Multiwavelength electronic speckle pattern ... - OSA Publishing

Multiwavelength electronic speckle pattern interferometry for surface shape measurement Eduardo A. Barbosa and Antonio C. L. Lino

Profilometry by electronic speckle pattern interferometry with multimode diode lasers is both theoretically and experimentally studied. The multiwavelength character of the laser emission provides speckled images covered with interference fringes corresponding to the surface relief in single-exposure processes. For fringe pattern evaluation, variations of the phase-stepping technique are investigated for phase mapping as a function of the number of laser modes. Expressions for two, three, and four modes in four and eight stepping are presented, and the performances of those techniques are compared in the experiments through the surface shaping of a flat bar. The surface analysis of a peach points out the possibility of applying the technique in the quality control of food production and agricultural research. © 2007 Optical Society of America OCIS codes: 120.3180, 120.4630, 120.6150, 120.6160.

1. Introduction

Interferometric methods are very useful in a wide range of metrology applications. Among them, electronic speckle pattern interferometry (ESPI) presents many interesting features, such as whole-field analysis, real-time display of interference fringes through standard TV cameras, and relaxed conditions for optical setup stability, when compared to other interferometric techniques. Since its development, ESPI has found several applications in displacement analysis, vibration measurement, material research, and surface analysis.1 Its reliability, simplicity, and applicability in noisy environments enables us to employ ESPI for scientific and industrial purposes with the development of commercial devices. Optical contouring by ESPI or holographic interferometry does permit whole-field, noncontacting measurements with applications in quality control, machine vision, and strain analysis in highly accuE. A. Barbosa ([email protected]) is with the Laboratório de Óptica Aplicada, Faculdade de Tecnologia de São Paulo, Centro Estadual de Educação Tecnológica “Paula Souza,” Universidade Estadual Paulista, Praça Cel Fernando Prestes, 30, CEP 01124 060, São Paulo, Brazil. A. C. L. Lino is with the Centro de Mecanização e Automação, Instituto Agronômico, Caixa Postal 16, 13201-970, Jundiaí, São Paulo, Brazil. Received 22 August 2006; revised 6 November 2006; accepted 8 November 2006; posted 9 January 2007 (Doc. ID 74240); published 23 April 2007. 0003-6935/07/142624-08$15.00/0 © 2007 Optical Society of America 2624

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rate quantitative and qualitative testing. The most used methods in this field are the two-wavelength method,2,3 the mirror-tilting method,4 the two-source method,5 and Fourier-domain interferometry.6 The first method consists of acquiring two images of a surface with different wavelengths ␭1 and ␭2. When both images are added, the resulting image is covered by interference fringes whose spatial period is related to the synthetic wavelength ␭S ⫽ ␭1␭2兾|␭1 ⫺ ␭2|.7,8 A variation of this technique is accomplished by illuminating the optical setup simultaneously by two properly tuned and aligned diode lasers. The synthetic wavelength is thus generated without the need for wavelength selection between the two exposures, and the condition of image addition is automatically fulfilled within one frame.9 A significant part of the current research in optical profilometry for real-life applications is based mainly on the design and construction of optical setups with good and reliable performance in noisy environments and on faster and simpler measurement procedures. With this objective, we study the use of a multimode, large, free spectral range (FSR) diode laser for surface contouring through ESPI. Because of the multiwavelength character of the laser emission, the object image appears covered with interference contour fringes with only one diode laser in single-exposure processes. The synthetic wavelength in this case is inversely proportional to the laser FSR. Whole-field optical interferometry using such lasers was already demonstrated in multiwavelength holography in sillenite Bi12TiO20 crystals.10,11 In that

case, multiple holographic gratings were stored in the medium, and the resulting diffraction efficiency was dependent on the optical path difference between the reference and the object beams, thus generating contour fringes in a single exposure. Low-coherence fiber-based ESPI using pulsed multimode diode lasers was applied for surface shaping. The coherence length of the lasers was approximately 150 ␮m, and the interferogram evaluation was carried out through the five-stepping technique.12 In the current study, the speckle pattern that originated from the object illuminated by a multimode diode laser is formed on the faceplate of a CCD camera, and the interferogram containing the information of the object relief is obtained without further laser tuning. Additive multiwavelength ESPI is performed in single-exposure processes with a single multimode laser. To enhance the interferogram visibility, two consecutive frames in the same setup configuration were subtracted. Thus the typical background noise from the speckle pattern was removed. The quantitative interferogram analysis was performed through phasestepping techniques. This was done by acquiring many interferogram frames and applying a discrete phase shift on the reference wave between each frame. The resulting phase map was unwrapped through the branch-cut method.13,14 The fringe evaluation was also studied by taking into account the number of oscillating modes. This parameter does affect the bright fringe width, leading to different expressions for phase retrieval. Expressions of the phase for two, three, and four equally intense laser modes considering four- and eight-step procedures were derived. In the experiments the performances of the different approaches were compared by measuring the relief of an opaque glass plate. The surface contouring of a peach shows also the possibility of applying this profilometry technique in quality control of vegetable production and storage. 2. Electronic Speckle Pattern Interferometry with a Large Free Spectral Range Multimode Laser A.

