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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 6 — Mar. 24, 2014
  • pp: 7087–7098
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When can temporally focused excitation be axially shifted by dispersion?

B. Leshem, O. Hernandez, E. Papagiakoumou, V. Emiliani, and D. Oron  »View Author Affiliations


Optics Express, Vol. 22, Issue 6, pp. 7087-7098 (2014)
http://dx.doi.org/10.1364/OE.22.007087


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Abstract

Temporal focusing (TF) allows for axially confined wide-field multi-photon excitation at the temporal focal plane. For temporally focused Gaussian beams, it was shown both theoretically and experimentally that the temporal focus plane can be shifted by applying a quadratic spectral phase to the incident beam. However, the case for more complex wave-fronts is quite different. Here we study the temporal focus plane shift (TFS) for a broader class of excitation profiles, with particular emphasis on the case of temporally focused computer generated holography (CGH) which allows for generation of arbitrary, yet speckled, 2D patterns. We present an analytical, numerical and experimental study of this phenomenon. The TFS is found to depend mainly on the autocorrelation of the CGH pattern in the direction of the beam dispersion after the grating in the TF setup. This provides a pathway for 3D control of multi-photon excitation patterns.

© 2014 Optical Society of America

1. Introduction

Nearly a decade ago, temporal focusing (TF) was suggested as an axially resolved multi-photon wide-field microscopy technique [1

1. D. Oron, E. Tal, and Y. Silberberg, “Scanningless depth-resolved microscopy,” Opt. Express 13, 1468–1476 (2005). [CrossRef] [PubMed]

, 2

2. G. Zhu, J. van Howe, M. E. Durst, W. Zipfel, and C. Xu, “Simultaneous spatial and temporal focusing of femtosecond pulses,” Opt. Express 13, 2153–2159 (2005). [CrossRef] [PubMed]

]. In recent years, many techniques such as multi-photon microscopy, optical lithography, tissue ablation and optogenetics have benefited from its ability to confine excitation axially [3

3. E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7, 848–854 (2010). [CrossRef] [PubMed]

7

7. E. Block, M. Greco, D. Vitek, O. Masihzadeh, D. A. Ammar, M. Y. Kahook, N. Mandava, C. Durfee, and J. Squier, “Simultaneous spatial and temporal focusing for tissue ablation,” Bio. Opt. Express 4, 831–841 (2013). [CrossRef]

].

In this work we consider beams formed with computer generated holography (CGH). CGH beams are generated by using a phase-only spatial light modulator (SLM) to modulate the phase of the incident light beam in a controlled manner. The SLM is located at the back focal plane of a lens so that the phase modulated beam is Fourier transformed to the front focal plane. The phase modulation is chosen in advance using an iterative algorithm such as the Gerchberg-Saxton algorithm [12

12. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

] so that after the Fourier transformation the desired amplitude modulated pattern is generated. The result can be either a predetermined, yet speckled, pattern or diffraction-limited spots configured in different axial planes.

CGH beams have found numerous applications in optical lithography, optical trapping and neuronal stimulation [13

13. M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999). [CrossRef]

19

19. F. Anselmi, C. Ventalon, A. Bègue, D. Ogden, and V. Emiliani, “Three-dimensional imaging and photostimulation by remote-focusing and holographic light patterning,” Proc. Natl. Acad. Sci. U. S. A. 108, 19504–19509 (2011). [CrossRef] [PubMed]

]. The combination of TF and CGH (TF-CGH) for the generation of extended excitation patterns was previously studied [20

20. E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express 16, 22039–22047 (2008). [CrossRef] [PubMed]

] and shown useful for optogenetics [21

21. A. Bègue, E. Papagiakoumou, B. Leshem, R. Conti, L. Enke, D. Oron, and V. Emiliani, “Multiphoton excitation in scattering media by holographic beams and their application in optogenetic stimulation,” Biomed. Opt. Express (to be published) (2013). [CrossRef]

]. Lately, it has been proposed that shifting of the temporal focus is useful for the study of neuronal networks [22

22. H. Dana and S. Shoham, “Remotely scanned multiphoton temporal focusing by axial grism scanning,” Opt. Lett. 37, 2913–2915 (2012). [CrossRef] [PubMed]

