High-speed, sub-Nyquist interferometry

The velocity measurement limit in dynamic interferometry is vNyq, the velocity at which the interferogram is sampled at the Nyquist limit. We show that vNyq can be exceeded by assuming continuity of the surface motion and unwrapping the velocity modulo 2vNyq. The technique was demonstrated in a high-speed speckle pattern interferometer with spatial phase stepping. Surface velocities of 4vNyq were measured experimentally. With a reduced exposure, high-speed sub-Nyquist interferometry could be implemented up to a maximum acceleration of vNyq/ts, where ts is the detector frame period. ©2011 Optical Society of America OCIS codes: (120.3180) Interferometry; (120.6160) Speckle interferometry. References and links 1. J. M. Kilpatrick, A. J. Moore, J. S. Barton, J. D. C. Jones, M. Reeves, and C. Buckberry, “Measurement of complex surface deformation by high-speed dynamic phase-stepped digital speckle pattern interferometry,” Opt. Lett. 25(15), 1068–1070 (2000). 2. W. N. MacPherson, M. Reeves, D. P. Towers, A. J. Moore, J. D. C. Jones, M. Dale, and C. Edwards, “Multipoint laser vibrometer for modal analysis,” Appl. Opt. 46(16), 3126–3132 (2007). 3. J. M. Huntley, G. H. Kaufmann, and D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38(31), 6556–6563 (1999). 4. X. Colonna de Lega and P. Jacquot, “Deformation measurement with object-induced dynamic phase shifting,” Appl. Opt. 35(25), 5115–5121 (1996). 5. P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, and G. H. Kaufmann, “Phase errors in low-frequency vibration measurement with high-speed phase-shifting speckle pattern interferometry,” Opt. Eng. 40(9), 1984– 1992 (2001). 6. T. Wu, J. D. Jones, and A. J. Moore, “High-speed phase-stepped digital speckle pattern interferometry using a complementary metal-oxide semiconductor camera,” Appl. Opt. 45(23), 5845–5855 (2006). 7. A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “Transient deformation measurement with electronic speckle pattern interferometry and a high-speed camera,” Appl. Opt. 38(7), 1159–1162 (1999). 8. A. Ettemeyer and Z. Wang, “Verfahren und Vorrichtung zur Bestimmung von Phasen und Phasendifferenzen,” Patent DE 195 13 234 (1995). 9. H. van Brug, “Temporal phase unwrapping and its application in shearography systems,” Appl. Opt. 37(28), 6701–6706 (1998). 10. J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26(24), 5245–5258 (1987). 11. M. Reeves, A. J. Moore, D. P. Hand, and J. D. C. Jones, “Dynamic shape measurement system for laser materials processing,” Opt. Eng. 42(10), 2923–2929 (2003). 12. A. J. P. Haasteren and H. J. Frankena, “Real-time displacement measurement using a multicamera phasestepping speckle interferometer,” Appl. Opt. 33(19), 4137–4142 (1994). 13. A. L. Weijers, H. van Brug, and H. J. Frankena, “Polarization phase stepping with a savart element,” Appl. Opt. 37(22), 5150–5155 (1998). 14. T. D. Upton and D. W. Watt, “Optical and electronic design of a calibrated multichannel electronic interferometer for quantitative flow visualization,” Appl. Opt. 34(25), 5602–5610 (1995). 15. B. B. García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, “Transient deformation measurement with electronic speckle pattern interferometry by use of a holographic optical element for spatial phase stepping,” Appl. Opt. 38(28), 5944–5947 (1999). 16. B. Barrientos García, A. J. Moore, C. Perez-Lopez, L. Wang, and T. Tschudi, “Spatial phase-stepped interferometry using a holographic optical element,” Opt. Eng. 38(12), 2069–2074 (1999). 17. J. Kranz, J. Lamprecht, A. Hettwer, and J. Schwider, “Fiber optical single frame speckle interferometer for measuring industrial surfaces,” Proc. SPIE 3407, 328–331 (1998). 18. A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39(4), 960–966 (2000). 19. J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub-Nyquist interferometry: Implementation and measurement capability,” Opt. Eng. 35(10), 2962–2969 (1996). 20. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer arraybased simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005). #143988 $15.00 USD Received 10 Mar 2011; revised 21 Apr 2011; accepted 21 Apr 2011; published 9 May 2011 (C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10111


Introduction
Full-field, transient deformation measurements with interferometry have become routine with the availability of inexpensive high-speed detectors. The maximum velocity that can be measured and the number of measurement points are limited by the data transfer rate of the detector: as the velocity increases, an increased frame rate is required to sample the interferogram, thus reducing the number of measurement points at a given data transfer rate. In this research, spatial phase stepping was introduced to a high-speed interferometer in a way to increase both the velocity measurement range and the effective data transfer rate. By assuming that the velocity is continuous, a further increase in both quantities was achieved. These methods were implemented for vibration measurement with speckle interferometry, but are applicable to other types of transient deformation measured with other optical systems.
