A dual-wavelength femtosecond laser pulse source and its application for digital holographic single-shot contouring are presented. The synthesized laser source combines sub-picosecond time scales with a wide reconstruction range. A center wavelength distance of the two separated pulses of only 15 nm with a high contrast was demonstrated by spectral shaping of the 50 nm broad seed spectrum centered at 800 nm. Owing to the resulting synthetic wavelength, the scan depth range without phase ambiguity is extended to the 100-μm-range. Single-shot dual-wavelength imaging is achieved by using two CMOS cameras in a Twyman-Green interferometer, which is extended by a polarization encoding sequence to separate the holograms. The principle of the method is revealed, and experimental results concerning a single axis scanner mirror operating at a resonance frequency of 0.5 kHz are presented. Within the synthetic wavelength, the phase difference information of the object was unambiguously retrieved and the 3D-shape calculated. To the best of our knowledge, this is the first time that single-shot two-wavelength contouring on a sub-ps time scale is reported.
©2009 Optical Society of America
Recent advances of reliable ultrashort-pulse light sources have enabled novel holographic recording techniques, allowing the investigation of time-resolved shape measurements and moving objects in rapid transient processes, even when they are not repeatable and sometimes difficult to control under laboratory conditions . In the early days of holography, high speed events were studied using the double exposure interferometry , allowing the reconstruction from separately acquired multiple frames, using spatial multiplexing techniques [3, 4, 5]. Generally in simple holographic measurements, the phase range is limited within modulo 2π of the laser wavelength λ because of the phase ambiguity of the optical interferometry. Thus, interferograms for largely deformed systems, e.g, exhibiting isolated plateau-like sections or other discontinuities in the height profile that exceed the wavelength, cannot be unambiguously analyzed using unwrapping routines . As a resort, multiple-wavelength holography [7,8] has been demonstrated to extend the unambiguous reconstruction range to, in principle, arbitrary values. Concerning the reconstruction range, the wavelength is now substituted by a synthetic wavelength, which grows with decreasing spectral proximity of the individual light sources involved. Certainly, such range extension is not unlimited and not without trade-off. Coverage of the range interesting for micro mechanical applications between 10 – 100 μm like microelectromechanical system (MEMS)  using only one laser source is a challenging task and one aspect of the presented work.
More than one decade ago, the introduction of digital holography has inspired substantial progress in the field of optical interferometry. In 1994, Schnars and Jueptner recorded off-axis holograms using a charge coupled device (CCD) and then reconstructed them on a personal computer . In particular, the numerical reconstruction process enables to retrieve the original object wavefront, granting access to the phase information.
Two-wavelength digital holography was first introduced by Wagner et al. , employing subtraction of two reconstructed phase maps obtained with stepwise wavelength tuning of a laser to enable millimeter contouring of the object. For real-time measurements, the concept of sequential acquisition certainly had to be overcome, as recently demonstrated by Kühn et al. , who used two continuous-wave (cw) diode lasers. However, these cw sources limit the temporal resolution to the camera global shutter time, which is contrasted to the use of laser pulses in holography, yielding sub-picosecond temporal resolution . As the latter demonstrations used only one laser pulse, they were again limited to a reconstruction of the optical wavelength. In the following, we will discuss a synthesized femtosecond dual-wavelength laser source that combines practically unlimited time resolution with a wide reconstruction range. In the latter respect, our source also clearly surpasses early approaches that were based on ruby lasers , which proved to be limited to wavelength separations below 0.05 nm. Attempts to improve on ultrafast dual-wavelength light sources for holography face two major challenges: insufficient spectral separation and below-optimum pulse energy [15, 16, 17, 18]. Meaningful applications in holography mandate a scan depth on the order of a few tens of microns, which translates into required spectral separations of 10 to 50 nm. It is important to note that the scan depth needs to be carefully adjusted according to the application, as excess depth also decreases the precision of the method. Moreover, the well depth of megapixel cameras automatically necessitates pulse energies on the order of one microjoule for optimum acquisition conditions.
One frequently ignored aspect of light sources for two wavelength contouring is their spectral stability . Using unseeded pulsed laser sources, spectra of subsequent laser shots often exhibit strong variations, which translates into serious noise issues for the task of two-wavelength contouring. In contrast, we here discuss a source that is featured by stable dual-wavelength emission independent on spectral instabilities of the seed laser and therefore virtually eliminates any such detrimental spectral variation.
