Table of contents

Structured light refers to the generation and application of custom light fields. As the tools and technology to create and detect structured light have evolved, steadily the applications have begun to emerge. This roadmap touches on the key fields within structured light from the perspective of experts in those areas, providing insight into the current state and the challenges their respective fields face. Collectively the roadmap outlines the venerable nature of structured light research and the exciting prospects for the future that are yet to be realized.

This review paper is a personal attempt to understand the current state of metamaterials science and its development directions, analyzing the main historical steps of its development from the late 19th century to our days.

Pigment based diffuse reflectors (DRs) have several advantages over metal reflectors such as good stability, high reflectivity, and low parasitic absorption. As such, DRs have the potential to be applied on high efficiency silicon solar cells and further increase the power conversion efficiency. In this paper, we perform a thorough review on the notable achievements to date of DRs' application for photovoltaics. We outline unique attributes of these technologies and discuss the theoretical and laboratory development working towards overcoming the challenges of transferring to high efficiency silicon solar cells. In order to understand the potential of DRs for high efficiency silicon solar cells, we provide a qualitative analysis of the impact of front reflection, rear absorption and the angular distribution on the useful light absorption in silicon wafers. By including this discussion, we provide an outlook for the application of DR in reaching maximum photo-current for high efficiency silicon solar cells.

Photoacoustic (PA) imaging offers depth-resolved images of optical absorbers with the spatial resolution of ultrasound imaging. To enhance tumour contrast, tumour-specific probes are used as contrast agents. We synthesised a colourless PA probe that is activated in the presence of γ-glutamyltranspeptidase, a cancer-associated enzyme, to show its original colour and fluorescence. We have acquired high specificity fluorescence images of small tumours, using a fluorescent probe based on similar enzymatic reactions. Here, we developed a PA imaging technique to detect the PA probe. In PA imaging, depending on the concentration and excitation wavelength of the probe, the intensities of the probe signals may be lower than those of the background signals produced by intrinsic optical absorbers such as haemoglobin. For probe imaging in the presence of strong background signals, multispectral photoacoustic (MS-PA) imaging was evaluated. In MS-PA imaging, the spectral fitting method, which distinguishes the probe signals from background signals using reference spectra, has been widely used. To compensate for the decrease of fluence due to optical attenuation in biological tissue, we used a simplified compensation method that calculates fluence inside biological tissues by the Monte-Carlo model using published data on optical properties of biological tissues. The validity of the method was confirmed using tissue-mimicking phantoms. Finally, MS-PA imaging of a mouse subcutaneous tumour injected with the activatable probe was demonstrated. In conclusion, our MS-PA imaging technique afforded successful detection of the activated probe in the tumour, and time-increase of PA signals were successfully observed.

Plasmon induced transparency (PIT) with graphene metamaterials is investigated with the finite-difference time-domain method. Interestingly, the modulation of the PIT transparency window can be achieved by changing not only the gap distance between the two resonators but also the polarization angle of the excitation light. The three-level plasmonic system is employed to clearly explain the formation mechanism of the PIT effect. The analytical results show good consistency with the numerical calculations. Moreover, the PIT resonant wavelength and group delays of incident waves can be dynamically tuned by varying the Fermi energy of the graphene. Our designed graphene nanostructure is promising for the development of compact elements such as tunable sensors, switches and slow-light devices.

In this article, a hybrid plasmonic waveguide (HPW) consisting of a thin metal film sandwiched between two identical cylindrical semiconductor nanowires is proposed and investigated numerically. With two air grooves carved symmetrically on the upper and lower surfaces of the metal film and two nanoscale semi-cylindrical ridges formed, the structured metal film and the semiconductor nanowires are embedded in a low-index silicon-dioxide medium. Based on the finite element method, our simulation results show that the proposed HPW can achieve a propagation length longer than 1000 μm in all circumstances, as well as a mode area as small as 5.21 × 10⁻⁴λ², and an excellent figure of merit. The high-performance of the novel HPW may provide theoretical guidance for further research of HPW and related applications in photonic integrated circuits.

