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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 2 — Jan. 17, 2011
  • pp: 1626–1640
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Spatially-chirped modulation imaging of absorbtion and fluorescent objects on single-element optical detector

Greg Futia, Philip Schlup, David G. Winters, and Randy A. Bartels  »View Author Affiliations


Optics Express, Vol. 19, Issue 2, pp. 1626-1640 (2011)
http://dx.doi.org/10.1364/OE.19.001626


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Abstract

Line imaging of fluorescent and absorptive objects with a single-pixel imaging technique that acquires one-dimensional cross-sections through a sample by imposing a spatially-varying amplitude modulation on the probing beam is demonstrated. The fluorophore concentration or absorber distribution of the sample is directly mapped to modulation frequency components of the spatially-integrated temporal signal. Time-domain signals are obtained from a single photodiode, with object spatial frequency correlation encoded in time-domain bursts in the electronic signal from the photodiode.

© 2011 Optical Society of America

Optical imaging, ubiquitous throughout science and technology, serves as a critical tool from biomedical applications [1

1. R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009). [CrossRef] [PubMed]

9

9. G. Stutzmann, “Seeing the brain in action: how multiphoton imaging has advanced our understanding of neuronal function,” Microsc. Microanal. 14, 482–491 (2008). [CrossRef] [PubMed]

] to industrial processes [10

10. S. Blonski and T. A. Kowalewski, “Piv analysis of turbulent flow in a micro-channel,” Theor. Appl. Mech. 45, 489–503 (2007).

14

14. S. T. Thoroddsen, T. G. Etoh, and K. Takehara, “High-speed imaging of drops and bubbles,” Annu. Rev. Fluid Mech. 40, 257–285 (2008). [CrossRef]

]. Imaging makes use of optical radiation spanning from the soft X-ray regime to THz frequencies. Spectral regions with photon energies in excess of the bandgap of silicon benefit from mature charge-coupled device (CCD) and complementary metal-oxide-semiconductor (CMOS) imagers [15

15. M. El-Desouki, M. J. Deen, Q. Y. Fang, L. Liu, F. Tse, and D. Armstrong, “Cmos image sensors for high speed applications,” Sensors 9, 430–444 (2009). [CrossRef]

], which are highly efficient and widely available as multi-element optical detectors. At wavelengths longer than those that can be detected by Si, detector arrays are available, but exhibit lower sensitivity, lower operational speed, and increased noise.

Many imaging methods rely on point-by-point scanning of an image acquired with a single-element optical detector. The ability of confocal microscopy to capture high contrast images in media that strongly scatter optical light has made it the most successful single-element optical imaging technique [16

16. J. J. Art and M. B. Goodman, “Rapid-scanning confocal microscopy,” in Methods in Cell Biology, B. Matsumoto, ed. (Academic Press, 1993), Vol 38, pp. 47–77. [PubMed]

20

20. G. Q. Xiao, T. R. Corle, and G. S. Kino, “Real-time confocal scanning optical microscope,” Appl. Phys. Lett. 53, 716–718 (1988). [CrossRef]

]. Laser-scanning microscopy [21

21. 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]

24

24. J. D. Lechleiter, D. T. Lin, and I. Sieneart, “Multi-photon laser scanning microscopy using an acoustic optical deflector,” Biophys. J. 83, 2292–2299 (2002). [CrossRef] [PubMed]

], in which a tightly focused laser beam is scanned through an object, and a signal is recorded with a single-element detector for each focal point is also widely used, particularly for nonlinear optical microscopy methods [21

21. 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]

23

23. W. Denk, J. H. Strickler, and W. W. Webb, “2-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990). [CrossRef] [PubMed]

, 25

25. J. A. Squier, M. Muller, G. J. Brakenhoff, and K. R. Wilson, “Third harmonic generation microscopy,” Opt. Express 3, 315–324 (1998). [CrossRef] [PubMed]

29

29. O. Masihzadeh, P. Schlup, and R. A. Bartels, “Label-free second harmonic generation holographic microscopy of biological specimens,” Opt. Express 18, 9840–9851 (2010). [CrossRef] [PubMed]

].

The serial acquisition nature of scanning imaging approaches means that extended time is required to form a full image – even in cases where high speed galvanometers, acousto-optic deflectors, or polygon mirrors are used for rapid scanning [1

1. R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009). [CrossRef] [PubMed]

]. Spatial multiplexing has been implemented, using arrays of beam foci, to increase the imaging rate of single-point laser-scanning methods [30

30. J. Bewersdorf, R. Pick, and S. W. Hell, “Multifocal multiphoton microscopy,” Opt. Lett. 23, 655–657 (1998). [CrossRef]

34

34. A. Vaziri and C. V. Shank, “Ultrafast widefield optical sectioning microscopy by multifocal temporal focusing,” Opt. Express 18, 19645–19655 (2010). [CrossRef] [PubMed]

]. These techniques include the generation of arrays of laser beams with microlens arrays for multiphoton and confocal imaging processes, but require scanning to fill in the gaps.

Another class of microscopic techniques has exploited spatial modulation of the beam, with data collection by a single-element photodetector is based on spatial structuring of an illumination or excitation beam [35

35. R. Juskaitis, T. Wilson, M. A. A. Neil, and M. Kozubek, “Efficient real-time confocal microscopy with white light sources,” Nature 383, 804–806 (1996). [CrossRef] [PubMed]

, 36

36. R. Heintzmann and P. A. Benedetti, “High-resolution image reconstruction in fluorescence microscopy with patterned excitation,” Appl. Opt. 45, 5037–5045 (2006). [CrossRef] [PubMed]

]. Compressed imaging is similar in that spatial patterns are used as masks to form overlap-integral projections between the image under study and a particular spatial mask representation. The series of inner product projections of the image is recorded on a single-element detector [37

37. J. Romberg, “Imaging via compressive sampling,” IEEE Signal Process Mag. 25, 14–20 (2008). [CrossRef]

]. The sequence of projections, in combination with a system model, is used to invert the projection data to compute the object image [38

38. W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008). [CrossRef]

] or through spatial correlation as in computational ghost imaging [39

39. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008). [CrossRef]

]. These methods have increased imaging speeds, and further optimization could potentially yield significant improvements in image acquisition rates.

