## Abstract

A theoretical and experimental study of the THG signal from a reference interface in confocal microscope allows precise analysis of beam propagation and optimization of the focusing objectives.

©2004 Optical Society of America

## 1. Introduction

Multiphoton laser scanning microscopy has become an important tool for the investigation of biological phenomena where high resolution, three-dimensional imaging is essential for understanding the underlying biological function [1–3] Two-photon laser scanning microscopy produces high-resolution images of thin optical sections that can provide complete three-dimensional information.

The success of the two-photon absorption fluorescence has generated substantial interest in the scientific imaging community in further developing and exploiting optical nonlinearities for new and enhanced image contrast. Some of these techniques such as second harmonic generation (SHG) [4,5] and coherent anti-Stokes Raman scattering microscopy (CARS) [6,7] have been around for some years, but have recently benefited from the development of compact, reliable ultrashort laser sources and thus are being looked at with renewed interest. Other techniques such as third-harmonic generation imaging have been recently proposed [8, 9] and exploited as a non resonant alternative, further case for the laser source selection [10]. Unfortunately the pulse characteristics in such extreme conditions (ultrashort and strongly focused) are deeply affected by all the optical components between the laser source and the sample under test [11].

A crucial aspect of multiphoton microscopy is the characterization of the spatial and temporal characteristics of the laser field at the focus of the microscope objective. Detailed knowledge of the field ensures optimum resolution performed at the highest efficiency. In addition, quantitative intensity measurements are necessary for accurately gauging the exposure limits that a living specimen can withstand. Among the proposed methods to fully characterize the focusing capabilities of a given setup, either spatial resolution (trough auto convolution under single and two photons illumination [12]) or temporal characterization (using in focus pulse autocorrelations [13]) have been proposed. Neither approaches offer a complete view, or even a unique test bench for the imaging process. In this paper we present *in situ* method by which the spatio temporal characteristics of the pulse can be measured in each point of field of view. Most importantly, it can be performed at the full numerical aperture NA of the objective, which is the conditions under which the system will be used for imaging. With minor changes in the apparatus to collect the forward beam, our method based on third harmonic generation can be applied to characterize non linear microscopes setup. Alternative techniques mentioned, although slightly more efficient in each field (spatial or temporal) than our global approach, they failed in the laser field characterization at the sample location.

## 2. Theory

In this part we assume a Gaussian behavior for the laser field. Obviously, for objectives with high NA (>0.7), full diffraction theory is more adequate than Gaussian model. According to Squier and Müller [10], the Gaussian model gives results close to reality with even smaller values for the FWHM of the spot size. Full diffraction theory is beyond the scope of this paper. Nevertheless, for high NA, this approach is still valid qualitatively to qualify a whole multi-photon microscopy setup.

We consider a laser beam propagating along the z direction focused within a medium. The field of the fundamental beam E_{ω} is defined by

where P_{ω} is the fundamental beam average power, T=1/F where F is the repetition rate of the laser source, c is the light celerity and τ_{p} is the pulse duration of the fundamental wave.

Then the third-harmonic field E_{3ω} is given by [14]:

$$\mathrm{with}\phantom{\rule{.2em}{0ex}}A=-i\frac{{\chi}^{3}}{8\lambda \sqrt{3}}$$

where χ^{3} is the third-order susceptibility of the material, w is the beam waist, *λ* is the fundamental wavelength, J is the effective interaction length and is roughly estimated in Gaussian beam approximation below:

where Δk=k_{3ω}-3k_{ω} is the phase mismatch, b is the confocal parameter of the fundamental beam and z_{0} and z_{1} are the input and output coordinates of the sample in the area of the focus plane of the objective, respectively (Fig. 1).

Other relations for J can be calculated and depend on the accuracy of the model. All the models are derived from propagation of coupled wave equations. In appendix, we describe a more realistic relation for the J parameter.

From Eq. (1) and Eq. (2) we can respectively have the shape of the fundamental beam and the third-harmonic beam intensities:

and

Replacing I_{ω} by its expression in Eq. (4) we have finally

K is a parameter depending on sample properties and experimental conditions.