Contour Fringe Generation

In this section the contour fringe formation on the object surface is described. We adopted the subtractive method for interferogram visualization.9 Let us consider the interference of the object 共SN兲 and reference 共RN兲 waves onto a CCD target. Both waves originate from a multimode laser with emission centered at ␭¯ and wavelength gap ⌬␭ between consecutive laser modes, so that SN and RN can be written as RN ⫽ R0

SN ⫽ S0

n⫽共N⫺1兲兾2

兺

n⫽⫺共N⫺1兲兾2 n⫽共N⫺1兲兾2

兺

n⫽⫺共N⫺1兲兾2

An exp兵i关共k៮ ⫹ n⌬k兲⌫R ⫹ ␾n兴其, An exp兵i关共k៮ ⫹ n⌬k兲⌫S ⫹ ␾n兴其, (1)

where N is the number of oscillating modes, ␾n is the phase of the nth mode at the laser output, k៮

⬅ 2␲兾␭¯ , ⌬k ⬅ 2␲⌬␭兾␭¯ , An is a real coefficient, and ⌫S and ⌫R are the optical paths of the object and the reference beams, respectively. For the first frame, the interference of both beams on the CCD is expressed by I1 ⫽ ⱍSNⱍ2 ⫹ ⱍRNⱍ2 ⫹ SN*RN ⫹ RN*SN.

(2)

Since different modes are not mutually coherent, the following orthonormality condition must be obeyed10: Sn*Rm ⫽ ␦n,mS0 R0 An Am exp兵⫺i关共k៮ ⫹ n⌬k兲⌫S ⫹ ␾n兴其 ⫻ exp兵i关共k៮ ⫹ m⌬k兲⌫R ⫹ ␾m兴其. (3) By applying condition (3), Eq. (2) yields I1 ⫽ S02 ⫹ R02 ⫹ S0R0 exp关ik៮ 共⌫S ⫺ ⌫R兲兴

n⫽共N⫺1兲兾2

兺

n⫽⫺共N⫺1兲兾2

An2

⫻ exp关in⌬k共⌫S ⫺ ⌫R兲兴 ⫹ R0S0 exp关⫺ik៮ 共⌫S ⫺ ⌫R兲兴 ⫻

n⫽共N⫺1兲兾2

兺

n⫽⫺共N⫺1兲兾2

An2 exp关⫺in⌬k共⌫S ⫺ ⌫R兲兴.

(4)

Since the sums in the third and the fourth terms of the right-hand side of the equation above are equal, Eq. (4) yields I1 ⫽ S02 ⫹ R02 ⫹ 2R0S0 cos关k៮ 共⌫S ⫺ ⌫R兲兴 ⫻ exp关in⌬k共⌫S ⫺ ⌫R兲兴.

n⫽共N⫺1兲兾2

兺

n⫽⫺共N⫺1兲兾2

An2 (5)

For fringe pattern visualization, the reference beam is then sinusoidally phase modulated, and a second frame is captured and stored. This modulation can be made by attaching the mirror, which delivers the reference beam to a transducer vibrating at tens of hertz with an amplitude of a few micrometers. If the amplitude of the phase modulation is large enough, the speckle pattern is decorrelated, and the intensity of the second frame is just I2 ⫽ S02 ⫹ R02. After the subtraction of both frames and rectification, the resulting frame is given by

冏

I ⫽ I2 ⫺ I1 ⫽ 2R0 S0 cos关k៮ 共⌫S ⫺ ⌫R兲兴

冏

n⫽共N⫺1兲兾2

兺

n⫽⫺共N⫺1兲兾2

⫻ exp关in⌬k共⌫S ⫺ ⌫R兲兴 .