An alternate single-element detector imaging technique uses a spinning modulator disk that maps spatial coordinates to modulation frequencies [40

40. J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, “Imaging with frequency-modulated reticles,” Opt. Eng. 30, 1720–1724 (1991). [CrossRef]

]. This imaging technique is based on earlier work for tracking point objects with frequency-modulated reticles [41

41. L. M. Biberman, Reticles in electro-optical devices, International series of monographs in infrared science and technology (Pergamon Press, 1966), Vol. 1.

, 42

42. J. K. Bae, Y. H. Doh, D. S. Noh, and S. J. Kim, “Imaging system using frequency modulation time division multiplexing hybrid reticle,” Opt. Eng. 37, 2119–2123 (1998). [CrossRef]

]. [43

43. R. G. Driggers, C. E. Halford, G. D. Boreman, D. Lattman, and K. F. Williams, “Parameters of spinning fm reticles,” Appl. Opt. 30, 887–895 (1991). [CrossRef] [PubMed]

] describes how a properly constructed reticle can be used to map a point objects radial position into frequency, and phase into angle to aquire 2D position. [40

40. J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, “Imaging with frequency-modulated reticles,” Opt. Eng. 30, 1720–1724 (1991). [CrossRef]

] used the spinning reticles to aquire ID wide field images. In these images, the pattern of [44

44. D. Lovell, “Electro-optic position indicator system,” U.S. Patent 2,997,699 (22Aug.1961).

] is used to linearly sweep the modulation frequency and the image is read directly from the spectrum of the electronic photodiode signal. This approach makes use of SPatIal Frequency modulation for Imaging, which we will abbreviate as SPIFI.

Here, we introduce a new approach to fluorescent imaging with a single-element detector. The basic setup, illustrated in Fig. 1, consists of an excitation beam that is spatially modulated with a linearly-chirped modulation across its spatial extent, providing a unique modulation frequency at each spatial point in the beam. To spatially and temporally modulate the excitation beam intensity, we make use of a phase modulating reticle [43

43. R. G. Driggers, C. E. Halford, G. D. Boreman, D. Lattman, and K. F. Williams, “Parameters of spinning fm reticles,” Appl. Opt. 30, 887–895 (1991). [CrossRef] [PubMed]

] that uses an amplitude mask printed onto a spinning disc to introduce a chirped intensity modulation on the excitation beam. Each spatial point is illuminated by an optical beam that excites fluorophores in an object. The emitted fluorescence oscillates at the frequency of the intensity modulation of the local excitation beam at a specific spatial point – provided that the modulation frequency is below the inverse of the fluorescence lifetime. This spatially-chirped modulation on the excitation beam encodes spatial fluorophore intensity in terms of the local spatial modulation frequency of the excitation beam. The fluorescent light filtered from the excitation beam is collected by a photodetector that spatially integrates the beam intensity across the detector area. The output of the photodetector provides a temporal signal with encoded spatial information about the object fluorophore concentration, which is directly recovered in the spectrum of the photodiode signal.

Fig. 1 Schematic of imaging method.

In this paper, we present theory and experimental demonstration of fluorescent SPIFI. The theory provides direct analogs to conventional optical imaging, so that expressions for numerical aperture, spatial resolution, and the number of resolvable imaging points can be obtained. In the experiments, the theoretical predictions are validated. While the exact speed we obtain is limited by our current hardware implementation, this technique can be readily scaled to higher imaging rates. In addition, the spatial correlation between modulation frequency and an image point suggests that SPIFI can be used for imaging methods that make use of laser scanning, including multiphoton fluorescence and coherent nonlinear optical microscopies. Leveraging the high-speed image acquisition for nonlinear optical microscopy could result in significant speed improvements. Of particular interest are applications where object are undergoing lateral translation – such as objects in a microfluidic chamber or particles in solution. In this scenario, images obtained from a pair of line images will provide image and velocity information. This technique should also prove valuable for THz and mid infrared imaging, where segmented detectors are not readily available.

1. Theory of Fluorescent SPIFI

In fluorescent SPIFI imaging, spatial information of the object fluorescence is imprinted onto the frequency components of a spatially-varying, frequency modulated excitation beam. The fluorescence from the object is collected by a single-element detector. A Fourier decomposition of the collected temporal signal reveals the spatial structure of the fluorescent object. In this section, we derive fluorescent SPIFI performance characteristics after developing the background theory. In the first subsection, we develop theory of imaging from absorption in the object (e.g., from fluorophores). The second subsection covers image formation from collection of fluorescence that exhibits frequency modulation in intensity that correlates with spatial variation in modulation frequency from the excitation beam. Experimental data compare SPIFI images formed from excitation beam absorption and fluorescence and show that the fluorescent images are nearly background-free and show reduction in overall noise in the image.

1.1. SPIFI Imaging Theory from Absorption

We write the transverse extent of the excitation beam, focused to a line along the x-direction, as Eex(x,t) = E0u(x)e0t, where u(x) is the normalized spatial field along the line focus and ω0 is the optical angular frequency. The excitation beam is modulated with a spatially and temporally varying modulation function m(x,t). On passing through, or reflecting from, the object, with transmission g(x), the object field is written Eobj(x,t) = E0u(x)m(x,t)g(x)e0t. Note that the reduced transmission can be either a dark absorber, or absorption from fluorophores in the sample under study. It is the function g(x) that is to be recovered by the absorptive SPIFI imaging process. The object beam is collected on the area of a single element photodetector, producing a time-dependent signal proportional to the spatially-integrated object beam, sobj(t) = γIobj(x,t)dx, multiplied by constant, γ that includes factors such as detector efficiency. The intensity of the object beam is given by Iobj(x,t)=12ncɛ0|Eobj(x,t)|2, which can be written as
Iobj(x,t)=I0|u(x)m(x,t)g(x)|2
(1)
where I0=12ncɛ0|E0|2. We assume that the temporal frequency bandwidth of the detector exceeds the highest modulation frequency in m(x,t) and that the area of the detector is larger than the area of the object field Eobj(x,t).

