Introduction

The observation of optical patterns is an excellent tool for studying the properties of quantum mechanics with or without chaos, quantized radiation fields, quantum optics, the concept of quantum-classical correspondence, etc.1,2. Recently, optical patterns possessing crystal- or quasicrystal-like structure have been widely investigated because they can be used to develop new materials and devices3,4,5 and employed to implement different types of solitons6,7,8,9,10, Anderson localization11 and other quasiperiodic structures12 by using the optical induction technique. Beside their implications in physics, they have attracted interest in other areas such as mathematics and art in both ancient and modern cultures13,14. Interference of multiple monochromatic point sources with specific symmetry13,14 and superposition of multiple laser modes such as high-order Hermite-Gaussian (HG) and/or Laguerre-Gaussian (LG) modes1,2 are two efficient methods for the generation of optical periodic and quasiperiodic structures.

It is well know that the paraxial wave equation for a laser propagating along z direction can be described by the Helmholtz equation: with , here two dimensional (2D) is a slowly varying complex function. The Helmholtz equation can also be expressed as: . One can find that the Helmholtz equation shown above has a form similar to the time dependent 2D Schrödinger equation15,16. Therefore, their solutions are similar and optical laser patterns which express the solution of the Helmholtz equation have been applied to investigate 2D quantum information17,18,19. In this field, the eigenfunction of the 2D quantum harmonic oscillator (QHO) represents a fundamental issue16 which can be used to illustrate the basic concepts and methods in quantum mechanics and has been applied in a variety of branches of modern physics, such as molecular spectroscopy, solid state physics, nuclear structure, quantum optics and so forth. By simple transformation, the wave functions of the 2D QHOs can be described by the HG functions which are eigensolutions for the oscillating modes in a laser cavity. In the process of frequency doubling, two photons would “collide” with each other and propagate as one and the resulted second-harmonic (SH) wave functions will be different even though they are dependent on the fundamental eigenfunctions. Therefore, the frequency doubling would induce the quantum state of the fundamental mode to become superposition of two macroscopically distinguishable states20.

The self-frequency-doubling (SFD) process is a combination of laser emission and frequency doubling. The laser oscillating modes are determined by the cavity and the overlap with the pump beam. Through the nonlinear process, the oscillating photons would “recombine” and become visible to the naked eye. Therefore, SFD process should be practicable for studying the interaction of fundamental and SH modes and the change of their quantum states. However, we are not aware of such investigations based on the SFD effect. In this work, we experimentally study the SH patterns of HG modes in a SFD microchip laser.

Results

The SFD experiments was performed with a non-centrosymmetric crystal of Nd:GdCOB as the SFD crystal and a fiber coupled laser diode (LD) as the pump source whose output intensity profile from the fiber had a doughnut shape distribution. When the Fresnel number is small (<5), which means the length between the front face of the SFD crystal and the focal spot of the re-imaging unit is short, only the optical vortices were observed21. At that time, the doughnut-shape intensity profile dominated the fundamental laser and overlapped well with the LG modes. However, at a position of the SFD crystal with the front surface about 3-mm behind the focal spot, where the doughnut spatial profile of the pump beam is well pronounced, the threshold of the SFD laser operating in a high-order HG mode was 1.1 W. A typical SH pattern observed in the far-field is shown in Fig. 2a. It can be clearly seen that the far-field pattern exhibits a periodic lattice structure with the intensity localized at the center, which is different with the usual HG modes. Analyzing the symmetry of the pattern, it can be seen that each of the two orthogonal bright lines can be considered as a two-fold axis of symmetry. Rotation by 180° about any of the two-fold axes brings the pattern into coincidence with the original one. Therefore, the 2D crystal-like pattern possesses symmetry of 22. The near-field pattern was also observed and it is shown in Fig. 3a, representing the intensity distribution in coordinate space. It is seen that the near-field pattern is different and the intensity concentrated along a classically rectangular shaped trajectory with four peaks. The changing with propagation of SFD patterns is different with traditional HG modes whose appearances are unchanged22.

