## Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator

Optics Express, Vol. 14, Issue 25, pp. 12109-12121 (2006)

http://dx.doi.org/10.1364/OE.14.012109

Acrobat PDF (848 KB)

### Abstract

As an alternative to the conventional optical frequency comb technique, a spatial frequency comb technique is proposed for dispersionfree optical coherence depth sensing. Instead of generating an optical frequency comb over a wide range of time spectrum, we generate a spatial frequency comb by modulating the incident angle of a monochromatic plane wave with a spatial light modulator (SLM). The use of monochromatic light combined with the SLM enables dispersion-free depth sensing that is free from mechanical moving components.

© 2006 Optical Society of America

## 1. Introduction

1. K. Hotate and T. Okugawa, “Selective extraction of a two-dimensional optical image by synthesis of the coherence function,” Opt. Lett. **17**, 1529–1531 (1992). [CrossRef] [PubMed]

4. K. Hotate and T. Okugawa, “Optical information processing by synthesis of the coherence function,” J. Lightwave Technol. **12**, 1247–1255 (1994). [CrossRef]

5. Z. He and K. Hotate, “Synthesized optical coherence tomography for imaging of scattering objects by use of a stepwise frequency-modulated tunable laser diode,” Opt. Lett. **24**, 1502–1504 (1999). [CrossRef]

6. S. Choi, M. Yamamoto, D. Moteki, T. Shioda,Y. Tanaka, and T. Kurokawa, “Frequency-comb-based interferometer for profilometry and tomography,” Opt. Lett. **31**, 1976–1978 (2006). [CrossRef] [PubMed]

7. P. A. Flournoy, R. W. McClure, and G. Wyntjes, “White-light interferometric thickness gauge,” Appl. Opt. , **11**, 1907–1915 (1972). [CrossRef] [PubMed]

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

13. J. Schwider and L. Zhou, “Dispersive interferometric profilometer,” Opt. Lett. **19**, 995–997 (1994). [CrossRef] [PubMed]

## 2. Principle

*k*

_{h}=

*k*cos

*θ*is the height component of the vector

**, and**

*k**θ*is the angle of incidence to the reference surface defined by the angle between the vector

**and the height vector**

*k***.**

*h**to be parallel to the height vector*

**k****so as to maximize the fringe sensitivity such that Δ**

*h**φ*=-2

*hk*with

*θ*=0. The OFC is formed with equally-spaced multiple line spectra corresponding to wavenumbers

*k*=

*nΔω*/

*c*, with

*n*and Δ

*ω*being an integer and the modefrequency separation, respectively. This characteristic of the OFC is illustrated in the k-vector space of Ewald sphere shown in Fig. 2.

**(0) is parallel to the height vector**

*k***such that**

*h**θ*=0. Under this illumination, the OFC corresponds to a set of collinear k-vectors aligned parallel to the vector

**(0) with their arrow tips equally spaced at an interval Δ**

*k**k*=Δ

*ω*/

*c*on the line through the center of the Ewald sphere. These radially distributed k-vectors inside the Ewald sphere cause the dispersion problems as they correspond to multiple optical frequencies. If we take a closer look at Eq. (1), we note an alternative solution in which we change the angle

*θ*while keeping the optical frequency constant. In the k-space shown in Fig. 2, this operation corresponds to changing the cone angle

*θ*of the

**(**

*k**θ*) vector while keeping the radius of the k-sphere unchanged. The projected height component -

*k*

_{h}of the

**(**

*k**θ*) vector plays the role of the

**(0) vector in the OFC. For example, if one can change**

*k**θ*over 0~30 degrees for the wavelength of 633nm, one can in principle realize the dispersion-free measurement with the performance comparable to the OFC with the wavelength range as wide as 98nm. We use a set of collimated monochromatic beams with different k-vectors angles whose longitudinal spatial frequency components

*k*

_{h}are equally spaced in the direction of

**, and refer to this technique as the spatial frequency comb (SFC) technique to differentiate it from the conventional optical frequency comb (OFC) technique. To generate the equally spaced SFC spectra equivalent to those of the OFC, the lateral component of the k-vector,**

*h**k*

_{⊥}(

*θ*), which is the projection of

**(**

*k**θ*) onto the plane normal to the

**(0) vector, should have a concentric Fresnel zone-plate structure as shown in Fig. 2(b). Just as the OFC generates a temporal coherence comb function with periodic coherence peaks with a time interval Δ**

