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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 19 — Sep. 18, 2006
  • pp: 8598–8603
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Quasi-ballistic imaging through a dynamic scattering medium with optical-field averaging using Spectral-Ballistic-Imaging

Er’el Granot, Shmuel Sternklar, Yossi Ben-Aderet, and Dan Schermann  »View Author Affiliations


Optics Express, Vol. 14, Issue 19, pp. 8598-8603 (2006)
http://dx.doi.org/10.1364/OE.14.008598


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Abstract

We measure the sub-picosecond optical impulse response of a system consisting of a varying 1D diffusive medium and a stationary hidden object. It is shown that by averaging the temporal optical field response of a diffusive medium (as opposed to the optical intensity response) the signal-to-noise ratio of the object’s reflection can be improved considerably. The Spectral-Ballistic-Imaging technique is used to reconstruct the optical-field impulse response with a 200fs temporal resolution.

© 2006 Optical Society of America

Optically turbid media such as biological tissue, ‘milky’ glass or milk in a glass, scatter illuminating light, hiding optical information embedded within or behind the medium. Various techniques have been studied to overcome this problem. For example, averaging is a well known method to improve the signal to noise ratio (SNR) of the image of a stationary object embedded in a dynamic noisy environment [1

1. See, for example, K.R. Castleman, “Digital Image Processing,”(Prentice Hall, New Jersey1996

4

4. J. Khoury, J.S. Kane, P.D. Gianino, P.L. Hemmer, and C.L. Woods, “Homodyne and heterodyne imaging through a scattering medium,” Opt. Lett. 26, 1433 (2001). [CrossRef]

]. Another well known technique is to employ some type of gating (e.g., “time gating”[5

5. L. Wang, P. P. Ho, F. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991) [CrossRef] [PubMed]

9

9. L. Wang, X. Liang, P. Galland, P. P. Ho, and R. R. Alfano, “True scattering coefficients of turbid matter measured by early-time gating,” Opt. Lett. 20, 913 (1995) [CrossRef] [PubMed]

], “spatial gating” [10

10. E. N. Leith, B. G. Hoover, S. M. Grannell, K. D. Mills, H. S. Chen, and D. S. Dilworth, “Realization of time gating by use of spatial filtering,” Appl. Opt. 38, 1370 (1999) [CrossRef]

11

11. K. D. Mills, L. Deslaurier, D. S. Dilworth, S. M. Grannell, B. G. Hoover, B. D. Athey, and E. N. Leith, “Investigation of ultrafast time gating by spatial filtering,” Appl. Opt. 40, 2282 (2001) [CrossRef]

], “polarization gating” [12

12. Yang Liu, Young L. Kim, Xu Li, and Vadim Backman, “Investigation of depth selectivity of polarization gating for tissue characterization,” Opt. Express 13, 601 (2005) [CrossRef] [PubMed]

15

15. X. Wang, L. V. Wang, C-W Sun, and C-C Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8, 608 (2003) [CrossRef] [PubMed]

], and “angle gating” [16

16. C. C. Yang, C-W Sun, C-K Lee, C-W Lu, M-T Tsai, C. C. Yang, and Y-W Kiang, “Comparisons of the transmitted signals of time, aperture, and angle gating in biological tissues and a phantom,” Opt. Express 12, 1157 (2004) [CrossRef] [PubMed]

]). In these techniques most of the noise is prevented from penetrating the measurement gate. For example, in the time-gating technique the first-arriving photons are separated from the rest (diffusive photons) by a fast shutter, giving a ballistic-image of the medium.