Figure 1
figure 1

Experimental configuration for the SFD laser.

The pump beam from the fiber coupled LD with doughnut spatial profile (inset) is used to excite higher order transversal modes in the Nd:GdCOB SFD crystal. The visible output at the second-harmonic is directly projected onto a flat screen.

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Figure 2
figure 2

Far-field crystal-structure patterns at the second harmonic.

(a) Experimental observation of the far-field pattern at a pump power of about 1.1 W. (b) Theoretical calculation of the SH far field for a fundamental mode (11,16) and (c) HG eigenfunction of the same mode as in (b), calculated for the fundamental.

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Figure 3
figure 3

Near-field transverse patterns.

(a) Near-field diamond-shaped pattern experimental observed in the visible for conditions corresponding to Fig.2a and (b) Theoretical calculation for the relative intensity distribution of the near field second harmonic pattern based on the HG eigenfunction of the same fundamental mode (11, 16).

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By tuning the setup and maintaining the pump power of 1.1 W almost unchanged, the pattern can be changed which corresponds to different orders related to different mode spatial profiles which are better overlapped with different pump beam sizes. The near-field patterns are similar to the one shown in Fig. 3a. However, under the far-field observation, patterns for different orders exhibit different numbers of bright points. Typical experimentally observed far-field patterns for different orders are presented in Figs. 4a and b. These patterns also possess crystal-like structure with 22 symmetry. It should be noted that, once the pattern with a certain order appeared, its structure was almost unchanged with time unless the setup was tuned. We believe that the stable structure may be generated by the good overlap between the pump and oscillating modes which only allows the optimal mode to oscillate near the threshold. As described in the Methods section, the output coupler of the present self-frequency-doubling microchip laser was high-transmission (HT, T>99%) in the 530-545 nm SH range, which means that the observed periodic visible modes did not oscillate in the cavity, their correlation coefficients should be zero and the gain distribution should be similar to the output pattern. The behavior of the present crystal-like structure patterns is different with the modes generated in the reported Talbot cavity23,24.

Figure 4
figure 4

Far-field pattern of the SH with different (m, n) fundamental mode indices.

(a) Experimental observation and (c) theoretical calculation of the far-field SH pattern for fundamental mode (7, 9); (b) Experimental observation and (d) theoretical calculation of the far-field SH pattern for the fundamental mode (8, 9).

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The measured SFD laser wavelength was 545 nm, corresponding to the SH of the fundamental wavelength at 1090 nm. The SH beam was linearly polarized, corresponding to the “fast” eigenmode in the biaxial SFD crystal while the fundamental was with perpendicular polarization corresponding to the “slow” eigenmode. It should be noted that, when using a pump source whose spatial profile in the focal plane corresponds to a Gaussian-like distribution, we were unable to observe high-order HG SH modes. It can be believed that the doughnut distribution of the pump spot which determines the gain distribution in the laser medium is essential for excitation of higher order HG modes through selective spatial overlap. Compared with the generated high-order LG modes which is well described by the cylindrical coordinate21, the symmetry of the emission cross-section of SFD crystal is also identified as an important factor for the generation of high-order HG modes. The symmetry of emission cross-section for Nd:GdCOB is with axis which is well described by the Cavendish coordinate system25 and agrees well with the symmetry of HG modes. Based on the analysis above, it can be obvious that, under small Fresnel number, the pump beam possesses small size whose symmetry dominates the generated fundamental laser with LG modes; however, when the pump size becomes large enough, the symmetry of the emission cross-section induces the generation of high-order HG modes, since the gain distribution in the laser medium is determined by both of the pump intensity distribution and emission cross-section. Therefore, the generating mechanism for the crystal-structure patterns in SFD lasers is totally different with the previously investigated nonlinear optical system, such as photorefractive media8 and vertical cavity surface emitting lasers26 which are due to the nonlinearity of refractive index and reflection by their bounding, respectively. It should be noted that, under the observation, there was no typical vortex patterns accompanying or competing as shown in Fig. 2.