Vibration amplitude and relative vibration phase have been measured simultaneously at a number of separated points with high-speed speckle interferometry [1]. The system used a line-scan camera (256 pixels) operating at 100,000 frames per second (fps) and temporal phase stepping. It operated as a multipoint vibrometer that measured velocities up to 3.2 mm/s [2]. The measurement of relative vibration phase at separated points is not possible with a conventional single point laser vibrometer due to the low bandwidth scanning of the measurement point; nor is it possible with standard full-field techniques such as timeaveraged speckle interferometry due to the low bandwidth of the detector. Speckle interferometry with temporal phase stepping has also been used to measure continuous surface deformations at lower speeds but at more measurement points. A maximum velocity of ~25 µm/s was measured over 239 × 192 pixels at 1,000 fps [3], and ~1 µm/s was measured over 512 × 512 pixels at 40 fps [4]. The data transfer rate from the camera was similar in each case, being 25.6, 45.9 and 10.5 Mpixels/s in [1], [3] and [4] respectively. The phase evolution with time can also be determined at each pixel independently by Fourier or wavelet analysis [4].
The maximum velocity is set by the Nyquist condition, that the intensity is sampled at least twice per interference fringe [1][2][3][4] , where N is the number of phase steps per interference fringe period. The velocity limit corresponding to the Nyquist fringe sampling limit with temporal phase stepping is therefore [1]: which has a maximum value of 0.5 Nyq v for 4 N  , i.e. π/2 radian phase step between frames. In practice, phase step errors associated with non-linear velocity (e.g. acceleration) introduce errors in the phase calculated from the N sequential frames which become unacceptable above a maximum velocity of approximately 0.3 Nyq v [5,6]. Spatial phase stepping, which separates phase-stepped interferograms to one or multiple detectors, eliminates PS v and potentially enables the velocity limit to be increased to Nyq v . Separating images to multiple detectors is not practical for most high-speed applications, where the additional hardware cost can quickly become prohibitive. Separating the images to distinct regions of a single detector is an alternative, although the resolution of each image is then reduced. If three (or more) phase-stepped channels are recorded on a single detector, the increase in measurement range to Nyq v will be offset by the reduced number of pixels at which the measurement is made. In other words, the same increase in maximum velocity could be achieved at fewer pixels by increasing the camera frame rate by three (or more) times in a temporal phase stepped system, and so the effective data transfer rate from the camera has not been improved.
In this paper, two spatial phase stepped channels were recorded on a single high-speed detector through the use of binary phase gratings. The maximum velocity range was increased to Nyq v and the effective data transfer rate from the camera was increased. It is shown in the following section that by assuming that the velocity is continuous, measurements beyond Nyq v can be made and so increase further the effective data transfer rate.

