## Selectable-wavelength low-coherence digital holography with chromatic phase shifter |

Optics Express, Vol. 20, Issue 18, pp. 19744-19756 (2012)

http://dx.doi.org/10.1364/OE.20.019744

Acrobat PDF (2006 KB)

### Abstract

We propose a new digital holography method using an ultra-broadband light source and a chromatic phase-shifter. The chromatic phase-shifter gives different frequency shifts for respective spectral frequencies so that the spectrum of the light reflected from the object can be measured to reveal the spectral property of the object, and arbitrary selection of signals in the temporal frequency domain enables single- and multi-wavelength measurements with wide dynamic range. A theoretical analysis, computer simulations, and optical experiments were performed to verify the advantages of the proposed method.

© 2012 OSA

## 1. Introduction

1. J. W. Goodman and R. W. Lawrence, “Digital image formation from electrically detected holograms,” Appl. Phys. Lett. **11**(3), 77–79 (1967). [CrossRef]

9. S. Tamano, Y. Hayasaki, and N. Nishida, “Phase-shifting digital holography with a low-coherence light source for reconstruction of a digital relief object hidden behind a light-scattering medium,” Appl. Opt. **45**(5), 953–959 (2006). [CrossRef] [PubMed]

10. P. S. Lam, J. D. Gaskill, and J. C. Wyant, “Two-wavelength holographic interferometer,” Appl. Opt. **23**(18), 3079–3081 (1984). [CrossRef] [PubMed]

13. S. Tamano, M. Otaka, and Y. Hayasaki, “Two-wavelength phase-shifting low-coherence digital holography,” Jpn. J. Appl. Phys. **47**(12), 8844–8847 (2008). [CrossRef]

10. P. S. Lam, J. D. Gaskill, and J. C. Wyant, “Two-wavelength holographic interferometer,” Appl. Opt. **23**(18), 3079–3081 (1984). [CrossRef] [PubMed]

12. N. Ninane and M. P. Georges, “Holographic interferometry using two-wavelength holography for the measurement of large deformations,” Appl. Opt. **34**(11), 1923–1928 (1995). [CrossRef] [PubMed]

13. S. Tamano, M. Otaka, and Y. Hayasaki, “Two-wavelength phase-shifting low-coherence digital holography,” Jpn. J. Appl. Phys. **47**(12), 8844–8847 (2008). [CrossRef]

14. C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. **39**(1), 79–85 (2000). [CrossRef]

16. B. P. Hildebrand and K. A. Haines, “Multiple-wavelength and multiple-source holography applied to contour generation,” J. Opt. Soc. Am. **57**(2), 155–157 (1967). [CrossRef]

10. P. S. Lam, J. D. Gaskill, and J. C. Wyant, “Two-wavelength holographic interferometer,” Appl. Opt. **23**(18), 3079–3081 (1984). [CrossRef] [PubMed]

13. S. Tamano, M. Otaka, and Y. Hayasaki, “Two-wavelength phase-shifting low-coherence digital holography,” Jpn. J. Appl. Phys. **47**(12), 8844–8847 (2008). [CrossRef]

17. Y. Kikuchi, D. Barada, T. Kiire, and T. Yatagai, “Doppler phase-shifting digital holography and its application to surface shape measurement,” Opt. Lett. **35**(10), 1548–1550 (2010). [CrossRef] [PubMed]

23. G. S. Kino and S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. **29**(26), 3775–3783 (1990). [CrossRef] [PubMed]

## 2. Principle

### 2.1 Dependence of complex amplitude spectrum on light source spectrum and velocity of reference mirror

*A*(

_{O}*x*,

*y*) and

*A*(

_{R}*x*,

*y*) are real amplitudes of object and reference wave,

*w*and

_{O}*w*are angular frequency of object and reference wave, respectively.