There are possibly many modulation patterns that can be used to recover the integrated spatial information of the field. In this work, we have implemented a spatial linear chirp in modulation frequency across the beam. We consider a real, positive semi-definite modulator with a linearly-chirped modulation frequency of the form
m(x,t)=w(t)2[1+cos(2πκxt)].
(2)
This type of intensity modulator is known as a frequency modulated reticle [41

41. L. M. Biberman, Reticles in electro-optical devices, International series of monographs in infrared science and technology (Pergamon Press, 1966), Vol. 1.

43

43. R. G. Driggers, C. E. Halford, G. D. Boreman, D. Lattman, and K. F. Williams, “Parameters of spinning fm reticles,” Appl. Opt. 30, 887–895 (1991). [CrossRef] [PubMed]

], which has previously been used for imaging of absorbing objects [40

40. J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, “Imaging with frequency-modulated reticles,” Opt. Eng. 30, 1720–1724 (1991). [CrossRef]

].

The finite time-window of the modulator is represented by the function w(t). This modulator leads to an object beam of
Eobj(x,t)=Eo2u(x)g(x)w(t)[1+cos(2πκxt)]eiω0t.
(3)

The collected intensity
Iobj(x,t)=I014|w(t)u(x)g(x)|2[32+2cos(2πκxt)+12cos(4πκxt)]
(4)
generates an electronic signal from the photodiode that can be separated into three relevant terms scaled by γ, and I0, sobj(t)=12I0γ[s0(t)+s1(t)+s2(t)]. The background term, s0(t)=|w(t)|234|u(x)g(x)|2dx is the overlap integration of the excitation beam and object, with a temproal variation reflecting the modulation window function. The signals s1 and s2 are respectively in the first and second harmonic bands of the fundamental modulation frequency, with
s1(t)=|w(t)|2|u(x)g(x)|2ej2πκtxdx+c.c.
(5)
s2(t)=14|w(t)|2|u(x)g(x)|2ej4πκtxdx+c.c.
(6)

The first and second harmonic sidebands contain the same information about the signal. We will concern ourselves with the analysis of the first harmonic sideband. The center frequency of the sideband is determined by the centroid of the intensity of the excitation beam, xc = ∫ x|u(x)|2dx, on the disc. Shifting the coordinate system to centroid of the beam, x′ = xxc, gives the first harmonic signal,
s1(t)=|w(t)|2ej2πκxct|u(x)g(x)|2ej2πκtxdx+c.c.
(7)

The integral of the fundamental harmonic modulation band s1(t) in Eq. (7) takes the form of a spatial Fourier transform, where the spatial frequency is identified as fxκt. With this identification, and defining the spatial Fourier transform and the Fourier transform operator 𝔉{},
𝒢(fx)=|u(x)g(x)|2ei2πfxxdx𝔉{|u(x)g(x)|2},
(8)
we obtain an expression for the time-varying photodiode signal of
s1(t)=2|w(t)|2|𝒢(κt)|cos(2πfct+𝒢(κt))
(9)
where ∠𝒢′ is the phase of the spatial frequency distribution, 𝒢′, and the center frequency is fc = κxc. The Fourier transform of the temporal signal, Ŝ1(f) = 𝔉{s1(t)}, shows a spectrum with sidebands decomposed as Ŝ1(f) = κ−1 Ŝ1+(f) + κ−1 Ŝ1–(f). The sidebands reflect the spatial extent of the illuminatation beam intensity, and limited in resolution by convolution with the temporal window. The upper side band can be written as
S^1+(x=fκ1)=𝒲(κx)|u(x)g(x)|2
(10)
where f is is the frequency of the electronic signal, and ⊛ is the convolution operator. The Fourier transform of the temporal window defines the equivalent “point spread function” of the system given by 𝒲(x′) = 𝔉 {|w(t)|2}f = x′κ−1.

1.2. SPIFI Imaging Theory from Collected Fluorescence

The molecular molar concentration distribution, [c(x)] = N(x)/NA is directly proportional to the number density of molecules, and thus the image provides a measure of the concentration distribution of the fluorescent molecules. We see that the expression for the intensity of the object is the same as found for the absorbing case, except where object power transmittance, |g(x)|2 has been replaced with the object power absorption |a(x)|2. Once this substitution is made, the remainder of the imaging theory calculations in the above section hold.

2. SPIFI Imaging Performance

For any imaging system, two of the most relevant system characteristics are the spatial resolution and the number of points resolved. The ability to resolve two closely spaced points in an image is characterized by the numerical aperture, NA, of the imaging system. The numerical aperture is defined through NA = n sin θmax, where n is the local index of refraction, and θmax is the largest angle collected by the imaging system. The definition of spatial frequencies, fx, of an object fxλ = sin θ [46

46. J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, 2004). [PubMed]

] means that the numerical aperture can also be given by NA = nλ fxmax, where fxmax is the maximum spatial frequency collected.