*k**τ*=

*π*/Δ

*ω*, the SFC generates a longitudinal spatial coherence comb function with periodic coherence peaks with an axial distance interval Δ

*h*=

*π*/Δ

*k*

_{h}(where the double optical path in the reflection interferometer has been taken into account). The Fourier-transform relation of Wiener-Khinchin theorem between the temporal coherence function and the optical frequency spectrum is now replaced by the Fourier-transform relation of the generalized van Cittert-Zernike theorem between the longitudinal spatial coherence function and the longitudinal component of the spatial frequency spectrum, similarly to the McChutchen theorem [24

24. C. W. McCutchen, “Generalized source and the van Cittert-Zernike theorem: A study of spatial coherence required for interferometry,” J. Opt. Soc. Am. **56**, 727–733 (1966). [CrossRef]

*u*

_{A}and

*u*

_{A′}, respectively. They are created as a superposition of the angular spectra of collimated beams from an extended quasimonochromatic source placed on the focal plane of the collimator lens:

*and*

**r**_{A}*are position vectors pointing at point A and point A’ from an arbitrarily chosen origin, and the integration is taken over the surface of the Ewald sphere assuming a quasi-monochromatic source. The mutual intensity between the two points A and A’ are given by*

**r**_{A′}*A*(

**)**

*k**A**(

**′)〉=|**

*k**A*(

**)|**

*k*^{2}

*Δ*(

**-**

*k***′) with 〈〉 being an ensemble average, and have used the relation**

*k**d*=2

**k***πk*sin

*θkdθ*=2

*πkdk*

_{h}about the ring surface element on the Ewald sphere. The complex degree of coherence is therefore given by

*k*

_{hmax}=

*k*and the comb spatial frequency interval Δ

*k*

_{h}

*π*/Δ

*k*

_{h}as in the Hotate scheme, and the depth resolution will be given by

*Δh*=2

*π*/(

*N*Δ

*k*

_{h}), which is inversely proportional to the total comb frequency bandwidth.

22. Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free absolute interferometry based on angular spectrum scanning,” Opt. Express **14**, 655–663 (2006). [CrossRef] [PubMed]

_{1}is normal to the reference mirror surface M

_{R}and is parallel to the height vector by virtue of reflection at beam splitter BS. Then the

*h*-component of the angular spectrum

*k*

_{h}can be written as

*r*is the radius of the ring source, and in arriving at the last expression, the paraxial assumption has been made that

*f*≫

*r*. One can generate a SFC with the maximum angular spatial frequency

*k*

_{hmax}=

*k*and the comb spatial frequency interval Δ

*k*

_{h}by adjusting the radius of the

*n*-th ring source as

22. Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free absolute interferometry based on angular spectrum scanning,” Opt. Express **14**, 655–663 (2006). [CrossRef] [PubMed]

_{R}has a tilt with the normal vectors

**, and**

*n***′ are not aligned to the optical axis. For this geometry,**

*n***·(**

*k***-**

*r*_{A}**) in Eq. (4) becomes**

*r*_{A′}**·2**

*k***′=2**

*h**k*

_{h′}

*h*′, with

**′ being a new vector**

*h***′, the projected spatial frequency spectrum |**

*h**A*(

*k*

_{h′})|

^{2}no longer has the spectral form of an equally spaced comb function. As the result of projection of the k-vectors of the beams from the ring sources, |

*A*(

*k*

_{h′})|

^{2}can even have a continuous broadband spectrum, as shown in green color in Fig. 4(a), and gives rise to low spatial coherence for all points except for the point

**′=0. In this way, the SFC generated by ring sources is extremely sensitive to a tilt alignment error of the reference mirror. This sensitivity to tilt may first look a disadvantage but our experience in experiments has revealed that this is in fact a great advantage in monitoring and adjusting the optical alignment. This characteristic of a ring source has also been used successfully for reducing the effects of coherent artifacts in interferometry by Kuechel [25]. Furthermore, this influence of mirror tilt can easily be compensated for by rotating the Ewald sphere to align its axis to the new height-vector**

*h***′, as shown in Fig. 4(b). To generate the equally spaced comb spectra in the direction of the**