In this work, we describe a method for improving the SNR of the image of an object embedded in a dynamic scattering medium by combining temporal gating and optical-field averaging. As can be seen from the list of references, the two techniques (time gating and field averaging) are well known in the literature; however, in this letter we show that the Spectral Ballistic Imaging (SPEBI) technique is a very good candidate to implement the two techniques simultaneously with a very high temporal resolution. With this technique it is possible to demonstrate the noise reduction in the entire impulse response. When a photon is scattered from a moving scatterer, not only is its path modified but so is its frequency [17

17. B. Cairns and E. Wolf, “Changes in the spectrum of light scattered by a moving diffuser plate,” J. Opt. Soc. Am. A 8, 1922 (1991) [CrossRef]

]. In fact, it is clear that the number of scattered photons that do not undergo a frequency shift is negligible. On the other hand, all the photons that were not scattered cannot experience this frequency (i.e., energy) shift. Thus, a spectrally narrow filter can separate the ballistic (non-scattered) photons from the scattered ones (in many respects this reasoning is similar to that of ref. [4

4. J. Khoury, J.S. Kane, P.D. Gianino, P.L. Hemmer, and C.L. Woods, “Homodyne and heterodyne imaging through a scattering medium,” Opt. Lett. 26, 1433 (2001). [CrossRef]

]). This is exactly temporal averaging.

It should be pointed out that this reasoning is valid even in the adiabatic case; so long as the measurements are taken at finite intervals. This can be illustrated for the simple 1D case shown in Fig. 1. In this figure the scatterer is represented as a local change in the index of refraction. The transmission coefficient of the scatterer is independent of its location (x 1 or x 2 in the figure), but the reflection coefficient is not, gaining a certain phase which is proportional to its location δ=2(x 2-x 1)k, where k=2π/λ is the wavenumber of the photons (λ is the wavelength). Therefore, when many measurements of the reflection and transmission coefficients are acquired at different scattering locations and averaged, the reflection coefficients average to zero, while the transmission coefficients are unaffected by the different locations.

Mathematically, this can be written

t(x)x=texp(iφt)x=texp(iφt)=t
(1)
r(x)x=rexp(iφr)x0
(2)

where the angular brackets and the subscript x denote averaging over the scatterer’s location, φt and φr are the phases of the transmission t and reflection r coefficient respectively.

Fig. 1. Illustration of dependence of the transmission and reflection coefficients on the scatterer’s location. r 1/t 1 and r 2/t 2 are the reflection/transmission coefficient which corresponds to the scatterer’s locations x 1 and x 2 respectively.

Therefore, this distinction between reflection and transmission is independent of the velocity of the scatterer so long as the averaging time is large enough. Since the transmission coefficient corresponds to photons which did not experience scattering, it is unaffected by averaging (as Eq. (1) suggests), while the reflection coefficient corresponds to the scattered photons and therefore averages to zero. It should be emphasized that in Eqs. (1) and (2) the averaging is done over the optical fields coefficients (〈Aexp(iφ)〉) and not separately on the amplitudes (〈A〉 or 〈A 2〉) and phases (〈φ〉)[2

2. E. Leith, P. Naulleau, and D. Dilworth, “Ensemble-averaged imaging through highly scattering media,” Opt. Lett. 21, 1691 (1996) [CrossRef] [PubMed]

3

3. P. Naulleau, E. Leith, H. Chen, B. Hoover, and J. Lopez, “Time-gated ensemble-averaged imaging through highly scattering media,” Appl. Opt. 36, 3889 (1997) [CrossRef] [PubMed]

].

We can generalize this reasoning to any system with multiple scatterers, without regard for the number of scatterers involved. If a scatterer is in a dynamic state during averaging over the fields, then it can be regarded as if its reflection scattering coefficients are zero. In the 1D case it means that the scatterer can be replaced with an equivalent object that has the same transmission coefficient but a zero reflection coefficient. Therefore, after averaging only the stationary scatterers are effectively contributing. Note, that unlike ref. 4, the scatterers’ motion can be random and not driven by an externally-controlled source.

Clearly, this averaging method can be used to image a turbid or a diffusive medium. Suppose we wish to image an object in a glass of milk. In principle, we can send a beam (or a train of pulses) of light into the medium and sample the reflection or transmission coefficient (depending on whether we desire reflection or transmission imaging) at a certain rate, with the time between successive measurements long enough for the medium to vary. Averaging over the coefficients will lead to the ballistic (non diffusive) image of the medium. The problem is, of course, that this averaging should take place over the electromagnetic field, while optical square-law detectors measure the beam’s power. To average over the field its amplitude and phase must be measured simultaneously. While the amplitude measurement is quite straightforward the phase measurement is more complicated, and in general requires interferometric measurements, which are relatively complicated and inherently noisy. Most time-gating experiments use ultra fast detectors (e.g., streak cameras); however they cannot measure the pulse’s phase. A technique that allows measuring both amplitude and phase simultaneously is the Frequency Resolved Optical Gating (FROG) method [18

