Second-order nonlinear holographic microscopy for highspeed, three-dimensional imaging is demonstrated. The use of harmonic generation allows image formation of endogenous features in biological samples such as muscle tissue. We have acquired holograms with acquisition times as short as 10 ms, limited by the switching speed of our shutter; frame rates of 100’s of Hz are expected to be possible. The samples are imaged with a Yb:KGW femtosecond laser oscillator, whose 1027 nm wavelength is well suited to minimize absorption and scattering. The low average power of the oscillator prevents damage to the sample.
© 2010 Optical Society of America
The combination of second-harmonic generation (SHG) microscopy and holographic image acquisition provides a unique combination of intrinsic contrast mechanism and imaging speed. On the one hand, nonlinear optical microscopy has proven to be a powerful technique for imaging biological samples such as collagen and myosin without the need of externally-introduced markers, and it offers additional benefits including nonlinear optical sectioning. On the other hand, the scanning techniques usually used in nonlinear microscopy limit the refresh rate of image acquisition, presenting a significant barrier for applications that need to capture dynamics of a process under study [1–9]. Since holography encodes 3D optical field information in single 2D images, it enables the sample to be imaged at vastly improved speeds. An application requiring very high speed SHG microscopy in three dimensions (3D) was presented in work describing a random-access SHG microscopy has been used to map out specific points of interest in a 3D neurological network . Here, a finite set of image points were studied to observe neurological network dynamics. In another application, time-resolved SHG imaging was used to look at 2D projections of calcium waves during egg fertilization  - a process that takes ~ 50–60 seconds. Calcium waves have also been observed in fluorescently labeled rat cardiac myocytes using SHG microscopy . Much more information of chemical waves, neurological networks, and other biological systems can be gleaned by measuring a 3D images at high speeds.
In this work, we present results of SHG microscopy of biological specimens, displaying image acquisition times as low as 10 ms. This frame rate is currently limited by our mechanical shutter, but with a faster shutter we anticipate a signal-to-noise ratio of 10 dB at 40 μs based on a signal-to-noise analysis of our 3D SHG images. This work demonstrates the capability of 3D images formed with SHG contrast in a volume with speeds far exceeding video rates, which will benefit numerous applications.
SHG microscopy was first introduced less than a decade after the first demonstration of the laser [13, 14]. Three-dimensional images were formed shortly after by scanning the focal spot of a continuous-wave laser through the sample to form 3D images . The nature of nonlinear interactions in biological specimens has motivated extensive investigations of both SHG and third harmonic generation (THG) optical microscopy [16–18]. Since SHG microscopy forms images for structures comprising ordered non-centrosymmetric molecules or tissues, such as microtubule, myosin and collagen [19–21], it provides a novel optical image contrast mechanism [22–24] that has found use in in-vivo biological studies.
Since the signal measured in SHG microscopy is proportional to the square of the illumination intensity, the signal is localized to a small volume near the focus, providing nonlinear optical sectioning and the ability to form high quality images in scattering media [25, 26]. Other nonlinear microscopies inherit these advantages while also offering rich possibilities for additional contrast mechanisms in biological imaging. THG microscopy [27–29] forms images at interfaces, and chemically-specific coherent nonlinear microscopy is obtained through use of coherent anti-Stokes Raman scattering [30, 31].
While nonlinear scanning microscopy is a valuable and powerful tool, it is inherently limited by low image acquisition rates. The standard implementation of this technique requires a tightly focused fundamental pulse to be mechanically scanned through each 3D volume element, where images are assembled by serially collecting a nonlinear signal from each voxel. By contrast, a hologram encodes the 3D optical field information in a 2D image. Recording the holographic interferograms with a CCD camera has opened a vast field of digital holography, leading to significant improvements in the speed of hologram processing. Using digital holograms, 3D microscopy images can be computed numerically from a single image capture [32–35]. In these techniques, a digital hologram is recorded by interference with a known reference wave, which allows the complex electric field to be determined at the CCD camera. The complex field can be numerically back-propagated to the sample region to obtain 3D image information . In this work, we make use of off-axis holography [37, 38], which provides a straight-forward algorithm for isolating the real and virtual images of the hologram.