*h**h*’- spatial frequency component, the lateral component of the k-vector

*k*

_{⊥}(

*θ*), which is the projection of

**(**

*k**θ*) onto the plane normal to the

**(0) vector, should now have a decentered ellipse structure as shown in Fig. 4(b); the center of the shifted ellipse for the**

*k**n*-th SFC spectrum is given by (

*k*-

*n*Δ

*k*

_{h′})

**′**

*n*_{⊥}, with

**′**

*n*_{⊥}being the lateral component of the unit normal vector of the virtual reference mirror M

_{R}’. It should be noted that by combining the tilt of the reference mirror with the adequately tailored spatially incoherent elliptical sources generated by SLM, we can control the direction of the depth sensing. This is one of the unique characteristics of the SFC technique as compared with the optical frequency comb technique.

## 3. Relation to other techniques

*k*

_{h}and the same bandwidth

*K*

_{h}=(

*N*-1) Δ

*k*

_{h}such that

*h*=0 and

*h*=±

*π*/Δ

*k*

_{h}. When projected onto the lateral plane normal to the

**vector as in Fig. 2(b), the SSFC becomes a zone plate in terms of the lateral component of the k-vector**

*h*26. J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. **37**, 4107–4111 (2000). [CrossRef]

27. M. Gokhler and J. Rosen, “Synthesis of a multiple-peak spatial degree of coherence for imaging through absorbing media,” Appl. Opt. **44**, 2921–2927 (2005). [CrossRef] [PubMed]

*k*

_{h}to these sectors, so that each sector generates coherence peaks at different locations. The advantage of the Gokhler-Rosen scheme is that the multiple coherence peaks need not be equally spaced as in the case of the SFC technique. However, since their technique adheres to the sinusoidal comb, the modulus value of the coherence peaks is halved as compared with the central peak, as seen in Eq. (11), and requires gray level control of SLM for its ideal performance. Furthermore, because of the sectored structure of the multiple zone plates, directional anisotropy is introduced to the sensitivity to tilt and the immunity to shading.

## 4. Experiments

### 4.1. Experimental setup

_{2}, L

_{3}and a diaphragm D

_{2}, which functions as a spatial filter to remove the effect of discrete pixels of DMD and to adjust the illumination level. A SFC displayed on DMD is relayed by the confocal lens pair and imaged onto a rotating ground glass GG placed on the front focal plane of lens L

_{4}, to generate a spatially incoherent source. The light from the source created on this rotating ground glass is introduced into a Michelson interferometer composed of a beam splitter BS, a reference mirror M

_{R}, and an object GB made of a pair of gauge brocks. Lens L

_{5}images the interference fringe pattern on the object surface onto CCD. A quarter-wave plate QWP is placed directly behind the laser to generate a circularly polarized light for uniformity of reflection at DMD. The object is a pair of gauge blocks with heights h

_{1}=1.600mm and h

_{2}=2.000mm, which are placed on an optical plate. Lens L

_{5}is focused on the surfaces of the gauge blocks. The magnification of the imaging lens L

_{5}is adjusted to ~1×. The focal length of lens L

_{2}, L

_{3}and L

_{4}are

*f*

_{2}=120mm,

*f*

_{3}=250mm and

*f*

_{4}=150mm, respectively. Before starting the measurement, we first performed alignment by placing the center of the ring sources on the axis of lens L

_{4}so as to ensure that the angular spectrum from each point source on the ring gives the same spatial comb-frequency component

*k*

_{h}=

*k*cos

*θ*. The detail of the alignment procedure has been described in our previous paper [22

22. Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free absolute interferometry based on angular spectrum scanning,” Opt. Express **14**, 655–663 (2006). [CrossRef] [PubMed]

### 4.2. Spatial frequency comb of fireworks

*A*(

*k*

_{h′})|

^{2}cannot create such an ideal SFC with uniform comb heights as modeled in Eq. (6), but it will generate a tapered comb whose comb height varies in proportion to the peripheral length of the ring sources. One could solve this problem by adjusting the irradiance of the ring sources by introducing the gray level control of SLM, but this would compromise the advantage of SFC over SSFC. One could also control the thickness of the ring sources, but inner rings would become too thick to function as the ideal sources for comb spectra. To get around this problem, we propose a new implementation scheme which we call a spatial frequency comb of fireworks (SFCF). As its name stands for, the source structure looks like a firework as shown in Fig. 6(a). In stead of a continuous ring structure, we adopted an array structure of discrete point sources of the same irradiance on a polar mesh. Specifically, we placed 18 discrete point sources at equal azimuthal angular intervals on the periphery of the ring. In Fig. 6(b) shows the radial distribution of the irradian ce on the source plane, and (c) shows the the corresponding comb spectrum in the spatial frequency domain. We excluded the central point source at

*r*

_{0}and its spectrum

*k*

_{0}, because 18 point sources on the ring degenerate into a single point source at the center, and the comb spectrum for this point is reduced down to 1/18.