, 23

23. D. Oron, E. Papagiakoumou, F. Anselmi, and V. Emiliani, “Two-photon optogenetics,” Prog. Brain Res. 196, 119–143 (2012). [CrossRef] [PubMed]

]. In particular, decoupling the temporal and spatial focus planes, will be useful for this purpose, since it is often preferable to photoactivate cells in one axial plane while optically observing the responses of cells in another plane (such as with calcium imaging). This is often a challenging task, since both photoactivation and imaging beams are focused to the sample through the same objective. Hence, independent axial movement of one in relation to the other is desirable. Here, we study the effect of applying GVD onto a TFCGH beam with the goal of identifying the conditions in which the temporal focal plane can be shifted despite the speckled excitation pattern. In the following we first present a theoretical analysis of the TFS for CGH beams as well as its physical interpretation, and then proceed to describe both numerical and experimental results.

2. Theoretical analysis

2.1. Calculation of the two-photon axial response

Here we calculate the two-photon axial response around the front focal plane of the objective for TF-CGH beams, as illustrated in the typical TF setup depicted in Fig. 1. Assuming a Gaussian spectral profile, the temporally focused pulse at the front focal plane of the objective can be described as a complex amplitude multiplied by a phase function:
E(x,y,Δω)=A(x,y)exp(iαΔωxiβΔω2Δω22δ2)
(1)
where A(x, y) is the complex amplitude function, α=2πMω0g, ω0 is the central frequency, Δω = ωω0 is the deviation from central frequency, M is the imaging system magnification, g is the diffraction grating’s groove spacing, 2β is the applied GVD and 2ln2 δ is the full width half maximum (FWHM) of the spectrum of the pulse, assumed to be of a Gaussian shape. The first term in the phase function represents the diffraction off the grating. This result can be derived simply by imposing on the grating equation the condition that the first order of diffraction for the central wavelength is perpendicular to the grating (here we neglect the geometrical scaling of the x coordinate). Equation (1) can be used for numerical simulations of temporally focused pulse propagation with or without imposing the Fresnel approximation, as was done for the numerical simulations below as well as previously [6

6. E. Papagiakoumou, A. Bègue, B. Leshem, O. Schwartz, B. M. Stell, J. Bradley, D. Oron, and V. Emiliani, “Functional patterned multiphoton excitation deep inside scattering tissue,” Nat. Photonics 7, 274–278 (2013). [CrossRef]

]. In the analytical derivation below we use the Fresnel approximation. Although not accurate for high-NA objectives [24

24. E. Yew, C. J. R. Sheppard, and P. T. C. So, “Temporally focused wide-field two-photon microscopy: Paraxial to vectorial,” Opt. Express 21, 12951–12963 (2013). [CrossRef] [PubMed]

], it captures the essence of the physical phenomenon. This assumption will be tested by comparing to both numerical and experimental results.

Fig. 1 Sketch of the TF-CGH setup. An Ultra-short pulse reflects off an SLM and is focused onto a diffraction grating (G) through a lens (L1). The first diffraction order is consequently imaged via a tube lens (L2) and an objective to the front focal plane (FFP) of the objective. The incident angle is such that the first order of diffraction is perpendicular to the grating. The SLM is imaged to the back focal plane (BFP) of the objective via lenses L1 and L2. Equation (1) describes the light distribution at the FFP of the objective.

In order to model a CGH beam we describe the complex amplitude A(x, y) as a Gaussian envelope of width W multiplied by a speckle pattern, S(x, y):
A(x,y)=S(x,y)exp(x2+y22W2)
(2)
and model the autocorrelation of the speckle pattern as a Gaussian of standard deviation σ:
S(ri)S(rj)=exp((rirj)22σ2)
(3)
where ri = (xi, yi) and the brackets denotes ensemble average over different realizations of the speckle pattern. We note that the exact functional form of the speckle pattern autocorrelation is determined mainly by the aperture of its Fourier transform [25

25. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company Publishers, 2007).

]. However, the results presented below are mostly affected by the autocorrelation width rather than by the details of the functional form. Furthermore, we assume that the speckle pattern is fully developed and thus obeys Gaussian statistics [25

25. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company Publishers, 2007).

