Selectable-wavelength low-coherence digital holography with ...

Selectable-wavelength low-coherence digital holography with chromatic phase shifter Quang Duc Pham, Satoshi Hasegawa, Tomohiro Kiire, Daisuke Barada, Toyohiko Yatagai, and Yoshio Hayasaki* Center for Optical Research and Education (CORE), Utsunomiya University 7-1-2 Yoto, Utsunomiya 321-8585, Japan * [email protected]

Abstract: We propose a new digital holography method using an ultrabroadband light source and a chromatic phase-shifter. The chromatic phaseshifter 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 Optical Society of America OCIS codes: (240.6700) Surfaces; (240.5770) Roughness; (090.1760) Computer holography; (100.5070) Phase retrieval.

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Received 5 Jul 2012; revised 31 Jul 2012; accepted 2 Aug 2012; published 13 Aug 2012

27 August 2012 / Vol. 20, No. 18 / OPTICS EXPRESS 19744

19. D. Barada, T. Kiire, J. Sugisaka, S. Kawata, and T. Yatagai, “Simultaneous two-wavelength Doppler phaseshifting digital holography,” Appl. Opt. 50(34), H237–H244 (2011). 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 16(16), 12227–12238 (2008). 21. P. Hariharan and M. Roy, “White-light phase-stepping interferometry for surface profiling,” J. Mod. Opt. 41(11), 2197–2201 (1994). 22. T. Dresel, G. Häusler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31(7), 919–925 (1992). 23. G. S. Kino and S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29(26), 3775–3783 (1990). 24. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005), 17–27. 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). 26. X. Y. Su and W. J. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42(3), 245–261 (2004).

1. Introduction Phase-shifting (PS) is a technique that allows removal of zero-order and twin images [1,2] through multi-exposure holographic recording while shifting the phase of the reference field [3–9]. Digital holography (DH) with PS (PSDH) by an integer fraction of 2π has been proposed. A 2π phase ambiguity that occurs when an illuminated surface has a height difference larger than the wavelength is an inherent problem in interference and digital holographic measurements. One method that has been proposed with the aim of solving this problem is two-wavelength digital holography (TW-DH) [10–13], which has been applied to both coherent DH [10–12] and low-coherence (LC) DH [13]. In this method, the modulo-2π phase distribution must be unwrapped to reconstruct three-dimensional (3D) shape of the object; therefore, if there is still any phase jump greater than π due to steps in the surface of the object, incorrect or ambiguous unwrapping may occur. Another approach is multiplewavelength (MW) DH, which uses more than two illumination wavelengths to overcome the 2π ambiguity of TW-DH [14–16]. When the TW- or MW-DH is used, the wavelength of the light source, at which the intensity of the reflected light from the object is desired to be highest, is commonly selected to reconstruct the 3D image. This selection is suitable for the monochromatic or homogeneous object in both cases of coherent or low-coherent DH [10–13]. However, the difficulty rises when the measurement is performed with chromatic, inhomogeneous or absorbing object. Because of the spectral property of the object, the spectrum of the reflected light from the object in this case differs from the one of the light source [2] and this difference is not easy to manage for all points on the surface of the object. In some cases, the selected wavelength may be out of the spectrum of the reflected light from the object and so the hologram corresponding to the selected wavelength is wrongly or not recorded, this causes the 3D image of the object reconstructed with a lot of error. In this paper, we describe an optical system based on a Michelson interferometer with an ultra-broadband light source and a reference mirror that is moved with constant velocity to modulate the phase of the reference wave. The variation of the interference pattern due to the motion of the reference wave is recorded as holographic images by a fast camera [17–23]. The recorded holographic images are analyzed to reveal the relation between the spectrum of the object wave and the temporal spectrum of the interference pattern so that the spectrum of the light reflected from the object can be measured to reveal the spectral property of the object. Moreover, the system has been shown to work as a chromatic phase-shifter, giving different frequency shifts for respective spectral temporal frequencies of the captured interference pattern, and arbitrary selection of signals in the temporal frequency domain enables intensity image reconstruction, single-wavelength (SW-) and multi-wavelength (MW) measurements with wide dynamic range. First, we examine the requirements for implementing the proposed idea by computer simulation, and then we demonstrate the idea and its advantages using an experimental digital holographic system.