18. R. Trebino, Frequency-resolved optical gating: the measurement of ultrashort lasers (Boston, Kluwer Academic Publishers2002) [CrossRef]

19

19. G. Stibenz and G. Steinmeyer, “Interferometric frequency-resolved optical gating,” Opt. Express , 13, 2617 (2005) [CrossRef] [PubMed]

]. However, this technique utilizes both interferometric equipment and nonlinear dynamics, which means that very intense beams are required, and therefore it cannot be implemented for diffusive medium imaging.

The idea to employ temporal averaging of the dynamic scatter to improve imaging is not new (see, for example Refs. [2

2. E. Leith, P. Naulleau, and D. Dilworth, “Ensemble-averaged imaging through highly scattering media,” Opt. Lett. 21, 1691 (1996) [CrossRef] [PubMed]

4

4. J. Khoury, J.S. Kane, P.D. Gianino, P.L. Hemmer, and C.L. Woods, “Homodyne and heterodyne imaging through a scattering medium,” Opt. Lett. 26, 1433 (2001). [CrossRef]

]), however we apply this method in conjunction with the SPEBI time-gating technique by averaging directly over the full optical fields to reconstruct the quasi-ballistic impulse response of the medium with a 0.2ps temporal resolution.

The SPEBI technique [20

20. E. Granot and S. Sternklar, “Spectral ballistic imaging: a novel technique for viewing through turbid or obstructing media,” J. Opt. Soc. Am. A , 20, 1595 (2003) [CrossRef]

,25

25. Er’el Granot, Shmuel Sternklar, Dan Schermann, Yossi Ben-Aderet, and Mordehai Hakham Itzhaq, “Spectral Ballistic Imaging with sub-picosecond temporal resolution”, 50th meeting of the Israel Physical Society (IPS2004)

,26

26. Er’el Granot, Shmuel Sternklar, Dan Schermann, Yossi Ben-Aderet, and Mordehai Hakham Itzhaq, “200 femtosecond Impulse Response of a Fabry-Perot Etalon with the Spectral Ballistic Imaging Technique,” Appl. Phys. B 82, 359 (2006) [CrossRef]

] operates in the frequency domain. Instead of measuring the impulse response, the spectral response of the medium H(ω) is measured. That is, for every frequency ω in the spectral domain of the pulse the amplitude A(ω) and phase φ(ω) of H(ω) are evaluated. However, instead of measuring the phase φ directly, the derivative τDdφ/dω, which can be interpreted as the time delay, is measured directly in a much more robust way. Since the phase difference Δφ is measured directly for every ω, the phase φ can easily be resolved by simple numerical integration.

Fig. 2. System Schematic.

In an experiment we implement the SPEBI technique to reconstruct a 200fs FWHM impulse response of a varying 1D diffusive medium. The sample in this experiment (the diffusive medium) consists of twelve 0.15mm glass plates with random orientation (see Fig. 2). Therefore, a lateral motion of the medium is equivalent to a random change in the plate’s locations.

To reconstruct the impulse response, we used a tunable laser, and scanned the entire optical telecommunication C-band (1535nm–1565nm) spectral range. Such a spectral range corresponds to a sinc-shape pulse whose FWHM is ~0.2ps. To measure the phase difference Δφ the laser beam is modulated with an external Mach-Zehnder modulator, controlled by a network analyzer. The spectral width of the laser (~200kHz) is considerably narrower than the modulation frequency Ω/2π=1GHz, therefore, the modulated beam (with the carrier frequency ω) that enters the etalon can be written Iin(t)I 0[1+cos(Ωt)].

