## When can temporally focused excitation be axially shifted by dispersion? |

Optics Express, Vol. 22, Issue 6, pp. 7087-7098 (2014)

http://dx.doi.org/10.1364/OE.22.007087

Acrobat PDF (1691 KB)

### 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

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

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]

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. 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]

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. 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]

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]

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]

## 2. Theoretical analysis

### 2.1. Calculation of the two-photon axial response

*A*(

*x*,

*y*) is the complex amplitude function,

*ω*

_{0}is the central frequency, Δ

*ω*=

*ω*−

*ω*

_{0}is the deviation from central frequency,

*M*is the imaging system magnification,

*g*is the diffraction grating’s groove spacing, 2

*β*is the applied GVD and

*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]

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]

*A*(

*x*,

*y*) as a Gaussian envelope of width

*W*multiplied by a speckle pattern,

*S*(

*x*,

*y*): and model the autocorrelation of the speckle pattern as a Gaussian of standard deviation

*σ*: where

*r*= (

_{i}*x*,

_{i}*y*) 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]. 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] so that moments of order higher than two can be separated into products of 2

_{i}^{nd}order moments. This assumption is required for the analytical calculation of the two photon response.

*A*(

*x*,

_{i}*y*)

_{i}*A*

^{★}(

*x*,

_{j}*y*) 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

_{j}*z*around the objective’s front focal plane is given by: where

*E*(

*r*,

*z*,

*t*) is the electromagnetic field,

*k*

_{0}is the wave number at the central wavelength

*λ*

_{0}[26].

*E*(

*x*,

*z*,

*t*) is given by the inverse Fourier transform to the temporal domain of Eq. (1) convolved with

*b*= 2

*β*

^{th}order moments into a product of 2

^{nd}order moments, we can write the two-photon signal in the x-direction as:

*σ*as the spatial autocorrelation width in the x direction, we get the two-photon axial response for the x-direction: where: Here the TFS is manifested in the z-direction shift of the two-photon axial response denoted as Δ

_{x}*z*. Solving for the y-coordinate we get similarly : where

*σ*is the autocorrelation width in the y direction. The overall two-photon axial response is given by:

_{y}*G*(

*x*,

*t*) in Eq. (5). Next, two dimensionless parameters are defined in Eq. (7):

*T*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

_{R}### 2.2. Limits of the results

*σ*→ ∞), Eqs. (7.b) and (7.c) reduces to: The TFS, Δ

_{x}*z*, for

*T*≪

_{S}*T*(which is typically the case) is proportional to the applied GVD, in accordance with [9

_{R}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]

*σ*does not directly effect the TFS. As

_{y}*σ*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 (

_{y}*b*→ 0) Eq. (9) simplifies to:

*z*, this describes the non-temporally focused y-direction axial response. The second is of similar form but with

_{R}*z*reduced by 1 +

_{R}*T*, this is the axial confinement introduced by TF.

_{R}*α*→ 0) Eq. (10.b) reduces to: which is the two-photon axial response for a Gaussian beam with Rayleigh range

*z*. As is evident from Eqs. (7) and (8),

_{R}*σ*and

_{x}*σ*play completely different roles in determining the TFS, Δ

_{y}*z*. While

*σ*also controls Δ

_{x}*z*directly through Eq. (7.b) the role of

*σ*is indirect. It attenuates the overall two-photon axial response as GVD is applied, thus practically limits the TFS.

_{y}### 2.3. Physical interpretation

*σ*and

_{x}*σ*. 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.

_{y}*σ*. This is because decreasing both

_{x}*σ*and

_{x}*σ*cause attenuation of the signal with applied GVD, but only

_{y}*σ*determines whether applying GVD is essentially equivalent to shifting the temporal focus. This crucial role of

_{x}*σ*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 (

_{x}*σ*→ ∞,

_{x}*σ*→ ∞) as a function of the applied GVD. In Figs. 3(b) and 3(c) either

_{y}*σ*or

_{y}*σ*are taken to be very small correspondingly. As can be seen, decreasing

_{x}*σ*attenuates the signal with the applied GVD thus effectively mildly decreasing TFS. In contrast, decreasing

_{y}*σ*eliminates the TFS altogether.