*M*×

*M*pixel matrix of the DMD, where

*M*=600, and the pixel size of the DMD is Δ

*x*Δ

*y*=13.7µm×13.7µ. Considering the magnification of the confocal lens system composed of L

_{2}(

*f*

_{2}=120mm) and L

_{3}(

*f*

_{3}=250mm), and the field angle of the Fourier transform lens L

_{4}(

*f*

_{4}=150mm), the full spatial frequency bandwidth of the SFCF can be calculated as

*k*(

*f*

_{3}/

*f*

_{2})

^{2}(

*M*Δ

*x*/2)

^{2}/(2

*N*rings, the spatial comb frequency interval is given by

*c*is light velocity. The corresponding coherence comb interval becomes

*N*in the firework source as a parameter specifying SFCF.

### 4.3. Characteristic of longitudinal coherence comb function

*h*by scanning the reference mirror. We carried out experiments for two different SFCFs with the ring parameters

*N*=8 and

*N*=16. The results are shown in Fig. 7. As predicted from Eq. (15), high contrast fringes were observed when the height becomes integer multiple of Δ

*h*, which means

*h*=1.6mm for the SFCF with

*N*=8, and

*h*=3.2mm for the SFCF with

*N*=16.

### 4.4. Depth ssensing by spatial comb frequency scanning

*N*of SFCF from 1 to 32. Several examples of fringe patters selected from the total 32 steps are shown in Fig. 8. During 32-step scanning, high-contrast fringes appeared at N

_{L}=2, 4, 8, and 16 on the left surface, and at N

_{R}=7 and 14 on the right surface of the gauge block set. From Eq. (15), the heights of the left and right surfaces,

*h*

_{L}and

*h*

_{R}, relative to the reference mirrors are given by the integer multiple of the coherence comb intervals for the largest SFCF parameters

*N*

_{L}=16 and

*N*

_{R}=14, such that

*h*

_{L}=3.2

*m*

_{L}[mm] and

*h*

_{R}=2.8

*m*

_{R}[mm], with

*m*

_{L}and

*m*

_{R}being integer numbers. Our a priori knowledge that the heights do not exceed 6.4mm and their difference

*h*

_{L}-

*h*

_{R}is less than 0.5mm permits us to uniquely determine the integers as

*m*

_{L}=1 and

*m*

_{R}=1. We therefore have

*h*

_{L}=3.2mm and

*h*

_{R}=2.8mm, and their height difference 0.4mm is in agreement to the nominal height-difference (400 µm) of the block gauge set.

### 4.5. Relation of SFC to angular spectrum scanning

*h*

_{L}=3.2mm with respect to the reference mirror, and generated a SFCF with eight rings

*N*=8, which is tuned to the left surface by its secondary coherence peak as shown in Fig. 7, but which is detuned to the right surface whose height is

*h*

_{R}=2.8mm. As seen in Fig. 9(a), the contrast of fringes on the left surface is high, but the fringe contrast on the right surface is much lower. Instead of illuminating all the rings simultaneously, we illuminated only the point sources on a selected single ring at a time. The selected ring is shifted sequentially from the inner ring to the outer ring, and the fringes on the left and right surfaces are observed at each step of the sequential ring shift. In Fig. 9(b) and 9(c) show fringe patterns observed by the illumination with the first and third ring sources of the SFCF (N=8), respectively. Note the phase shift of the fringes on the right surface between (b) and (c). As shown in Fig. 9(d), the fringe intensity on the left surface does not change significantly with the shift of the ring source because the SFCF (

*N*=8) is tuned on this surface; the slight decrease of the intensity is due to the vignetting of the light beam by the limited acceptance angle of the interferometer. On the other hand, the fringe intensity on the SFCF-detuned right surface varies significantly with the shift of the ring source, as seen in Fig. 9(e), serving as the evidence for the predicted large phase change for the SFCF-detuned height. Next we increased the comb parameter of

*N*of the SFCF to 32, which is tuned to the distance 6.4mm by its first coherence peak, with 1/4 comb frequency interval. Fig. 9(g) shows the sinusoidal variation of the fringe intensity on the left surface with the shift of the ring source by the (four times finer) comb frequency interval for the SFCF (

*N*=32); this operation is similar to the angular spectrum scanning technique in our previous paper. The red circles indicate the intensity at the ring position for the SFCF (

*N*=8) tuned to the left surface. Note that the fringes are all in phase at the ring position of the tuned SFCF (

*N*=8), which is in agreement to our theoretical observation.