If the transmitted beam (the beam that passed through the etalon) is attenuated and phase shifted, it can be written approximately as [20

20. E. Granot and S. Sternklar, “Spectral ballistic imaging: a novel technique for viewing through turbid or obstructing media,” J. Opt. Soc. Am. A , 20, 1595 (2003) [CrossRef]

] Iout(t)=η(Ω,ω)I 0{1+cos[Ωt+ΔΦ(Ω,ω)]}. After measuring Iout (t) with the electronic detector, the network analyzer compares the input and output signals to evaluate both η(Ω,ω) and ΔΦ(Ω,ω).

There is a simple relation between the phase shift ΔΦ of the intensity wave to the phase φ of the EM field [20

20. E. Granot and S. Sternklar, “Spectral ballistic imaging: a novel technique for viewing through turbid or obstructing media,” J. Opt. Soc. Am. A , 20, 1595 (2003) [CrossRef]

]: ΔΦ(Ω,ω) is equal to the phase difference between the two side bands of the modulated beam, which means that ΔΦ/Ω is an approximation of dφ/dω, or, more accurately, τD=dφdω=limΩ0[ΔΦ(Ω,ω)Ω]. Therefore, the frequency response H(ω) of the system is simply (up to an arbitrary global phase)

H(ω)=η(Ω,ω)exp[iωdωτD(ω)]=η(Ω,ω)exp[ilimΩ0ωdωΔΦ(Ω,ω)Ω]
(3)

In practice the modulation frequency is Ω/2π=1GHz and not Ω→0, which would result in extremely small phase differences and low signal-to-noise ratio. We then scan the entire spectral range (from 1535nm to 1565nm) with spectral steps of δλ=0.01nm (which corresponds to frequency steps of δω~1.25GHz), and measure both η and ΔΦ for the entire spectral range. Under these conditions the spectral response of the medium can be approximated at the discrete frequencies ωN≡ω0+Nδω for every integer N, by

H(ωN)η(Ω,ωN)exp[iδωΩn=0NΔΦ(Ω,ωn)]
(4)

where N≡(ωN0)/δω, and ω0 is the lower boundary of the spectral range. After evaluating the spectral response H(ω) of the medium we reconstruct the impulse response by a simple fast-Fourier-transform. We repeat the experiment at ten different locations on the sample. By doing so, we simulate a varying medium. Ten impulse responses of the medium are acquired in this fashion.

In Fig. 3 all of the ten impulse responses are superimposed with different colors. The reflection response from the mirror is marked by an ellipse. All the other peaks are due to reflections from the glasses. Clearly, we can use this impulse response to separate between the mirror reflection and the surrounding noise. This method is equivalent to the well known time-gating technique, although here the measurements were taken in the frequency domain.

Fig. 3. The impulse responses acquired for ten different locations on the medium.

However, since in this experiment we measure all the parameters of the electromagnetic field (both amplitude and phase) we can do more. In fig. 4 the field of the impulse responses is plotted. As can be seen, in the region of the impulse response that corresponds to the reflection from the mirror (the stationary component) – all the pulses are in-phase, and therefore do not cancel each other by averaging their fields. On the other hand, the reflections from the other non-stationary components (we focus on one of them in this figure) are out-of-phase and therefore cancel-out by averaging.

In Fig. 5 we show how such averaging can improve the SNR. Not only do the phases of the peaks vary from one pulse to the next, but their temporal locations vary as well. Therefore, even intensity averaging reduces the noise peaks more than it reduces the reflection from the mirror (see the lower panel of Fig. 5). However, this kind of averaging cannot eliminate the noise, it only smears it. On the other hand, if we average over the fields (see the upper panel of Fig. 5) the noise, which correspond to the reflections from the varying parts of the system is reduced considerably. In this figure the SNR is increased by a factor of 3 (which concurs with the square root of the number of measured impulse responses).

Fig. 4. top: the field of the impulse response super-imposed for the ten measurements. As can be seen from the two bottom figures, the region that corresponds to the reflection from the mirror (right) consists of in-phase field components and therefore will not vanish after averaging, as opposed to other regions (left).
Fig. 5. The difference between averaging over the intensities of the impulse responses (lower figure) and the fields (upper figure).

In this work we simulated a diffusive medium by a relatively simple structure; however, this is a proof of principle, which can easily be generalized to any varying diffusive medium.