] so that moments of order higher than two can be separated into products of 2nd order moments. This assumption is required for the analytical calculation of the two photon response.

We note that even though in general A(xi, yi)A(xj, yj) is not a separable function, its ensemble average is separable. Hence, we can solve separately for the x and y coordinates. The ensemble average of the two-photon signal at axial position z around the objective’s front focal plane is given by:
I2p(z)=|E(r,z,t)|4drdt=|E(r,z=0,t)*h(r,z)|4drdt
(4)
where E(r, z, t) is the electromagnetic field, h(r,z)=eik0ziλ0zexp(ik02zr2) is the convolution kernel in the Fresnel approximation and k0 is the wave number at the central wavelength λ0 [26

26. J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005).

].

For simplicity, we first calculate for the x-coordinate. E(x, z, t) is given by the inverse Fourier transform to the temporal domain of Eq. (1) convolved with h(x,z)=eik0ziλ0zexp(ik02zx2). Using Eq. (2) we get:
E(x,z,t)=FT(Δωt)1{S(x)exp(x22W2iαΔωxiβΔω2Δω22δ2)}*h(x,z)=S(x)G(x,t)*h(x,z)
(5)
where: G(x,t)=(a+ib)12exp((tαx)22(a+ib)x22W2), a=1δ2, b = 2β

Using Eqs. (4) and (5) and separating 4th order moments into a product of 2nd order moments, we can write the two-photon signal in the x-direction as:
Ix2p(z)=|E(x,z,t)|4dxdt=2dxdt[dx1dx2S(x1)S(x2)G(x1,t)G(x2,t)h(xx1,z)h(xx2,z)]2
(6)

Substituting Eq. (3) and denoting σx as the spatial autocorrelation width in the x direction, we get the two-photon axial response for the x-direction:
Ix2p(z)=Ax((zΔz)2+zTF2)12
(7.a)
Δz=TR(σxW)2TS(1+TS2+TR2)+(σxW)2(TS2+(1+TR)2)zR
(7.b)
zTF2=(1+TS2+TR)[2(1+TS2)(σxW)2+(1+TS2+TR)(σxW)4][((1+TR)2+TS2)(σxW)2+2(1+TR+TS2)]2zR2
(7.c)
where:
TR=(Wα)2a,TS=ba,zR=k0W2,Ax=πσxzR2a32[(1+TS2+TR2)+(σxW)2(TS2+(1+TR)2)]12.
Here the TFS is manifested in the z-direction shift of the two-photon axial response denoted as Δz. Solving for the y-coordinate we get similarly :
Iy2p(z)=Ay(2W2z2σy2+z2+zR2)12
(8)
where Ay=π2WzR and σy is the autocorrelation width in the y direction. The overall two-photon axial response is given by:
I2p(z)=Ix2p(z)Iy2p(z)
(9)

The parameters in our results have a straightforward physical interpretation. First, we note that in time domain, TF might be viewed as an ultra-fast line scanning. The parameter α=2πMω0g has units of timelength and can be interpreted as the inverse of the ultra-fast line scanning velocity. This can be seen from the expression for G(x, t) in Eq. (5). Next, two dimensionless parameters are defined in Eq. (7): TR=(Wα)2a can be interpreted as the square of the ratio between the ultra-fast line scanning duration and the pulse duration. TR governs the two photon axial response of the TF pulse as is evident in Eq. (10.b) below. The GVD appears only in the dimensionless parameter TS=ba which is the ratio between the GVD and the square of the pulse duration.

2.2. Limits of the results

In the case of smooth pattern (in the x-direction) such as a Gaussian beam (i.e. σx → ∞), Eqs. (7.b) and (7.c) reduces to:
zTF1+TS2+TR(1+TR)2+TS2zR,ΔzTSTRTS2+(1+TR)2zR
(10.a)
The TFS, Δz, for TSTR (which is typically the case) is proportional to the applied GVD, in accordance with [9

9. M. E. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing for axial scanning,” Opt. Express 14, 12243–12254 (2006). [CrossRef] [PubMed]

], with the proportionality constant given by the parameters of the experimental setup. We note that σy does not directly effect the TFS. As σy increases it merely reduces Eq. (8) towards a two-photon axial response of a Gaussian beam in one lateral dimension. If, for a Gaussian beam, no GVD is applied (b → 0) Eq. (9) simplifies to:
I2p(z)1z2+zR21z2+zR2(1+TR)2
(10.b)

In this case, the result is a product of two terms. The first is the axial response of a 1D Gaussian beam with a Rayleigh range zR, this describes the non-temporally focused y-direction axial response. The second is of similar form but with zR reduced by 1 + TR, this is the axial confinement introduced by TF.