_{x}*σ*, Fig. 4(a) shows the two-photon axial profile given by Eq. (9) as a function of

_{x}*σ*, with a GVD of 40000

_{x}*fs*

^{2}. The superimposed dashed line is the peak position for a TF Gaussian beam with the same amount of GVD applied and beam waist of

*σ*. As can be seen, for large values of

_{x}*σ*the TFS is close to 20

_{x}*μ*m. In stark contrast, for

*σ*smaller than 3

_{x}*μ*m the TFS reduces, i.e. the axial profile peak shifts towards

*z*= 0. Furthermore, the two-photon response attenuates rapidly as

*σ*decreases. In Fig. 4(b), each point shows the value of the TFS for which the two-photon signal decreases with applied GVD, to half its initial value without GVD. This value of the TFS is denoted as

_{x}*σ*.

_{x}*σ*and decreases towards zero as

_{x}*σ*is decreased. In both Figs. 3 and 4 we used

_{x}*W*= 10

*μ*m as the width of the Gaussian envelope described in Eq. (2) and a pulse duration of 170 fs, the other parameters were set as in the experiments described below. In Fig. 6 we present the analytical results that are compared with the experimental and numerical ones. The comparison is discussed in the next section.

## 3. Numerical and experimental results

*μ*m 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.

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]

## 4. Conclusions

## Acknowledgments

## References and links

1. | D. Oron, E. Tal, and Y. Silberberg, “Scanningless depth-resolved microscopy,” Opt. Express |

2. | G. Zhu, J. van Howe, M. E. Durst, W. Zipfel, and C. Xu, “Simultaneous spatial and temporal focusing of femtosecond pulses,” Opt. Express |

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 |

4. | A. Vaziri, J. Tang, H. Shroff, and C. V. Shank, “Multilayer three-dimensional super resolution imaging of thick biological samples,” Proc. Natl Acad. Sci. U. S. A. |

5. | E. Y. S. Yew, H. Choi, D. Kim, and P. T. C. So, “Wide-field two-photon microscopy with temporal focusing and hilo background rejection,” in “ |

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. | 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 |

8. | H. Suchowski, D. Oron, and Y. Silberberg, “Generation of a dark nonlinear focus by spatio-temporal coherent control,” Opt. Commun. |

9. | M. E. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing for axial scanning,” Opt. Express |

10. | M. E. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing in nonlinear microscopy,” Opt. Commun. |

11. | O. Martinez, “3000 times grating compressor with positive group velocity dispersion: Application to fiber compensation in 1.3–1.6 μm region,” IEEE J. Quantum Electron. |

12. | R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik |

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. |

14. | C. Lutz, T. S. Otis, V. de Sars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods |

15. | P. Wang and R. Menon, “Three-dimensional lithography via digital holography,” in “ |

16. | J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. |

17. | I. Reutsky-Gefen, L. Golan, N. Farah, A. Schejter, L. Tsur, I. Brosh, and S. Shoham, “Holographic optogenetic stimulation of patterned neuronal activity for vision restoration,” Nat. Commun. |

18. | S. Yang, E. Papagiakoumou, M. Guillon, V. de Sars, C.-M. Tang, and V. Emiliani, “Three-dimensional holographic photostimulation of the dendritic arbor,” J. Neur. Eng. |

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. |

20. | E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express |

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. |

23. | D. Oron, E. Papagiakoumou, F. Anselmi, and V. Emiliani, “Two-photon optogenetics,” Prog. Brain Res. |

24. | E. Yew, C. J. R. Sheppard, and P. T. C. So, “Temporally focused wide-field two-photon microscopy: Paraxial to vectorial,” Opt. Express |

