## Third-harmonic generation microscopy with Bessel beams: a numerical study |

Optics Express, Vol. 20, Issue 22, pp. 24886-24902 (2012)

http://dx.doi.org/10.1364/OE.20.024886

Acrobat PDF (4098 KB)

### Abstract

We study theoretically and numerically third-harmonic generation (THG) from model geometries (interfaces, slabs, periodic media) illuminated by Bessel beams produced by focusing an annular intensity profile. Bessel beams exhibit a phase and intensity distribution near focus different from Gaussian beams, resulting in distinct THG phase matching properties and coherent scattering directions. Excitation wave vectors are controlled by adjusting the bounding aperture angles of the Bessel beam. In addition to extended depth-of-field imaging, this opens interesting perspectives for coherent nonlinear microscopy, such as extracting sample spatial frequencies in the *λ*/8 - *λ* range in the case of organized media.

© 2012 OSA

## 1. Introduction

1. W. Mohler, A. C. Millard, and P. J. Campagnola, “Second harmonic imaging of endogenous structural proteins,” Methods **29**, 97–109 (2003). [CrossRef] [PubMed]

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7. D. Débarre, W. Supatto, and E. Beaurepaire, “Structure sensitivity in third-harmonic generation microscopy,” Opt. Lett. **30**, 2134–2136 (2005). [CrossRef] [PubMed]

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4. D. Yelin and Y. Silberberg, “Laser scanning third-harmonic generation microscopy in biology,” Opt. Express **5**, 169–175 (1999). [CrossRef] [PubMed]

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19. N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyriéras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using label-free nonlinear microscopy,” Science **339**, 967–971 (2010). [CrossRef]

14. 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]

*χ*

^{(3)}(−3

*ω; ω*,

*ω*,

*ω*) inhomogeneities, with an efficiency depending on the relative axial sizes of the inhomogeneity and of the focused Gaussian beam [7

7. D. Débarre, W. Supatto, and E. Beaurepaire, “Structure sensitivity in third-harmonic generation microscopy,” Opt. Lett. **30**, 2134–2136 (2005). [CrossRef] [PubMed]

20. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**, 1499–1501 (1987). [CrossRef] [PubMed]

23. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. **46**, 15–28 (2005). [CrossRef]

24. A. Boivin, “On the theory of diffraction by concentric arrays of ring-shaped apertures,” J. Opt. Soc. Am. **42**, 60–64 (1952). [CrossRef]

25. W. T. Welford, “Use of annular apertures to increase focal depth,” J. Opt. Soc. Am. **50**, 749–752 (1960). [CrossRef]

27. S. W. Hell, P. E. Henninen, A. Kuusisto, M. Schrader, and E. Soini, “Annular aperture two-photon excitation microscopy,” Opt. Commun. **117**, 20–24 (1995). [CrossRef]

30. T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods **8**, 417–423 (2011). [CrossRef] [PubMed]

31. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. **151**, 207–211, (2008). [CrossRef]

32. F.O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics **4**, 780–785, (2010). [CrossRef]

33. K. Shinozaki, X. Chang-Qing, H. Sasaki, and T. Kamijoh, “A comparison of optical second-harmonic generation efficiency using Bessel and Gaussian beams in bulk crystals,” Opt. Commun. **133**, 300–304 (1997). [CrossRef]

34. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A **60**, 2438–2441 (1999). [CrossRef]

35. C. F. R. Caron and R. M. Potvliege, “Optimum conical angle of a bessel–gauss beam for low-order harmonic generation in gases,” J. Opt. Soc. Am. B **16**, 1377–1384 (1999). [CrossRef]

39. S. Yang and Q. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. A **10**, 125103 (2008). [CrossRef]

6. J.-X. Cheng and X. S. Xie, “Green’s function formulation for third harmonic generation microscopy,” J. Opt. Soc. Am. B **19**, 1604–1610 (2002). [CrossRef]

11. N. Olivier and E. Beaurepaire, “Third-harmonic generation microscopy with focus-engineered beams: a numerical study,” Opt. Express **16**, 14703–14715 (2008). [CrossRef] [PubMed]

## 2. Modeling of signal generation and properties of the excitation field distribution

### 2.1. Model

11. N. Olivier and E. Beaurepaire, “Third-harmonic generation microscopy with focus-engineered beams: a numerical study,” Opt. Express **16**, 14703–14715 (2008). [CrossRef] [PubMed]

40. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanetic system,” Proc. Royal Soc. A **253**, 358–379 (1959). [CrossRef]

*χ*

^{(3)}. The resulting nonlinear field is finally propagated into the far field using Green’s functions [6