To summarize, in this paper the SPEBI technique was used to measure the impulse-response of a diffusive medium with a 200fs temporal resolution. Since the SPEBI technique measures the pulses’ amplitude and phase simultaneously, it is possible to apply averaging over intensities of the response as well as over the fields. It is shown that when field-averaging is executed over a varying diffusive medium, the reflections from the moving scatterers are reduced considerably with respect to the reflections from the stationary objects. This property of field averaging increases the SNR of the measurement and can be used to image objects in dynamic diffusive media.

This research was supported by the Israel Science Foundation (grant no. 144/03-11.6).

References and links

1.

See, for example, K.R. Castleman, “Digital Image Processing,”(Prentice Hall, New Jersey1996

2.

E. Leith, P. Naulleau, and D. Dilworth, “Ensemble-averaged imaging through highly scattering media,” Opt. Lett. 21, 1691 (1996) [CrossRef] [PubMed]

3.

P. Naulleau, E. Leith, H. Chen, B. Hoover, and J. Lopez, “Time-gated ensemble-averaged imaging through highly scattering media,” Appl. Opt. 36, 3889 (1997) [CrossRef] [PubMed]

4.

J. Khoury, J.S. Kane, P.D. Gianino, P.L. Hemmer, and C.L. Woods, “Homodyne and heterodyne imaging through a scattering medium,” Opt. Lett. 26, 1433 (2001). [CrossRef]

5.

L. Wang, P. P. Ho, F. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991) [CrossRef] [PubMed]

6.

J. C. Hebden and D.T. Delpy, “Enhanced time-resolved imaging with a diffusion model of photon transport,” Opt. Lett. 19, 311 (1994) [CrossRef] [PubMed]

7.

G. M. Turner, G. Zacharakis, A. Soubret, J. Ripoll, and V. Ntziachristos, “Complete-angle projection diffuse optical tomography by use of early photons,” Opt. Lett. 30, 409 (2005) [CrossRef] [PubMed]

8.

A. Ya. Polishchuk, J. Dolne, F. Liu, and R. R. Alfano, “Average and most-probable photon paths in random media,” Opt. Lett. 22, 430 (1997) [CrossRef] [PubMed]

9.

L. Wang, X. Liang, P. Galland, P. P. Ho, and R. R. Alfano, “True scattering coefficients of turbid matter measured by early-time gating,” Opt. Lett. 20, 913 (1995) [CrossRef] [PubMed]

10.

E. N. Leith, B. G. Hoover, S. M. Grannell, K. D. Mills, H. S. Chen, and D. S. Dilworth, “Realization of time gating by use of spatial filtering,” Appl. Opt. 38, 1370 (1999) [CrossRef]

11.

K. D. Mills, L. Deslaurier, D. S. Dilworth, S. M. Grannell, B. G. Hoover, B. D. Athey, and E. N. Leith, “Investigation of ultrafast time gating by spatial filtering,” Appl. Opt. 40, 2282 (2001) [CrossRef]

12.

Yang Liu, Young L. Kim, Xu Li, and Vadim Backman, “Investigation of depth selectivity of polarization gating for tissue characterization,” Opt. Express 13, 601 (2005) [CrossRef] [PubMed]

13.

C-W Sun, C-Yu Wang, C. C. Yang, Y-W Kiang, I-J Hsu, and C-W Lin, “Polarization gating in ultrafast-optics imaging of skeletal muscle tissues,” Opt. Express. 26, 432 (2001)

14.

C-W Sun, C-Yu Wang, C. C. Yang, Y-W Kiang, C-W Lu, I-J Hsu, and C-W Lin, “Polarization-Dependent Characteristics and Polarization Gating in Time-Resolved Optical Imaging of Skeletal Muscle Tissues,” IEEE J. Sel. Top. Quantum Electron. 7, 924 (2001) [CrossRef]

15.

X. Wang, L. V. Wang, C-W Sun, and C-C Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8, 608 (2003) [CrossRef] [PubMed]

16.

C. C. Yang, C-W Sun, C-K Lee, C-W Lu, M-T Tsai, C. C. Yang, and Y-W Kiang, “Comparisons of the transmitted signals of time, aperture, and angle gating in biological tissues and a phantom,” Opt. Express 12, 1157 (2004) [CrossRef] [PubMed]

17.