Second harmonic generation holography was recently introduced to acquire 3D SHG microscopy images . This experiment used a 1 mJ, 10 Hz repetition rate femtosecond laser amplifier to image the distribution of 100 nm diameter nanoparticle clusters made of the non-centrosymmetric BaTiO3 crystal. A subsequent experiment employed a Ti:sapphire oscillator and recorded SHG holograms from (HeLa) cells labeled with BaTiO3 nanoparticles . The extrinsic nanoparticle markers were introduced to compensate for the biological samples under study, which did not themselves yield SHG signals. In another work, Shaffer et al.  used SHG holography to study SHG generated by the longitudinal field at an air-to-glass interface.
The advantages of the intensity-dependent response in nonlinear microscopy require high intensities to obtain images with an adequate signal-to-noise ratio. Peak intensity and average power thermal damage to the specimen limits the usable intensity . These issues can be partly mitigated by the use of femtosecond illumination pulses and ultra-sensitive EMCCD detectors. In the experiments presented here, we use ultrashort pulses from a 1027 nm Yb:KGW oscillator, whose wavelength is in a biological window where the combination of scattering and absorption are exceptionally low . The combination of oscillator-only, SHG digital holographic microscopy allows us to obtain 3D images without extrinsic markers at an unprecedentedly high speed.
In our experiments, off-axis digital holograms are formed by recording, using a digital CCD camera, the interference between SHG radiation generated by ordered structure in a specimen, and an off-axis reference wave. To describe the recorded hologram, we will consider the electric field of the incident beam illuminating the sample, which takes the form 𝓔0(r 0,t) = E 0(r 0)e i(ω0t-k0z0), where E 0(r 0) describes the spatial distribution of the field strength and ω 0 is the incident field optical radial frequency. The fundamental field drives a polarization density oscillating at the second harmonic of the incident field given by
The nonlinear optical susceptibility χ (2)(r 0) tensor elements are dictated by the material of the object. Here, r 0 = x 0 ĩ+y 0j̃+z 0 k̃ is the spatial coordinate in the object space.
The second harmonic radiation emerging from the object, 𝓔SHG(x,y,t;z) = E obj(x,y;z)e i(ω2t-k2z), contains information encoding the 3D spatial structure of the specimen, and can be described by a Green’s function formalism .
where we assume z ≫ z 0, ω 2 = 2ω 0, k 2 = ω 2 n/c. The dyadic Green’s function is given by
The radiated SHG field is collected by a microscope objective and a hologram is formed by interference with an independently frequency-doubled reference, E ref(x,y)e i2ω0t. The hologram is recorded with a lateral magnification Ml, and its intensity pattern is given by
The first two terms are the intensities of the reference and the object waves, and the last two terms are the real and the virtual images, respectively . Here, (x,y) are the transverse coordinates in the plane where the magnified hologram is recorded. The linear polarization of the reference beam selects co-polarized object beam components to form interference.
Off-axis holography enables complete extraction of the complex field of the object beam, provided that the spatial carrier frequency of the interference fringes is sufficiently high . In digital holography, a two-dimensional Fourier transform of the hologram IH is used to separate the real and virtual images from the static terms in the transformed space (kx, ky) . The angle of interference between the object and reference waves is chosen to provide a sufficiently high carrier spatial frequency to prevent aliasing of these three terms.
Numerical reconstruction of the field requires isolation of the complex object field, followed by back propagation through the object space to recover the three-dimensional object distribution. The object field recorded at the CCD is obtained by filtering the real image sideband of the numerically 2D Fourier transformed digital hologram. An inverse 2D transform of this sideband yields the quantity E * ref(x,y) · E obj(x,y). The object field is obtained by multiplying by the conjugate of plane wave reference field, whose intensity is independently measured for each acquired hologram. Once the object field E obj(x,y) has been retrieved, we use a Fresnel propagation kernel in the spatial frequency domain to yield the field at a propagation distance z .
