1. Introduction

In 1987, Durnin first proposed a class of ideal nondiffracting beam solutions to the Helmholtz equation in circular cylindrical coordinates [1], which can perfectly maintain its lateral beam profile without being subject to diffractive light spreading. After that, exact solutions of the Helmholtz equation in other coordinates were found, for example, the Mathieu beams in elliptic coordinates [2] and the parabolic beams in parabolic coordinates [3]. However, an ideal nondiffracting beam of an infinite lateral field extent and power is not physically realizable. Therefore, several kinds of modified nondiffracting beams of finite power that can be experimentally realized have been proposed. For example, the generalized Bessel–Gauss beam is the ideal Bessel beam modified by flat-top Gaussian function [4], and the Helmholtz-Gauss beams are ideal nondiffracting Helmholtz beams modified by Gaussian envelope [5]. All the realized nearly nondiffracting beams (or say quasi-nondiffracting beams) can propagate over a large range without a significant spread of the light field, however, their beam intensity changes along beam propagating direction due to its finite lateral power extent.

For the nearly nondiffracting characteristics and the abundant field structures, the nearly nondiffracting/quasi-Mathieu beams have been used in several research and applications, such as optical trapping and tweeter [6–8], laser processing [9–11], and soliton lattice manipulation [12]. The generation of the quasi-Mathieu beam was mostly using the extra-cavity method. The zero-order Mathieu beam was first observed by adding a Gaussian aperture before the annual slit of Durnin’s experiment setup [13,14]. Volke-Sepúlveda et al. generated Mathieu-Gauss beams using a spatial light modulator (SLM) to display the phase of Mathieu-Gauss beams and following spatial filtering with an annular aperture [15]. Ren et al. generated a quasi-Mathieu beam by a combination of an axicon and an amplitude modulation [16]. In Ren et al.’s work, the amplitude-type spatial light modulator was employed to produce the amplitude of the angular Mathieu function, which was then transferred into a Mathieu beam with an axicon. The SLM provides the flexibility in the control of the transverse profile of the resulting quasi-Mathieu beam, and the method can generate the family of all kinds of nearly nondiffracting Mathieu beams is determined. Gutíerrez-Vega et al. proposed the first and the only intra-cavity method to generate quasi-Mathieu beams with an axicon-based laser resonator [17]. By breaking the resonator symmetry with tilting output coupler and using of one or more intra-cavity metal wire/wires, they successfully observed several kinds of even and odd types of Mathieu–Gauss beams. In general, the intra-cavity beam shaping method provides higher energy conversion efficiency than extra-cavity methods. However, in terms of application, this intra-cavity method [17] cannot provide a fast and controlled way to the generation of an on-demand type of quasi-Mathieu beam due to its beam selection of mechanical adjusting. A recently invented laser system, the digital laser [18,19], due to the use of an intra-cavity SLM, has a high capability of generating dynamically controlled laser fields. However, there are no research reports on the direct generation of nondiffracting Mathieu beams from digital lasers. It is speculated that it is difficult to provide the high conical phase of the nondiffracting quasi-Mathieu beam by the intra-cavity SLM of a limited phase modulation range. In this paper, an intra-cavity method for generating nondiffracting quasi-Mathieu beams with a digital laser was proposed. In the method, the high conical phase of quasi-Mathieu beams was excluded from the digital laser cavity and provided by an axicon outside the cavity. The selection of the lasing quasi-Mathieu beams is controlled by the projection phase of the intra-cavity SLM of digital lasers, which provides flexibility in dynamically controlled selection/manipulation of quasi-Mathieu beams.

The paper is organized as follows: In Section 2, the formalism to the generating nondiffracting beam and the beam characteristics are detailed; in Section 3, the experiment result of quasi-Mathieu beam generation are presented and discussed; finally, in Section 5, a summary of this study is given.

2. Beam generation mechanism and formalism [as applied] to the quasi-Mathieu beams

This study was inspired by the extra-cavity quasi-Mathieu beam generation method, “a combination of axicon and amplitude modulation” [16]. It was analytically and experimentally proved that a light field of the amplitude of angular Mathieu function can be converted into nearly nondiffracting Mathieu beams by passing it through an axicon. In other words, once one can generate an intra-cavity light field of the amplitude of angular Mathieu function, it can lead to a successful generation of high-efficiency quasi-Mathieu beams with a conversion of an extra-cavity axicon.