The number of spatially resolved points, N, is given by the beam width, W divided by the spatial resolution, δx. Making the substitutions for SPIFI, the number of spatial points is N = W/δx = κWTm, which is the same as the space-spatial frequency bandwidth product, N = 2W fxmax. For the temporal signal, the number of points is the the modulated bandwidth, Δfm, divided by the resolution bandwidth, δ f. The resolution bandwidth δ f is the inverse of the acquisition duration, Tm, leading to the time-bandwidth product, N = ΔfmTm. For SPIFI the signal bandwidth is Δfm = κW, and the resolution bandwidth of the trace is Tm1, giving a temporal number of points κWTm, identical to the spatial-spatial frequency bandwidth product. The limit on both the number of points and the resolution is seen to be based only on the modulator design and is independent of the wavelength of the illumination.

3. Rotating Frequency Modulated Reticle Modulator

Frequency modulated spinning retical discs [40

40. J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, “Imaging with frequency-modulated reticles,” Opt. Eng. 30, 1720–1724 (1991). [CrossRef]

43

43. R. G. Driggers, C. E. Halford, G. D. Boreman, D. Lattman, and K. F. Williams, “Parameters of spinning fm reticles,” Appl. Opt. 30, 887–895 (1991). [CrossRef] [PubMed]

] provide a convenient method for generating the modulation m(x,t). We use a family of patterns defined first by [44

44. D. Lovell, “Electro-optic position indicator system,” U.S. Patent 2,997,699 (22Aug.1961).

]
m(R,θ)=12+12cos[(k0+ΔkR)θ]
(11)
where R is the radial, and θ the angular, coordinate relative to the center of the disc. The beam to be modulated is brought to a line focus and samples the modulation pattern across a radial line such that R = x. Spinning the mask at a constant angular velocity dθ/dt = 2π fr so that we can make a substitution θ → 2π frt in Eq. (2). provides the requisite linear chirp in modulation frequency expressed in Eq. (2), where we can make the identification κ = frΔk, with an additional offset of the modulator frequency given by κ0 = frk0. In the masks used, the pattern spatial frequency offset was k0 = 0. Adjusting this parameter, in conjunction with fr, controls the center frequency, fc, of the modulation band of s1(t) in Eq. (9). This flexibility allows the frequency modulation band to be moved away from low-frequency noise, but be kept below the sampling limit of the digitization board. Additional shifting of the frequencies to be detected can also be achieved by mixing the detected signal with a separate reference oscillator. An example of a mask pattern is shown in Fig. 2 for the parameters k0 = 0 and Δk = 3 mm−1.

Fig. 2 Example mask pattern of Eq. (11) with parameters k0 = 0, Δk = 3 mm−1, and Δθ = 6 rad.

The time-window of the modulator is given by Tm = (2π fr)−1 Δθ, where Δθ defines the angular width of the linearly-chirped modulator pattern. This gives a spatial resolution of δx = 2πθ Δk)−1, a modulation bandwidth of Δfm = frW Δk, the number of resolved points N = (2π)−1 ΔθW Δk, and a center modulation frequency of fc = fr(k0 + Δkxc), where xc is the beam center position on the modulator disk.

4. Experimental Results

Images using both absorptive and fluorescent SPIFI images are presented below. The experimental imaging performance is is compared in detail to imaging theory presented. Comparison of the images taken with absorption of a fluorescent dye based object with that obtained from collection of the fluorescence shows the advantages of fluorescent imaging, which has negligible background light.

4.1. Absorptive SPIFI Imaging

We present experimental results using an implementation of SPIFI using a spinning-disc modulator. The laser source was based on a 1550-nm, 5-mW fiber-coupled cw DFB laser. The seed was amplified in a single-stage home-built single-mode erbium-doped fiber amplifier, pumped by a 400-mW diode at 976 nm, run at 8% of maximum power producing 10 mW of light at the output of the fiber collimator. A sequence of knife-edge measurements through the optical system yielded a beam quality of M2 ≈ 1.02 at the output of the fiber. The beam profile was expanded in a 1:3 telescope, then focused with a 50-mm focal length cylindrical lens onto the spinning disk modulator. The modulation plane was re-imaged with a 3:2 telescope to make object insertion more convenient. After passing through the object, the beam was collected on a 5 mm active area Ge photodiode (Thorlabs, DET50B). Using a low-noise current preamplifier (Stanford Research Systems, SR570), the photodiode output was transimpedance amplified at 1 mA/V and low-pass filtered with a −6 dB, corner frequency of 100 kHz. The amplified signal was digitized at 250 kHz with a 16-bit data acquisition board (National Instruments, 6251-USB). The amplified, filtered signal, as well as a timing signal from the disc speed controller, were collected simultaneously using custom-written C# software, that Fourier transformed in real-time to recover the spatial image along the line focus direction, and exported the signals for post-processing with Matlab.

4.1.1. Modulator Masks

We used rotating disc modulator mask patterns following the form of Eq. (11). High-resolution TIFF-format image files, generated in Matlab, were printed directly onto clear, unpressed polycarbonate CD-ROM substrates (U-TECH, Dallas, Texas) with a commercial thermal printer (courtesy MAM-A, Colorado Springs, Colo.). To prevent halftone dithering, we printed only monochrome patterns such as the one shown in Fig. 2, where regions of m(R,θ)12 were overprinted in black and areas where m(R,θ)<12 remained transparent.

The printer’s resolution limits the value of Δk that can be applied to a mask, since the patches of m(R,θ) at θ = ±π must remain distinct. In our case, the the 400 dots-per-inch (DPI) resolution limited Δk to 7 mm−1. In principle, higher resolution printing or lithographic patterning techniques could push the local spatial frequency to near the wavelength limit of 2/λ, which could yield Δk > 1000 mm−1.

4.1.2. Acquisition Speed

The modulator discs were mounted to a brushless DC motor (Faulhaber 1628) with an actively feedback-stabilized angular velocity (Faulhaber SC1801 controller). The motor has a rated maximum speed of 477 rotations per second, but with the load of the modulator discs, we found the rotation speed was limited to 130 Hz. For the experiments presented, the motor was operated at speed of fr ≈ 50 Hz. Motors with a higher load capacity are expected to scale this speed by an order of magnitude, with a spin velocity limited by material fracture limits. Further increased speeds could be possible by the use of inertia-free systems, such as acousto-optic deflectors (AODs).