6. J.-X. Cheng and X. S. Xie, “Green’s function formulation for third harmonic generation microscopy,” J. Opt. Soc. Am. B **19**, 1604–1610 (2002). [CrossRef]

41. L. Novotny and B. Hecht, *Principles of Nano-Optics* (Cambridge Univ. Press, 2006). [CrossRef]

*θ*| <

_{THG}*π*/2) since backward THG is in most cases orders of magnitude weaker than forward THG because of larger phase mismatch [42

42. D. Débarre, N. Olivier, and E. Beaurepaire, “Signal epidetection in third-harmonic generation microscopy of turbid media,” Opt. Express **15**, 8913–8924 (2007). [CrossRef] [PubMed]

### 2.2. Non-paraxial focusing of Bessel beams

*i.e.*the intensity does not depend on z) and is characterized by a rapid on-axis phase variation.

*x*-polarized Gaussian beam through an annular aperture at the back aperture of a high NA objective, and we neglect diffraction at the aperture edges. The phase and amplitude at the back aperture are supposed to be constant over the annulus. Such beams can be described using two parameters:

*NA*and Δ

_{B}*NA*with

*NA*=

_{B}*n.sin*(

*θ*) and Δ

_{B}*NA*≈ 2

*n.cos*(

*θ*).Δ

_{B}*θ*, where

*θ*depends on the radius of the annular aperture, and Δ

_{B}*NA*represents the width of this aperture. The focal field

*E*can be expressed as a function of

_{f}*θ*and Δ

_{B}*θ*as: with:

*J*

_{0},

*J*

_{1}and

*J*

_{2}are the zero, first and second order Bessel functions.

*NA*is obtained when setting

*NA*= 1), and three different Bessel beams (

*NA*= 0.8 ± 0.2,

_{B}*NA*= 0.8 ± 0.05,

_{B}*NA*= 0.95 ± 0.05), assuming

_{B}*n*= 1.33 and

*λ*= 1.2

*μm*. The two-dimensional intensity distributions are shown in Fig. 2A.

*NA*, three phenomena happen as Δ

_{B}*NA*decreases :

- The axial extension increases as 1/Δ
*NA*. - The lateral extension of the central intensity peak decreases.
- Secondary rings become conversely more important so that the lateral extent of the beam, defined as the root mean square width of its intensity distribution in the lateral plane, remains approximately constant.

27. S. W. Hell, P. E. Henninen, A. Kuusisto, M. Schrader, and E. Soini, “Annular aperture two-photon excitation microscopy,” Opt. Commun. **117**, 20–24 (1995). [CrossRef]

28. E. J. Botcherby, R. Juškaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. **268**, 253–260 (2006). [CrossRef]

*NA*= 0.95 ± 0.05), for which the axial extension is increased by a factor ≈ 10 for a similar maximum NA. The secondary rings are generally an issue in linear microscopy because they generate considerable background. However their influence on THG contrast is expected to be more limited due to the nonlinear dependence of the signal generation on the excitation intensity. In addition, since the lateral extension of the third-order point spread function (PSF) is mostly determined by the principal excitation focus, its reduced lateral extension is a side benefit corresponding to a gain in the lateral resolution of the image. This improvement comes at the cost of energy being redistributed in the secondary rings, implying that higher excitation power may be required experimentally.

_{B}*z*and in

*x*as a function of

*NA*and Δ

_{B}*NA*. Whereas the lateral resolution is principally determined by

*NA*, the axial extension strongly depends on Δ

_{B}*NA*, as can be seen on the two graphs showing the lateral and axial FWHM as a function of Δ

*NA*for

*NA*= 0.95. For a given lateral resolution, it is therefore possible to increase the axial extension by decreasing Δ

_{B}*NA*.

43. Q. Zhan “Second-order tilted wave interpretation of the Gouy phase shift under high numerical aperture uniform illumination,” Optics Commun. **242**, 351–360 (2004). [CrossRef]

*NA*is increased, and increases weakly as Δ

_{B}*NA*is decreased. This phenomenon results in different phase-matching properties for Bessel beams as compared to Gaussian beams.