3. Experimental Setup
The ultrashort laser pulses used in this experiment are derived from a home-built Yb:KGW laser oscillator. Briefly, a 20 W diode array at 975 nm (Apollo) is free-space imaged and focused into a 2 mm long, 5%-at doped Yb:KGW crystal (NovaPhase). The standing-wave cavity has one 105 μm (1/e 2) waist overlapping the pump focus inside the crystal, and a second 100 μm waist on a 4% modulation depth SESAM (BATOP). The SESAM and an intracavity compressor lead to stable modelocked pulses with 4.5 nm optical bandwidth centered at 1027 nm. We obtain a 70 MHz pulse train with energies up to 14 nJ per pulse.
The holographic microscopy setup is based on a modified Mach–Zehnder interferometer as shown in Fig. 1(a). A half wave plate (Tower Optical) followed by polarizing beamsplitter cube (PB) controls the balance between the beam energies in the reference and object arms. In the reference arm, the fundamental beam is focused into a 100 μm thick KDP crystal (C, EKSPLA) with a f 1 = 50 mm lens to generate the SHG reference beam. This reference is collected and collimated with a f 2 = 125 mm lens, to fill the back aperture of the focusing objective (f 3) of a spatial filter (SF, Newport). The reference beam is spatially filtered with a 15 μm pinhole and recollimated with a f 4 = 100 mm lens. This results in a ~20 mm diameter reference beam with high spatial phase purity. In the object arm, the beam is focused into the sample with a Meiji S-Plan 4× 0.1 NA objective. The 1/e 2 focal spot radius is measured to be 35 μm, in good agreement with the expected value of 32 μm. The SHG signal from the sample is collected with a Zeiss Epiplan 50× 0.5 NA objective. Both focusing and collection objectives are mounted in tip-tilt mounts (Thorlabs) on three-axis translation stages (OptoSigma) to allow for precise alignment. The specimens are held in a custom microscope slide holder on a three-axis stage. The sample and reference SHG beams are combined with a non-polarizing beam splitter (BS, Thorlabs). The combined signal is first filtered by a 10 nm bandpass filter centered at 510 nm and a colored glass bandpass filter (Thorlabs). To record the holograms, we use an Andor Newton electron-multiplying charge-coupled device (EMCCD) camera, thermo-electrically cooled to −65°C, that has 1600×400 pixels with a 16 μm pixel pitch. The high sensitivity of the electron-multiplying gain, along with its low noise cooled CCD, allows us to record low intensity SHG holograms with high signal-to-noise and short integration times. When processing the holograms presented below, we crop a relevant 400×400 pixel area.
The normalized spectrum of the second harmonic reference beam is shown in Fig. 1(b) with SHG signals from several of the samples under study. The spectra are centered around 513 nm, with an optical bandwidth of ≈2.3 nm. Without a sample in the microscope, we confirmed that the microscope glass slide interfaces did not yield any measurable second-harmonic signal.
Due to the short coherence length of the pulses used to form the hologram (ℓc ≈ 80μm), the object and reference beams must be temporally overlapped. This is accomplished with a delay stage in the reference arm. In order to co-locate the pulses in time, the object fundamental beam is independently frequency-doubled before the microscope in a 4 mm KDP crystal (Ct), and combined with the reference without a sample. The interference fringes between the object and reference beams are observed as the length of the reference arm is changed by the translation stage. Time overlap is optimized by maximizing the depth of modulation of the interference pattern. Figure 1(c) shows the Fresnel interference pattern for an on-axis hologram setup. The holographic data shown for the rest of this paper are taken in an off-axis configuration.
In Fig. 2, we show holographic reconstructions of a cluster of potato starch granules. Figure 2(a) shows a linear microscopy image of the starch illuminated with white light, while Fig. 2(b) shows the SHG generated purely in the sample collected near the image plane. The SHG signal is concentrated where the intensity of the incident fundamental beam is greatest, near three starch granules labeled A, B and C for comparison with Fig. 3. The hologram, recorded at distance dH = 25 cm from the image plane, is shown in Fig. 2(c). The hologram was reconstructed using the analysis described in the theory section and is shown in Fig. 2(d).