In the absence of TF (α → 0) Eq. (10.b) reduces to:
I2p(z)1z2+zR2
(10.c)
which is the two-photon axial response for a Gaussian beam with Rayleigh range zR. As is evident from Eqs. (7) and (8), σx and σy play completely different roles in determining the TFS, Δz. While σx also controls Δz directly through Eq. (7.b) the role of σy is indirect. It attenuates the overall two-photon axial response as GVD is applied, thus practically limits the TFS.

2.3. Physical interpretation

Fig. 2 a) Illustration of TF beam impinging on a grating when GVD is applied. The temporal shift between “blue” and “red” portions of the temporal spectrum generates a spatial shift as the pulse scans the grating. This is equivalent to shifting the grating in the z-direction as long as the spectral portions can be considered as replicas of one another. b) The same effect but with TF-CGH beams. The autocorrelation width in the x-direction determines the lateral shift for which the spectral portions can be considered to be replicas of one another, which in turn determines the amount of TFS that can be achieved.
Fig. 3 Axial profile as a function of the applied GVD for the case: a) σx, σy → ∞. b) σx → ∞, σy = 0.3 μm. c) σy → ∞, σx = 0.3 μm.
Fig. 4 a) Axial profile of a TF-CGH beam when applying GVD of 40000 fs2 vs. σx (with σy → ∞). The superimposed dashed line is the peak position of a TF Gaussian beam with the same amount of GVD applied and beam waist varying as σx. b) The TFS for which the two-photon signal decreases to half its value with no GVD (denoted Δz12), as a function of σx. In both plots the minimum value of σx is 0.3 μm

The second aspect, attenuation of the signal with the TFS, depends mainly on the spatial frequency content of the TF beam. The temporal focused signal attenuates as it is shifted outside of the beam’s Rayleigh range. This corresponds to shift between the two axial profiles terms in Eq. (9). The higher the spatial frequency content of the beam, the shorter it’s Rayleigh range. Hence, beams with higher frequency content will cause faster attenuation of the signal as the temporal focus is shifted and vice versa. In the calculation above, the frequency content depends on the autocorrelation widths σx and σy. For TF-CGH beams this means that the signal will decrease faster with TFS when the speckles are smaller. Therefore, passing the TF-CGH beam through a spatial low-pass filter (in both lateral directions) can be used to mitigate the attenuation of the signal with TFS. An alternative solution is to pass the beam through a spatial low-pass filter in the x-direction while shifting the spatial focus plane along with the temporal one by applying a quadratic spatial phase on the SLM.

Of the two effects described above the more significant one is the dependence of the TFS on σx. This is because decreasing both σx and σy cause attenuation of the signal with applied GVD, but only σx determines whether applying GVD is essentially equivalent to shifting the temporal focus. This crucial role of σx is illustrated in Fig. 3. We note that since the peak of the two-photon signal is generated at the TF plane, the TFS is given by the location of the axial profile peak. Figure 3(a) shows the axial profile of a TF Gaussian beam (σx → ∞, σy → ∞) as a function of the applied GVD. In Figs. 3(b) and 3(c) either σy or σx are taken to be very small correspondingly. As can be seen, decreasing σy attenuates the signal with the applied GVD thus effectively mildly decreasing TFS. In contrast, decreasing σx eliminates the TFS altogether.

3. Numerical and experimental results

In order to verify the above analytical calculation we perform numerical simulations and experimental investigation. Below we describe both and compare to the analytical results.