Discussion

The eigenfunctions of the HG modes in a laser cavity are given by22

with , and . Here, w0 is the beam waist, zR is the Rayleigh range, wm,n,l is the resonance frequency, m and n are the transverse mode indices, l is the longitudinal mode index and Hm(x) is the m-th order Hermite polynomial. For a single HG mode, the phase does not affect the intensity distribution but it is essential for the Fourier transformation relating the near- and far-field distributions. In the near-field (z~0), the amplitude distribution is similar to that in the beam waist (z = 0) and can be expressed in terms of HG functions:

As we know, the Hamiltonian for a 2D QHO can be shown as19:

with the 2D momentum operator as: . The eigenfunctions for 2D QHO are given by:

with , where μ is the oscillator mass and ω is the angular frequency of the harmonic oscillator. Equating , the HG transverse eigenmodes in the near field are equivalent to the eigenfunctions of the 2D QHO and can be used to simulate the mechanics of the 2D QHO. For the 2D QHO, the Eq. (4) represents the eigenfunction in the coordinate space. By Fourier transformation, the eigenfunction in the momentum space can be achieved from Eq. (4)16. Considering the Fourier transforming relationship between the transverse eigenmodes in the near and far fields based on the light propagating theory and the equivalence between the eigenfunctions of 2D QHO in the coordinate space shown in Eq. (4) and HG modes in the near field shown in Eq. (2), it can be reduced that the far-field HG transverse eigenmodes are analogous with the eigenfunctions of 2D QHO in the momentum space.

In SH generation, two photons collide with each other and combine into one with doubled photon energy and half wavelength. Based on the second-harmonic theory, the field amplitude for the fundamental Em,n and SH ESH can be expressed as:

with , where deff is the efficient nonlinear coefficient, n(f) and n(2f) are the refractive indexes for the fundamental and SH wavelength, respectively, f is the frequency of the fundamental light, k1 and k2 are the respective wave vectors of the fundamental and SH lights. For small signal approximation, the SH amplitude can be shown as:

Since the other parameters are constant for the phase matching , the resulted SH field amplitude of the SH is proportional to the square of the fundamental field distribution27:

It should be noted that the large angular acceptance bandwidth and short length (6 mm) of the nonlinear crystal used28 induce the ignorable effect generated by the divergence of high-order fundamental HG modes. If the SH field is decomposed to a linear combination of eigenfunctions of the 2D QHO or HG modes29, the physics contained in the SH field may be more obvious. Since

with

The SH field amplitude may be shown as:

here is the decomposed coefficient corresponding to the probability density of the 2D QHO in state. Compared with Eq.s (2) and (10), it can be found that the SH light is composed by and localized on a superposed state of or , with and which results in the eigenfunctions and localization of the SH light becoming different with the fundamental modes but possessing similar symmetry.

Using Eq.s (2) and (7) or (8), we calculated the intensity distribution in coordinate space for m = 11, n = 16, simplified as (11, 16) and the results is shown in Fig. 3b. It can be seen that light density is localized at four high intensity spots and decreases as x and y tend to 0, in good agreement with the experimental near-field pattern and similar with the fundamental HG mode whose structure would be unchanged with propagation. Based on an analysis using the Bohr correspondence principle and the classical areal velocity law30, the probability density exhibits peaks at the points corresponding to the highest potential, just like a classical particle which would reside more time near the apogee of its motion.