### 2.3. Phase-matching mechanisms

36. S. P. Tewari, H. Huang, and R. W. Boyd, “Theory of third-harmonic generation using bessel beams, and self-phase-matching,” Phys. Rev. A **54**, 2314–2325 (1996). [CrossRef] [PubMed]

**k**

*(resp.*

_{ω}**k**

_{3}

*) is the wave vector at the fundamental (resp. harmonic) frequency,*

_{ω}*i.e.*|

**k**

*| = 2*

_{ω}*πn*/

_{ω}*λ*. The phase matching condition is thus satisfied in the case of a non-dispersive medium for which

*n*=

_{ω}*n*

_{3ω}along a THG scattering cone with angle

*θ*=

_{THG}*θ*. As a result, THG is obtained from an homogeneous medium. This is in contrast with the case of Gaussian excitation, which in the case of a homogeneous medium and a Gaussian-Gaussian interaction yields: where

_{B}**k**

*(resp.*

_{g,}_{ω}**k**

_{g,}_{3}

*) is a vector oriented along the z axis opposite to the beam propagation, which describes the Gouy phase shift around the focus of the excitation (resp. scattered) beam plotted on Fig. 2D [6*

_{ω}6. J.-X. Cheng and X. S. Xie, “Green’s function formulation for third harmonic generation microscopy,” J. Opt. Soc. Am. B **19**, 1604–1610 (2002). [CrossRef]

*b*, |

**k**

*| ≈*

_{g,}_{ω}*π*/4

*b*. Because of this additional wave vector, phase matching is only achieved in media with negative dispersion, and no THG is obtained in the case of non dispersive or positively dispersive media. Instead, when considering samples with infinite transverse (

*x*−

*y*) extension, THG is obtained from interfaces and objects whose longitudinal size is of the order of the coherent construction length (or effective coherence length)

*l*defined as:

_{fw}*NA*≠ 0, a similar behaviour is obtained. In this situation, THG can be described as a non-perfect Bessel - Bessel interaction and the phase matching condition follows Eq. (4), with (see derivation in Appendix): where Δ

*NA*is the THG angular spread. From these equations, it can be seen that no bulk emission is obtained in the case of positive dispersion, or of zero dispersion unless

_{THG}*k*=

_{g,}_{ω}*k*

_{g,}_{3}

*= 0, which corresponds to the perfect Bessel - Bessel interaction for which Δ*

_{ω}*NA*= Δ

*NA*= 0. Instead, coherent emission is obtained from structures with longitudinal size of

_{THG}*l*or less.

_{fw}*k*

_{3ω}−

*k*) <<

_{ω}*k*), emission is dominated by the phase mismatch due to tripling of the excitation Gouy phase shift. The longitudinal effective coherence length is then determined by the numerical aperture and angular spread of the excitation beam. Conversely in the case of large positive dispersion or Bessel beams with very long axial extension (i.e. (

_{g}*k*

_{3ω}−

*k*) >>

_{ω}*k*), emission is dominated by the material index dispersion between the fundamental and harmonic frequencies (

_{g}*n*

_{3ω}−

*n*), and depends little on the excitation beam parameters (numerical aperture, angular dispersion).

_{ω}36. S. P. Tewari, H. Huang, and R. W. Boyd, “Theory of third-harmonic generation using bessel beams, and self-phase-matching,” Phys. Rev. A **54**, 2314–2325 (1996). [CrossRef] [PubMed]

*k*

_{3ω}sin(

*θ*) > 3

_{THG}*k*sin(

_{ω}*θ*), so that emission can only be obtained for sin(

_{B}*θ*)

_{THG}*=*sin(

*θ*)

_{B}*n*/

_{ω}*n*

_{3ω}, thereby preventing longitudinal phase matching for

*n*

_{3ω}>

*n*. This function also reaches a maximum for

_{ω}*k*

_{3ω}sin(

*θ*) =

_{THG}*k*sin(

_{ω}*θ*), which corresponds to sin(

_{B}*θ*) = sin(

_{THG}*θ*)

_{B}*n*/(3

_{ω}*n*

_{3}

*). This situation has been termed self-phase matching [36*

_{ω}36. S. P. Tewari, H. Huang, and R. W. Boyd, “Theory of third-harmonic generation using bessel beams, and self-phase-matching,” Phys. Rev. A **54**, 2314–2325 (1996). [CrossRef] [PubMed]

*θ*does not however simultaneously satisfy the longitudinal phase matching condition for non dispersive or positively dispersive media. Coherent construction of the signal for the self-phase matching angle is thus limited to a finite length, so that the magnitude of the signal created at this angle oscillates as a function of sample thickness.

_{THG}*θ*≈

_{THG}*θ*, which satisfies the longitudinal phase-matching condition; and

_{B}*θ*≈

_{THG}*θ*/3, which satisfies the lateral self-phase matching condition.