The experimental setup is described in Fig. 5. The laser source used in the experiments is a Ti:Sapphire laser (MaiTai Deep-See, Spectra-Physics) producing pulses of 8 nm FWHM at a central wavelength of 800 nm. The pulses are passed through a grating compressor/stretcher in order to apply GVD. Next, the beam is magnified with a 10x beam expander and shined upon a phase-only SLM (LCOS-SLM, X10468-02, Hamamatsu Photonics), the resultant beam is Fourier transformed by focusing through a 100 cm lens. The phase imprinted on the SLM is predetermined using the Gerchberg-Saxton algorithm in order to generate a circle of 15 μm diameter. Control of the autocorrelation in x and y directions is achieved by passing the beam through a spatial low-pass filter, to which we refer below as low-passing. It is realized using a cylindrical telescope and a slit or, alternatively, an iris, that are set before the SLM. The slit used for x-direction low-passing (Fig. 6(b)) is a 1 mm slit located before the SLM. Since the SLM is imaged to the back aperture of the objective with 2x demagnification the effective low-passing is 0.5 mm of an overall objective back aperture of 5.4 mm. The y-direction low-passing (Fig. 6(c)) is performed similarly in the perpendicular direction. The iris used for low-passing in both x and y directions (Fig. 6(d)) is of 2.5 mm diameter which corresponds to 1.25 mm in the back aperture of the objective.

Fig. 5 Schematics of the experimental setup. Laser beam from a Ti:Sapphire laser is passed through a grating compressor/stretcher where GVD is applied. The beam then impinges on an SLM and focused into the TF setup, constituted of a diffraction grating, G, and an imaging system. L: Lens, M: Mirror, BE: Beam Expander, OBJ: Microscope Objectives, FFP: Front Focal Plane.
Fig. 6 Each plot depicts the analytical results for the two-photon axial profiles for different values of the applied GVD. Plots a–d are for different slit widths corresponding to different spatial lowpass filters. a) No slit. b) 1 mm slit with its short axis in the x-direction. c) 1 mm slit with its short axis in the y-direction. d) 2.5 mm iris. The overlayed TF-CGH patterns are presented here for illustration, they are calculated for a single realization of a speckle pattern with a Gaussian envelope with the same parameters as the analytical calculation.

The TF setup is composed of a 300 lines/mm diffraction grating that is imaged with a 50 cm tube lens and a water immersion objective (Olympus, LUMPLFLN 60XW, 0.9NA). The front focal plane of the objective coincides with the TF plane when no GVD is applied. In the TF focal plane the beam excites a thin layer of Rhodamine 6G and the fluorescent light is imaged onto a CCD camera (CoolSNAP HQ2, Roper Scientific) through a second objective (Olympus UPLSAPO60XW, NA 1.0) and a tube lens.

In the numerical calculations we use Eq. (1) as our initial light distribution with the complex amplitude of a CGH pattern generated via the Gerchberg-Saxton algorithm. We then numerically calculate its propagation within the framework of vectorial theory [24

24. E. Yew, C. J. R. Sheppard, and P. T. C. So, “Temporally focused wide-field two-photon microscopy: Paraxial to vectorial,” Opt. Express 21, 12951–12963 (2013). [CrossRef] [PubMed]

], and calculate the two-photon signal.

Figures 6 and 7 summarize the comparison between the analytical, numerical and experimental results. Figure 6 shows the analytical results with the same parameters as the experiments. The plots compare between the axial profiles without or with low-passing in either x,y or both x and y directions. Again, the difference in TFS between low-passing in the x and y directions is immediately apparent when comparing Fig. 6(a) with Figs. 6(b) and 6(c) respectively. In Fig. 6(d) the iris size was chosen to produce speckles with approximately the same area as those produced by the slit. Despite the fact the speckles area is similar, low-passing in both directions results in less TFS than x-direction low-passing. This supports our theoretical results and heuristic explanation according to which low-passing in the x-direction controls the amount of TFS while in the y-direction it serves merely to mitigate the attenuation of the axial response with GVD. We also note that the axial response profile width increases for low-passing in the x-direction but not in the y-direction. This can be explained noting that the two-photon axial response, depicted in Eq. (9), is the product of x-direction and y-direction axial responses. Hence the narrower axial profile, which is the x-direction one, determines the width of the overall axial response.