As is well known, the far-field pattern corresponds to the momentum-space wave function which is the Fourier transformation of the coordinate-space wave function. The near-field pattern corresponds to the coordinate-space wave function. The far-field pattern can provide more useful information than the near-field one31. By Fourier transformation of Eq. (7), the far-field SH pattern can be calculated for (11, 16), as shown in Fig. 2b and it resembles the experimentally obtained visible pattern. The theoretically calculated far-field patterns for (7, 9) and (8, 9) are shown in Fig. 4c and d, respectively. From these figures, it can be clearly seen that the far-field pattern exhibits a periodic lattice structure with the same symmetry as the near-field pattern. For comparison, a calculated transverse mode distribution at the fundamental wavelength is also presented in Fig. 2c whose intensity distribution keeps unchanged and localized on four peaks similar with the observation of SFD in the near-field. It should be noted that the frequency-doubling resulted in intensity changed and localized at the center of the pattern and the number of bright points along each transversal coordinate is larger but less than two times of the fundamental ones, although the symmetry is unchanged.

We proposed that, by mode conversion, angular momentum interactions between photons and nonlinear amplifying medium such as “angular momentum Raman-like scattering” may be realized and studied through the generated SFD pattern32. Because the HG modes can be employed to define an infinitely dimensional discrete Hilbert space33 and SH generation is an important mechanism for realizing entangled states of lights34, this approach provides a route to investigate the entanglement involving many orthogonal quantum states. Similar to the nonlinear optical lattices which are dominated by the initial nonlinearity of the laser system and possess phase singularity and periodic structures35,36, the present linear SFD patterns could also be used for the generation of periodic trapping fields due to the crystal-structure distribution of their intensity. Finally, the present method for direct generation of crystal structure patterns is based on a compact and simple design with easy control of the pattern and intensity distribution characteristics which could be applied for development of new materials and devices using the optical induction technology3,4,5.

In conclusion, we observed 2D crystal-structure patterns with 22 symmetry analogous to a superposed state of the quantum harmonic oscillator in the near- and far-fields of the SH emission from a SFD microchip laser. Patterns of different orders are easy obtained by tuning. The laser modes based on the HG functions were calculated and compared with the experimental results. We believe that SFD visible lasers could become useful candidate besides two-wave mixing in photorefractive media and high-Fresnel laser cavities for testing quantum mechanical theories while the directly generated crystal-structure patterns could play an important role in developing new materials and devices.

Methods

A non-centrosymmetric crystal of Nd:GdCOB grown by the Czochralski method with a Nd concentration of ~8 at.% was employed as an active element. This crystal is monoclinic biaxial and maximum effective nonlinearity for type-I SH generation with a fundamental wavelength of 1090 nm occurs in a phase-matched direction outside the principal planes; the 6-mm long sample was cut along this direction37,38. The 3 mm × 3 mm faces were polished and the front one was anti-reflection (AR) coated for 808 nm (pump) and high-reflection (HR, R>99.8%) coated for 1060–1091 nm (fundamental) and 532–545 nm (SH). The other face was HR coated for 808 nm and 1060–1091 nm, with high-transmission (HT, T>99%) in the 530–545 nm SH range. The two opposite faces of the crystal form a laser cavity which amplifies the light and generates the laser. The cavity ensures double pass pumping, the fundamental is not out-coupled and the generated SH is completely out-coupled through the exit crystal face after one round trip. Thus, the self-frequency-doubling microchip laser consists of the pump source, cavity and Nd:GdCOB crystal. The experimental configuration is shown in Fig. 1. The pump source employed was a LD with a central wavelength around 808 nm coupled into a fiber with a core diameter of 200 μm and numerical aperture of 0.22. The output intensity profile from the fiber had a doughnut shape distribution. The pump light was focused onto the Nd:GdCOB crystal by a 1:1 re-imaging unit. To remove the heat generated, the Nd:GdCOB crystal was wrapped in indium foil and mounted in a water-cooled copper block. The laser spectrum was measured with an optical spectrum analyzer (AQ-6315, YOKOGAWA). The SH pattern was recorded with a 21 Megapixel digital camera (EOS 5D Mark II, Canon).