High-speed sub-Nyquist interferometry
For an interferometer using continuous-wave illumination, the interference intensity recorded by a detector at pixel ( , ) xy and at time n t , integrated over the frame exposure e t , is given by [7]: , and the random speckle phase in a speckle interferometer.
For temporal phase stepping [1][2][3][4] I I  I  I  I I  I  I  tt  I I  I It is assumed that the frame rate is sufficiently high that the velocity is approximately constant during any four consecutive frames so that the phase at a given pixel can be calculated from 1 n I  to 2  n I . When the data include some known or stationary reference, recovery of the absolute spatiotemporal profile from the phase is possible. For the general case of continuous motion with unknown initial deformation, phase changes between frames represent velocity: where at a given pixel represents the change in phase due to the deformation between frames. Normalizing the velocity by the Nyquist velocity limit in Eq. (5) gives an immediate indication of the interferometer performance without referring to the particular laser wavelength and camera frame rate [6].
For spatial phase stepping, the intensity at corresponding pixels N y x ) , ( recorded on N separate regions of a single detector (or N separate detectors) at time n t is given by: phase difference at each pixel ( , ) xy can be calculated using the relation [8,9]: Greivenkamp introduced sub-Nyquist interferometry to measure large aspheric profiles in which the magnitude of the phase difference between adjacent pixels exceeded π radians [10]. Provided that the phase was measured accurately, the Nyquist limit was due to the reconstruction (spatial unwrapping) algorithm. A large increase in the measurement range of the interferometer was achieved by assuming that the spatial derivative of the aspheric surface shape (i.e. its slope) was continuous. The phase gradient was used to interpolate between pixels during spatial unwrapping of the phase. By analogy, the magnitude of the phase difference between adjacent frames in high-speed interferometry can exceed π radians if the temporal derivative of the deformation (i.e. velocity) is continuous. The measured normalized velocity will be wrapped in the range  (9) where at a given pixel (9) yields an acceleration limit for unwrapping the normalized velocity of:

Experimental demonstration
A schematic of the experimental system is shown in Fig. 1. The output from a diode-pumped, frequency-doubled Nd:YVO 4 laser (single-frequency continuous wave output at 532 nm at power levels up to 500 mW) was divided by a polarizing beam splitter (PBS) into orthogonally linearly polarized object and reference beams. Each beam was launched into the fast axis of a highly birefringent (hi-bi) optical fibre. Light from the object fibre illuminated the test object directly or through a cylindrical lens (L1) depending on the area under test. The distal end of the object fibre was rotated by 90° about its optical axis to align its fast axis (and hence the linear polarization of the emerging light) with that of the reference fibre. The test object was a centre-pinned 14 cm diameter circular aluminium plate with retroreflective coating, driven by a piezoelectric element attached to its rear face. The object was imaged to the 1024 × 1024 pixel array of a CMOS camera (Photonfocus, MV-D1024, 8-bit resolution). Light from the reference fibre was collimated, passed through an electro-optic phase modulator (PM), and then relaunched into a hi-bi fibre to path match the object beam. Light emerging from the reference fibre was collimated (by L2) and directed to the combining beam splitter (BS) to produce an on-axis reference beam on the CMOS camera. The phase modulator was used only to calibrate the size of the spatial phase steps introduced. Two identical binary phase gratings (G1, G2) of 40 µm pitch were used to divide the object and reference wavefronts. The gratings were designed with a phase modulation depth to suppress the zero and even diffracted orders at the wavelength of the laser [11]. The ± 1 diffracted orders were recorded by the detector. A π/2 phase step was produced between the diffracted orders by introducing a relative lateral translation of G2 with respect to G1 of a quarter of the grating pitch. To the best of our knowledge, it is the first time binary gratings have been used to separate the images and to introduce the phase step. Polarisation [12,13], diffraction gratings [14], and holographic optical elements [15,16], have been proposed to introduce spatial phase steps. The implementation of [14] is most relevant to this paper. A pair of diffraction gratings produced three phase-shifted interferograms on a single detector and the phase step between channels was calibrated with temporal phase stepping. The system was used for transient flow visualisation with a camera operating at 82 frames per second. A binary grating has been used to separate the images for spatial phase stepping, but ancillary polarizing elements were used to introduce the phase step [17,18].
