rate by using an alternative device such as a galvanometer. Improving the scanning rate will allow attaining measurements in real time of grating type objects, as it is the case of optical storage devices.

An additional parameter that should be considered is the tilt of the sample while scanning. This effect can be estimated as follows. The FZPT introduces a tilt of approximately 9 μ rad in an overall scan of 100 μm. When measuring a flat, this tilt will behave as a slope whose maximum vertical height will be approximately 0.9 nm. In the experimental results reported above, the effect of this tilt is very low as compared to the maximum vertical height of the sample that is approximately 110 nm.

Finally the method used to calibrate the system is described. In principle, the system could be absolutely calibrated by measuring each one of the parameters involved in the mathematical formulation given by Eq. (15) in an independent manner. However, this method can present some difficulties. Alternatively, the measurements obtained by means of the AFM were used for calibration purposes (although the comparison with respect to the measurements obtained with the AFM may not be considered an absolute method of calibration, this is a relatively good calibration procedure). For this to be done, several measurements were taken with the AFM, and similar measurements were taken with the proposed technique in approximately the same zone of the surface under test. As the lateral resolution of the methods is different, the average rms values of the different measurements were taken and fitted. Once the system had been calibrated, the repeatability of the system was considered. The standard deviation calculated after several measurements was 2 nm. With this result and considering the maximum effect of the tilt discussed above, the vertical sensitivity of the system can be estimated to a value of the order of λ/100.

7. Conclusions

A simple heterodyne system based on the propagation of Gaussian beams was presented. The system uses a focussed probe beam that is reflected from the surface under test. The reflected beam, collected by the focussing lens is directed to a detection plane where it is coherently superimposed with a reference beam at the sensitive plane of a photodiode. The reference beam is the first diffracted order obtained at the output of a Bragg-cell. The sensitive area of the photodiode integrates the overall intensity of the beams that present a Gaussian intensity profile. We showed analytically, by means of the Fresnel diffraction integral, that by appropriately choosing the parameters involved in the operation of the system, at the output of the photodetector a temporal signal is obtained whose amplitude is a linear function of the local vertical height of the object under test. The object is scanned to obtain a linear profile. We show experimental measurements of a sample consisted in a blazed-reflective grating with a pitch of 300 lines/mm. The profiles obtained with the proposed technique were compared to the measurements performed by means of an atomic force microscope. Although the lateral resolution of the methods is different, the experimental results indicate that the system exhibits a good approximation to the shape of the surface under test with respect to the measurements obtained by the AFM.

Acknowledgments

The authors thank CONACYT for financial support.

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W. J. Goodman, Introduction to Fourier Optics, Second ed., Mc Graw-Hill, New York, 2000. Chap. 4, 5.

23.

A. Kühle, B. Rosén, and J. Garnaes, “Comparison of roughness measurement with atomic force microscopy and interference microscopy,” Proc. SPIE 5188, 154–161 (2003). [CrossRef]

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.3940) Instrumentation, measurement, and metrology : Metrology
(240.6700) Optics at surfaces : Surfaces

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: March 2, 2007
Revised Manuscript: May 10, 2007
Manuscript Accepted: June 1, 2007
Published: June 19, 2007

Citation
J. Mauricio Flores, Moisés Cywiak, Manuel Servín, and Lorenzo Juárez P., "Heterodyne two beam Gaussian microscope interferometer," Opt. Express 15, 8346-8359 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8346


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References

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  18. B. Barrientos, M. Cywiak and M. Servín, "Profilometry of optically smooth surfaces by a Gaussian probe beam," Opt. Eng. 42, 3004-3012 (2003). [CrossRef]
  19. M. Cywiak, J. F. Aguilar and B. Barrientos, "Low-numerical-aperture Gaussian beam confocal system for profiling optically smooth," Opt. Eng. 44, 1-7 (2005). [CrossRef]
  20. J. Murakowski, M. Cywiak, B. Rosner, and D. van der Weide, "Far field optical imaging with subwavelength resolution," Opt. Commun. 185, 295-303 (2000). [CrossRef]
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  22. W. J. Goodman, Introduction to Fourier Optics, Second ed., (Mc Graw-Hill, New York, 2000). Chaps. 4, 5.
  23. A. Kühle, B. Rosén and J. Garnaes, "Comparison of roughness measurement with atomic force microscopy and interference microscopy," Proc. SPIE 5188, 154-161 (2003). [CrossRef]

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