4.1.3. Measured Signals

Data from the SPIFI setup running at an spin velocity of fr = 52 Hz is shown in Fig. 3(a). Equation (9) shows that the envelope of the temporal signal is the product of the temporal window intensity, |w(t)|2 and the autocorrelation of the transverse spatial profile of the line focus. The transverse Gaussian intensity profile of the modulated beam puts a Gaussian modulation envelope on the modulation frequency spectrum, as is evident by inspection of Eq. (10) since the temporal window does not restrict the signal bandwidth. In Fig. 3(a), the square pulses bracketing the coherence burst arise from the blank area in the disc patterns visible in Fig. 2, and can be used for timing synchronization. The inset shows the burst in more detail.

Fig. 3 (a) Digitized time-domain signal collected for an unperturbed excitation beam, with the inset showing the coherence burst near t = 0. (b) Fourier transformed frequency domain signal of the shaded area of the time trace.

In order to recover the object, we take a Fourier transformation of the time domain signal excluding the timing pulses, shown as the shaded area in Fig. 3(a). The result, plotted normalized to the DC component on a logarithmic scale, is shown in Fig. 3(b). The first harmonic s1 modulation signals are clearly evident centered at fc ≈ ±18 kHz. The second harmonic peaks at s2 are visible around 36 kHz. Their amplitude is lower than that predicted by Eq. (6); we ascribe this phenomenon to the square-pulse shape of the amplitude modulation mask, which comprises mainly odd-order spatial frequency harmonics.

4.1.4. Spatial Calibration

In order to determine the mapping of modulation frequency to the spatial coordinate in the object plane, we scanned a 2-μm width slit through the re-imaged modulated beam profile. Data showing the spectrum of the positive frequency sideband of the FFT of the sampled SPIFI temporal trace are plotted as a function of displacement of the slit along the x-direction in Fig. 4. Data were acquired in an automated fashion using a stepper motor (Newport LTA-HS) and acquiring SPIFI data at each spatial position. The slope of the distribution obtained by plotting the spectral centroid as a function of x-displacement provides a direct measurement of the modulator chirp parameter, κ. These data show a value of κ = 570 Hz/mm, in good agreement with the expected value of 578 Hz/mm.

Fig. 4 Space-to-frequency calibration, obtained by scanning a 2-μm slit across the modulated beam at the object plane. The beam intensity distribution at the object plane (black line, bottom axis) is found by summing the temporal trace at each slit position. The frequency bandwidth at a single position (left axis) reveals the system’s PSF. When the slit is removed, the resulting frequency distribution, mapped to the spatial coordinate (blue dashed line) shows excellent agreement with the beam profile.

These data also provide a measurement of the transverse line-focus modulated beam along x, shown by the black line in Fig. 4 on the lower axes. From these data, the 1/e2 diameter of the beam is determined to be W ∼ 5.0 mm. This, in combination with the chirp rate, provides an expected modulation bandwidth of Δ fm ∼ 2.6 kHz. For comparison, we have plotted the first sideband of the trace in Fig. 3 as a dashed blue line in Fig. 4, scaled with the determined value of κ. The excellent agreement confirms the validity of the space-to-modulation frequency mapping.

By transforming a single time trace at the center of the beam, we can estimate the spatial resolution of the system. As an estimation for the expected width of the point spread function (PSF) of the SPIFI imaging configuration, we use the resolution limit of δx ∼ Δk−1. This value is 142 μm, or 80 Hz. This estimation agrees well with the PSF data measurement shown in Fig. 4, where the 2-μm slit produces a 75-Hz wide peak. We thus estimate that our infrared configuration is capable of resolving N ≈ 38 points.

By displacing the slit away from the imaging plane along the beam propagation direction, we found that κ varies by ∼ ±6% and the spectral width of the PSF varies by ∼ ±16% over a range of optic axis displacement of Δz ∼ 30 mm. This is a range that spans well beyond the Rayleigh range of line focus, which demonstrates that system alignment is not critically sensitive to image planes.

4.1.5. Modulation Frequency Correction

Under some conditions, the signal measured for a narrow object, such as a 150μm diameter pinhole placed in the object plane, resulted in additional structure as shown in Fig. 5(a). The spectral profile is asymmetric containing a side peak, which can introduce artifacts and distortions in imaging. A spectrogram of the temporal trace reveals that the distortions arise due to imperfect frequency stability. Figure 5(b) shows a spectrogram of the time-domain SPIFI trace, found by applying a Gaussian temporal window and sliding it along the time axis of the acquired data.(Gabor Transform) The frequency “wobble” was found to be consistent across each revolution of the spinning disc and ascribed to imperfect centering of the disc relative to the motor rotation axis. The frequency wobble can be removed with an algorithm that takes the centroid of the spectrogram as a function of time and computes the fractional change in the local time frequency compared to the average measured frequency, δ f(tc)/. A new time axis is generated by constructing an non-uniform temporal spacing Δt(tc) = Δtsδ f(tc)/, where Δts is the original temporal step of the data sampling. A new time-axis is constructed by integrating the locally corrected temporal steps, tnew=0tnewΔt(tc)dtc. Re-interpolating the data with this warped time axis corrects for the effects of the frequency wobble, as can be seen in Fig. 5(c). The frequency wobble can also be determined from a separate point-focus sampling of the spinning disc, so it could be acquired synchronously and be used to correct for other imperfections, such as motor rotation velocity.

Fig. 5 (a) The signal obtained from a 150-μm pinhole contains additional structure, arising from variations in the modulation frequency as the disc revolves. (b) Imperfections in the modulation frequency are revealed by a spectrogram. (c) Frequency wobble correction removes structure on the slit transmission measurement.