_{B}## 3. Results

### 3.1. THG from slabs

*e*) with an infinite extension perpendicularly to the beam propagation direction (

*z*). This geometry has been extensively studied in the case of Gaussian beams. The dependence of THG on slab width

*e*gives straightforward information about the phase-matching efficiency along the z-axis [6

**19**, 1604–1610 (2002). [CrossRef]

7. D. Débarre, W. Supatto, and E. Beaurepaire, “Structure sensitivity in third-harmonic generation microscopy,” Opt. Lett. **30**, 2134–2136 (2005). [CrossRef] [PubMed]

**16**, 14703–14715 (2008). [CrossRef] [PubMed]

*i.e.*we assume that

*n*=

_{ω}*n*

_{3ω}=

*n*. We obtain therefore

*k*

_{3ω}= 3

*k*, and the effective coherence length along the optical axis for a Gaussian-Gaussian interaction (or along

_{ω}*θ*=

_{THG}*θ*for an imperfect Bessel-Bessel interaction) can be expressed as:

_{B}*e*increases from zero, we first expect a coherent increase of the signal (THG scales as the number of emitters squared,

*i.e.*as

*e*

^{2}for a slab sample) until the effective coherence length is reached, and then a decrease as a function of

*e*because of destructive interferences until the THG signal reaches zero for a homogeneous sample, as illustrated in Fig. 3A (black curve).

*NA*decreases. In addition, THG exhibits oscillations as a function of the slab thickness, which correspond to the oscillations of the self-phase matched and on-axis emission with periods ≈

*k*(cos(

*θ*/3) − cos(

_{B}*θ*)) and ≈

_{B}*k*(1 − cos(

*θ*)), respectively. The oscillations of the on-axis scattered component are illustrated on Fig. 3B. We normalized the simulations by considering a constant power at the back aperture of the objective. While the expected THG signal is significantly decreased (by a factor ≈15 for

_{B}*NA*= 0.95 ± 0.05) when using Bessel beams, due to energy being redistributed in the secondary rings, it is by no means negligible.

_{B}*NA*= 0.95 ± 0.05) as a function of slab width

_{B}*e*and scattering angle. Three regimes are highlighted: on-axis scattering (red), intermediate-angle scattering (orange) and off-axis scattering (green) corresponding to three detection NA ranges, namely 0–0.2, 0.2–0.4 and 0.8–1. The integrated signal over these detection ranges is shown in Fig. 4B, along with the signal detected over the whole NA range (0–1.33). These different ranges can be linked to the three phase matching situations as discussed in the previous section: indeed in the absence of dispersion, THG scattering mostly occurs along the longitudinal (axial phase matching,

*θ*=

_{THG}*θ*, off-axis scattering, green) and lateral (self-phase matching,

_{B}*θ*≈

_{THG}*θ*/3, intermediate-angle scattering, orange) phase matching directions. Along the self-phase matching direction, we observe as expected dampened oscillations with a larger amplitude and period than along the on-axis direction. In contrast, at large angles we obtain emission from large

_{B}*e*, indicating that the effective coherence length is larger than the slab widths investigated here. The diminishing total THG signal observed in Fig. 4A (black) on the considered range of slab thicknesses is therefore mostly due to the poorly phase-matched directions (red, orange). The signal created around

*θ*=

_{THG}*θ*(green curve) is however also expected to decrease as the slab width is further increased, on a length scale given by the imperfect Bessel-Bessel coherence length (see Eqs. (5)–(7)), consistently with previous predictions [36

_{B}**54**, 2314–2325 (1996). [CrossRef] [PubMed]

### 3.2. Influence of dispersion

*i.e.*we assume that

*n*≠

_{ω}*n*

_{3ω}. Indeed, while we focus mostly on non-dispersive media in this study to simplify the discussion of phase-matching mechanisms, most biological samples are positively dispersive (typically Δ

*n*≈ 0.02 − 0.03) and dispersion cannot be neglected in practice. In the case of THG with a tightly focused Gaussian beam, phase-matching is dominated by the Gouy phase mismatch, meaning that dispersion has only a minor effect. However, we have seen that in the case of Bessel-Bessel coupling, the geometrical phase matching conditions are more favorable and the interaction length is larger, so the influence of dispersion may become significant when the medium coherence length defined as

*l*=

_{c}*π*/Δ

*n*is shorter than the dispersion-free imperfect Bessel-Bessel interaction length.