25. | J. W. Goodman, |

26. | J. W. Goodman, |

**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/vjbo/abstract.cfm?URI=oe-22-6-7087

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### References

- D. Oron, E. Tal, Y. Silberberg, “Scanningless depth-resolved microscopy,” Opt. Express 13, 1468–1476 (2005). [CrossRef] [PubMed]
- G. Zhu, J. van Howe, M. E. Durst, W. Zipfel, C. Xu, “Simultaneous spatial and temporal focusing of femtosecond pulses,” Opt. Express 13, 2153–2159 (2005). [CrossRef] [PubMed]
- E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7, 848–854 (2010). [CrossRef] [PubMed]
- A. Vaziri, J. Tang, H. Shroff, C. V. Shank, “Multilayer three-dimensional super resolution imaging of thick biological samples,” Proc. Natl Acad. Sci. U. S. A. 105, 20221–20226 (2008). [CrossRef] [PubMed]
- E. Y. S. Yew, H. Choi, D. Kim, P. T. C. So, “Wide-field two-photon microscopy with temporal focusing and hilo background rejection,” in “SPIE BiOS” (International Society for Optics and Photonics, 2011), p. 79031O.
- E. Papagiakoumou, A. Bègue, B. Leshem, O. Schwartz, B. M. Stell, J. Bradley, D. Oron, V. Emiliani, “Functional patterned multiphoton excitation deep inside scattering tissue,” Nat. Photonics 7, 274–278 (2013). [CrossRef]
- E. Block, M. Greco, D. Vitek, O. Masihzadeh, D. A. Ammar, M. Y. Kahook, N. Mandava, C. Durfee, J. Squier, “Simultaneous spatial and temporal focusing for tissue ablation,” Bio. Opt. Express 4, 831–841 (2013). [CrossRef]
- H. Suchowski, D. Oron, Y. Silberberg, “Generation of a dark nonlinear focus by spatio-temporal coherent control,” Opt. Commun. 264, 482–487 (2006). [CrossRef]
- M. E. Durst, G. Zhu, C. Xu, “Simultaneous spatial and temporal focusing for axial scanning,” Opt. Express 14, 12243–12254 (2006). [CrossRef] [PubMed]
- M. E. Durst, G. Zhu, C. Xu, “Simultaneous spatial and temporal focusing in nonlinear microscopy,” Opt. Commun. 281, 1796–1805 (2008). [CrossRef] [PubMed]
- O. Martinez, “3000 times grating compressor with positive group velocity dispersion: Application to fiber compensation in 1.3–1.6 μm region,” IEEE J. Quantum Electron. 23, 59–64 (1987). [CrossRef]
- R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
- M. Reicherter, T. Haist, E. U. Wagemann, H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999). [CrossRef]
- C. Lutz, T. S. Otis, V. de Sars, S. Charpak, D. A. DiGregorio, V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods 5, 821–827 (2008). [CrossRef]
- P. Wang, R. Menon, “Three-dimensional lithography via digital holography,” in “Frontiers in Optics” (Optical Society of America, 2012).
- J. E. Curtis, B. A. Koss, D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002). [CrossRef]
- I. Reutsky-Gefen, L. Golan, N. Farah, A. Schejter, L. Tsur, I. Brosh, S. Shoham, “Holographic optogenetic stimulation of patterned neuronal activity for vision restoration,” Nat. Commun. 4, 1509 (2013). [CrossRef] [PubMed]
- S. Yang, E. Papagiakoumou, M. Guillon, V. de Sars, C.-M. Tang, V. Emiliani, “Three-dimensional holographic photostimulation of the dendritic arbor,” J. Neur. Eng. 8, 046002 (2011). [CrossRef]
- F. Anselmi, C. Ventalon, A. Bègue, D. Ogden, 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]
- E. Papagiakoumou, V. de Sars, D. Oron, V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express 16, 22039–22047 (2008). [CrossRef] [PubMed]
- A. Bègue, E. Papagiakoumou, B. Leshem, R. Conti, L. Enke, D. Oron, V. Emiliani, “Multiphoton excitation in scattering media by holographic beams and their application in optogenetic stimulation,” Biomed. Opt. Express (to be published) (2013). [CrossRef]
- H. Dana, S. Shoham, “Remotely scanned multiphoton temporal focusing by axial grism scanning,” Opt. Lett. 37, 2913–2915 (2012). [CrossRef] [PubMed]
- D. Oron, E. Papagiakoumou, F. Anselmi, V. Emiliani, “Two-photon optogenetics,” Prog. Brain Res. 196, 119–143 (2012). [CrossRef] [PubMed]
- E. Yew, C. J. R. Sheppard, P. T. C. So, “Temporally focused wide-field two-photon microscopy: Paraxial to vectorial,” Opt. Express 21, 12951–12963 (2013). [CrossRef] [PubMed]
- J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company Publishers, 2007).
- J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005).

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