Finally, real-time or single shot two-wavelength contouring requires the simultaneous and separable recording of the two holograms. In general, one CCD camera is used and the spectral separation is implemented via different reference wave tilts , which requires challenging interferometric setups. Here we introduce two cameras for digital holographic recording, which allows a simplified interferometric setup with only one propagation direction of the reference wave. The two holograms are separated by a polarization multiplexing technique.
2. Basic considerations
Let us introduce some fundamental terminology for discussing properties of ultrashort light sources for holography. One important aspect is their coherence properties. For the generation of a hologram, two electrical wave fields, a reference Eref and an object wave Eobj, have to be superimposed, yielding
An interference pattern appears due to the different optical path lengths of the object wave with respect to the reference wave. As the consequence of temporal coherence it can be described introducing the delay parameter τ:
The intensity of the interference in Eq. (1) is encoded in the intensity pattern of the hologram
The function G(τ) denotes the first order cross correlation known as the complex self coherence function (SCF). The modulus of the normalized SCF is called complex degree of coherence ∣g(τ)∣ . It is commonly used to describe the coherence properties of light sources most important for holographic applications. Single frequency lasers yield ∣g(τ)∣ close to 1, i.e., ideal coherence properties. Quite naturally, ultrafast light sources employ many longitudinal modes, and can therefore only exhibit partial coherence. Furthermore, coherence is a spectral property, which can be described using statistical optics , both, for cw sources and for ultrashort laser pulse sources. Assuming a Gaussian spectral distribution, the coherence time τc is estimated as
where Δν is the spectral width (FWHM) of the light source. Equation (4) clearly illustrates that coherence is essentially dictated by the spectral width. The coherence time is readily translated into a coherence length
with the speed of light c. Consequently, as interference can only be expected within the coherence length, one has to ultimately trade the available reconstruction range for an increased temporal resolution when using short pulses for holographic applications. For example, a coherence length of 85 μm requires a 5 nm spectral width at 800 nm, which, in turn, translates into a minimum (Fourier-limited) pulse duration of about 200 fs.
As in all interferometric methods, phase retrieval in digital holography becomes ambiguous when the optical path difference between the object and the reference light is larger than the used optical wavelength. Noisy phase maps, in particular those exhibiting a large number of phase discontinuities, typically cause failure of phase unwrapping routines. Calculating the difference phase map of two separately reconstructed holograms captured at two wavelengths λ1 and λ2
where l is the optical path length, the range of phase measurement is extended to a longer periodicity than the optical wavelength. The optical path difference of the object wave field with respect to the reference wave field, i.e., the axial shape information of the object, is now given in units of the synthetic wavelength
Optimum evaluation of the reconstructed phase information requires the synthetic wavelength to be matched the expected surface dynamics of the object shape. This results in a preferential choice for λ1 and λ2 for any given laser source . Ideally, one laser source with variable spectral separation of λ1 and λ2 is required.
For spectral shaping a commercial Ti:sapphire chirped-pulse amplifier laser system (Femtolasers FemtoPro) was used. The laser system delivers 650-μJ pulses at 1 kHz repetition rate and is widely similar to the one described in Ref. 23. The amplified pulse spectrum has a spectral width of about 50 nm (FWHM) and is centered at 790 nm. Normally, the pulses are compressed to 25 fs. However, we use the spectral bandwidth of the pulse to isolate two separated wavelengths for our holographic contouring application. Consequently, the compressor section of the laser system was modified by implementing a spectrograph setup. The pulse shaping is depicted in Fig. 1.
The amplified chirped 7 ps pulses are launched into a four SF11 prism sequence to introduce sufficient angular dispersion and are subsequently imaged by a cylindrical lens into the Fourier plane for spectral filtering. In the latter, a mask with two 180 μm wide slits is placed directly in front of the retro mirror to cut out two distinct parts of the spectrum. The spectral separation was set to 15 nm yielding a synthetic wavelength of 40 μm [Eq. (9)]. Translating the slit mask in the Fourier plane, the spectral peak positions are adjustable within the entire seed spectrum between 750 and 840 nm. We chose λ1 = 772 nm and λ2 = 787 nm, as shown in Fig. 2(a) together with the seed pulse spectrum. The spectrographic setup ensures the absolute dual-wavelength stability independent on seed laser fluctuations, which is essential for precise two-wavelength contouring. The two 2 μJ pulses exhibit a high spectral contrast [Fig. 2(a)], with each pulse displaying very similar spectral and temporal characteristics. The spectral width of the individual pulses amounts to 4 nm (FWHM), corresponding to a coherence length of 100 μm, which defines the maximum scan depth of the object. Figure 2(b) shows the autocorrelation trace behind the spectrograph, indicating two pulses with a temporal separation of 4 ps, setting our temporal recording window. The single pulse duration of the wavelength λ2 = 787 nm is shown in the inset of Fig. 2(b). Assuming a sech2-pulse shape the duration is about 800 fs.