An2 (6)

The cosine term in Eq. (6) represents the high spatial frequency modulation of the interference pattern and the random behavior of the speckle pattern. This term is highly sensitive to phase shifts of the order of ␭¯ in one of the interfering beams. Since typically k៮ ⬎⬎ ⌬k, the second term has a low spatial frequency and is modulated according to the relief of the object surface, leading to interference contour fringes. The intensity in Eq. (6) above must be averaged by lowpass filtering and then squared in order to suppress the high spatial frequency term and provide only pos10 May 2007 兾 Vol. 46, No. 14 兾 APPLIED OPTICS

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itive values. Thus from Eq. (6) one gets the interferogram signal V:

冉

V ⬅ 具I典2 ⫽ 2R0 S0

n⫽共N⫺1兲兾2

兺 n⫽⫺ N⫺1 兾2 共

兲

冊

2

An2 exp关in⌬k共⌫S ⫺ ⌫R兲兴 . (7)

For interferogram analysis Eq. (7) can be simplified by assuming that all modes oscillate with equal amplitudes, i.e., An ⫽ 1, so that V becomes V ⫽ 2R0 S0

sin2关N⌬k共⌫S ⫺ ⌫R兲兾2兴 sin2关⌬k共⌫S ⫺ ⌫R兲兾2兴

.

(8)

With the help of Eq. (8) one obtains the distance ⌬z between two neighboring planes of constant elevation: ␭2 ␭S ⌬z ⫽ ⬅ , 2⌬␭ 2

(9)

where ␭S is the so-called synthetic wavelength.15 For a laser with a resonator length L the FSR is ⌬v ⫽ c兾2L.16 The wavelength gap ⌬␭ in turn is related to the FSR as ⌬v ⫽ c⌬␭兾␭2. Hence the depth difference between two adjacent fringes is simply ⌬z ⫽ L; i.e., the shorter the laser cavity, the higher the spatial frequency of the fringe pattern. B.

Phase Stepping

The four-frame phase stepping procedure was applied for fringe evaluation. This technique is carried out by sequentially phase shifting one of the interfering beams in the optical setup by discrete phase values. The number of oscillating modes N as well as the central emission wavelength ␭¯ is strongly dependent on the operating laser current17 and clearly affects the bright fringe width.10 Thus it is worthwhile to study the influence of N on phase retrieval. In the following analysis different expressions for the phase will be obtained for different values of N considering four- and eight-frame procedures. Depending upon how the laser is tuned, the number of oscillating modes ranges from 1 to 5. The equations for phase stepping cannot be obtained directly from Eq. (7), but from Eq. (8), which in turn is valid only for modes with the same intensity. Since real laser modes usually oscillate with different intensities, an effective value Neff of equally intense modes must be determined from the experimental data to permit phase retrieval. For instance, if the diode laser emits five modes enveloped by a Lorentzian line shape, a possible set of values for the Ai2 coefficients is A02 ⫽ 1, A12 ⫽ A⫺12 ⫽ 0.43, and A22 ⫽ A⫺22 ⫽ 0.16. For this laser configuration, the solid curve in the graph of Fig. 1 shows the interferogram signal V as a function of ⌫S (for ⌫R ⫽ 0) calculated through Eq. (7), while the dashed curve shows V according to Eq. (8) for Neff ⫽ 4 modes. For both curves, ⌬k ⫽ 1.18 rad兾mm and 2626

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Fig. 1. Signal V as a function of ⌫S. The solid curve was obtained through Eq. (7), while the dashed curve was calculated by Eq. (8) for Neff ⫽ 4 modes.

␭¯ ⫽ 670 nm. Thus from Eq. (8) the intensity of the lth frame at a point (x, y) can be written as

再

Vl ⫽ 2R0S0

sin关Neff共⌬k⌫S共x, y兲 ⫹ l␾兲兾2兴 sin关共⌬k⌫S共x, y兲兾2 ⫹ l␾兲兾2兴

冎

2

, (10)

where l is an integer and ␾ is a discrete phase shift. The four-frame procedure is the most employed phase-stepping technique for interferogram analysis. The intensities of the frames are obtained from Eq. (10) for l ⫽ 0, 1, 2, and 3 and for ␾ ⫽ ␲兾2 rad. For Neff ⫽ 2, Eq. (10) provides the well-known formula

冉

␾4⫺step共Neff ⫽ 2兲 ⫽ arctan

冊

V3 ⫺ V1 . V0 ⫺ V2

(11)

For Neff ⫽ 3 modes, the phase is calculated from Eq. (10) to be ␾4⫺step共Neff ⫽ 3兲 ⫽

冉

冊

1 V1 ⫺ V3 . arctan 2 V0 ⫺ V2

(12)

Finally, the retrieved phase for Neff ⫽ 4 is given by

冋 冉

␾4⫺step共Neff ⫽ 4兲 ⫽ arctan

V0V2 V1 ⫺ V3 V1V3 V0 ⫺ V2

冊册

. (13)