4.1.6. 2D Image Acquisition

Line-focus SPIFI is a one-dimensional (1D) imaging method. A number of methods can be employed to produce two-dimensional (2D) images, including lateral translation, rotation, scanning galvanometers, or AODs. The easiest of these is to translate the object in the direction perpendicular to the line focus, and assemble the sequence of 1D slices into a 2D image. We have used this approach to image a standard 1951 U.S. Air Force test target (Thorlabs, R3L3S1P) mounted on a motorized translation stage. The image shown in Fig. 6(a) is a sequence of nine scans paths captured by translating along y, then repeating the scan at a new x position. The images have been normalized relative to the excitation intensity measured in the absence of a sample, and thresholding the images edge at points where noise from low excitation beam intensity becomes important. The wiggles visible on several of the vertical bars arise from inconsistent modulator disc motor rotation speeds, which cause the measured object slice to appear at a shifted position coordinate.

Fig. 6 (a) Image of 1951 USAF test pattern, assembled from a series of scanned SPIFI acquisitions. (b)–(d) Slices across bars of varying width showing image resolution performance.

The spatial resolution of an imaging system can be measured by observing the ability to resolve lines in the test pattern. Figure 6(a) shows several segments taken from the test pattern, and these lineouts are plotted on Fig. 6(b)–(d) for a number of different patch widths. In the test pattern, the width or spacing of the line sets in mm is given by (2G+16E)1 where G and E are the group and element numbers, respectively. The resolution is determined by the line spacing periodicity that can no longer be resolved by the imaging system. From this figure, we see that this limit is around G = +2, E = 3. This indicates that the spatial resolution of our SPIFI imaging system is at least 177 μm, which is consistent with our theory and measurements. The ability to resolve bars across vertical position scans is limited by the y extent of the line focus at the object plane, and we estimate that it is ∼150 μm.

4.2. Fluorescent SPIFI Imaging

The samples used for fluorescent SPIFI were excited using a 532 nm solid-state laser (Coherent DPSS 532). The 23 mW beam was propagated through the same optical setup as used for absorptive SPIFI up until the object plane. After the object, the fluorescence and remaining excitation beam were collected using a 50 mm cylindrical lens. The pump beam was separated from the fluorescence by a 600 mm−1 grating and the desired light focused with a 125 mm lens onto a reverse-biased Si photodiode. A long-pass filter with cut-on at 600 nm was used before the detector in the case of the fluorescence to suppress scattered 532 nm light.

The fluorescent object consisted of a glass slide with an image imprinted on it in fluorescent ink. The slide was prepared by drying a coat of aerosol hairspray to its surface to aid in adherence of the ink. The felt tip of a orange highlighting marker was rubbed onto the image area of a rubber stamp, which as then then pressed onto the slides surface transferring the pattern to the prepared slide. Images of the florescence dye pattern, taken in absorptive and emission configurations, can be seen in Fig. 7. Inhomogeneity of the dyes application to the slide can be seen in the blotchiness of the heads of the flower. Each image is the composite of three 2D SPIFI image segments as described in §4.1.6, with a manual translation stage used to offset the image relative to the line focus for each scan. The images were captured on a the surface of reversed biased Si diode (ThorLabs DET100A), and amplified again using a transimpedance amplifier. For the absorptive images the tranimpedance gain was set to 1 mA/V while for the fluorescent images the gain was increased to 20 nA/V. The necessary increase in gain for the fluorescence case is due in part to inefficient collection of the generated fluorescence. In both cases the amplifier’s bandpass filter was set with −6 dB corner frequencies of 3 and 100 kHz.

Fig. 7 Images obtained using (a) absorptive and (b) fluorescent SPIFI of a fluorescent sample image.

The images have been normalized relative to the leak through excitation or illumination intensity measured in the absence of a sample. As a measure of image quality, the ratio has been calculated between the average value over a small area within the object and in the background. This ratio has been measured at 10 different locations within each image, and the average and standard deviation taken of those measures. In the fluorescent image, the average is 13.9 with a standard deviation of 5.5. For the absorptive image, the ratio average is 2.68 with a standard deviation of 0.28. Calculating the standard deviation of the pixel values in all of the selected background regions for the two images and taking the ratio, σAbs/σFluor = 11.1, indicating more background noise in the case of the absorptive image. Both these metrics demonstrate a reduction in the background and background noise in the fluorescence image as compared to the absorption.

5. Conclusion

In this paper, we have theoretically developed and experimentally demonstrated fluorescent line imaging with chirped spatially modulated excitation beam. A linear modulation chirp imposed across the beam allows capture of spatial images with a single-element photodiode detector. Collection of the entire beam encodes the object spatial information in the measured time-domain signal, which represents the spatial frequency distribution of the product of the excitation beam and the object. We have implemented this technique using a transparent rotating FM reticle modulator. The experimental results agree very well with values predicted by our theory. Using this pattern, we have acquired line scan images at rates as high as 130 Hz, and extended the technique to 2D images by laterally scanning the object through the beam. Substantial imaging speed increases are expected with higher speed motors, or by implementing alternative, inertia free modulation methods.

Using a standard resolution test target, we determine the spatial resolution of our specific implementation to be 177 μm in the direction of the line focus, and 150 μm in the orthogonal axis. The resolution can be improved by changing the magnification of the telescope that images the modulation plane to the object. The number of points N = 38 is currently limited by the printing resolution and the size of our optics. Optimization of modulator parameters for SPIFI could allow for greatly improved imaging speed for many imaging modalities, and may be particularly beneficial for wavelengths in which segmented imaging detectors are less common, such as in the mid-infrared and terahertz spectral regions.