However, the infinite lateral field extent of the angular Mathieu function makes the light field physically non-realizable in digital lasers. Therefore, this study aimed to generate a Gaussian-modulated angular Mathieu function. Figures, 1(a), 1(b), and 1(c), respectively plot example light fields of an angular Mathieu function, a Gaussian-modified angular Mathieu function, and the spatial Fourier-transform of the Gaussian-modified angular Mathieu function. As Fig. 1(a) shows, the angular Mathieu functions have an infinite lateral field extent which makes the light field is not physically realizable in digital lasers. As Fig. 1(b) shows, different from angular Mathieu functions, the angular Mathieu functions modulated/multiplied by a Gaussian envelope, named as angular Mathieu-Gauss function, is of finite lateral extent. However, the low energy distribution and the high spatial resolution of its center field leads to the failure of its intra-cavity generation in digital lasers. This can be illustrated by two points. First, the Angular Mathieu-Gauss function has a relatively low energy distribution at the center. In the intra-cavity beam generation, the specified laser field needs to compete with other resonant cavity modes through modal gain, diffraction loss, etc., and then determines the final laser field. Without the use of non-spatially homogeneous laser pumping, it is difficult for the laser system to avoid the present of on-axis cavity eigenmodes of relatively low diffraction loss. Therefore, it is difficult to directly generate angular Mathieu-Gauss function of no significant distortion in digital lasers. Second, the center of the angular Mathieu-Gauss function has extreme fine pattern. In the angular spectrum interpretation of the propagation of light wave in free space [20], the intracavity formation of the fine pattern in the center of an angular Mathieu-Gauss function requires the superposition of more light waves of a higher (angular) frequency, and this part corresponds to the plane wave propagation term with a larger slope of the light propagation direction in the laser cavity. A finite NA (numerical aperture) value of a laser cavity essentially limits the light waves of high angular frequency to survive in the laser cavity, and it leads to the distortion and the failure in the resulting center pattern of an angular Mathieu-Gauss function. Instead, this study generated the light field of the spatial Fourier spectrum of the angular Mathieu-Gauss function in digital lasers and then converted it into an angular Mathieu-Gauss function field with an extra-cavity lens. Figure 1(c) plots the spatial spectrum of the angular Mathieu-Gauss function that Fig. 1(b) shows. As Fig. 1(c) shows, the spatial spectrum of the angular Mathieu-Gauss function is of both a finite lateral extent and a low spatial resolution to be capable of generating in digital lasers.

 

Fig. 1. Example light fields of the amplitude of (a) angular Mathieu function, (b) angular Mathieu-Gauss function (b), and (c) the Fourier spectrum of the field in Fig. (b).

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The following section shows both analytically and numerically that passing a light field of the amplitude of the angular Mathieu-Gauss function through an axicon produces a nondiffracting quasi-Mathieu beam. Moreover, the beam characteristics were also discussed. The resulting light field by passing light field through an axicon can be calculated using the Fresnel diffraction integral in cylindrical coordinates,

The U₀ is the light field just behind the axicon. In this study, it is the angular Mathieu-Gauss function, A(ϕ)G(r), multiplied by the amplitude transmittance of the axicon, T(r). That is,

Here, the even kind angular Mathieu function of the first class was selected as an example to show the proof. Substitute Eq. (4.1) into Eq. (2), then into Eq. (1), and then use the following integral relation,

Equation (6) is of the integral form that the stationary phase method can be dealt with [22,23]. That is, $\mathop \int \nolimits_0^R g(r )exp[{ikf(r )} ]dr$ as k is a large number. Here, $g(r )\; = \; J{e_m}({\xi ,{q_z}} )c{e_m}({\eta ,{q_z}} )G(r )r$, and $f(r )\; = \; \frac{{{r^2}}}{{2z}} - ({n - 1} ){\theta _0}r$. The stationary phase method is a typical and effective way to solve the propagating field behind the axicon [16,24,25]. According to the stationary phase method, the significant contribution for the integral is satisfied ${ {f^{\prime}(r )} |_{r\; = \; {r_0}}} = 0$. Thus, the stationary-phase point of Eq. (6) can be obtained as ${r_0}\; = \; ({n - 1} ){\theta _0}z$. Using the stationary phase method, it is found that when $r\; = \; {r_0} \in [{0,R} ]$, the approximated analytical result of the complex amplitude distribution behind the axicon at the range $z < {z_{max}}$, is

Referring to Eq. (8), the light field behind the axicon in the range $z < {z_{max}}$ is quasi-Mathieu beams, which is the first even kind ideal Mathieu beam, $J{e_m}({\xi ,q} )c{e_m}({\xi ,q} )$, with its lateral extent is modulated by a gaussian term, ${|{G({{r_0}} )} |^2} = exp ({ - 2{r_0}^2/{w_0}^2} )\textrm{ = }exp [{ - 2{{({n - 1} )}^2}\theta_0^2{z^2}/{w_0}^2} ]$. The proof of the other three types of quasi-Mathieu beams can be obtained by replacing Eq. (4.1) with Eqs. (4.2)–(4.3) and repeat the above derivation progress.

