1. Introduction

With the increase of intensity, laser propagation in optically transparent media will undergo from linear to nonlinear, and even extremely nonlinear processes, which are usually accompanied by abundant physical phenomena [1–4]. Many applications are also derived during the intensity dependent propagation [5–7]. For example, the spatial distribution of laser intensity changes, namely focusing and defocusing, during the propagation process. Filamentation is such a laser intensity dependent propagation phenomenon which is due to the dynamic balance between the Kerr-induced self-focusing effect and plasma-induced defocusing effect of the laser beam [8–10]. During filamentation the intense ultrashort laser pulses can excite some accompanying radiation phenomena, such as new frequency generation through four-wave mixing [1–3,11], THz radiation [12–14], third harmonic radiation [15–18] and supercontinuum generation etc [19–21].

Laser intensity cross-sectional distribution and its longitudinal evolution along laser filament are crucial and important not only to fully understand laser filamentation but also to influence its applications. The intensity in the filament is ∼$5 \times {10^{13\; }}$W/$\textrm{c}{\textrm{m}^2}$ for free propagation [22] and ∼${10^{14}}$ W/$\textrm{c}{\textrm{m}^2}$ under external focusing condition [23], corresponding electron density of ${10^{16}}$-${10^{18}}\textrm{c}{\textrm{m}^{ - 3}}$[24]. The formation of the filaments is a dynamically temporal evolution process simultaneously [25,26]. It is impossible to directly measure the intensity distribution by inserting any traditional detector in the channels because of the damage. Optical imaging is an efficient method to record the intensity distribution of beam cross-section at different positions, which has been used to study the intensity distribution and evolution of filament in details [27–29]. Moll et al [30] used a CCD to image the beam profile through frosted glass to characterize whether the beam collapsed. Ting et al. [31] used CCD imaging method to directly measure the fluence distribution of a laser pulse that has undergone filamentation in air. The energy, size and other characteristics of the filament have been clearly determined. Chen et al. [32] recorded the evolution of femtosecond laser filament in air by a CCD camera from a wedge positioned inside the filament. The diameter and energy distribution of the filament at different positions were characterized. In these imaging systems, it is generally necessary to move the imaging lens and CCD step by step so that to obtain multiple cross-sections of the object of the filament. The multiple operations not only make the experimental process cumbersome, but also induce the uncertainty on the experimental results. A three-dimensional (3D) imaging method is highly welcome which can perfectly solve this problem.

F-P cavity was invented by Charles Fabry and Alfred Pérot in 1897 [33]. It is composed of two parallel mirrors with high reflection film on the inner surface, so that the beam is continuously reflected and refracted between the two mirrors. Multiple parallel beams transmitted by incident light through F-P cavity will produce multiple beam interference at infinity or focal plane. According to the sharpness and filtering characteristics of interference fringes, F-P cavity has been used in many fields. In the field of spectroscopy, F-P cavity can select wavelength and narrow linewidth of input non-monochromatic light, which can improve the monochromaticity of laser [34]. The fine sharpness of interference fringes can improve the spectral resolution, which can be used for the analysis of spectral hyperfine structures [35]. In addition, F-P cavity has been used for frequency stabilization technology because it can distinguish small changes in frequency [33]. F-P cavity has been used in auxiliary imaging. Tsia et al[36] applied the principle of F-P cavity to two-dimensional (2-D) spatial disperser, aiming to map the spectrum into a 2-D space and generating a 2-D spectral shower on which the spatial information is encoded. Thus, the serial time-encoded amplified microscopy (STEAM) can perform 2-D imaging. Dresselhaus-Cooper et al[37] used a F-P cavity to generate a pulse train and applied in various imaging configurations with the purpose of visualizing the shock with sequences of up to 16 images.

In this work, by utilizing the feature of the beam reflection and refraction multiple times in an F-P cavity, we applied an F-P cavity on imaging the intensity transverse distribution at different longitudinal positions of femtosecond laser filament in air. As a proof-of-principle demonstration of 3D imaging, multiple cross-sectional images of the filament along propagation were obtained by a CCD camera in single exposure mode. Longitudinal resolution of the 3D image along filament was tuned by adjusting the spacing distance and incident angles of the F-P cavity. The intensity evolution along filament propagation measured by the F-P imaging method was consistent with that of the fluorescence measurement of the filament from the side.