It is important to remark, however, that the fourstep processes obtained through Eqs. (12) and (13) are very sensitive to optical noise and fringe discontinuities owing to the narrow fringe width. An alternative, which is by far much more interesting though more cumbersome, is sequentially taking eight ␲兾4-phase-shifted frames. From Eq. (10), ␾ ⫽ ␲兾4 rad, and l ranges from 0 to 7, so that the phase is

given by

冉

␾8⫺step ⫽ arctan

冊

V7 ⫹ V3 ⫺ V5 ⫺ V1 . V4 ⫹ V0 ⫺ V6 ⫺ V2

(14)

The equation above is valid for both Neff ⫽ 3 and Neff ⫽ 4. Notice from Eq. (10) that since Vl ⫹ Vl⫹4 is a cos2-like function, the pattern resulting from Eq. (14) is actually equivalent to a phase map that originated from cos2 interferograms in a standard ␲兾2four-frame technique with a synthetic wavelength ␭S兾2. The advantage in this case lies in the fact that smaller synthetic wavelengths provide less noisy measurements,18 as will be seen in the experiments in Section 4. 3. Optical Setup

The experiments were performed on the optical setup shown in Fig. 2. The beam is divided into reference and object beams by beam splitter BS1, while beam splitter BS2 does couple both beams to interfere on the CCD target. After BS2 the directions of the reference and the object beams are roughly parallel to each other. The emission of the 30 mW diode laser is centered at 670 nm, and its FSR is 53 GHz, corresponding to ⌬␭ ⫽ 0.082 nm. A stop placed at the aperture of lens L1 adjusts the speckle size to the pixel size of the CCD. A micrometric screw supporting mirror M3 introduces the phase shift on the reference beam. Beam splitter BS3 ensures normal incidence on M3 throughout the phase-stepping process. Mirror M3 was supported by a piezoelectric transducer excited by a 30 Hz dither signal for phase modulation of the reference beam. The first frame was acquired and stored, and after reference-beam phase modulation, a speckle-decorrelated second frame was obtained. 4. Results and Discussion

Figure 3(a) shows the multiwavelength speckle pattern of a glass bar 20° tilted with respect to the CCD faceplate for ␭¯ ⫽ 670 nm with output power of 29.5 mW. The bar with a flatness of ⬃2␭¯ was covered with

Fig. 2. Optical setup: BS1–BS3, beam splitters; L1–L3, lenses; M1–M3, mirrors; SF, spatial filter; CCD, camera.

Fig. 3. Multiwavelength speckle profilometry of a glass bar through the four-stepping technique. (a) Speckle pattern, (b) lowpass-filtered intensity distribution of the interference pattern (dotted curve) and fitting curve for V after Eq. (8) for Neff ⬵ 2 (solid curve), (c) 3D plot.

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an opaque tape to enhance reflection, and its image appears covered with the expected vertical parallel interference fringes. Figure 3(b) shows the intensity distribution of the horizontal profile of the resulting interference fringes after low-pass filtering (dotted curve) and the fitting curve for V according to Eq. (8) (solid curve). This fitting is valid once the relation between x and ⌫S is linear for the flat plate. The best fit was achieved for Neff ⫽ 2.001 ⫾ 0.008. Figure 3(c) shows the 3D reconstruction of the tilted glass bar after applying the phase mapping and phase unwrapping processes. The laser was then tuned to ␭¯ ⫽ 671 nm (with 31 mW of output power), and a new measurement pro-

Fig. 4. Profilometry of the glass bar through the four-stepping technique. (a) Speckle pattern, (b) intensity distribution of the interference pattern (dotted curve) and fitting curve for V after Eq. (8) for Neff ⬵ 3 (solid curve), (c) resulting 3D plot.

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Fig. 5. Profilometry of the glass bar through the eight-stepping technique. (a) Measured intensity of the fringe pattern (dotted curve) and fitting curve for Neff ⬵ 4 (solid curve), (b) resulting 3D plot.

cedure of the same bar at the same position was carried out. The resulting speckle pattern of the tilted bar is shown in Fig. 4(a). The clearly narrower fringes evidence that the laser emitted more modes, when compared with the previous measurement shown in Fig. 3. Indeed, the intensity profile of the low-passfiltered fringes [dotted curve of Fig. 4(b)] was best fit for Neff ⫽ 3.00 ⫾ 0.01 [solid curve of Fig. 4(b)]. The object contour was then obtained in a four-frame process with the help of Eq. (12). The 3D reconstruction of the glass plate is shown in Fig. 4(c). The drive current of the diode laser was varied again until ␭¯ ⫽ 671.5 nm and an output power of 31.4 mW. Figure 5(a) shows the intensity profile of the low-pass filtered fringes, showing the best fit for Neff ⫽ 3.99 ⫾ 0.01. In this case, the eight-step process was adopted for phase map calculation through Eq. (14). The corresponding 3D reconstruction after phase unwrapping is shown in Fig. 5(b). Figure 6 compares the z coordinate of the same bar cross section reconstructed through the unwrapped