_{R}*k*is the wave number,

*θ*is the initial phase, and

_{O}*z*(

*x,y*) is the depth information of the object. The optical intensity signal sampled at an arbitrary camera pixel can be expressed aswhere

*I*=

_{o}*A*(

_{O}*x*,

*y*)

^{2}+

*A*(

_{R}*x*,

*y*)

^{2}and

*M =*2

*A*(

_{O}*x*,

*y*)

*A*(

_{R}*x*,

*y*)/

*I*are the intensity and the fringe contrast. It is known that when constant relative motion occurs between the object and the reference mirror in the optical system, the angular frequency of the reference wave is modulated according to the following formulawhere

_{o}*v*is the velocity of the reference mirror [17

_{R}17. Y. Kikuchi, D. Barada, T. Kiire, and T. Yatagai, “Doppler phase-shifting digital holography and its application to surface shape measurement,” Opt. Lett. **35**(10), 1548–1550 (2010). [CrossRef] [PubMed]

*w*, representing the difference between the angular frequencies of the object wave and the reference wave, is defined asThe intensity of the pixel in Eq. (1) is then expressed as

_{b}*N*sections in which the intensity and interference fringe contrast on a section respectively denoted as

*I*and

_{oi}*M*(1≤

_{i}*i*≤

*N*) are considered as constant as shown in Fig. 2 . Therefore, integrating over

*k*in a section yieldsSumming up all

*I*[

_{i}*z*(

*x*,

*y*),

*t*], we have

*I*[

_{WL}*z*(

*x,y*)

*,t*] is a function of time; therefore, its temporal Fourier transform

*δ*(

*w*) is delta function of

*w*. From Eq. (9), it can be seen that the spectrum of

*FI*[

_{WL}*z*(

*x,y*)

*,w*] is in the range ofwhere and

*f = w/*2π. Because the temporal frequency of the optical intensity of the interference pattern

*I*[

_{WL}*z*(

*x,y*)

*,t*] is a positive value, when the reference mirror is moved toward the beam splitter (

*v*>0), the maximum and minimum temporal frequencies of the optical intensity are given by Eqs. (11) and (12). For each specific value of

_{R}*f*selected in the optical intensity spectrum, the phase information of the object can be extracted as the second term of Eq. (9). On the contrary, when the reference mirror is moved away from the beam splitter (

*v*<0), the maximum and minimum temporal frequencies of the optical intensity are the absolute values given by Eqs. (11) and (12), and the phase information of the object is identified by the third term of Eq. (9).

_{R}### 2.2. Sampling complex amplitude

*I*[

_{WL}*z*(

*x,y*)

*,t*] and

*FI*[

_{WL}*z*(

*x,y*)

*,w*]. In order to reconstruct the form of

*I*[

_{WL}*z*(

*x,y*)

*,t*] exactly, the frame rate of the camera, denoted as

*f*, should be much larger than two times the maximum frequency of the complex amplitude [24], which is specified by Eq. (11). This condition can be generalized as:When the condition in Eq. (14) is satisfied, it is possible to reconstruct the form of

_{s}*FI*[

_{WL}*z*(

*x,y*)

*,w*] from the sampled holographic images, and the discrete Fourier transform (DFT) of

*I*[

_{WL}*z*(

*x,y*)

*,t*] is given bywhere

*N*is the number of samples,

_{s}*t*/

_{n}= n*f*,

_{s}*w*2

_{m}=*Пmf*, and

_{s}/N_{s}*m =*0,1,2

*,…,N*1. With a specific value of

_{s}-*w*, the complex amplitude of the object wave at each pixel is estimated by Eq. (13) so that the phase of the complex amplitude derived from Eq. (15) can be used to reconstruct a 3D image of the object by using the angular spectrum method [25

_{m}25. S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, “Angular spectrum method with correction of an aphorism for numerical reconstruction of digital holograms on tilted planes,” Opt. Express **13**(24), 9935–9940 (2005). [CrossRef] [PubMed]