In this study, to prevent the on-axis beam intensity fluctuation due to the diffraction from the axicon edges [22,23,26], the beam size passing through the axicon was less than half of the axicon size. The analytical result indicates the nondiffracting distance of the resulting quasi-Mathieu beam, ${z_{max}}\textrm{ = }R/[{({n - 1} ){\theta_0}} ]$, that depended on the effective pupil radius of the axicon, R. Within the situation of this study, the effective pupil radius R of the axicon was estimated by the waist size of the modified Gaussian-term G(r), w₀. Therefore, the nondiffracting distance of the resulting quasi-Mathieu beam of this study was ${z_{max}}\textrm{ = }{w_0}/[{({n - 1} ){\theta_0}} ]$. The w₀ value this study used is 1.28 mm, which gave the z_max value of 0.08 m.

The analytical results, Eqs. (7) and (8), confirm that passing a light field of the amplitude of angular Mathieu-Gauss function through an axicon produces a quasi-Mathieu beam of a Gaussian-modified Mathieu beam field distribution. Some of the beam characteristics are addressed here. Like other nearly nondiffracting beams [5], the intensity of this quasi-Mathieu beam also varies along the propagation direction due to the limited power and lateral light field distribution. There is a z-dependent intensity term due to the Gaussian term, |G(r₀)|², which also exists in other kinds of nearly nondiffracting beams of Gaussian-form lateral energy limit [23,26].

Figure 2 provides a numerical verification of the quasi-Mathieu beam generation method of this study. The resulting light fields behind the axicon by passing a light field of the amplitude of angular Mathieu-Gauss function (m = 1, q = 10) through an axicon were numerically calculated using the Fresnel diffraction integral [27] with Fast-Fourier-Transform (FFT) method. Figures 2(a) and 2(b) plot resulted in transverse intensity distributions behind the axicon of a first kind even quasi-Mathieu beam at different z-planes, z =0.5z_max and z = z_max, respectively. Figures 2(c) and 2(d) plot the propagation of beam intensity along the y-z (ϕ=π) plane and x-z (ϕ=0) plane in the range [0, z_max]. The numerical results confirm the analytical result of this section and reveal that within the nondiffracting range, $z < {z_{max}}$. The quasi-Mathieu beams exhibited a nondiffracting characteristic, i.e., the resulting quasi-Mathieu beams maintain the lateral beam profile without being subject to diffractive light spreading.

 

Fig. 2. (a, b) Transverse intensity distribution of a first-class even quasi-Mathieu beam (m = 2, q = 10) at different z planes, z =0.5z_max and z = z_max. (c, d) Propagation of the intensity pattern along the transverse planes (y, z) and (x, z) in the range [0, z_max]

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3. Experiments and discussion

3.1 Experimental setup

Figure 3 shows the experimental setup of this study to verify the intra-cavity generation of the light field of the spatial Fourier spectrum of an angular Mathieu-Gauss function and its conversion to the quasi-Mathieu beam using an axicon.

 

Fig. 3. A schematic of the experimental setup.