The plot of the normalized third-harmonic intensity is displayed on Fig. 2. The third-harmonic signal is localized around the interface between two media. The curve represents the usual third harmonic signal as the beam waist is shifted across the interface. As the beam spot explores a sample plane at a z_{0} depth with an optical scanner, a three dimensional data must be recorded for a full description of femtosecond laser properties on the focused plane.

The curve in Fig. 2 is defined using only three parameters. These can be seen as a complete characterization of the microscope objective: The lag of the maximum signal with respect to its reference position, the FWHM and the peak amplitude.

#### 2.1 Intensity Lag with respect to its original position

From the THG theory [15], the maximum signal is recorded for focalization of the beam at an interface between two media, in our experiment this condition is reached when the optical flat silica coverslip have its position z_{0} placed at the beam waist of the fundamental laser beam (z_{0} = 0). As we record the intensity profile for each beam waist position (x, y), we can analyze the main quality of the microscope objective. A lag evolution across the scanned region can be directly understood as deviation of the objective from the flat field conditions and directly linked to the spherical aberration.

#### 2.2 FWHM

From Eq. (6) one can identify the width of the third-harmonic intensity around z_{0} position. The origin of this signal width (FWHM) is simply driven by the confocal parameter b (i.e. the Rayleigh range). This parameter is unfortunately not an analytical function of b. A numerical computation (Gaussian model) gives indeed a linear relationship versus b:

As the confocal parameter b is given by b= 2π ${{\mathrm{w}}_{0}}^{2}$/ λ :

The experimental measure of Δz at each spot position (x, y) gives information on the confocal parameter and thus the local beam waist w_{L}. While scanning the whole focal plane, the objective performances can be retrieved. Full diffraction theory calculations [16] predict even smaller values for the FWHM than with Gaussian model. However here, a measure at each (x,y) point of the field of view is proposed.

#### 2.3 Amplitude

If the beam waist is located at the coverslip interface (z_{0}) the signal magnitude is also driven by the confocal parameter b through the effective interaction length for a given peak power. The numerical computation for the J function at maximum is

This numerical approximation is derived from the appendix 6 and is verified for a Gaussian model. From Eq.(5) one can normalize value for the maximum signal, the experimental signal is multiplied by local values of ${{\mathrm{w}}_{\mathrm{L}}}^{4}$ and J_{L}:

To enhance the eventual (small) distortion, we can further use a logarithmic scale of the local intensity to the intensity at centre.

We expand (P_{ω} / τ_{p}) by this following relation:

One finally obtains:

This expression represents the variations of the harmonic laser beam across the scanned region as a function of the ratio P_{ω} to τ_{p}. However, in our experiment, the pulse duration in the IR is sufficiently large to neglect the pulse broadening in the objective in such a way that Eq. (11) reflects only the local fundamental beam power.

## 3. Experiment

Our experiment is developed around a Zeiss Axiovert 200M microscope, modified for multi-photon microscopy. IR laser pulse is injected into the microscope and, in this particular experiment, the transmitted THG signal needed for setup optimization is efficiently collected. The experimental setup is represented in Fig.3. Two laser sources have been evaluated: first a synchronously pumped OPO (Spectra-Physics Tsunami - Opal System) which delivers 150 fs pulses around 1.5 μm, at a repetition rate of 80 MHz and an average power of 350mW and a T-pulse laser (Amplitude Systemes- France), extremely compact, which delivers 200 fs pulses around 1.03 μm, at a repetition rate of 50 Mhz and an average power of 1.1 W. These two sources have sufficient peak power for non linear excitation. The THG signal is strong enough to be easily collected by a PMT. Both lasers present TEM00 mode and M^{2} less than 1.2. Each laser beam diameter is adapted by a spatial filter before the entrance to microscope.

Two scanning mirrors stir the focal point of the laser in x-y plane of the sample. The microscope objectives are motorized along the z axis. The THG light at the wavelength at 0.5 μm (or 343nm) is collected through a specific condenser (NA=0.55), and, after filtering, recorded by a photomultiplier tube (PMT Hamamatsu, H7732-01).The photocurrent is amplified and is collected by data acquisition system (Imtec, T114). The computer synchronises both the scanning process and data collection.

The sample, a silica flat coverslip, was explored across an area of 450 ^{*} 450 μm at 30 different slices of the objective (Zeiss Plan-NeoFluar x40 NA=0.75) within the 15 μm range depth location. The experiment is realized at the second (glass-to-air) interface of the coverslip. At a laser average power of 5 mW, each slice of the 3D set of data requires 20 s for 512 ^{*} 512 pixels.