2. Experimental setup

The schematic of the experimental setup is shown in Fig. 1. In the Experiments, a diode-pumped solid-state femtosecond laser system (CB3-40 W, Light Conversion) was used which delivered 1 kHz/196 fs laser pulses. The maximum laser pulse energy was 400 µJ with central wavelength at 1030 nm. The femtosecond laser pulse was focused by a plano-convex lens (Lens, focal length of 17.5 cm) to form filament in air. A digital camera was used to take the fluorescence image from the side. The filament cross sections were imaged by a 2-inch lens (Imaging lens, focal length of 20 cm) onto a wide dynamic range CCD (R1, Qimaging) in the forward. The imaging lens worked in the 4-f configuration. A 1030 nm HR mirror and neutral density filters (NDFs) were inserted before the Imaging lens to avoid light saturation to the CCD. A pair of parallel high-reflection mirrors at 800 nm was chosen to work as the F-P cavity at filamenting central wavelength of 1030 nm which was inserted before the CCD. In our imaging system, the signal is reduced after a few reflections. The dynamic range can be significantly enlarged by using high reflectivity-to-transmission ratio mirrors for the F-P cavity together with a large dynamic range CCD camera.

 

Fig. 1. Schematic of the experimental setup. The focal lengths of Lens and Imaging lens are 17.5 and 20 cm, respectively. 1030 nm HR is a high-reflection mirror at 1030 nm. NDF, neutral density filter. F-P cavity, placed in front of the CCD, is a Fabry-Perot cavity worked at the filamenting laser wavelength. The reflection and refraction of imaging beam in F-P cavity are shown in the inset figure.

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The working principle of the F-P cavity is shown in Fig. 1. A, B and C represent cross-sections at different positions of the filament. The filament beam undergoes multiple times reflection and refraction in the F-P cavity which results in the output multiple beam patterns of A’, B’ and C’ recorded on the CCD. The output beams A’, B’ and C’ correspond to the imaging beams of cross-sections A, B and C, respectively. Therefore, the CCD camera can image different cross-sections of the filament in single exposure mode. All the images obtained in the following measurements were averaged for 10 times.

3. Results and discussion

For the pulse energy of 395.9 µJ, the raw image recorded on the CCD is shown in Fig. 2(a) with F-P spacing distance of 1 mm and incident angle of 15 deg. Five separated intensity distribution patterns from laser propagation at different positions were experimentally recorded in an imaging picture. Note that by adjusting the spacing distance and incident angle of F-P cavity and NDFs different number of cross-sectional images along laser filament can be obtained in the CCD imaging plane. To retrieve the intensities at different propagation positions from recorded patterns, the intensity calibration for the patterns is crucial. The procedure of intensity calibration is followed. The freely propagated laser is directly imaged on the CCD through the F-P cavity (without focusing) while keeping other conditions same as filamentation case. The peak power of our laser pulse is 2.04GW which is less than the critical power for self-focusing reported in [38]. Assuming that there is no change on laser intensity distribution at different imaging propagation distance and the intensity of the n^th image is I_n, the calibration value of adjacent images is q_n= I_n + 1/I_n. I_nwas defined as peak value of the n^th image. Thus, the calibration value of the n^th image regarding to the first image is Q_n= 1${\times} $q₁${\times} $…${\times} $q_n-1(q₀= 1). Therefore, the intensity calibration values can be obtained based on the images from free propagation, which are used to calibrate the filament images by F-P. The magnification effect was tested to be very small and can be safely ignored. The filament cross-sectional images at different propagation distances after calibrating are shown in Fig. 2(b-f). The corresponding 3D intensity profiles are shown in Fig. 2(g-k). The corresponding propagation toward 3D filament imaging is shown in Fig. 2(l) by the F-P assisted single shot image. The working principle of F-P assisted imaging method is no doubt valid for asymmetric imaging although the validity was demonstrated using a Gaussian beam as shown in Fig. 2(b)-(k). Using the method to image structured light filament are interesting and under investigation.

 

Fig. 2. (a) The raw image with F-P spacing distance of 1 mm and incident angle of 15 deg. (b)-(f) are the images of filament cross section at different positions. (g)-(k) are the corresponding 3D intensity profiles. (l) is the corresponding nonlinear propagation toward 3D filament imaging.

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Spacing distance and incident angle of the F-P cavity are two key factors to control the images. When the angle of the F-P cavity is constant, the smaller the spacing distance is, the closer the distance between the adjacent images is. Thus, a better longitudinally resolved evolution of cross section images should be obtained along laser filament. To avoid the imaging spot overlap, the product of the distance between the two mirrors and the angle of F-P cavity should be greater than the filament diameter. Based on the intensity calibrated images, the longitudinal evolution of filament intensity and area are shown in Fig. 3(a) and Fig. 3(b) when the F-P cavity spacing distance are 1 mm (black) and 0.6 mm (red) respectively. Here the geometrical focus of the focusing lens for filamentation is defined as “0”. The negative values indicate the propagation position before the geometric focus. The sum of all pixels with intensity greater than 1/e of the maximum intensity in Fig. 2 is defined as the laser cross-sectional area. At shorter F-P spacing distance of 0.6 mm, more images were recorded at shorter propagation distance as compared with the images at F-P spacing distance of 1 mm. Therefore, a better longitudinal resolution of filament images at shorter F-P cavity distance is confirmed. Moreover, filament intensity and cross-sectional area at F-P distance of 0.6 mm nicely follows the results of the F-P spacing of 1 mm, which confirms the validity of the method.