The lateral optical resolution of the SHG holographic microscopy system is determined by a number of factors. The first limitation is set by the NA of the collection objective. Note that the recorded wavelength relevant for imaging resolution is the second harmonic wavelength of λ SHG = 513 nm. To evaluate the imaging resolution or our collection objective, we measured the lateral resolution due to this objective by collecting SHG images from an 80 nm diameter gold nanosphere with the camera at the focal image plane. Under optimal conditions, the Rayleigh resolution is given by r 0 = 0.61λ SHG/NA = 0.62 μm and the FWHM of the point spread function is slightly smaller, with a value of ∆r FWHM = 0.52λ SHG/NA = 0.53 μm. The measured PSF of 0.85 μm, in relatively good agreement with the expected value. The resolution of holography is further constrained by the recorded fringes, which can be limited by the temporal coherence, the spatial coherence, and the resolution and digitization of the recording medium . To characterize the lateral magnification of our holographic recording system, we reconstructed holograms of 12.5 μm pitch wire calibration mesh, illuminated by SHG generated before microscope, that were recorded with the camera in the hologram plane. The lateral magnification was determined to be ∣M ℓ∣ ≈ 67, which implies that the holographic resolution δr CCD = 1.22∆pixel/M ℓ, where ∆pixel is the CCD pixel size, yields δr CCD ~ 0.29μm. The last relevant constraint on spatial resolution is that imposed by the coherence length, for which the resolution is given by , where zH is the distance between the object image formed by the collection objective and the CCD camera. This resolution evaluates to ∆r coh ~ 0.2μm. From this analysis, it is clear that the NA of the collection objective provides the dominant limitation on our spatial resolution, which could be improved with the use of higher-NA objectives including water- and oil-immersion.
Translating the starch cluster shown in Fig. 2 allows us to characterize the axial magnification of the microscope. In Fig. 3, we show the maximum reconstructed intensity, in a restricted region around the granules labeled A, B and C, for different reconstruction distances dH. The local peak intensities reconstructed from the hologram in Fig. 2(c) are shown with a solid blue line. From the data, we see that the granules come into focus at distances dH = 190,198,180 mm for A,B,C. Next, we translate the sample by dS = 10 μm and record a new hologram. The reconstruction peak intensities are shown by the red dashed line, and the reconstruction distances for the granule foci are observed to shift by ∆dH ≈ 47 mm. We attribute the decrease in the peak intensity for the ds = 10 μm data to a reduced light collection angle of the collection objective when the sample is translated. The change in the width may be due to different lateral magnification when the sample is reconstructed at a different image plane. The large width and characteristic structure of peak C as compared to A and B arises because the interference rings from granules A and B enter into the region around C. The axial magnification is calculated by Ma = ∆dH/dS ≈ 4700, in good agreement with the theoretical value of Ma = M 2 ℓ ≈ 4500.
The validity of our reconstruction algorithm was investigated by comparing second harmonic images to holographic images back propagated to the same SHG image acquisition plane, using a prepared corn seed slide (The Microscope Store). Figure 4 shows second harmonic images taken at two different camera positions Z1 = 14 cm and Z2 = 6 cm from the image plane, corresponding to 17 μm displacement in sample space. A hologram was recorded, numerically reconstructed, and propagated to the plane at which the two reference SHG images were acquired. It can be seen that different features (marked by circles in Fig. 4) come into focus for different camera positions, corresponding to different slices within the sample. The small discrepancies between the images and reconstructions may be due to potential phase variations across the reference beam, which we assume to be a plane wave. These variations can be compensated for with existing algorithms [45–47].