We employed a Twyman-Green interferometer to record the holograms and to perform digital holography (Fig. 3). A central cubic beam splitter (BS) separates reference and object beam for illumination and brings them to interference in the camera arm. The object under investigation is placed in one arm of the interferometer; the other arm contains the reference mirror mounted on a piezo translation stage, which allows for a precise adjustment of the interferometer arm length difference.
For our approach of single shot two-wavelength contouring, two spectrally separated holograms are to be simultaneously yet independently recorded by two CCD cameras. To separate both interferograms, we employed polarization encoding with two multi-order half-wave plates behind the beam splitter in the camera arm (Fig. 3). These waveplates rotate the polarization of λ1 by 90 degrees, directing the respective hologram on to camera 1, whereas λ2 preserves its polarization and is transmitted to camera 2. The polarization beam splitter (PBS) serves as a spectral separator. The suppression ratios in the two camera arms are shown in Fig. 4, revealing a 10:1 ratio between desired and suppressed wavelength component in each arm without cross-talk during the digital holographic reconstruction routine.
Two identical 8 bit monochromatic complementary metal-oxide-semiconductor (CMOS) cameras (AVT Marlin F-131B) with a 1280×1024 pixel sensor and a pixel size of 6.7 μm are used. The minimum global shutter time amounts to 20 μs while the maximum frame rate is 25 frames per second. The camera recording routine is triggered by the laser source. Given a laser pulse repetition rate of about 1 kHz, the short shutter time of the camera therefore still enables capturing of individual pulses. Consequently, the recording time for the hologram is now set by the pulse duration itself and no longer by the camera shutter time.
Single-shot holography requires the extraction of the object information from a single exposure. For this purpose, off-axis holography is commonly used [24, 25]. The reference wave ri(x,y) is tilted with respect to the object wave during recording, inducing a spatial carrier in the hologram. This technique permits filtering the hologram in its Fourier plane, where the additional spatial carrier frequency contains the object information. The both pre-processed object informations hi(x,y) in the camera plane are then independently focused back into the origin object plane along the reconstruction distance di by numerically solving the Fresnel-Kirchhoff diffraction integral
The angles θS and θB indicate the illumination and the observation direction with respect to the optical axis. For a linear space-invariant system the diffraction integral can be written as a convolution integral of the product of hologram and reference wave with the convolution kernel f(x′ - x, y′ - y). For solving the integral the Fourier convolution theorem is used. This method is called convolution approach [22, 26]. Taking advantage of the discrete fast Fourier transformation reconstruction rates higher than 1 frame/s are demonstrated in the literature . In contrast to the Fresnel-method no wavelength-dependent object scaling occurs with the convolution approach during reconstruction. As a consequence, the independently reconstructed object information has the same lateral spatial resolution as the camera. In principle, the phase difference can be calculated without any scale correction.
Using two cameras for independent hologram recording mandates two main issues. First, due to the polarization beam splitter, one hologram appears vertically mirrored compared to the other one. Second, positioning of the two cameras with sub-pixel precision with respect to the laser beam is a difficult task. Thus, the holograms are slightly shifted with respect to each other. Both problems can be numerically solved using image processing tools and Fourier theorems. The amount of displacement is determined by the cross correlation of the both images and is corrected using the Fourier shift theorem . This lateral correction is the necessary precondition for the accurate phase difference calculation, i.e., a precise contouring result.
To study our approach of single-shot digital holographic two-wavelength contouring, a MEMS single axis scanner served as test object (Fig. 3, inset). The scanner oscillates around one axis with a specific tilt angle by the suspension of the mirror plate on two torsion beams. The oscillating scanner mirror in the center of the conductor board has a size of 4×4 mm2 and is electrostatically driven at its mechanical resonance frequency of 540 Hz. The driver parameter bias and the sinusoidal modulating voltage can be varied to realize specific tilt angles during the mirror oscillation. To detect fast shape modifications, a synthetic wavelength larger than these dynamics is essential. The MEMS single axis scanner investigations have been performed at a maximum mirror elongation that can be completely captured, as imposed by the size of the CMOS-camera chip. The maximum tolerable elongation angle amounts to roughly 0.25°, which fits the 20 μm unambiguous evaluable depth of object shape dictated by the generated synthetic wavelength. Figure 5 shows simultaneously recorded holograms of the oscillating mirror, as stroboscopic recorded with both cameras. The state near the reversal point was chosen for further evaluation in this regime with limited elongation. The phase maps of the wave field in the object plane are reconstructed for λ1 = 772 nm and λ2 = 787 nm.