*n*

_{3ω}−

*n*for a Gaussian beam (Fig. 5A,

_{ω}*NA*= 1.2) and a Bessel beam (Fig. 5B,

*NA*= 0.95 ± 0.05). As expected, THG is obtained in the case of a homogeneous negatively dispersive medium, both for the Gaussian and for the Bessel excitation, but this bulk emission disappears for zero or positively dispersive media. Nevertheless, the influence of dispersion on the curves of THG as a function of the slab thickness differs significantly for the two excitation geometries. In the case of the Gaussian excitation, apart from the bulk emission efficiency, the THG dependence on the sample size does not change much with dispersion. In particular, the slab thickness for which the maximum THG emission is obtained is little affected. On the contrary, the location of the maximum of the signal curve obtained for a Bessel excitation depends strongly on Δ

_{B}*n*. This is a straightforward consequence of the change in the non-dispersive effective coherence length of the beams compared to the dispersion coherence length

*l*=

_{c}*π*/Δ

*n*. In particular, increasing the positive dispersion of the nonlinear medium reduces the slab width for which the maximum THG signal is obtained in the case of a Bessel excitation.

*θ*for a slab perpendicular to the propagation axis, and dispersion results in reduced phase-matching for waves scattered at this angle. This component therefore exhibits a bell-shaped variation as

_{B}*e*increases, corresponding to the first period of an oscillating behavior with period

*l*=

_{c}*π*/Δ

*n*. Intermediate-angle scattering retains its oscillating behavior in the presence of dispersion, although the frequency and amplitude of the oscillations are shifted.

### 3.3. Quasi phase-matching Bessel beams

44. A. Piskarskas, V. Smilgevičius, A. Stabinis, V. Jarutis, V. Pašiškevičius, S. Wang, J. Tellefsen, and F. Laurell, “Noncollinear second-harmonic generation in periodically poled KTiOPO(4) excited by the bessel beam,” Opt. Lett. **24**, 1053–1055 (1999). [CrossRef]

*χ*

^{(3)}oscillates along

*z*with a period

*e*. For simplicity, we neglect the changes in linear indices. Figure 6 illustrates the calculated THG signal as a function of the spatial period for the same Bessel and Gaussian beams considered in Fig. 3. We point out several phenomena:

- In each case, there is a resonant spatial frequency providing QPM, which is narrower for Bessel beams than for Gaussian beams, and gets narrower as the Bessel NA spread (Δ
*NA*) gets smaller. - The enhancement of the total integrated signal for a given axial period (Fig. 6B) mostly stems from the resonance at the self-phase matching angle (
*θ*≈_{THG}*θ*/3), for which (quasi)-phase matching is then achieved both in the longitudinal and axial directions._{B} - Higher NA yields a smaller spatial resonance period, which is consistent with a smaller effective coherence length along the optical axis for the self-phase matching angle.
- In the case of Bessel beams, the signal for large axial periods goes to zero only for large sample thicknesses, consistent with the calculations shown in Fig. 4A.
- Since QPM can be achieved at the self-phase matching angle for which the secondary emission ring contribute efficiently to THG emission, the increased QPM efficiency yields a signal at resonance comparable to the Gaussian case.

### 3.4. Probing organized media using Bessel beams

*NA*=

_{d,Max}*n sin*(

*θ*) = 0.2 to retain only the near-axis propagating THG (which would be simple to implement experimentally by using a low NA collection lens or adding a diaphragm after the collection optics), and we investigate the influence of excitation NA.

_{d,Max}*NA*. As

*NA*increases, the on-axis coherence length decreases (see section 3.1) so the spatial resonance period also decreases. Therefore, a very interesting property of Bessel beams is that they provide a way to probe the axial organization of a sample, since the spatial resonance period can be tuned from ≈ 500

_{B}*nm*to ≈ 3

*μm*by changing the excitation NA. We note that the resonance broadening observed for smaller

*NA*s comes from the fact that we considered a constant Δ

*NA*, and that it could be avoided by setting Δ

*NA*∝

*NA*.

_{B}*NA*is increased in order to also detect the self-phase matched contribution, the signal is increased by a factor ≈ 4, and the resonance range is slightly shifted and broadened. This is illustrated by the the dashed curves in Fig. 7B and by the difference between the two theoretical resonant spatial periods in Fig. 7E, shown as black and blue curves). Finally, the use of higher detection

_{d,Max}*NA*would result in the detection of additional off-axis components and in a more complex signal (as in Fig. 6).

*NA*considered.

*NA*, the Bessel beam turns into a plane wave for which the coherence length is

_{B}*l*=

_{c}*λ*/12

*n*, and hence

*p*=

*λ*/6

*n*. In the other limiting case of

*NA*≈

_{B}*n*, the axial component of the excitation wave vector becomes negligible and we have:

*l*=

_{c}*λ*/6

*n*, and hence

*p*=

*λ*/3

*n*.