Biological tissues such as collagen, myosin, and muscle fibrils exhibit a strong second harmonic signal due to their highly organized molecular structures [20, 21, 48–52]. This allows us to record second harmonic holographic microscope images without the need for additional labels. The incident pulse energy was adjusted to 4 nJ, corresponding to a peak intensity of around 300 MW/cm2. A 3D reconstruction of a sample of muscle tissue is shown in Fig. 5. The animation shows the series of reconstruction planes, showing the intensity source data from which the isosurface is calculated. In particular, we see two closely-spaced fibrils at reconstruction distances dS = 25,35 μm. These results demonstrate the utility of this laser source and microscopy setup to biologically relevant samples.
As muscle fiber is highly ordered, SHG microscopy images are sensitive to the incident polarization [49, 50, 53]. To study these effects,we insert a half wave plate in the object arm, before the condenser. At the image plane, we record the second harmonic signal, while rotating the polarization of the field incident on the sample by increments of 10°. The result is shown in Fig. 6. As can be seen in the movie, the intensity of the fibrils changes versus the polarization angle. Moreover, depending on the orientation of the fibrils with respect to the polarization angle, different part of the sample, corresponding to tissue regions with changing orientational directions can be observed.
The images presented above were recorded with a 500 ms integration time, to maximize the signal-to-noise ratio (SNR) of the data and minimize readout artifacts in the camera, where illumination of the CCD during electron shifting causes a smearing of the image. In order to investigate the relationship between integration time and SNR, and assess the feasibility of this microscopy system for high-speed applications, we measured and reconstructed holograms of a single starch bead and a muscle tissue sample, using a mechanical shutter to restrict the illumination time. The results are shown in Fig. 7. As a measure of the signal, we determine the maximum signal strength within a small region about the center of the image, shown in the inset by the blue rectangle; the noise was estimated by taking the intensity root-mean square deviation in a region at the edge of the hologram, marked by the yellow square. We observe the SNR ratio to decrease approximately linearly, as expected for a variation of CCD integration time, with shorter integration times, until the integration time falls below the 10 ms shutter switching speed. CCD acquisition times below the shutter speed still accumulate charge while the image is clocked out, leading to distortion of the object. Extrapolating the trend measured above this limit, we anticipate a 10 dB SNR at 40 μs integration time. The deterioration of the signal-to-noise could be further mitigated by more aggressively cooling the camera (down to -100° with water cooling).
In conclusion, label-free second harmonic holographic microscopy has been performed with biological samples for the first time with an ultrashort Yb:KGW laser oscillator. The holograms were recorded with a CCD camera and numerically reconstructed with a numerical Fresnel propagation kernel. Importantly, these images required pulse energies below 5 nJ, and using SHG obviates the need for external contrast markers. No tissue damage or loss of image contrast was observed, even after continued exposures. We used the microscope to image various samples including corn seeds, potato starch, and human muscle fibrils. The resolution could be increased by the use of a more powerful collection objective, a higher-resolution camera, or by employing a more sophisticated reconstruction algorithm that includes aberration compensation. A key motivator for SHG holographic microscopy is to increase the speed of 3D SHG image acquisition. In this work, we demonstrate the capture of a 3D image volume of a biological specimen with an exposure time of ~ 10 ms with an SNR > 30 dB. This exposure time will allow for imaging of volumes at nearly 100 cubes per second rates using a non-mechanical shutter. Extrapolating our current measurement SNR data, we anticipate the sub-ms exposure time will be possible, opening the possibility of SHG volume images to be captured at rates nearing 1000 cubes per second, which will enable real-time observation of the full 3D spatio-temporal dynamics of neurological circuits , among other applications. Sensitivity and potentially speed of 3D SHG holography can be further optimized with reference waves. We anticipate that label-free SHG holography will enable numerous applications for studying high speed 4D spatio-temporal, particularly in biological systems.
The authors gratefully acknowledge support from the National Science Foundation Award ECCS-0901870.
