## Heterodyne two beam Gaussian microscope interferometer

Optics Express, Vol. 15, Issue 13, pp. 8346-8359 (2007)

http://dx.doi.org/10.1364/OE.15.008346

Acrobat PDF (535 KB)

### 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

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

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

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]

## 2. Experimental setup and analytical description

_{2}) excited at 80 MHz. Only the diffracted orders zero and one are used.

*x*

_{0},

*y*

_{0}). The photodetector is located at a coordinate plane (ξ,η).

*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*.

*z*/2 from the focal plane II of lens L1. This distance is introduced in this way to allow simplification of the mathematical analysis as will be apparent in the next section.

_{p}*z*

_{2}measured from the focal plane I.

*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.

_{l}+ω

_{s}. 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 ω

_{l}. Figure 2 depicts the absolute amplitude distribution of both Gaussians at the plane of detection for beams with a power equal to one.

### 2.1. *Propagation of the beam from the output of the Bragg-cell*, (*x*_{0}, *y*_{0}), *to the focal plane I of lens L1* (*x*, *y*).

*P*

_{0}is the beam power and

*r*

_{0}is the beam radius of the Gaussian beam.

*x*,

*y*,

*z*=

*z*), we use the Fresnel diffraction integral [22] as,

*ω*

_{l}is the temporal angular frequency of the laser light, λ is the beam wavelength and

*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, (x_{1}, y_{1}).

### 2.3. Propagation from the focal plane II, (x_{1}, y_{1}), of lens L1 towards the object plane and back to this plane, (x_{2}, y_{2}).

*z*this overall optical path length of propagation, the reflected amplitude distribution, precisely at the focal plane II of L1, located at a coordinated plane (

_{p}*x*

_{2},

*y*

_{2},

*z*=

*z*), can be written as

_{p}### 2.4. Fourier transform of the beam performed by lens L1. (This corresponds to the
propagation from focal plane II, (*x*_{2}, *y*_{2}), to focal plane I of lens L1, (*x*_{F}, *y*_{F})).

*x*,

_{F}*y*,

_{F}*z*=

*f*) as

*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,

### 2.5. Propagation of the beam from the focal plane I of L1, (*x*_{F}, *y*_{F}), to the photodetector plane (ξ,η).

_{F}

_{F}

*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

*B*, the semi-width

*r*and the radius of curvature

_{p}*R*of the probe beam at this plane are given by,

_{p}*r*,

*R*are functions of

*z*as given by Eq. (8),

_{p}*r*,

_{p}*R*given by Eqs. (9) are also functions of

_{p}*z*.

_{p}*x*

_{0},

*y*

_{0}), to the photodetector plane, (ξ, η), a distance

*z*

_{3}. Thus,

_{s}is the excitation frequency applied to the acousto-optical cell,

*C*is a complex amplitude,

*r*

_{m}the semi-width and

*R*the radius of curvature given as,

_{m}*z*

_{3}remains constant.

## 3. Calculation of the power detected by the photodiode

_{s}.

_{s}.

## 4. Selection of the operating point

*z*, 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

_{p}*z*, according to the setup shown in Fig. 1, when all the other parameters of the set-up are considered fixed.

_{p}*z*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

_{p}*z*varies. When

_{p}*z*= 0, the reflective object is placed precisely at the focal plane of lens L1 (Fig. 1). Negative values of

_{p}*z*indicate that the object is closer to the lens, while positive values of

_{p}*z*indicate that the object is moving away. The smallest semi-width of the probe beam does not coincide with the value

_{p}*z*= 0 due to the geometry of the set-up.

_{p}*z*. This collected power as a function of the distance

_{p}*z*can be calculated by using Eq. (15), as can be seen in Fig. 4, where a plot of

_{p}*P*as a function of the distance

_{AC}*z*is shown.

_{p}*P*, represent temporal signals with low amplitude, resulting in low signal to noise ratios. To obtain higher electrical temporal signals, high numerical absolute values of

_{AC}*P*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

_{AC}*z*= 3

_{p}*μ*

*m*, as shown in Fig. 4.

*nm*) in the measurements, the collected power becomes a linear function of the profile under measurement.

## 5. Experimental results

*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).

*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 24×10

^{3}

*V/W*.

*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.

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]

## 6. Experimental considerations

*μm*.

*μs*. This would represent a scanning rate of approximately 2.5

*μm*/

*ms*.

*μm*/

*s*can safely be selected. This mechanical scanning rate is so low that it does not affect the lateral resolution.

*μm*, being of the order of λ. As seen in Fig 5., the lateral resolution limitation smooths the measured profile of the surface under test with reference to the AFM showed in Fig 6.

*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*.