*p*=

*λ*/6

*n*(low NA Bessel, backward scattering) to almost infinitely large periods. What limits this range is the geometrical focusing requirement: even at very high

*NA*(

*e.g.*1.27 in water), the angle is still limited to 72.7 degrees which means that cos(

*arcsin*(

*NA*/

_{B}*n*)) can only vary from ≈ 0.3 to 1, excluding from the attainable range the region [

*λ*/3

*n*(1+0.3)

*λ*/3

*n*(1−0.3)] ≈ [

*λ*/4

*n*

*λ*/2

*n*]. One possible workaround for this limit could be to use two different excitation wavelengths with non-overlapping exclusion ranges. While measuring large periods using QPM seems of limited interest, the possibility to measure periodicity in the range [

*λ*/6

*n*

*λ*/4

*n*] gives access to sub-wavelength information in ordered media.

### 3.5. Comparison of the THG imaging properties of focused Gaussian and Bessel beams

*NA*= 0.7 and a Bessel beam defined by

*NA*= 1.15 ± 0.05. These two beams have similar axial extensions and therefore have fairly similar properties for incoherent imaging, and this axial extension is small enough that it may be used for microscopy imaging.

_{B}*μm*, corresponding to the effective coherence length in the forward direction, while the THG produced by a focused Bessel beam exhibits a much slower decay because of the favorable off-axis phase-matching conditions. Figure 9C illustrates the average emission angle obtained from the previous geometry, and shows that THG scattering from a Gaussian beam resembles a Gaussian beam while THG from a Bessel beam is close to a Bessel beam. The last panel (Fig. 9D) shows the axial (

*NA*= 0.2) forward-QPM of the two beams with a period of twice the coherence length in both cases, and a much sharper spatial resonance when using Bessel beams due to the larger interaction distance.

_{d,Max}## 4. Conclusion and outlook

45. V. V. Krishnamachari and E. O. Potma, “Multi-dimensional differential imaging with fe-cars microscopy,” Vib. Spectrosc. **50**, 10–14 (2009). [CrossRef]

- We considered optimal beam shaping by normalizing the power at the back focal plane of the objective. Even if there are efficient strategies to produce annular illumination before the focusing objective (
*e.g.*[28]), some excitation energy will be lost in the shaping process, which is an important issue for a third-order process. Moreover, except in the case of QPM, the energy redistribution in the secondary rings implies a reduced nonlinear signal.28. E. J. Botcherby, R. Juškaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun.

**268**, 253–260 (2006). [CrossRef] - We only considered homogeneous media and neglected differences in optical indices. This model obviously needs to be refined in the case of extended depth-of-field imaging of heterogeneous media.

*χ*

^{(3)}. For example, since lipid droplets are one important source of contrast in biological tissues [18

18. D. Débarre, W. Supatto, A.-M. Pena, A. Fabre, T. Tordjmann, L. Combettes, M.-C. Schanne-Klein, and E. Beaurepaire, “Imaging lipid bodies in cells and tissues using third-harmonic generation microscopy,” Nat. Methods **3**, 47–53 (2006). [CrossRef]

46. D. Débarre and E. Beaurepaire, “Quantitative characterization of biological liquids for third-harmonic generation microscopy,” Biophys. J. **92**, 603–612 (2007). [CrossRef]

*χ*

^{(3)}more easily than with interference techniques [46

46. D. Débarre and E. Beaurepaire, “Quantitative characterization of biological liquids for third-harmonic generation microscopy,” Biophys. J. **92**, 603–612 (2007). [CrossRef]

13. V. V. Krishnamachari and E. O. Potma, “Detecting lateral interfaces with focus-engineered coherent anti-stokes raman scattering microscopy,” J. Raman Spectr. **39**, 593–598 (2008). [CrossRef]

45. V. V. Krishnamachari and E. O. Potma, “Multi-dimensional differential imaging with fe-cars microscopy,” Vib. Spectrosc. **50**, 10–14 (2009). [CrossRef]

47. S. Carrasco, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B **23**, 2134–2141 (2006). [CrossRef]

50. F. Lu, W. Zheng, and Z. Huang, “Coherent anti-Stokes Raman scattering microscopy using tightly focused radially polarized light,” Opt. Lett. **34**, 1870–1872 (2009). [CrossRef] [PubMed]

## Appendix A: derivation of the phase shift wavevector dependence on *NA*_{B} and Δ*NA*

_{B}

**k**

*with*

_{g}*NA*and Δ

_{B}*NA*, we calculate the electric field distribution on axis (

*ρ*= 0) around the focal point (

*z*<< 1/|

**k**

*|). Because we are only interested in a scaling law, we use a number of approximations that are detailed below. We have:*