phase patterns of the cases presented above, including also an eight-step measurement for Neff ⫽ 3. The solid curve is the expected value of z. Figures 6(a) and 6(b) show the plots (circles) employing the fourframe technique for Neff ⫽ 2 and Neff ⫽ 3, respectively, while Figs. 6(c) and 6(d) show the corresponding results using the eight-frame technique for Neff ⫽ 3 and Neff ⫽ 4. Those plots show clearly that the best results were obtained in the eight-step procedures. The average rms deviation from the expected plate shape was approximately 1.5 mm for Neff ⫽ 3 in the fourstep procedure, while this value does not exceed 0.3 mm for Neff ⫽ 4 using the eight-step technique. This can be attributed mainly to the smaller effective synthetic wavelength and to the low sensitivity of the eight-step method with respect to fringe discontinuities. Such discontinuities in turn usually arise in the low-pass-filtering process, and techniques to avoid this problem have already been studied elsewhere.19 –21 We realized that those irregularities frequently spoil the phase map calcu-

Fig. 6. Reconstructed z coordinate of the bar cross section. (a) Four-frame technique for Neff ⫽ 2, (b) four-frame technique for Neff ⫽ 3, (c) eight-frame technique for Neff ⫽ 3, and (d) eight-frame technique for Neff ⫽ 4. The solid curve is the expected z value. 10 May 2007 兾 Vol. 46, No. 14 兾 APPLIED OPTICS

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lation and the unwrapping process, most notably in the case of narrower bright fringes (i.e., larger N values). Hence for Neff ⫽ 4 Eq. (13) is applicable only for very smooth, low-noise, and regular fringe patterns. We also accomplished the surface shaping of a partially illuminated peach by multiwavelength ESPI in order to test our technique with more complex shapes

with positive and negative derivatives. Figures 7(a)– 7(c) show the speckle pattern, the phase map, and the 3D reconstruction of the peach. The laser in this case was tuned to ␭¯ ⫽ 670 nm, and the phase mapping was carried out through Eq. (11) for Neff ⫽ 2. The reconstructed surface was in very good agreement with the actual object surface concerning its shape and dimensions. This measurement can be useful in the study of vegetable and fruit growing and blooming processes, as well as in bruising assessment. Moreover, profilometry by multiwavelength ESPI of living surfaces potentially allows for the analysis of the viability of seeds as carried out through other speckle techniques.22 5. Conclusion

Contour interference fringes have been obtained from the electronic speckle pattern by using a multimode diode laser. The simultaneous emission of several longitudinal modes does permit the generation of synthetic wavelength fringe patterns in a single exposure. The subtractive technique was employed to enhance the fringe visibility. We have shown that the synthetic wavelength is half the distance ⌬z between consecutive planes of constant elevation (i.e., the contour interval), which in turn equals the laser resonator length. This work has shown that the choice of the most suitable phase-stepping method for fringe pattern evaluation relies on the proper knowledge of the laser emission spectrum, mainly the number of oscillating modes and their relative intensities. Concerning the influence of those parameters on the bright fringe width, the effective number Neff of equally intense modes was introduced. The phase map was calculated for two, three, and four modes in four- and eight-step processes. The eight-step technique has been shown to be less sensitive to optical and electronic noise and to fringe discontinuities. The four-mode eight-step procedures result in smaller synthetic wavelength interferograms, leading to more accurate and precise measurements. Through the relief measurement of a flat opaque glass plate the performance of the different phase step techniques were compared, showing that the rms deviations from the expected plate shape is approximately five times smaller in the eight-step four-mode process if compared to the four-step twomode measurement. The surface shaping of a peach was also performed, showing the possibilities of measuring the relief of more complex structures. The result shows the potentialities of qualitative and quantitative nondestructive testing of smooth and delicate tissues, with promising applications in areas such as artificial vision and quality control in agriculture and the diagnosis of biological activity.

Fig. 7. Multiwavelength speckle profilometry of a peach for Neff ⫽ 2. (a) Speckle pattern, (b) phase map, and (c) 3D reconstruction. 2630

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The authors thank Sonia Tatumi of the Faculdade de Tecnologia de São Paulo for fruitful discussions. This work was supported in part by the Fundação de Apoio à Tecnologia (Brazil).

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