26. X. Y. Su and W. J. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. **42**(3), 245–261 (2004). [CrossRef]

### 2.3. Low-coherence digital holography with selectable wavelengths

*λ*is desired to measure a steeper object shape [4

_{e}4. I. Yamaguchi, T. Ida, M. Yokota, and K. Yamashita, “Surface shape measurement by phase-shifting digital holography with a wavelength shift,” Appl. Opt. **45**(29), 7610–7616 (2006). [CrossRef] [PubMed]

*λ*and

_{s1}*λ*should be very similar. It is assumed that

_{s2}*λ*and

_{s1}*λ*are respectively derived from

_{s2}*f*and

_{1}*f*by Eq. (11). Because the minimum difference between

_{2}*f*and

_{1}*f*, denoted as

_{2}*Δf*, is given bywhere

_{min}*N*is the total number of samples of the holograms recorded by a high-speed CCD camera, the maximum value of

_{s}*λ*will be

_{e}*λ*, which is given byFrom Eqs. (17) and (18), a greater value of

_{emax}*λ*can be achieved by increasing the velocity of the PZT or expanding the motion range of the PZT to record more samples of the optical intensity.

_{emax}## 3. Experimental results

*f*= 1000 Hz). The camera was set to record for 1 s (

_{s}*N*= 1000). The frequency step calculated using Eq. (15) was 1 Hz.

*f*was set to a constant value of 82 Hz, and

_{1}*f*was varied. The results are shown in Table 1 .

_{2}*λ*were also calculated to evaluate the phase errors. The experiment results showed that, as the speed of the PZT was set higher, a larger error occurred. This can be explained by the increasing vibrations with increasing PZT velocity.

_{emax}*Δt*= 1 ms), as in the first experiment. The intensity spectrum of the hologram was also from 54 Hz to 82 Hz.

*f*= 66 Hz and

_{1}*f*= 64 Hz, were selected in the temporal spectrum of the optical intensity. Two corresponding wavelengths,

_{2}*λ*= 606 nm and

_{1}*λ*= 625 nm, were estimated by using Eq. (13), and the equivalent wavelength,

_{2}*λ*20 μm, was estimated by using Eq. (18). The surface of the object reconstructed by TW-LCDH, a cross-section, and error profile are shown in Figs. 9(a) , 9(b) and 9(c), respectively. The value of RMS error calculated from the error profile was 0.024 μm, which was about 833 times smaller than the equivalent wavelength. Again the frequency of 53 Hz was selected in the intensity spectrum, and SW-LCDH was applied to reconstruct a 3D image of the object with less error, as shown in Fig. 10 . In this case, the profile of the object obtained by TW-LCDH was used to detect and compensate for the 2π ambiguity error of the phase. The value of RMS error calculated from the error profile was 0.009 μm, which was about 84 times smaller than the wavelength.

_{e}=*f*= 62 Hz and

_{1}*f*= 63 Hz corresponding to two wavelength

_{2}*λ*= 645 nm and

_{1}*λ*= 635 nm in the red light region, were selected. The equivalent wavelength,

_{2}*λ*= 40 μm, was estimated by using Eq. (18). The surface of the object reconstructed by TW-LCDH was shown in Fig. 15(a) . The cross-sections along the areas marked by red, green and blue lines on the surface of the object were respectively shown in Figs. 15(b), 15(c), and 15(d).

_{e}## 4. Conclusion and discussion

## References and links

1. | J. W. Goodman and R. W. Lawrence, “Digital image formation from electrically detected holograms,” Appl. Phys. Lett. |

2. | C. Depeursinge, “Digital holography applied to microscopy,” in |

3. | S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Phase shifting by a rotating polarizer in white-light interferometry for surface profiling,” J. Mod. Opt. |

4. | I. Yamaguchi, T. Ida, M. Yokota, and K. Yamashita, “Surface shape measurement by phase-shifting digital holography with a wavelength shift,” Appl. Opt. |

5. | T. Zhang and I. Yamaguchi, “Three-dimensional microscopy with phase-shifting digital holography,” Opt. Lett. |