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First, an end-pumped L-shaped digital laser was used to generate a target light field, U_T, of the spatial Fourier-transform of an on-demand angular Mathieu-Gauss function, i.e., U_T =FT{A(ϕ)G(r)}. The laser crystal, 45-degree-tilted plate dichroic mirror, and SLM panel constituted an L-shaped laser cavity. The SLM-projected phase boundary was specified to match with the chosen target light field at the plane of the laser crystal [28] to form a stable resonator for the target light field. The calculation of the SLM projection phase in this study is explained as follows. When the cavity length of the L-shaped laser resonator is represented by the symbol L and the light field at the crystal position (i.e., z = 0) is set as the target light field U(0)=U_T, the target light field propagates to the SLM panel (i.e. z = L) can be calculated numerically [27] and expressed as U(L) = U_SLM = A exp(iϕ_L). Where symbols, A and ϕ_L, are the amplitude and phase value of the light field U_SLM, respectively. The projection phase of the SLM panel was given by ϕ_SLM = -2ϕ_L, which enables the propagating specified target light field to become its conjugate light after being reflected by the SLM. Therefore, the on-demand target light field can reciprocate stably in the laser cavity, which then becomes a stable laser cavity for the specified target light field. The gain medium is a 5×5 mm², 1 $\textrm{mm}$ thick, 3%-doped, a-cut $\textrm{Nd}:\textrm{GdV}{\textrm{O}_4}$ crystal, which was pumped by a quasi-continuous wave (QCW) 808 nm pumping light source focused by a microscope objective lens (20x, numerical aperture 0.4). The microscope objective lens was mounted on a three-axis stage to control the range and position of the pump beam on the crystal. The use of a-cut $\textrm{Nd}:\textrm{GdV}{\textrm{O}_4}$ crystal forced the polarization of the intra-cavity light field to match with the requirement of the SLM. The gain crystal was attached to the copper sheet to dissipate heat. A quasi-continuous pump mechanism (with a duty cycle of 3% and a frequency of 100 Hz) was adopted to reduce the thermal effects and to prevent possible damage to the laser crystal from focusing pumping. The incident surface of the laser crystal and dichroic mirror had transmission of > 95% and > 85% at 808 nm, reflectance > 99.8%, and > 97% at 1064 nm, respectively. The SLM is a phase-type liquid crystal modulator (LCOS-SLM X13138-03WR) of Hamamatsu Photonics. The size of the liquid crystal panel was 16.0×12.8 mm, the single-pixel size was 12.5×12.5 $\mathrm{\mu}{\textrm{m}^2}$, the pixel number was 1280×1024, the phase from 0 to 2π was divided into 216 steps, the fill factor was 96%, and the light adopting efficiency was 97%.

After that, the output laser field was Fourier-transformed to light field of angular Mathieu-Gauss function using a lens of focal length 17.5 cm. The light field of angular Mathieu-Gauss function was observed at the back focal plane of the lens after passing through an attenuator to be attenuated to an appropriate intensity. A long-pass filter (FEL0900) of Thorlabs, Inc. is added before the CCD to remove the remaining pump beam with a wavelength of 808 nm. At the front focal plane of the lens, an iris was added to filter out faint interference fringes in the light field of the resulting angular Mathieu-Gauss function, which was caused by the inner reflection of laser output from two-flat surfaces of a dichroic mirror. Then, the quasi-Mathieu beams could be obtained by directly passing this light field through an axicon situated at the back focal plane of the lens. Here, for measuring the propagating beam pattern, the resulting angular Mathieu-Gauss light field was beam-expanded through a 2× beam expander and then passed through axicon to be observed on a charge-coupled device (CCD). The CCD was installed on the linear translation stage, which enabled the measurement of the beam field pattern along the beam propagation direction. The axicon is a product of Thorlabs, whose radius is 8 mm, the index of refraction is 1.457, and the base angle is 2°.

3.2 Results and discussion

Figures 4 (a) and 4(b), respectively, show three experimentally measured quasi-Mathieu beams and the light field of its corresponding angular Mathieu-Gauss distributions. The beam parameters (q, m) of three quasi-Mathieu beams were (0,1), (10,1), and (0,3), respectively. The result indicates that the proposed method can freely adjust either parameter q or m of the generated quasi-Mathieu beams (i.e., q: 0→10 and m:1→3). Figure 5 shows more generated quasi-Mathieu beams. The size of each picture was 1 mm×1 mm. Figure 5(a) shows the resulting even type Mathieu beams with the elliptic number of q = 0. Particularly, the circular symmetric q = 0 and m = 0 quasi-Mathieu beam is the quasi-Bessel beam. Figures 5(b) and 5(c) respectively show three even and odd type quasi-Mathieu beams of the same beam parameters (q, m). The proposed method can stringently generate any kind of quasi-Mathieu beam as desired, included both even and odd types with the proposed method, the selection of the generated quasi-Mathieu beam was simply achieved by changing the SLM-projected phase without any mechanical adjustment of the experimental setup. That is, this method can directly and quickly realize the change of the quasi-Mathieu beam shape as compared with the previous mechanical adjustment method [17].

 

Fig. 4. Measured field intensity: (a) three quasi-Mathieu-beams and (b) its corresponding light field of angular Mathieu-Gauss function. The beam parameters (q, m) of three quasi-Mathieu-beams are: (0,1), (10,1), and (0,3) respectively.