Fig. 7 The experimental and numerical results corresponding to the ones presented in Fig. 6. Each plot depicts comparison between experimental (circles) and numerical (solid line) results. The overlayed images are the corresponding experimental TF-CGH patterns.

Figure 7 shows the experimental and numerical plots corresponding to the plots in Fig. 6. There is a good agreement between the experimental and numerical results. Also, as is evident from comparing Figs. 6 and 7 there is a good agreement between the analytical and experimental results. We note that small discrepancies are to be expected due to assumptions taken throughout the calculation which are not completely accurate, such as the Fresnel approximation and Gaussian statistics of the holographic pattern.

4. Conclusions

We have investigated the effect of applying GVD on TF-CGH beams theoretically and experimentally. We have shown that the equivalence between applying GVD and generating TFS depends mainly on the autocorrelation width of the CGH pattern in the x-direction, which is the direction in which the ultrashort pulse diffracts off the grating in the TF setup. Correspondingly, low-passing in the x-direction was shown to increase the TFS, in contrast to low-passing in the perpendicular direction. Therefore, temporally focused excitation can be axially shifted by dispersion only when the autocorrelation width of the illumination in the x-direction is sufficiently large. Signal attenuation with applied GVD was shown to be dependent on the autocorrelation in both lateral directions, due to the shift between the temporal and spatial focal planes. Although low-passing in both lateral dimensions can serve to increase the TFS, it also increases the speckle size which may not be desirable for some applications such as multi-photon optogenetics. The investigated effect, in which the two-photon excitation is controlled in 3D by manipulating both temporal and spatial frequencies of the incident light, can be useful for optogenetics, nonlinear microscopy and micro-machining.

Acknowledgments

DO acknowledges financial support by the European Research Council starting investigator grant SINSLIM 258221, by the I-CORE Program of the Planning and Budgeting Committee and the Israel Science Foundation and from the Laboratoire Européen Associé NaBi between the CNRS and the Weizmann Institute. VE acknowledges financial support by the Human Frontier Science Program ( RGP0013/2010), the ‘Fondation pour la Recherche Mèdicale’ (FRMÈquipe) and the ‘Agence Nationale de la Recherche’ (grants ANR-12-BSV5-0011-01, Neurholog) and by the France-BioImaging infrastructure supported by the French National Research Agency ( ANR-10-INSB-04, Investments for the future). OH acknowledges the program Nanotechnologies France-Israël for financial support.

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E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express 16, 22039–22047 (2008). [CrossRef] [PubMed]

21.

A. Bègue, E. Papagiakoumou, B. Leshem, R. Conti, L. Enke, D. Oron, and V. Emiliani, “Multiphoton excitation in scattering media by holographic beams and their application in optogenetic stimulation,” Biomed. Opt. Express (to be published) (2013). [CrossRef]

22.

H. Dana and S. Shoham, “Remotely scanned multiphoton temporal focusing by axial grism scanning,” Opt. Lett. 37, 2913–2915 (2012). [CrossRef] [PubMed]

23.

D. Oron, E. Papagiakoumou, F. Anselmi, and V. Emiliani, “Two-photon optogenetics,” Prog. Brain Res. 196, 119–143 (2012). [CrossRef] [PubMed]

24.

E. Yew, C. J. R. Sheppard, and P. T. C. So, “Temporally focused wide-field two-photon microscopy: Paraxial to vectorial,” Opt. Express 21, 12951–12963 (2013). [CrossRef] [PubMed]

25.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company Publishers, 2007).

26.

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005).

OCIS Codes
(170.6900) Medical optics and biotechnology : Three-dimensional microscopy
(320.7110) Ultrafast optics : Ultrafast nonlinear optics
(090.1995) Holography : Digital holography

ToC Category:
Holography

History
Original Manuscript: December 3, 2013
Revised Manuscript: February 6, 2014
Manuscript Accepted: February 11, 2014
Published: March 19, 2014

Virtual Issues
Vol. 9, Iss. 5 Virtual Journal for Biomedical Optics

Citation
B. Leshem, O. Hernandez, E. Papagiakoumou, V. Emiliani, and D. Oron, "When can temporally focused excitation be axially shifted by dispersion?," Opt. Express 22, 7087-7098 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-7087


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  26. J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005).

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