B. Cairns and E. Wolf, “Changes in the spectrum of light scattered by a moving diffuser plate,” J. Opt. Soc. Am. A 8, 1922 (1991) [CrossRef]

18.

R. Trebino, Frequency-resolved optical gating: the measurement of ultrashort lasers (Boston, Kluwer Academic Publishers2002) [CrossRef]

19.

G. Stibenz and G. Steinmeyer, “Interferometric frequency-resolved optical gating,” Opt. Express , 13, 2617 (2005) [CrossRef] [PubMed]

20.

E. Granot and S. Sternklar, “Spectral ballistic imaging: a novel technique for viewing through turbid or obstructing media,” J. Opt. Soc. Am. A , 20, 1595 (2003) [CrossRef]

21.

W.H. Knox, N.M. Pearson, K.D. Li, and C.A. Hirlimann, “Interferometric measurements of femtosecond group delay in optical components,” Opt. Lett. 13, 574 (1988) [CrossRef] [PubMed]

22.

R.C. Moore and F.H. Slaymaker, “Direct measurement of phase in a spherical-wave Fizeau interferometer,” Appl. Opt. 19, 2196 (1980) [CrossRef] [PubMed]

23.

M. Beck and I.A. Walmsley, “Measurement of group delay with high temporal and spectral resolution,” Opt. Lett. 15, 492 (1990) [CrossRef] [PubMed]

24.

S.E. Mechels, J.B. Schlager, and D.L. Franzen, “Accurate Measurements of the Zer-Dispersion Wavelength in Optical Fibers,” J. Res. Natl. Inst. Stand. Technol. 102, 333 (1997) [CrossRef]

25.

Er’el Granot, Shmuel Sternklar, Dan Schermann, Yossi Ben-Aderet, and Mordehai Hakham Itzhaq, “Spectral Ballistic Imaging with sub-picosecond temporal resolution”, 50th meeting of the Israel Physical Society (IPS2004)

26.

Er’el Granot, Shmuel Sternklar, Dan Schermann, Yossi Ben-Aderet, and Mordehai Hakham Itzhaq, “200 femtosecond Impulse Response of a Fabry-Perot Etalon with the Spectral Ballistic Imaging Technique,” Appl. Phys. B 82, 359 (2006) [CrossRef]

OCIS Codes
(110.3080) Imaging systems : Infrared imaging
(170.5280) Medical optics and biotechnology : Photon migration
(290.4210) Scattering : Multiple scattering

ToC Category:
Imaging Systems

History
Original Manuscript: September 28, 2005
Revised Manuscript: May 9, 2006
Manuscript Accepted: May 15, 2006
Published: September 18, 2006

Virtual Issues
Vol. 1, Iss. 10 Virtual Journal for Biomedical Optics

Citation
Er'el Granot, Shmuel Sternklar, Yossi Ben-Aderet, and Dan Schermann, "Quasi-ballistic imaging through a dynamic scattering medium with optical-field averaging using Spectral-Ballistic-Imaging," Opt. Express 14, 8598-8603 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-19-8598