### 4.6. Comparison between SFCF and SSFC

*N*=4, 8, 16, and then switched DMD from a binary to a grey mode of operation using an ALP controlling board (ViALUX, GmbH) and sequentially generated a SSFC of the corresponding parameters

*N*=4, 8, 16. The result is shown in Fig. 10. Whereas the SFCF produces the multiple coherence peaks (the primary coherence peak for

*N*=8, and the secondary coherence peak for

*N*=4), the SSFC produces only one coherence peak (the primary coherence peak for

*N*=8), as predicted from Eq. (11).

## 5. Conclusion

## Acknowledgments

1. | K. Hotate and T. Okugawa, “Selective extraction of a two-dimensional optical image by synthesis of the coherence function,” Opt. Lett. |

2. | K. Hotate and O. Kamatani, “Optical coherence domain reflectometry by synthesis of coherence function,” J. Lightwave Technol. |

3. | S.-J. Lee, B. Widiyatmoko, M. Kourogi, and M. Ohtsu, “Ultrahigh scanning speed optical coherence tomography using optical frequency comb generators,” Jpn. J. Appl. Phys. Part II , |

4. | K. Hotate and T. Okugawa, “Optical information processing by synthesis of the coherence function,” J. Lightwave Technol. |

5. | Z. He and K. Hotate, “Synthesized optical coherence tomography for imaging of scattering objects by use of a stepwise frequency-modulated tunable laser diode,” Opt. Lett. |

6. | S. Choi, M. Yamamoto, D. Moteki, T. Shioda,Y. Tanaka, and T. Kurokawa, “Frequency-comb-based interferometer for profilometry and tomography,” Opt. Lett. |

7. | P. A. Flournoy, R. W. McClure, and G. Wyntjes, “White-light interferometric thickness gauge,” Appl. Opt. , |

8. | N. Tanaka, M. Takeda, and K. Matsumoto, “Interferometrically measuring the physical properties of test object,” United States Patent 4,072,422 (filed October 20 1976, 1978). |

9. | M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in |

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

11. | T. Dresel, G. Hausler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. |

12. | D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science |

13. | J. Schwider and L. Zhou, “Dispersive interferometric profilometer,” Opt. Lett. |

14. | M. Takeda and H. Yamamoto, “Fourier-transform speckle profilometry: Three-dimensional shape measurements of diffuse objects with large height steps and/or spatially isolated surfaces,” Appl. Opt. |

15. | S. Kuwamura and I. Yamaguchi, “Wavelength scanning profilometry for real-time surface shape measurement,” Appl. Opt. |

16. | H. J. Tiziani, B. Franze, and P. Haible, “Wavelength-shift speckle interferometry for absolute profilometry using mode-hope free external cavity diode laser,” J. Mod. Opt. |

17. | M. Kinoshita, M. Takeda, H. Yago, Y. Watanabe, and T. Kurokawa, “Optical frequency-domain microprofilometry with a frequency-tunable liquid-crystal Fabry-Perot etalon device,” Appl. Opt. |

18. | D. S. Mehta, M. Sugai, H. Hinosugi, S. Saito, M. Takeda, T. Kurokawa, H. Takahashi, M. Ando, M. Shishido, and T. Yoshizawa, “Simultaneous three-dimensional step-height measurement and high-resolution tomographic imaging with a spectral interferometric microscope,” Appl. Opt. |

19. | M. Takeda, “The philosophy of fringes: Analogies and dualities in fringe generation and analysis,” in |

20. | M. Takeda, “Space-time analogy in absolute optical profilometry using frequency-scan techniques: A tutorial review,” in |

21. | J. Ch. Vienot, R. Ferriere, and J. P. Goedgebur, “Conjugation of space and time variables in optics,” in |

22. | Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free absolute interferometry based on angular spectrum scanning,” Opt. Express |

23. | P. D. Ruiz and J. M. Huntley, “Depth-resolved displacement measurement using tilt scanning speckle interferometry,” in |

24. | C. W. McCutchen, “Generalized source and the van Cittert-Zernike theorem: A study of spatial coherence required for interferometry,” J. Opt. Soc. Am. |

25. | M. Kuechel, “Apparatus and method(s) for reducing the effects of coherent artifacts in an interferometry,” US Patent 6804011 B2, (2004) or US Patent Application 2003/0030819 A1, (2003). |

26. | J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. |

27. | M. Gokhler and J. Rosen, “Synthesis of a multiple-peak spatial degree of coherence for imaging through absorbing media,” Appl. Opt. |

**OCIS Codes**

(120.2650) Instrumentation, measurement, and metrology : Fringe analysis

(120.2830) Instrumentation, measurement, and metrology : Height measurements

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: August 24, 2006

Revised Manuscript: September 22, 2006

Manuscript Accepted: September 22, 2006

Published: December 11, 2006

**Citation**

Zhihui Duan, Yoko Miyamoto, and Mitsuo Takeda, "Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator," Opt. Express **14**, 12109-12121 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-12109

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### References

- K. Hotate and T. Okugawa, "Selective extraction of a two-dimensional optical image by synthesis of the coherence function," Opt. Lett. 17, 1529-1531 (1992). [CrossRef] [PubMed]
- K. Hotate and O. Kamatani, "Optical coherence domain reflectometry by synthesis of coherence function," J. Lightwave Technol. 11, 1701-1709 (1993). [CrossRef]
- S.-J. Lee, B. Widiyatmoko, M. Kourogi, and M. Ohtsu, "Ultrahigh scanning speed optical coherence tomography using optical frequency comb generators," Jpn. J. Appl. Phys. Part II, 40, L878-L880 (2001). [CrossRef]
- K. Hotate and T. Okugawa, "Optical information processing by synthesis of the coherence function," J. Lightwave Technol. 12, 1247-1255 (1994). [CrossRef]
- Z. He and K. Hotate, "Synthesized optical coherence tomography for imaging of scattering objects by use of a stepwise frequency-modulated tunable laser diode," Opt. Lett. 24, 1502-1504 (1999). [CrossRef]
- S. Choi, M. Yamamoto, D. Moteki, T. Shioda,Y. Tanaka, and T. Kurokawa, "Frequency-comb-based interferometer for profilometry and tomography," Opt. Lett. 31, 1976-1978 (2006). [CrossRef] [PubMed]
- P. A. Flournoy, R. W. McClure, and G. Wyntjes, "White-light interferometric thickness gauge," Appl. Opt., 11, 1907-1915 (1972). [CrossRef] [PubMed]
- N. Tanaka, M. Takeda, and K. Matsumoto, "Interferometrically measuring the physical properties of test object," United States Patent 4,072,422 (filed October 20 1976, 1978).
- 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 775, 233-247 (1987).
- B. S. Lee and T. C. Strand, "Profilometry with a coherence scanning microscope," Appl. Opt. 29, 3784-3788 (1990). [CrossRef] [PubMed]
- T. Dresel, G. Hausler, and H. Venzke, "Three-dimensional sensing of rough surfaces by coherence radar," Appl. Opt. 31, 919-925 (1992). [CrossRef] [PubMed]
- D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991). [CrossRef] [PubMed]
- J. Schwider and L. Zhou, "Dispersive interferometric profilometer," Opt. Lett. 19, 995-997 (1994). [CrossRef] [PubMed]
- M. Takeda, and H. Yamamoto, "Fourier-transform speckle profilometry: Three-dimensional shape measurements of diffuse objects with large height steps and /or spatially isolated surfaces," Appl. Opt. 33, 7829-7837 (1994). [CrossRef] [PubMed]
- S. Kuwamura and I. Yamaguchi, "Wavelength scanning profilometry for real-time surface shape measurement," Appl. Opt. 36, 4473-4482 (1997). [CrossRef] [PubMed]
- H. J. Tiziani, B. Franze, and P. Haible, "Wavelength-shift speckle interferometry for absolute profilometry using mode-hope free external cavity diode laser," J. Mod. Opt. 44, 1485-1496 (1997). [CrossRef]
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