Nearly background-free fluorescence images have been obtained using fluorescent test targets. Direct comparison of images created by absorption of the excitation beam and emitted fluorescence demonstrates lower noise operation for the fluorescent case. As expected, the number of points and resolution remains the same for both images – demonstrating the achromatic operation of the system.

Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant No. 0901870. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. We thank MAM-A (Colorado Springs, Colorado) for printing the modulation mask patterns on the blank CD substrates.

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J. R. Royer, D. J. Evans, L. Oyarte, Q. Guo, E. Kapit, M. E. Mobius, S. R. Waitukaitis, and H. M. Jaeger, “High-speed tracking of rupture and clustering in freely falling granular streams,” Nature 459, 1110–1113 (2009). [CrossRef] [PubMed]

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W. Denk, J. H. Strickler, and W. W. Webb, “2-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990). [CrossRef] [PubMed]

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J. D. Lechleiter, D. T. Lin, and I. Sieneart, “Multi-photon laser scanning microscopy using an acoustic optical deflector,” Biophys. J. 83, 2292–2299 (2002). [CrossRef] [PubMed]

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J. A. Squier, M. Muller, G. J. Brakenhoff, and K. R. Wilson, “Third harmonic generation microscopy,” Opt. Express 3, 315–324 (1998). [CrossRef] [PubMed]

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A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-stokes raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999). [CrossRef]

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O. Masihzadeh, P. Schlup, and R. A. Bartels, “Control and measurement of spatially inhomogeneous polarization distributions in third-harmonic generation microscopy,” Opt. Lett. 34, 1090–1092 (2009). [CrossRef] [PubMed]

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O. Masihzadeh, P. Schlup, and R. A. Bartels, “Enhanced spatial resolution in third-harmonic microscopy through polarization switching,” Opt. Lett. 34, 1240–1242 (2009). [CrossRef] [PubMed]

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O. Masihzadeh, P. Schlup, and R. A. Bartels, “Label-free second harmonic generation holographic microscopy of biological specimens,” Opt. Express 18, 9840–9851 (2010). [CrossRef] [PubMed]

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J. Bewersdorf, R. Pick, and S. W. Hell, “Multifocal multiphoton microscopy,” Opt. Lett. 23, 655–657 (1998). [CrossRef]

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K. Bahlmann, P. T. C. So, M. Kirber, R. Reich, B. Kosicki, W. McGonagle, and K. Bellve, “Multifocal multiphoton microscopy (mmm) at a frame rate beyond 600 hz,” Opt. Express 15, 10991–10998 (2007). [CrossRef] [PubMed]

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R. Carriles, K. E. Sheetz, E. E. Hoover, J. A. Squier, and V. Barzda, “Simultaneous multifocal, multiphoton, photon counting microscopy,” Opt. Express 16, 10364–10371 (2008). [CrossRef] [PubMed]

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J. Romberg, “Imaging via compressive sampling,” IEEE Signal Process Mag. 25, 14–20 (2008). [CrossRef]

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W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008). [CrossRef]

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J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008). [CrossRef]

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J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, “Imaging with frequency-modulated reticles,” Opt. Eng. 30, 1720–1724 (1991). [CrossRef]

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OCIS Codes
(110.2970) Imaging systems : Image detection systems
(110.2990) Imaging systems : Image formation theory
(180.0180) Microscopy : Microscopy
(180.5810) Microscopy : Scanning microscopy
(110.3010) Imaging systems : Image reconstruction techniques

ToC Category:
Microscopy

History
Original Manuscript: November 18, 2010
Revised Manuscript: December 10, 2010
Manuscript Accepted: January 11, 2011
Published: January 13, 2011

Virtual Issues
Vol. 6, Iss. 2 Virtual Journal for Biomedical Optics

Citation
Greg Futia, Philip Schlup, David G. Winters, and Randy A. Bartels, "Spatially-chirped modulation imaging of absorption and fluorescent objects on single-element optical detector," Opt. Express 19, 1626-1640 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-1626