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2. Principle 2.1 Dependence of complex amplitude spectrum on light source spectrum and velocity of reference mirror For a typical setup is shown as in Fig. 1, the complex amplitudes of the object and reference wave on image sensor of a specific wavelength are respectively expressed by U O [ z ( x , y ), t , k ] = AO ( x , y ) e

− j [ wO t − 2 z ( x , y ) k −θO ]

U R [ z ( x , y ), t , k ] = AR ( x , y ) e

− jwR t

,

,

(1) (2)

where, AO(x,y) and AR(x,y) are real amplitudes of object and reference wave, wO and wR are angular frequency of object and reference wave, respectively. k is the wave number, θO is the initial phase, and z(x,y) is the depth information of the object. The optical intensity signal sampled at an arbitrary camera pixel can be expressed as I [ z ( x , y ), t ] = {U O [ z ( x , y ), t , k ] + U R ( t , k )}

2

(3)

= I {1 + M cos[( wR − wO )t + 2 z ( x , y ) k + θ o ]}, O

where Io = AO(x,y)2 + AR(x,y)2 and M = 2AO(x,y)AR(x,y)/Io 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 formula 1 + 2vR

wR (vR ) = w

O

1 − 2vR

v /c ≈ w (1 + 2 R ), c /c O

(4)

where vR is the velocity of the reference mirror [17]. Quantity wb, representing the difference between the angular frequencies of the object wave and the reference wave, is defined as wb =

wR (vR ) − w ≈ 2 O

vR c

wO

= 2kvR .

(5)

Fig. 1. Experimental setup.

The intensity of the pixel in Eq. (1) is then expressed as (6) Because the light source which is used to record holography is broad–band light source, so the interference patterns on sensor image is summing up monochromatic interference patterns of all wavelengths, which is reflected from the object. For simplicity, the spectrum of the light reflected from the object is separated into N sections in which the intensity and interference I [ z ( x , y ), t ] = I {1 + M cos[θ o + wb t + 2 z ( x , y ) k ]}. O

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fringe contrast on a section respectively denoted as Ioi and Mi (1≤i≤N) are considered as constant as shown in Fig. 2. Therefore, integrating over k in a section yields ki+1

I i [ z ( x , y ), t ] =

∫ I [ z ( x, y ), t ]dk .

(7)

ki

Summing up all Ii[z(x,y),t], we have N −1

IWL [ z ( x, y ), t ] =

∑ I [ z( x, y), t , k ] i =1 N −1 k

=

i +1

∑∫I i =1

Oi

{1 + M i cos[θ o + wb t + 2 z ( x, y ) k ]}dk

(8)

ki

N −1

=

∑ (k

N −1

i +1

− ki ) I 0 +

i =1

∑M I

i 0 i ( k i +1

− ki ) sin c{

ki +1 − ki

i =1

π

[vR t + z ( x , y )]}

× cos[θ 0i + z ( x, y )(ki +1 + ki ) + vR t ( ki +1 + ki )]. From Eq. (8), it can be seen that IWL[z(x,y),t] is a function of time; therefore, its temporal Fourier transform FIWL [ z ( x , y ), w] can be expressed as FIWL [ z ( x, y ), w] =

N −1

∑ (k

i +1

− ki ) I Oi δ ( w)

i =1 N −1

π Mi

i =1

2 vR

+ ∑ I Oi

{e jθo + jz ( x , y ) w /( 2vR ) rect[

+e − jθo − jz ( x , y ) w /(2 vR ) rect[

w − vR (ki +1 + ki ) ] 2(ki +1 − ki )vR

(9)

w + vR (ki +1 + ki ) ]}, 2(ki +1 − ki )vR

where δ(w) is delta function of w. From Eq. (9), it can be seen that the spectrum of FIWL[z(x,y),w] is in the range of f min

≤

f ≤ f max ,

(10)

where

f max = vR

kN

f min = vR

k1

π π

,

(11)

(12)

and f = w/2π. Because the temporal frequency of the optical intensity of the interference pattern IWL[z(x,y),t] is a positive value, when the reference mirror is moved toward the beam splitter (vR>0), the maximum and minimum temporal frequencies of the optical intensity are given by Eqs. (11) and (12). For each specific value of 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 (vR