_{g}*θ*≪

*θ*and neglect the variations of the amplitude terms in this integral in comparison with phase variations, to obtain:

_{B}*θ*)≈ cos(

*θ*) − (

_{B}*θ*−

*θ*) sin(

_{B}*θ*), and thus:

_{B}*sinc*term that governs the intensity distribution corresponds to an axial width that varies as ∝ 1/(sin(

*θ*)Δ

_{B}*θ*) ∝ 1/(

*NA*× Δ

_{B}*NA*). The phase term should be evaluated after subtraction of the propagation phase term, which scales as ≈

*e*

^{ikz(cos(θB−Δθ))}. It is indeed easily seen that this phase term correctly becomes ≈

*e*for

^{ikz}*θ*= Δ

_{B}*θ*, which corresponds to the Gaussian case. The remaining on-axis phase therefore becomes : and hence :

## Acknowledgments

## References and links

1. | W. Mohler, A. C. Millard, and P. J. Campagnola, “Second harmonic imaging of endogenous structural proteins,” Methods |

2. | J.-X. Cheng and X. S. Xie, “Coherent anti-Stokes Raman scattering microscopy: instrumentation, theory, and applications,” J. Phys. Chem. B |

3. | A. Volkmer, “Vibrational imaging and microspectroscopies based on coherent anti-Stokes Raman scattering microscopy,” J. Phys. D: Appl. Phys. |

4. | D. Yelin and Y. Silberberg, “Laser scanning third-harmonic generation microscopy in biology,” Opt. Express |

5. | J. Mertz and L. Moreaux, “Second-harmonic generation by focused excitation of inhomogeneously distributed scatterers,” Opt. Commun. |

6. | J.-X. Cheng and X. S. Xie, “Green’s function formulation for third harmonic generation microscopy,” J. Opt. Soc. Am. B |

7. | D. Débarre, W. Supatto, and E. Beaurepaire, “Structure sensitivity in third-harmonic generation microscopy,” Opt. Lett. |

8. | C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. |

9. | M. R. Foreman, S. S. Sherif, P. R. T. Munro, and P. Török, “Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express |

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

11. | N. Olivier and E. Beaurepaire, “Third-harmonic generation microscopy with focus-engineered beams: a numerical study,” Opt. Express |

12. | K. Yoshiki, K. Ryosuke, M. Hashimoto, T. Araki, and N. Hashimoto, “Second-harmonic-generation microscope using eight-segment polarization-mode converter to observe three-dimensional molecular orientation,” Opt. Lett. |

13. | V. V. Krishnamachari and E. O. Potma, “Detecting lateral interfaces with focus-engineered coherent anti-stokes raman scattering microscopy,” J. Raman Spectr. |

14. | Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. |

15. | M. Müller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D-microscopy of transparent objects using third-harmonic generation,” J. Microsc. |

16. | D. Oron, D. Yelin, E. Tal, S. Raz, R. Fachima, and Y. Silberberg, “Depth-resolved structural imaging by third-harmonic generation microscopy,” J. Struct. Biol. |

17. | C. K. Sun, S.-W. Chu, S.-Y. Chen, T.-H. Tsai, T.-M. Liu, C.-Y. Lin, and H.-J. Tsai, “Higher harmonic generation microscopy for developmental biology,” J. Struct. Biol. |

18. | D. Débarre, W. Supatto, A.-M. Pena, A. Fabre, T. Tordjmann, L. Combettes, M.-C. Schanne-Klein, and E. Beaurepaire, “Imaging lipid bodies in cells and tissues using third-harmonic generation microscopy,” Nat. Methods |

19. | N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyriéras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using label-free nonlinear microscopy,” Science |

20. | J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

21. | J. Durnin, “Exact solutions for nondiffracting beams. i. the scalar theory,” J. Opt. Soc. Am. A |

22. | J. E. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. |

23. | D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. |

24. | A. Boivin, “On the theory of diffraction by concentric arrays of ring-shaped apertures,” J. Opt. Soc. Am. |

25. | W. T. Welford, “Use of annular apertures to increase focal depth,” J. Opt. Soc. Am. |

26. | C. J. R. Sheppard, “The use of lenses with annular aperture in scanning optical microscopy,” Optik |

27. | S. W. Hell, P. E. Henninen, A. Kuusisto, M. Schrader, and E. Soini, “Annular aperture two-photon excitation microscopy,” Opt. Commun. |

28. | E. J. Botcherby, R. Juškaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. |

29. | N. Olivier, A. Mermillod-Blondin, C. B. Arnold, and E. Beaurepaire, “Two-photon microscopy with simultaneous standard and extended depth of field using a tunable acoustic gradient-index lens,” Opt. Lett. |