First experiment confirms that I_{3ω} is power-dependent (Fig. 4) of fundamental beam. Indeed this plot (in log scale) shows that I_{3ω} grows like ${{\mathrm{P}}_{\mathrm{\omega}}}^{3}$.

## 4. Experimental section: data analysis

#### 4.1 Peak position: Lag

Figure 5 presents the focus plane overall quality for this objective.

At a wavelength of 1.5 μm (Fig.5.A) the flat field condition is effective on an area of only 200μm in diameter. At 200μm away from the centre, the deviation reaches as such as 3μm.

At a wavelength of 1.03 μm (Fig.5.B), we can see that the flat field condition is effective on an area of more than 250 μm in diameter. Moreover at 250 μm away from the centre of this circle, the deviation reaches as such as 2 μm.

We can conclude that this objective is more corrected in flat field condition for λ = 1.03 μm than for λ = 1.5 μm. We also think that the flat field condition is optimal with a visible laser source. Indeed this microscope objective is usually used in a confocal single-photon microscope.

#### 4.2 FWHM of the third-harmonic

The FWHM of the third harmonic gives the confocal parameter and the beam waist at each (x, y) point (Fig. 6 @1.5μm (A) and @ 1.03μm (B)).

As measured in the case A, the depth resolution is not optimum at centre but rather on a ring of 200 μm diameter where as measured in the case B, the depth resolution is homogenous along all the field of view of the objective.

By Eq.(8), we can deduce the local beam waist w_{L} for the laser beam at each wavelength.

The optical resolution is traditionally defined through the Rayleigh criterion. By Kirchloff, Debye laws and paraxial approximation one can define the beam waist as function of the wavelength of the incident laser beam λ and the numerical aperture NA of the objective [10]:

By Eq. (12) we obtain for each wavelength:

We can observe that, at 1.5 μm, theoretical and experimental values of beam waist are similarly equal. We can deduce that laser beam have a good diameter before Zeiss Plan-NeoFluar x40 objective.

However, at 1.03 μm, experimental value of beam waist is larger than his theoretical value. Experimental numerical aperture is less than theoretical numerical aperture because at this wavelength, the laser underfills the entrance objective aperture. Figure 6-B demonstrate clearly that the experimental beam diameter where not optimized, nevertheless we can conclude that spatial resolution is more homogenous over the whole field of view of the objective at this wavelength.

#### 4.3 Peak intensity

Figure 7(a) and Fig. 7(b) present (in log scale) the normalized intensity with respect to b parameter and relative to the signal amplitude at centre.

As our lasers use relative long pulse (150 fs @ 1.5 μm and 200 fs @ 1.03 μm) all the fluctuations on peak intensity seem as power capability of the objective itself. As a consequence of paragraph 4.2, a shyly better transmittance across a ring can be detected on Fig. 7(a) for a wavelength of 1.5 μm where as we can see that on Fig. 7-B the transmittance is homogenous is all the field of view.

If we use other laser source which provide short pulse duration it will be possible to determinate perfectly the pulse broadening in each (x, y) point of the objective.

Following this paper, we can derive some useful and practical informations.

A standard procedure to qualify a whole multi-photon microscopy setup will be described as follow:

First collect the forward beam- if not available- and filter out the laser wavelength to extract the third harmonic.

To mimic standard illumination, set a cover slip

Scan off, locate the second interface (highest THG signal); Make a rough estimate of the practical depth of view.

Explore and record a 3D image spanning the whole field of view by this depth of view.

Exploit this 3D data as described to qualify your setup.

## 5. Conclusion

THG intensity signal on a simple interface is a powerful tool for beam diagnostic in non linear microscopy. The optical aberration, expected resolution and pulse with broadening can be completely determined with this simple technique, enabling the efficient selection of microscope objective’s specifications for any multiphoton imaging process.

## Appendix

We present in this appendix a more realistic relation for J parameter used in our Gaussian model:

Ei is an exponential integer function given by:

When the beam waist of the fundamental laser beam is placed at the interface of the silica coverslip, i.e., at z_{0}=0, the J parameter can be expressed like this following relation:

with

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