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Fig. 5. Measured field intensity pattern of (a) three even types (q = 0, m) quasi-Mathieu-beams, (b) three even types (q = 10, m) quasi-Mathieu-beams, and (c) three odd type (q = 10, m) quasi-Mathieu-beams at plane z = z_max

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Figure 6 shows propagation characteristics of the resulting quasi-Mathieu beam. The blue line shows the analytical result of the propagation of the peak intensity of a q = 0 and m = 0 quasi-Mathieu beam of Eq. (8) with superimposed experimental data points (red dots). The experiment measurement shows a good agreement with the analytical result. The embedded four figures of Fig. 6 show resulting quasi-Mathieu beam intensity patterns at z = 0.5 z_max, z = z_max, z = 1.5 z_max, and z = 1.98 z_max, in sequence. Experiment results show that the central bright spot of the quasi-Mathieu beam appeared at z = 0.5 z_max behind the axicon and was fading after z = 1.98 z_max. The measurement was stopped at z = 1.98 z_max since the on-axis intensity was no longer the peak value of the light field after the z position. In the experiment, the quasi-Mathieu beam could not be formed before the z = 0.5 z_max plane behind the axicon. It is speculated that the disappearance of the resulting quasi-Mathieu beam in this range is due to the use of the iris. In experiments, the iris is added to filter out faint interference fringes caused by the inner reflection from the parallel flat surface of the dichroic mirror. The use of the iris at the same time led to the loss of high-frequency component in the resulting angular Mathieu-Gauss function so that the loss of high-inclination refracted light of forming the quasi-Mathieu beam is close to the axicon. Replacing the sheet-type output coupler with a prism-type output coupler should be able to avoid the resulting interference fringes due to plate output coupler, and thus can eliminate the use of iris in the experimental setup to avoid the disappearance of the quasi-Mathieu beam adjacent to the axicon. Besides, as Fig. 6 shows, as the propagation distance exceeds the estimated nondiffracting distance z_max, CCD can still observe the quasi-Mathieu beam. This result is reasonable since the angular Mathieu-Gauss amplitude passing through the axicon still contains energy at the region r > w₀.

 

Fig. 6. The analytical value of the propagation of peak intensity of a q = 0 and m = 0 quasi-Mathieu beam with experiment data points superimposed (red dots). Four embedded figures, a, b, c, and d, show measured quasi-Mathieu beam intensity fields at different z-plane: z = 0.5z_max, z = z_max, z = 1.5 z_max, and z = 1.98 z_max, in sequent.

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Figure 7 shows three pictures of resulting quasi-Mathieu laser beams captured from a recording movie (see Visualization 1) from CCD. Figure 7 and the supplementary movie (see Visualization 1) show the flexibility in intracavity-controlling of the resulting quasi-Mathieu beam with the proposed method, which cannot be achieved by the previous intra-cavity method of mechanical adjusting [17]. The supplementary movie shows a dynamic control of q = 10 and m = 1 Mathieu beam in rotating 360°. The 3-seconds video was made by the measured resulting beam transverse intensity patterns of CCD. The video was made as the following approaches. Ninety different SLM projection phases matched with the spatial Fourier spectrum of rotated angular Mathieu-Gauss functions were calculated at first, and then the phases were loaded to the SLM panel in a sequence at a rate of one frame per second. The resulting lasering pattern was cautiously recorded by CCD, and then the recorded pictures was used to make the video. In theory, the rate of intra-cavity modulation of the lasering field is limited by the SLM projection frame rate. The real-time rotating modulation is beneficial to the application in optical micromanipulation [7].

 

Fig. 7. Resulting q = 10 and m = 1 quasi-Mathieu beam of controlled rotation angles as loading different phases to the SLM (see Visualization 1). The rotating angles of quasi-Mathieu beam in three figures respectively are: (a) 44°, (b) 92°, and (c) 136°.

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4. Conclusion

In this paper, an intra-cavity method to generate any kind of nearly nondiffracting quasi-Mathieu beam was proposed. The spatial Fourier spectrum of the angular Mathieu-Gauss function of a finite lateral extent and low spatial frequency was generated with the digital laser, and then the laser field was converted into the quasi-Mathieu beam by passing the laser field through an extra-cavity lens and an axicon. The resulting quasi-Mathieu beams of this study were both numerically and experimentally shown to exhibit nondiffracting characteristics. By excluding the conical phase from the laser cavity and providing it with an axicon outside the cavity, this method can select and manipulate the resulting quasi-Mathieu laser beam with simply a control of the intra-cavity SLM projection phase. For practical applications, this method provides better flexibility in real-time controlling/manipulating quasi-Mathieu laser beam than the mechanical adjusting method. The real-time modulation and the selection of the resulting quasi-Mathieu beams are beneficial for applications, such as optical trapping, laser processing, and micromanipulation. In the future, the method can be explored for the intra-cavity generation of the parabolic beam, the fourth kind of fundamental non-diffraction beam, which never generated intra-cavity due to its unique asymmetric transverse amplitude distribution and non-homogeneous phase.