References and links
1. M. Antkowiak, N. Callens, C. Yourassowsky, and F. Dubois, “Extended focused imaging of a microparticle field with digital holographic microscopy,” Opt. Lett. 33, 1626–1628 (2008). [CrossRef] [PubMed]
2. Y. Otsu, V. Bormuth, J. Wong, B. Mathieu, G. P. Dugue, A. Feltz, and S. Dieudonne, “Optical monitoring of neuronal activity at high frame rate with a digital random-access multiphoton (RAMP) microscope,” J. Neurosci. Meth. 173, 259–270 (2008). [CrossRef]
3. M. S. Amin, Y. Park, N. Lue, R. R. Dasari, K. Badizadegan, M. S. Feld, and G. Popescu, “Microrheology of red blood cell membranes using dynamic scattering microscopy,” Opt. Express 15, 17001–17009 (2007). [CrossRef] [PubMed]
5. N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Quantitative phase imaging of live cells using fast Fourier phase microscopy,” Appl. Opt. 46, 1836–1842 (2007). [CrossRef] [PubMed]
6. V. Crepel, D. Aronov, I. Jorquera, A. Represa, Y. Ben Ari, and R. Cossart, “A parturition-associated nonsynaptic coherent activity pattern in the developing hippocampus,” Neuron 54, 105–120 (2007). [CrossRef] [PubMed]
7. O. Garaschuk, R. I. Milos, C. Grienberger, N. Marandi, H. Adelsberger, and A. Konnerth, “Optical monitoring of brain function in vivo: from neurons to networks,” Pflügers Archiv European Journal of Physiology 453, 385–396 (2006). [CrossRef] [PubMed]
8. T. A. Ooms, R. Lindken, and J. Westerweel, “Digital holographic microscopy applied to measurement of a flow in a T-shaped micromixer,” Exp. Fluids 47, 941–955 (2009). [CrossRef]
10. L. Sacconi, J. Mapelli, D. Gandolfi, J. Lotti, R.-P. O’Connor, E. D’Angelo, and F.-S. Pavone, “Optical recording of electrical activity in intact neuronal networks with random access second-harmonic generation microscopy,” Opt. Express 16, 14910–14921 (2008). [PubMed]
12. C. Soeller and M.-B. Cannell, “Construction of a two-photon microscope and optimisation of illumination pulse duration,” Pflgers Archiv European Journal of Physiology 432, 555–561 (1996). [CrossRef]
13. R. Hellwarth and P. Christensen “Nonlinear microscopic examination of structure in polycrystalline ZnSe,” Opt. Commun. 12, 318–322 (1974). [CrossRef]
15. J. N. Gannaway and C. J. R. Sheppard, “Second harmonic imaging in the scanning optical microscope,” Opt. Quantum Electron. 10, 435–439 (1978). [CrossRef]
16. C. Sun, “Higher harmonic generation microscopy,” Adv. Biochem. Eng. Biot. 95, 17–56, (2005).
17. J. Squier and M. Müller, “High resolution nonlinear microscopy: A review of sources and methods for achieving optimal imaging,” Rev. Sci. Instrum. 72, 2855–2867 (2001). [CrossRef]
18. K. E. Sheetz and J. Squier, “Ultrafast optics: Imaging and manipulating biological systems,” J. Appl. Phys. 105, 051101 (2009). [CrossRef]
19. P. J. Campagnola, A. C. Millard, M. Terasaki, P. E. Hoppe, C. J. Malone, and W. A. Mohler, “Three-dimensional high-resolution second-harmonic generation imaging of endogenous structural proteins in biological tissues,” Biophys. J. 82, 493–508 (2002). [CrossRef]
20. P. J. Campagnola and L. M. Loew, “Second-harmonic imaging microscopy for visualizing biomolecular arrays in cells, tissues and organisms,” Nat. Biotechnol. 21, 1356–1360 (2003). [CrossRef] [PubMed]
22. Y. Guo, P. P. Ho, A. Tirksliunas, F. Liu, and R. R. Alfano, “Optical harmonic generation from animal tissues by the use of picosecond and femtosecond laser pulses,” Appl. Opt. 35, 6810–6813 (1996). [CrossRef] [PubMed]
23. S. W. Chu, I. H. Chen, T. M. Liu, P. C. Chen, C. K. Sun, and B. L. Lin, “Multimodal nonlinear spectral microscopy based on a femtosecond Cr:forsterite laser,” Opt. Lett. 26, 1909–1911 (2001). [CrossRef]
24. G. Peleg, A. Lewis, M. Linial, and L. M. Loew, “Nonlinear optical measurement of membrane potential around single molecules at selected cellular sites,” P. Natl. Acad. Sci. U.S.A. 96, 6700–6704 (1999). [CrossRef]
25. P. J. Campagnola, M. D. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear optical imaging of live cells by second harmonic generation,” Biophys. J. 77, 3341–3349 (1999). [CrossRef] [PubMed]
27. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997). [CrossRef]