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References

  1. See, for example, K. R. Castleman, "Digital Image Processing," (Prentice Hall, New Jersey 1996
  2. E. Leith, P. Naulleau, and D. Dilworth, "Ensemble-averaged imaging through highly scattering media," Opt. Lett. 21, 1691 (1996) [CrossRef] [PubMed]
  3. P. Naulleau, E. Leith, H. Chen, B. Hoover, and J. Lopez, "Time-gated ensemble-averaged imaging through highly scattering media," Appl. Opt. 36, 3889 (1997) [CrossRef] [PubMed]
  4. J. Khoury, J.S. Kane, P.D. Gianino, P.L. Hemmer, and C.L. Woods, "Homodyne and heterodyne imaging through a scattering medium," Opt. Lett. 26, 1433 (2001). [CrossRef]
  5. L. Wang, P. P. Ho, F. Liu, G. Zhang, and R. R. Alfano, "Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate," Science 253,769-771 (1991) [CrossRef] [PubMed]
  6. J. C. Hebden and D. T. Delpy, "Enhanced time-resolved imaging with a diffusion model of photon transport," Opt. Lett. 19, 311 (1994) [CrossRef] [PubMed]
  7. G. M. Turner, G. Zacharakis, A. Soubret, J. Ripoll, V. Ntziachristos, "Complete-angle projection diffuse optical tomography by use of early photons," Opt. Lett. 30, 409 (2005) [CrossRef] [PubMed]
  8. A. Ya. Polishchuk, J. Dolne, F. Liu, and R. R. Alfano, "Average and most-probable photon paths in random media," Opt. Lett. 22, 430 (1997) [CrossRef] [PubMed]
  9. L. Wang, X. Liang, P. Galland, P. P. Ho, and R. R. Alfano, "True scattering coefficients of turbid matter measured by early-time gating," Opt. Lett. 20, 913 (1995) [CrossRef] [PubMed]
  10. E. N. Leith, B. G. Hoover, S. M. Grannell, K. D. Mills, H. S. Chen, and D. S. Dilworth, "Realization of time gating by use of spatial filtering," Appl. Opt. 38, 1370 (1999) [CrossRef]
  11. K. D. Mills, L. Deslaurier, D. S. Dilworth, S. M. Grannell, B. G. Hoover, B. D. Athey, and E. N. Leith, "Investigation of ultrafast time gating by spatial filtering," Appl. Opt. 40, 2282 (2001) [CrossRef]
  12. Yang Liu, Young L. Kim, Xu Li, Vadim Backman, "Investigation of depth selectivity of polarization gating for tissue characterization," Opt. Express 13, 601 (2005) [CrossRef] [PubMed]
  13. C-W Sun, C-Yu Wang, C. C. Yang, Y-W Kiang, I-J Hsu, C-W Lin, "Polarization gating in ultrafast-optics imaging of skeletal muscle tissues," Opt. Express. 26, 432 (2001)
  14. C-W Sun, C-Yu Wang, C. C. Yang, Y-W Kiang, C-W Lu, I-J Hsu, and C-W Lin, "Polarization-Dependent Characteristics and Polarization Gating in Time-Resolved Optical Imaging of Skeletal Muscle Tissues," IEEE J. Sel. Top. Quantum Electron. 7, 924 (2001) [CrossRef]
  15. X. Wang, L. V. Wang, C-W Sun, C-C Yang, "Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments," J. Biomed. Opt. 8, 608 (2003) [CrossRef] [PubMed]
  16. C. C. Yang, C-W Sun, C-K Lee, C-W Lu, M-T Tsai, C. C. Yang, and Y-W Kiang, "Comparisons of the transmitted signals of time, aperture, and angle gating in biological tissues and a phantom," Opt. Express 12, 1157 (2004) [CrossRef] [PubMed]
  17. B. Cairns and E. Wolf, "Changes in the spectrum of light scattered by a moving diffuser plate," J. Opt. Soc. Am. A 8, 1922 (1991) [CrossRef]
  18. R. Trebino, Frequency-resolved optical gating: the measurement of ultrashort lasers (Boston, Kluwer Academic Publishers 2002) [CrossRef]
  19. G. Stibenz, G. Steinmeyer, "Interferometric frequency-resolved optical gating," Opt. Express,  13, 2617 (2005) [CrossRef] [PubMed]
  20. E. Granot and S. Sternklar, "Spectral ballistic imaging: a novel technique for viewing through turbid or obstructing media," J. Opt. Soc. Am. A,  20, 1595 (2003) [CrossRef]
  21. W.H. Knox, N.M. Pearson, K.D. Li, and C.A. Hirlimann, "Interferometric measurements of femtosecond group delay in optical components," Opt. Lett. 13, 574 (1988) [CrossRef] [PubMed]
  22. R. C. Moore and F. H. Slaymaker, "Direct measurement of phase in a spherical-wave Fizeau interferometer," Appl. Opt. 19, 2196 (1980) [CrossRef] [PubMed]
  23. M. Beck and I. A. Walmsley, "Measurement of group delay with high temporal and spectral resolution," Opt. Lett. 15, 492 (1990) [CrossRef] [PubMed]
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