To identify the deformation of the mirror surface all linear parts of the reconstructed phase are subtracted [Figs. 6(a) and 6(b)], generating a phase ramp by taking the carrier location in the Fourier transformation map. The mirror slope disappears during this procedure. But the slope information remains in the linear phase part of the unchanged background of the MEMS module, visible in the rim zones of Figs. 6(a) and 6(b). By turning back the background in a plane position as a reference the original mirror slope is recognizable. Retrieving such a contour with the phase information of only one wavelength (e.g. λ1), more than thirty 2π-phase jumps appear, which will most likely cause immediate failure of typical phase unwrapping algorithms. Therefore the mirror slope is retrieved by performing our two-wavelength contouring method by calculating the difference of the reconstructed phases.
Prior to the phase map subtraction of the two holograms, the difference of its spatial positions has to be corrected with sub-pixel accuracy. For this reason a cross correlation between the reconstructed amplitudes is performed. Using a center of mass routine the coordinates of the maximum are found with the required accuracy yielding the relative spatial position of the two reconstructed object fields. Now, the difference of the spatial position of the holograms is corrected by taking advantage of the Fourier shift theorem. In a subsequent step, the reconstructed phase maps are taken to calculate the phase difference information. This calculation delivers a phase map with phase values between -π and π, converted to a gray level scale as displayed in Fig. 6(c). As the phase dynamics slightly exceeds the 2π range of the synthetic wavelength, an unwrapping routine is applied for phase demodulation. The phase values Δφ are converted into optical path differences using
Additionally, 3D plots were calculated from the unwrapped and converted phase difference maps of the static and the oscillating mirror at its turning point shown in Fig. 7(a) and Fig. 7(b), respectively. From a cut along the slope of the 3D contour map of the moving mirror in Fig. 7(b), the tilt angle and the elongation of the mirror can be readily extracted [Fig. 7(c)]. A linear fit to the data indicates a distance between the upper and lower edge of the oscillating mirror of 17.7 μm and an average slope of 0.25°. The same slope is independently confirmed by monitoring the angular displacement of a cw laser beam for identical mirror driver parameters.
For the calculated phase difference map, several error sources must be considered. The influence of spectral crosstalk was examined by recording the hologram for only one wavelength and blocking the other wavelength, e.g. by sequentially capturing the holograms. The calculated phase difference was the same as in the presented case of simultaneous recording. The largest contribution is probably from the cemented PBS leading to errors in the phase image as visible in Fig. 6(b). Also, the accuracy of the cross correlation of the two reconstructions calculated from holograms stored with different cameras is certainly limited. Accounting for inaccuracies related to the mask setting in the Fourier domain and phase distortions arising from setup components like the beam splitters lead to an estimated error of approximately 3 μm, corresponding to about a tenth of the synthetic wavelength Λ. This means that waviness of the mirror surfaces seen in Fig. 7 are artifacts rather than a real deformation of the single axis scanner.
To reliably measure the surface deformation of the oscillating mirror the evaluation of the single phase images is sufficient as obvious in Fig. 6. The resulting deformation of the mirror based on the reconstructed single phase from the hologram captured at λ2 = 787 nm is shown in Fig. 8(a). The deduced rms-value of the surface shape of the oscillating mirror is only ~50 nm, corresponding to a surface flatness of better than λ/10. This is an excellent result, keeping in mind that the mirror of the single axis scanner is only 50 μm thick, yet has an area of 16 mm2. This detail information from one interferogram can now be combined with the coarse shape information deduced from the phase difference map of the two interferograms captured at different wavelengths. Using two-wavelength contouring only for extracting the linear slope of the mirror and combining this with the detail information of Fig. 8(a) yields the picture in Fig. 8(b). This combined reconstruction provides a much more realistic picture of the virtually vanishing deformations of the single axis scanner operated at its resonance. We are not aware of any other method that could provide equally detailed information on such a MEMS structure while simultaneously capturing such a large amplitude of the dynamics.
We presented a new approach for recording two spectrally separated holograms simultaneously to perform single-shot digital holographic two-wavelength contouring of fast moving objects. For this purpose, a laser pulse source was developed that generates two spectrally separated ultrashort pulses. Furthermore, the spectral positions of the generated wavelengths are fixed and independent on seed laser instabilities. The sub-ps duration desensitizes the holographic setup toward environmental impacts, like shocks or vibrations. A center wavelength distance of only 15 nm with a high contrast was demonstrated by spectral shaping of the 50 nm broad seed spectrum centered at 800 nm. A Twyman-Green interferometer was extended by a polarization encoding sequence to separate the interferograms for the recording process. The two holograms were captured simultaneously introducing two CMOS-cameras in the interferometer setup for the first time to the best of our knowledge.