30. | T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods |

31. | Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. |

32. | F.O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics |

33. | K. Shinozaki, X. Chang-Qing, H. Sasaki, and T. Kamijoh, “A comparison of optical second-harmonic generation efficiency using Bessel and Gaussian beams in bulk crystals,” Opt. Commun. |

34. | J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A |

35. | C. F. R. Caron and R. M. Potvliege, “Optimum conical angle of a bessel–gauss beam for low-order harmonic generation in gases,” J. Opt. Soc. Am. B |

36. | S. P. Tewari, H. Huang, and R. W. Boyd, “Theory of third-harmonic generation using bessel beams, and self-phase-matching,” Phys. Rev. A |

37. | V. E. Peet and R. V. Tsubin, “Third-harmonic generation and multiphoton ionization in bessel beams,” Phys. Rev. A |

38. | V. E. Peet and S. V. Shchemeljov, “Spectral and spatial characteristics of third-harmonic generation in conical light beams,” Phys. Rev. A |

39. | S. Yang and Q. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. A |

40. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanetic system,” Proc. Royal Soc. A |

41. | L. Novotny and B. Hecht, |

42. | D. Débarre, N. Olivier, and E. Beaurepaire, “Signal epidetection in third-harmonic generation microscopy of turbid media,” Opt. Express |

43. | Q. Zhan “Second-order tilted wave interpretation of the Gouy phase shift under high numerical aperture uniform illumination,” Optics Commun. |

44. | A. Piskarskas, V. Smilgevičius, A. Stabinis, V. Jarutis, V. Pašiškevičius, S. Wang, J. Tellefsen, and F. Laurell, “Noncollinear second-harmonic generation in periodically poled KTiOPO(4) excited by the bessel beam,” Opt. Lett. |

45. | V. V. Krishnamachari and E. O. Potma, “Multi-dimensional differential imaging with fe-cars microscopy,” Vib. Spectrosc. |

46. | D. Débarre and E. Beaurepaire, “Quantitative characterization of biological liquids for third-harmonic generation microscopy,” Biophys. J. |

47. | S. Carrasco, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B |

48. | N. Olivier, F. Aptel, K. Plamann, M-C. Schanne-Klein, and E. Beaurepaire, “Harmonic microscopy of isotropic and anisotropic microstructure of the human cornea,” Opt. Express |

49. | O. Masihzadeh, P. Schlup, and R. A. Bartels, “Enhanced spatial resolution in third-harmonic microscopy through polarization switching,” Opt. Lett. |

50. | F. Lu, W. Zheng, and Z. Huang, “Coherent anti-Stokes Raman scattering microscopy using tightly focused radially polarized light,” Opt. Lett. |

**OCIS Codes**

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(190.4160) Nonlinear optics : Multiharmonic generation

(180.4315) Microscopy : Nonlinear microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: August 3, 2012

Revised Manuscript: October 3, 2012

Manuscript Accepted: October 3, 2012

Published: October 16, 2012

**Citation**

Nicolas Olivier, Delphine Débarre, Pierre Mahou, and Emmanuel Beaurepaire, "Third-harmonic generation microscopy with Bessel beams: a numerical study," Opt. Express **20**, 24886-24902 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-24886

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### References

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- A. Piskarskas, V. Smilgevičius, A. Stabinis, V. Jarutis, V. Pašiškevičius, S. Wang, J. Tellefsen, and F. Laurell, “Noncollinear second-harmonic generation in periodically poled KTiOPO(4) excited by the bessel beam,” Opt. Lett.24, 1053–1055 (1999). [CrossRef]
- V. V. Krishnamachari and E. O. Potma, “Multi-dimensional differential imaging with fe-cars microscopy,” Vib. Spectrosc.50, 10–14 (2009). [CrossRef]
- D. Débarre and E. Beaurepaire, “Quantitative characterization of biological liquids for third-harmonic generation microscopy,” Biophys. J.92, 603–612 (2007). [CrossRef]
- S. Carrasco, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B23, 2134–2141 (2006). [CrossRef]
- N. Olivier, F. Aptel, K. Plamann, M-C. Schanne-Klein, and E. Beaurepaire, “Harmonic microscopy of isotropic and anisotropic microstructure of the human cornea,” Opt. Express5, 5028–5040 (2010). [CrossRef]
- 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]
- F. Lu, W. Zheng, and Z. Huang, “Coherent anti-Stokes Raman scattering microscopy using tightly focused radially polarized light,” Opt. Lett.34, 1870–1872 (2009). [CrossRef] [PubMed]

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