6. | I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. |

7. | P. Hariharan, “Achromatic phase-shifting for white-light interferometry,” Appl. Opt. |

8. | S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “White-light interferometry with polarization phase-shifter at the input of the interferometer,” J. Mod. Opt. |

9. | S. Tamano, Y. Hayasaki, and N. Nishida, “Phase-shifting digital holography with a low-coherence light source for reconstruction of a digital relief object hidden behind a light-scattering medium,” Appl. Opt. |

10. | P. S. Lam, J. D. Gaskill, and J. C. Wyant, “Two-wavelength holographic interferometer,” Appl. Opt. |

11. | J. C. Wyant, “Testing aspherics using two-wavelength holography,” Appl. Opt. |

12. | N. Ninane and M. P. Georges, “Holographic interferometry using two-wavelength holography for the measurement of large deformations,” Appl. Opt. |

13. | S. Tamano, M. Otaka, and Y. Hayasaki, “Two-wavelength phase-shifting low-coherence digital holography,” Jpn. J. Appl. Phys. |

14. | C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. |

15. | A. Wada, M. Kato, and Y. Ishii, “Multiple-wavelength digital holographic interferometry using tunable laser diodes,” Appl. Opt. |

16. | B. P. Hildebrand and K. A. Haines, “Multiple-wavelength and multiple-source holography applied to contour generation,” J. Opt. Soc. Am. |

17. | Y. Kikuchi, D. Barada, T. Kiire, and T. Yatagai, “Doppler phase-shifting digital holography and its application to surface shape measurement,” Opt. Lett. |

18. | J. Park and S. W. Kim, “Vibration-desensitized interferometer by continuous phase shifting with high-speed fringe capturing,” Opt. Lett. |

19. | D. Barada, T. Kiire, J. Sugisaka, S. Kawata, and T. Yatagai, “Simultaneous two-wavelength Doppler phase-shifting digital holography,” Appl. Opt. |

20. | T. Yamauchi, H. Iwai, M. Miwa, and Y. Yamashita, “Low-coherent quantitative phase microscope for nanometer-scale measurement of living cells morphology,” Opt. Express |

21. | P. Hariharan and M. Roy, “White-light phase-stepping interferometry for surface profiling,” J. Mod. Opt. |

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

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

24. | J. W. Goodman, |

25. | S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, “Angular spectrum method with correction of an aphorism for numerical reconstruction of digital holograms on tilted planes,” Opt. Express |

26. | X. Y. Su and W. J. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. |

**OCIS Codes**

(090.1760) Holography : Computer holography

(100.5070) Image processing : Phase retrieval

(240.5770) Optics at surfaces : Roughness

(240.6700) Optics at surfaces : Surfaces

**ToC Category:**

Holography

**History**

Original Manuscript: July 5, 2012

Revised Manuscript: July 31, 2012

Manuscript Accepted: August 2, 2012

Published: August 13, 2012

**Citation**

Quang Duc Pham, Satoshi Hasegawa, Tomohiro Kiire, Daisuke Barada, Toyohiko Yatagai, and Yoshio Hayasaki, "Selectable-wavelength low-coherence digital holography with chromatic phase shifter," Opt. Express **20**, 19744-19756 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-19744