29. M. Müller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc.-Oxford 191, 266–274 (1998). [CrossRef]
31. A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Biophys. J. 82, 4142–4145 (1999).
32. E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38, 6994–7001 (1999). [CrossRef]
33. F. Palacios, J. Ricardo, D. Palacios, E. Goncalves, J. L. Valin, and R. De Souza, “3D image reconstruction of transparent microscopic objects using digital holography,” Opt. Commun. 248, 41–50 (2005). [CrossRef]
35. U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms”, Meas. Sci. Technol. 13, R85–R101 (2002). [CrossRef]
36. J. W. Goodman, Fourier Optics (3rd ed., Roberts & Company, Greenwood Village, CO, 2005).
37. E. N. Leith and J. Upatniek, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52, 1123–1128 (1962). [CrossRef]
38. E. N. Leith and J. Upatniek, “Microscopy by wavefront reconstruction,”. J. Opt. Soc. Am. 55, 569–570 (1965). [CrossRef]
39. Y. Pu, M. Centurion, and D. Psaltis, “Harmonic holography: a new holographic principle,” Appl. Opt. 47, 103–110 (2008). [CrossRef]
40. C. L. Hsieh, R. Grange, Y. Pu, and D. Psaltis, “Three-dimensional harmonic holographic microcopy using nanoparticles as probes for cell imaging,” Opt. Express 17, 2880–2891 (2009). [CrossRef] [PubMed]
41. E. Shaffer, N. Pavillon, J. Kühn, and C. Depeursinge, “Digital holographic microscopy investigation of second harmonic generated at a glass/air interface,” Opt. Lett. 34, 2450–2452 (2009). [CrossRef] [PubMed]
42. V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis (2nd ed., SPIE, Belingham, WA, 2007).
43. L. Novotny and B. Hecht, Principles of nano-optics (Cambridge University Press, Cambridge, UK, 2007).
44. R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y. W. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297, 376–378 (2002). [PubMed]
45. T. Colomb, J. Kühn, F. Charrière, and C. Depeursing, “Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram,” Opt. Express 14, 4300–4306 (2006). [CrossRef] [PubMed]
46. S. D. Nocola, P. Ferraro, A. Finizio, and G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” Opt. Laser Eng. 37, 331–340 (2002). [CrossRef]
47. T. Colomb, E. Cuche, F. Charrière, J. Hühn, N. Aspert, F. Montfort, P. Marquet, and C. Depeursinge, “Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation,” Appl. Opt. 45, 851–863 (2006). [CrossRef] [PubMed]
49. S. Roth and I. Freund, “Coherent optical harmonic generation in rat-tail,” Opt. Commun. 33, 292–296 (1980). [CrossRef]
50. I. Freund, M. Deutsch, and A. Sprecher, “Optical second-harmonic microscopy, crossed-beam summation, and small-angle scattering in rat-tail tendon,” Biophys. J. 50, 693–712 (1986). [CrossRef] [PubMed]
52. A. T. Yeh, N. Nassif, A. Zoumi, and B. J. Tromberg, “Selective corneal imaging using combined second-harmonic generation and two-photon excited fluorescence,” Opt. Lett. 27, 2082–2084 (2002). [CrossRef]
53. F. Tiaho, G. Recher, and D. Rouede, “Estimation of helical angles of myosin and collagen by second harmonic generation imaging microscopy,” Opt. Express 15, 12286–12295 (2007). [CrossRef] [PubMed]