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References

  1. R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009). [CrossRef] [PubMed]
  2. W. Gobel, B. M. Kampa, and F. Helmchen, “Imaging cellular network dynamics in three dimensions using fast 3d laser scanning,” Nat. Methods 4, 73–79 (2007). [CrossRef]
  3. M. Gora, K. Karnowski, M. Szkulmowski, B. J. Kaluzny, R. Huber, A. Kowalczyk, and M. Wojtkowski, “Ultra high-speed swept source oct imaging of the anterior segment of human eye at 200 khz with adjustable imaging range,” Opt. Express 17, 14880–14894 (2009). [CrossRef] [PubMed]
  4. N. Ji, H. Shroff, H. N. Zhong, and E. Betzig, “Advances in the speed and resolution of light microscopy,” Curr. Opin. Neurobiol. 18, 605–616 (2008). [CrossRef]
  5. R. A. Niesner, V. Andresen, and M. Gunzer, “Intravital two-photon microscopy: focus on speed and time resolved imaging modalities,” Immunol. Rev. 221, 7–25 (2008). [CrossRef] [PubMed]
  6. C. Nitschke, A. Garin, M. Kosco-Vilbois, and M. Gunzer, “3D and 4D imaging of immune cells in vitro and in vivo,” Histochem. Cell Biol. 130, 1053–1062 (2008). [CrossRef] [PubMed]
  7. G. D. Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci. 11, 713–720 (2008). [CrossRef]
  8. P. J. Scherz, J. Huisken, P. Sahai-Hernandez, and D. Y. R. Stainier, “High-speed imaging of developing heart valves reveals interplay of morphogenesis and function,” Development 135, 1179–1187 (2008). [CrossRef] [PubMed]
  9. G. Stutzmann, “Seeing the brain in action: how multiphoton imaging has advanced our understanding of neuronal function,” Microsc. Microanal. 14, 482–491 (2008). [CrossRef] [PubMed]
  10. S. Blonski and T. A. Kowalewski, “Piv analysis of turbulent flow in a micro-channel,” Theor. Appl. Mech. 45, 489–503 (2007).
  11. S. Gekle, J. M. Gordillo, D. van der Meer, and D. Lohse, “High-speed jet formation after solid object impact,” Phys. Rev. Lett. 102, 034502 (2009). [CrossRef] [PubMed]
  12. D. Hessman, M. Lexholm, K. A. Dick, S. Ghatnekar-Nilsson, and L. Samuelson, “High-speed nanometer-scale imaging for studies of nanowire mechanics,” Small 3, 1699–1702 (2007). [CrossRef] [PubMed]
  13. J. R. Royer, D. J. Evans, L. Oyarte, Q. Guo, E. Kapit, M. E. Mobius, S. R. Waitukaitis, and H. M. Jaeger, “High-speed tracking of rupture and clustering in freely falling granular streams,” Nature 459, 1110–1113 (2009). [CrossRef] [PubMed]
  14. S. T. Thoroddsen, T. G. Etoh, and K. Takehara, “High-speed imaging of drops and bubbles,” Annu. Rev. Fluid Mech. 40, 257–285 (2008). [CrossRef]
  15. M. El-Desouki, M. J. Deen, Q. Y. Fang, L. Liu, F. Tse, and D. Armstrong, “Cmos image sensors for high speed applications,” Sensors 9, 430–444 (2009). [CrossRef]
  16. J. J. Art and M. B. Goodman, “Rapid-scanning confocal microscopy,” in Methods in Cell Biology, B. Matsumoto, ed. (Academic Press, 1993), Vol 38, pp. 47–77. [PubMed]
  17. K. B. Im, S. M. Han, H. Park, D. Kim, and B. M. Kim, “Simple high-speed confocal line-scanning microscope,” Opt. Express 13, 5151–5156 (2005). [CrossRef] [PubMed]
  18. G. J. Tearney, R. H. Webb, and B. E. Bouma, “Spectrally encoded confocal microscopy,” Opt. Lett. 23, 1152–1154 (1998). [CrossRef]
  19. R. Wolleschensky, B. Zimmermann, and M. Kempe, “High-speed confocal fluorescence imaging with a novel line scanning microscope,” J. Biomed. Opt. 11, 064011 (2006). [CrossRef]
  20. G. Q. Xiao, T. R. Corle, and G. S. Kino, “Real-time confocal scanning optical microscope,” Appl. Phys. Lett. 53, 716–718 (1988). [CrossRef]
  21. 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]
  22. 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]
  23. W. Denk, J. H. Strickler, and W. W. Webb, “2-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990). [CrossRef] [PubMed]
  24. J. D. Lechleiter, D. T. Lin, and I. Sieneart, “Multi-photon laser scanning microscopy using an acoustic optical deflector,” Biophys. J. 83, 2292–2299 (2002). [CrossRef] [PubMed]
  25. J. A. Squier, M. Muller, G. J. Brakenhoff, and K. R. Wilson, “Third harmonic generation microscopy,” Opt. Express 3, 315–324 (1998). [CrossRef] [PubMed]
  26. A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-stokes raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999). [CrossRef]
  27. O. Masihzadeh, P. Schlup, and R. A. Bartels, “Control and measurement of spatially inhomogeneous polarization distributions in third-harmonic generation microscopy,” Opt. Lett. 34, 1090–1092 (2009). [CrossRef] [PubMed]
  28. O. Masihzadeh, P. Schlup, and R. A. Bartels, “Enhanced spatial resolution in third-harmonic microscopy through polarization switching,” Opt. Lett. 34, 1240–1242 (2009). [CrossRef] [PubMed]
  29. O. Masihzadeh, P. Schlup, and R. A. Bartels, “Label-free second harmonic generation holographic microscopy of biological specimens,” Opt. Express 18, 9840–9851 (2010). [CrossRef] [PubMed]
  30. J. Bewersdorf, R. Pick, and S. W. Hell, “Multifocal multiphoton microscopy,” Opt. Lett. 23, 655–657 (1998). [CrossRef]
  31. K. Bahlmann, P. T. C. So, M. Kirber, R. Reich, B. Kosicki, W. McGonagle, and K. Bellve, “Multifocal multiphoton microscopy (mmm) at a frame rate beyond 600 hz,” Opt. Express 15, 10991–10998 (2007). [CrossRef] [PubMed]
  32. R. Carriles, K. E. Sheetz, E. E. Hoover, J. A. Squier, and V. Barzda, “Simultaneous multifocal, multiphoton, photon counting microscopy,” Opt. Express 16, 10364–10371 (2008). [CrossRef] [PubMed]
  33. D. N. Fittinghoff, P. W. Wiseman, and J. A. Squier, “Widefield multiphoton and temporally decorrelated multifocal multiphoton microscopy,” Opt. Express 7, 273–279 (2000). [CrossRef] [PubMed]
  34. A. Vaziri, and C. V. Shank, “Ultrafast widefield optical sectioning microscopy by multifocal temporal focusing,” Opt. Express 18, 19645–19655 (2010). [CrossRef] [PubMed]
  35. R. Juskaitis, T. Wilson, M. A. A. Neil, and M. Kozubek, “Efficient real-time confocal microscopy with white light sources,” Nature 383, 804–806 (1996). [CrossRef] [PubMed]
  36. R. Heintzmann, and P. A. Benedetti, “High-resolution image reconstruction in fluorescence microscopy with patterned excitation,” Appl. Opt. 45, 5037–5045 (2006). [CrossRef] [PubMed]
  37. J. Romberg, “Imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 14–20 (2008). [CrossRef]
  38. W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008). [CrossRef]
  39. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008). [CrossRef]
  40. J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, “Imaging with frequency-modulated reticles,” Opt. Eng. 30, 1720–1724 (1991). [CrossRef]
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