 

Fig. 3. Calculated intensity (a) and cross-sectional area (b) of filament as a function of the laser longitudinal position according to F-P assisted images. Laser energy was 395.9 µJ. The F-P cavity spacing distances were of 1 mm (black) and 0.6 mm (red). The incident angle of the F-P cavity was 15 deg. “0” is defined as the geometrical focus of the focusing lens for filamentation. The distances with negative values are before the geometrical focus.

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In order to verify the effectiveness of the imaging method based on the F-P cavity, traditional filament imaging method [39] based fluorescence from the side was taken by a digital camera. The camera exposure time was 0.25s. The longitudinal evolution of fluorescence intensity was measured by integrating the pixel intensity of fluorescence image and averaged for 250 shots as shown in Fig. 4 (black). As for a comparison, we also plotted the laser intensities at multiple filament longitudinal positions taken from the images by the F-P method (red). It is clear that the intensity by the F-P method nicely follows the fluorescence intensity which confirms the effectiveness of the 3D imaging method based on F-P cavity.

 

Fig. 4. The fluorescence intensity recorded from the side (black) and the laser intensity taken from the F-P image (red) as a function of propagation position.

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As an application of the F-P assisted 3D imaging, we investigated the evolution of laser propagation under the same focusing condition by tuning laser pulse energy from 74.5 µJ to 395.9 µJ. Figure 5(a) shows the intensity of laser imaged by the F-P method as a function of the propagation position for laser energies of 74.5 µJ (black), 99.4 µJ (red), 198.7 µJ (blue), 296.5 µJ (pink) and 395.9 µJ (green), respectively. In addition, Fig. 5(b) presents the cross-sectional area of laser as a function of the propagation position for laser energies used in Fig. 5(a). When laser energy is not more than 198.7 µJ (e.g., 74.5 µJ, 99.4 µJ, and 198.7 µJ), the obtained laser intensity is the strongest while the area is the smallest at the geometrical focus. The laser intensity and laser beam area are symmetrical along propagation with respect to the geometrical focus [Fig. 5(a) and Fig. 5(b)], which confirms the linear focusing law of the beam at low energy. When the laser energy is higher than 198.7 µJ (e.g., 296.5 µJ and 395.9 µJ), the laser intensities increase at before geometrical focus. Correspondingly, the beam sizes decrease. This is a clear indicator of nonlinear Kerr self-focusing, which contributes to the laser beam focus in advance. The phenomenon is consistent with many other experimental results [40,41].

 

Fig. 5. (a) Intensity of laser (or filament) as a function of the propagation position for laser energies of 74.5 µJ (black), 99.4 µJ (red), 198.7 µJ (blue), 296.5 µJ (pink) and 395.9 µJ (green) respectively. (b) the corresponding cross-sectional area of laser (or filament) as a function of the spatial position for laser energies used in (a).

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The imaging method in this work relies on linearly propagating fields, namely the imaging wavelength. The method is good to look into the filamentation process by using filamenting wavelength. In order to look into the nonlinear process which is working at other wavelength, the imaging system working at other wavelengths has to be employed. To overcome the limited spatial reshaping, J.C. Diels and co-workers [42], reported using a hydrodynamic window to truncate the filament, hence measuring the beam profile during the nonlinear propagation. This method also requires multiple window slice truncation to obtain multiple cross-sections of the filament.

4. Conclusion

In summary, by applying an F-P cavity in the traditional imaging system, we demonstrated the ability to record the cross-sectional intensity patterns at different longitudinal positions along filament within a single exposure of CCD. As a proof-of-principle demonstration, 3D imaging of laser filament was obtained according to multiple intensity cross patterns of laser filament at different positions recorded in an imaging picture. When keeping the incident angle of the F-P cavity as a constant and reducing its spacing distance, a better longitudinal resolved evolution of cross-sectional filament intensity patterns was obtained. The intensity evolution along laser filament obtained based on F-P cavity imaging method was consistent with the filament fluorescence intensity from the side. Then the F-P cavity assisted 3D imaging was used to test the self-focusing by tuning the laser energy from low to high. The Kerr self-focusing was confirmed by the increase of laser intensity and the reduction of beam size before the geometrical focus. Although the F-P cavity assisted 3D imaging in a single exposure was demonstrated at 1030 nm femtosecond laser for filamentation, this imaging method does in principle work for 3D propagation characterization of other wavelength lasers and their accompanying radiations which is in particular useful to record the propagation of structured light within a single exposure.

In our experiment, the peak power of the laser is only 2.04GW, less than the self-focusing peak power of 1030 nm femtosecond laser of 5.3GW [38]. The filament produced in our work is relatively short. Small spacing distance of F-P cavity allows 3D reconstruction of short plasma filament. The principle of 3D imaging is successfully demonstrated. We believe the method is applicable for characterizing more complicated filamentation driven by more intense, structured femtosecond laser pulses.