Alignment of the imaging system required the two diffracted orders of the object and reference beams to be sampled equally by the pixel array of the detector with sub-pixel correspondence, and the relative phase step between the diffracted orders to be π/2 radians at each pixel. To achieve equal sampling of the diffracted object beams, grating G1 was rotated in its plane about the observation optical axis and translated along the optical axis for fine control of the magnification. Image correlation was used to quantify alignment errors between the two images of the object, without the reference beam [13,14]. Firstly, the correlation coefficient was maximised in the vertical direction as the grating was rotated and then maximized in the horizontal direction as the grating was translated along the observation axis. Reference beam alignment was achieved in the same way, by rotation of G2 in its plane about the reference beam optical axis and translation along the optical axis. The spatial phase step between the diffracted orders was calibrated using temporal phase stepping. With the object stationary and thermal effects reduced to a minimum by appropriate shielding around the interferometer, the phase in both diffracted orders was calculated from four consecutive phase-stepped frames using Eq. (4). A histogram of the phase difference (modulo 2π) between corresponding pixels in the diffracted orders showed a normal distribution centred on the size of the spatial phase step. G2 was translated in its plane in the horizontal direction to centre the distribution on the desired spatial phase step. Figure 2 shows the spatial phase step calculated in this way plotted against translation of the reference grating. The gradient of the best fit line is 9°/μm, corresponding to a 10 μm relative translation between G1 and G2 for a 90° phase step, i.e. a quarter of the pitch of the gratings.
A qualitative demonstration of the alignment of the diffracted orders is shown in Fig. 3. Full-field speckle interferograms of the test object were recorded before and after an out-ofplane deformation, Figs. 3(a) and 3(b). Direct subtraction of the images produced excellent quality fringes but the relative phase step between N and 1  N is lost, Fig. 3(c). Crosssubtraction between the two diffracted orders requires sub-pixel alignment. The fringe visibility was reduced due to small errors in the alignment, but the relative phase step between the diffracted orders is seen, Fig. 3(d). Clearly fringes are not required for calculating the phase with Eqs. (4) or (7), but demonstrate qualitatively the alignment of the diffracted orders. Dynamic measurements were recorded for the centre-clamped circular target vibrating at natural frequencies of 250 and 518 Hz. Full-field time-averaged fringe patterns were recorded at 25 fps to identify the vibration modes and regions of interest for the subsequent high-speed measurements. The cylindrical lens L1 was then inserted into the object beam to illuminate a horizontal region of interest which was interrogated at 20,000 fps. The ability to switch rapidly between regions of interest is an advantage of the CMOS detector [6]. The vibration amplitude was varied to produce a range of nominal velocities at these frequencies, estimated from the full-field fringe patterns.
The spatial phase stepped system was initially tested for nominal velocities of  Fig. 4(b). The velocity for one pixel is shown in Fig. 4(c). The same data were then analysed using Eq. (7), combining the spatial phase-stepped data from the diffracted orders, Fig. 4(c). The spatial phase stepping approach worked successfully to the theoretical limit of 0.5 Nyq v , whilst errors in the temporal phase stepped data increased as the velocity increased, Fig. 4(d).
x (Pixels)   The normalized velocity was calculated from the simulated interferograms using the procedure described in Section 2. The maximum error between the calculated normalized velocity and the value used to calculate the simulated interferograms is shown in Fig. 6(a) for a range of normalized velocities. The maximum error (which occurs at the maximum velocity), rather than the rms error over the whole vibration period, was used so as to keep the results general to other types of deformation. For each