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### References

- J. W. Goodman and R. W. Lawrence, “Digital image formation from electrically detected holograms,” Appl. Phys. Lett.11(3), 77–79 (1967). [CrossRef]
- C. Depeursinge, “Digital holography applied to microscopy,” in Digital Holography and Three-Dimensional Display, T. C. Poon, ed. (Springer, 2006), 104–147.
- S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Phase shifting by a rotating polarizer in white-light interferometry for surface profiling,” J. Mod. Opt.46, 993–1001 (1999).
- I. Yamaguchi, T. Ida, M. Yokota, and K. Yamashita, “Surface shape measurement by phase-shifting digital holography with a wavelength shift,” Appl. Opt.45(29), 7610–7616 (2006). [CrossRef] [PubMed]
- T. Zhang and I. Yamaguchi, “Three-dimensional microscopy with phase-shifting digital holography,” Opt. Lett.23(15), 1221–1223 (1998). [CrossRef] [PubMed]
- I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett.22(16), 1268–1270 (1997). [CrossRef] [PubMed]
- P. Hariharan, “Achromatic phase-shifting for white-light interferometry,” Appl. Opt.35(34), 6823–6824 (1996). [CrossRef] [PubMed]
- S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “White-light interferometry with polarization phase-shifter at the input of the interferometer,” J. Mod. Opt.47(6), 1137–1145 (2000). [CrossRef]
- S. Tamano, Y. Hayasaki, and N. Nishida, “Phase-shifting digital holography with a low-coherence light source for reconstruction of a digital relief object hidden behind a light-scattering medium,” Appl. Opt.45(5), 953–959 (2006). [CrossRef] [PubMed]
- P. S. Lam, J. D. Gaskill, and J. C. Wyant, “Two-wavelength holographic interferometer,” Appl. Opt.23(18), 3079–3081 (1984). [CrossRef] [PubMed]
- J. C. Wyant, “Testing aspherics using two-wavelength holography,” Appl. Opt.10(9), 2113–2118 (1971). [CrossRef] [PubMed]
- N. Ninane and M. P. Georges, “Holographic interferometry using two-wavelength holography for the measurement of large deformations,” Appl. Opt.34(11), 1923–1928 (1995). [CrossRef] [PubMed]
- S. Tamano, M. Otaka, and Y. Hayasaki, “Two-wavelength phase-shifting low-coherence digital holography,” Jpn. J. Appl. Phys.47(12), 8844–8847 (2008). [CrossRef]
- C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng.39(1), 79–85 (2000). [CrossRef]
- A. Wada, M. Kato, and Y. Ishii, “Multiple-wavelength digital holographic interferometry using tunable laser diodes,” Appl. Opt.47(12), 2053–2060 (2008). [CrossRef] [PubMed]
- B. P. Hildebrand and K. A. Haines, “Multiple-wavelength and multiple-source holography applied to contour generation,” J. Opt. Soc. Am.57(2), 155–157 (1967). [CrossRef]
- Y. Kikuchi, D. Barada, T. Kiire, and T. Yatagai, “Doppler phase-shifting digital holography and its application to surface shape measurement,” Opt. Lett.35(10), 1548–1550 (2010). [CrossRef] [PubMed]
- J. Park and S. W. Kim, “Vibration-desensitized interferometer by continuous phase shifting with high-speed fringe capturing,” Opt. Lett.35(1), 19–21 (2010). [CrossRef] [PubMed]
- D. Barada, T. Kiire, J. Sugisaka, S. Kawata, and T. Yatagai, “Simultaneous two-wavelength Doppler phase-shifting digital holography,” Appl. Opt.50(34), H237–H244 (2011). [CrossRef] [PubMed]
- T. Yamauchi, H. Iwai, M. Miwa, and Y. Yamashita, “Low-coherent quantitative phase microscope for nanometer-scale measurement of living cells morphology,” Opt. Express16(16), 12227–12238 (2008). [CrossRef] [PubMed]
- P. Hariharan and M. Roy, “White-light phase-stepping interferometry for surface profiling,” J. Mod. Opt.41(11), 2197–2201 (1994). [CrossRef]
- T. Dresel, G. Häusler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt.31(7), 919–925 (1992). [CrossRef] [PubMed]
- G. S. Kino and S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt.29(26), 3775–3783 (1990). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005), 17–27.
- S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, “Angular spectrum method with correction of an aphorism for numerical reconstruction of digital holograms on tilted planes,” Opt. Express13(24), 9935–9940 (2005). [CrossRef] [PubMed]
- X. Y. Su and W. J. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng.42(3), 245–261 (2004). [CrossRef]

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