*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

*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

## References and links

1. | J. M. Bennett, “Comparison of techniques for measuring the roughness of optical surfaces,” Opt. Eng. |

2. | J. M. Bennett and J. H. Dancy, “Stylus profiling instrument for measuring statistical properties of smooth optical surfaces,” Appl. Opt. |

3. | D. Walker, H. Yang, and S. Kim, “Novel hybrid stylus for nanometric profilometry for large optical surfaces,” Opt. Express |

4. | H. J. Tiziani, “Optical methods for precision measurements,” Opt. Quantum Electron. |

5. | S. R. Clark and J. E. Greivenkap, “Optical reference profilometry,” Opt. Eng. |

6. | G. S. Kino and S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. |

7. | W. Zhou, Z. Zhou, and G. Chi, “Investigation of common-path interference profilometry,” Opt. Eng. |

8. | M. B. Suddendorf, C. W. See, and M. G. Somekh, “Combined differential amplitude and phase interferometer with a single probe beam,” Appl. Phys. Lett , |

9. | Z. F. Zhou, T. Zhang, W. Zhou, and W. Li “Profilometer for measuring superfine surfaces,” Opt. Eng. |

10. | G. E. Sommargren, “Optical heterodyne profilometry,” Appl. Opt. |

11. | C-C. Huang, “Optical heterodyne profilometer,” Opt. Eng. |

12. | J. C. Wyant, “Optical profilers for surface roughness,” Proc. SPIE |

13. | M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection and Process Control, K. M. Monahan, ed., Proc. SPIE |

14. | P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. |

15. | G. W. Johnson, D. C. Leiner, and D. T. Moore, “Phase-locked Interferometry,” Proc. SPIE |

16. | K. Creath and J. C. Wyant, “Absolute measurement of surface roughness,” Appl. Opt. |

17. | B. S. Lee and T. C. Strand, “Profilometry with a coherence scanning microscope,” Appl. Opt. |

18. | B. Barrientos, M. Cywiak, and M. Servín, “Profilometry of optically smooth surfaces by a Gaussian probe beam,” Opt. Eng. |

19. | M. Cywiak, J. F. Aguilar, and B. Barrientos, “Low-numerical-aperture Gaussian beam confocal system for profiling optically smooth,” Opt. Eng. |

20. | J. Murakowski, M. Cywiak, B. Rosner, and D. van der Weide, “Far field optical imaging with subwavelength resolution,” Opt. Commun. |

21. | M. Cywiak, J. Murakowski, and G. Wade., “Beam blocking method for optical characterization of surfaces,” IJIST |

22. | 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 |

**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

- J. M. Bennett, "Comparison of techniques for measuring the roughness of optical surfaces," Opt. Eng. 24, 380-387 (1985).
- J. M. Bennett and J. H. Dancy, "Stylus profiling instrument for measuring statistical properties of smooth optical surfaces," Appl. Opt. 20, 1785 (1981). [CrossRef] [PubMed]
- D. Walker, H. Yang and S. Kim, "Novel hybrid stylus for nanometric profilometry for large optical surfaces," Opt. Express 11, 1793-1798 (2003). [CrossRef] [PubMed]
- H. J. Tiziani, "Optical methods for precision measurements," Opt. Quantum Electron. 21, 253-282 (1989). [CrossRef]
- S. R. Clark, and J. E. Greivenkap, "Optical reference profilometry," Opt. Eng. 40, 2845 (2001). [CrossRef]
- G. S. Kino and S. S. C. Chim, "Mirau correlation microscope," Appl. Opt. 29, 3775-3783 (1990). [CrossRef] [PubMed]
- W. Zhou, Z. Zhou and G. Chi, "Investigation of common-path interference profilometry," Opt. Eng. 36, 3172-3175 (1997). [CrossRef]
- M. B. Suddendorf, C. W. See, M. G. Somekh, "Combined differential amplitude and phase interferometer with a single probe beam," Appl. Phys. Lett, 67, 28-30 (1995). [CrossRef]
- Z. F. Zhou, T. Zhang, W. Zhou and W. Li "Profilometer for measuring superfine surfaces," Opt. Eng. 40, 1646-1652 (2001). [CrossRef]
- G. E. Sommargren, "Optical heterodyne profilometry," Appl. Opt. 20, 335-343 (1981). [CrossRef]
- C-C. Huang, "Optical heterodyne profilometer," Opt. Eng. 23, 365-370 (1984).
- J. C. Wyant, "Optical profilers for surface roughness," Proc. SPIE 525, 174-180 (1985).
- M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, "An application of interference microscopy to integrated circuit inspection and metrology," Proc. SPIE 775, 233-247 (1987).
- P. J. Caber, "Interferometric profiler for rough surfaces," Appl. Opt. 32, 3438-3441 (1993). [CrossRef] [PubMed]
- G. W. Johnson, D. C. Leiner and D. T. Moore, "Phase-locked Interferometry," Proc. SPIE 126, 152-160 (1977).
- K. Creath and J. C. Wyant, "Absolute measurement of surface roughness," Appl. Opt. 29, 3823-3827 (1990). [CrossRef] [PubMed]
- B. S. Lee and T. C. Strand, "Profilometry with a coherence scanning microscope," Appl. Opt. 29, 3784-3788 (1990). [CrossRef] [PubMed]
- B. Barrientos, M. Cywiak and M. Servín, "Profilometry of optically smooth surfaces by a Gaussian probe beam," Opt. Eng. 42, 3004-3012 (2003). [CrossRef]
- 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]
- 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]
- M. Cywiak, J. Murakowski, and G. Wade., "Beam blocking method for optical characterization of surfaces," Int. J. Imaging Syst. Technol. 11, 164-169 (2000).
- W. J. Goodman, Introduction to Fourier Optics, Second ed., (Mc Graw-Hill, New York, 2000). Chaps. 4, 5.
- 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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