Single-shot digital holographic two-wavelength contouring was performed with an off-axis setup using a single axis MEMS operating at a resonance frequency of 0.5 kHz. The phase difference information of the object within the synthetic wavelength of 40 μm was unambiguously generated and the 3D-shape calculated. Different elongation states up to 18 μm of the oscillating mirror were clearly identified. The surface dynamics during operation are in the order of 50 nm, corresponding to a surface flatness of better than λ/10.
The results open up the perspective for using digital holography as a tool for recording rapid deformations on depth scales in the micrometer range. We are confident that our setup can be extended to single-shot multi-wavelength contouring. Future experiments will include high-speed CCD cameras in the setup to improve the temporal resolution.
This work has been supported by the Deutsche Forschungsgemeinschaft under the grant no. GR2115/1-1. The authors are very grateful to Markus Fromm (ETH Zürich) for initial dual-wavelength source experiments.
References and links
1. Z. Liu, M. Centurion, G. Panotopoulus, J. Hong, and D. Psaltis, “Holographic recording of fast events on a CCD camera,” Opt. Lett. 27, 22–24 (2002). [CrossRef]
2. L. O. Heflinger, R. F. Wuerker, and R. E. Brooks, “Holographic Interferometry,” J. Appl. Phys. 37, 642–649 (1966). [CrossRef]
3. T. Tschudi, C. Yamanaka, T. Sasaki, K. Yoshida, and K. Tanaka, “A study of high-power laser effects in dielectrics using multiframe picosecond holography,” J. Phys. D11, 177–180 (1978).
4. W. Hentschel and W. Lauterborn, “High-Speed Holographic Movie Camera,” Opt. Eng. 24, 687–691 (1985).
6. U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques, (Springer Verlag, 2005).
7. B. P. Hildebrand and K. A. Haines,”Multiple-wavelength and multiple-source holography applied to contour generation,” J. Opt. Soc. Am. 57, 155–162 (1967). [CrossRef]
9. T. Gessner, J. Bonitz, C. Kaufmann, S. Kurth, and H. Specht, “MEMS based micro scanners: Components, Technologies and Applications,” Actuator 2006, 10th Intern. Conf. on New Actuators, 193–198 (2006).
11. C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000). [CrossRef]
12. J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Realtime dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15, 7231–7242 (2007). [CrossRef] [PubMed]
14. G. Pedrini, P. Froening, H. J. Tiziani, and M. E. Gusev, “Pulsed digital holography for high-speed contouring that uses a two-wavelength method, ” Appl. Opt. 38, 3460–3467 (1999). [CrossRef]
15. J. F. Xia, J. Song, and D. Strickland, “Development of a dual-wavelength Ti:sapphire multi-pass amplifier and its application to intense mid-infrared generation,” Opt. Commun. 206, 149–157 (2002). [CrossRef]
16. Z. Zhang, A. M. Deslauriers, and D. Strickland, “Dual-wavelength chirped-pulse amplification system,” Opt. Lett. 25, 581–583 (2000). [CrossRef]
17. C. P. J. Barty, G. Korn, F. Raksi, C. Rose-Petruck, J. Squier, A.-C. Tien, K. R. Wilson, V. V. Yakovlev, and K. Yamakawa, “Regenerative pulse shaping and amplification of ultrabroadband optical pulses,” Opt. Lett. 21, 219–221 (1996). [CrossRef] [PubMed]
20. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52, 1123–1130 (1962). [CrossRef]
21. W. Lauterbach and T. Kurz, Coherent Optics: fundamentals and applications, (Springer Verlag, 2003).
22. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (Wiley & Sons Inc., 1991).
23. M. Hentschel, Z. Cheng, F. Krausz, and C. Spielmann, “Generation of 0:1-TWoptical pulses with a single stage Ti:sapphire amplifier at a 1-kHz repetition rate,” Appl. Phys. B 70, 161–164 (2000).
24. M. Takeda, I. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982). [CrossRef]
25. U. Schnars, T. M. Kreis, and W. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996). [CrossRef]
26. T. Kreis, Holographic Interferometry: Principles and Methods (Akademie Verlag, 1996).
27. J. W. Goodman, Introduction to Fourier Optics, 3rd ed (Roberts & Company Publishers, 2005).