We demonstrate the possibility of decelerating chirped soliton formation at  femtosecond pulse propagation in a medium with gold nanoparticles. We take into account the dependence of one-photon absorption on the nanorod aspect ratio and time-dependent nanorod aspect ratio changing due to nanorod reshaping because of laser energy absorption. The soliton formation occurs due to laser radiation trapping by the nanorod reshaping front. We show analytically that a chirp induced by the negative phase grating is crucial for this trapping.

We have developed a semi-analytical approach to the modulation transfer function (MTF) for negative-index flat lenses based on photonic crystals (PhCs). Contributions of various PhC modes to the MTF have been identified and analyzed. With a certain surface termination, a high-order PhC surface mode can be tamed to produce a broad angular resonance. As such, the isotropy of the image field can be significantly enhanced, resulting in an ideal image formation with nearly perfect outgoing circular wavefronts. Ray-optics analysis has also been utilized to assist the design of a negative-index flat lens. Finite-difference time domain simulations confirm the effectiveness of PhC lens designed by this semi-analytic approach to the MTF.

In this work, we analytically find that perfect waveguide mode conversion could be realized in a waveguide structure with zero index metamaterials (ZIMs) by constructing air grooves inside ZIMs. Such a result is not only obtained in a straight waveguide, but may extend to bending or even multi-port waveguide systems. The proposed ZIM waveguides could also simultaneously work as optical diodes, which can effectively achieve unidirectional transmission. All phenomena are well verified by numerical simulations.

We investigate a broadband perfect absorber for microwave frequencies, with a wide incident angle, using resistive sheets, based on both simulation and experiment. The absorber uses periodically-arranged meta-atoms, consisting of snake-shape metallic patterns and metal planes separated by three resistive sheet layers between four dielectric layers. We demonstrate the mechanism of the broadband by impedance matching with free space, and the distribution of surface currents at specific frequencies. In simulation, the absorption was over 96% in 1.4–6.0 GHz. The corresponding experimental absorption band over 96% was 1.4–4.0 GHz, however, the absorption was lower than 96% in the 4.0–6.0 GHz range because of the rather irregular thickness of the resistive sheets. Furthermore, it works for wide incident angles and is relatively independent of polarization. The design is scalable to smaller sizes in the THz range. The results of this study show potential for real applications in prevention of microwave frequency exposure, with devices such as cell phones, monitors, and microwave equipment.

We propose and demonstrate a novel refractometric sensor based on optofluidic technology in photonic crystal fibers with a composite core. The composite core consisting of a ring-like fluid channel around the refractive index matching core is architected within photonic crystal fibers. A different refractive index of water-like analyte is filled into the same channel in turn to form steady microflows around the matching core, and the refractive index of analyte can be detected by observing the resonant coupling between the composite and solid-core modes. The sensitivity of water-like analyte around 1.33 is about −1.11 × 10³ nm per refractive index unit. Simulations indicate that analyte refractive index sensing possesses a dynamic range of 1 to 1.4. We also analyze the matching core with different refractive indices and optimize the structure. Since this kind of refractomeric sensor can be reused with high sensitivity by controlling the refractive index of matching core at different temperatures, it is a good candidate for bio-sensing.

In this paper, we report the tunable and enhanced SERS activity of magneto-plasmonic Ag–Fe₃O₄ nanocomposites that are synthesized by a one pot method. Crystal violet (CV), rhodamine 6 G (R6G) and 4-mercaptobenzoic acid (4-MBA) molecules are used to investigate the SERS optical activity of Ag–Fe₃O₄ nanocomposites under different external magnetic fields (1500, 2000, 2500, 3500 and 5000 gauss). The experimental results demonstrate the enhanced Raman effects that can be obtained by increasing the magnitude of external magnetic field. This is because the electromagnetic hot spots located between the neighboring Ag–Fe₃O₄ nanocomposites can be tuned by utilizing the external magnetic field. The bigger density of the hot-spots and amplitude of the electric field in the hot-spot are responsible for the enhanced SERS effect. The detection limit of CV molecule can be at least down to 10⁻⁹ M. The spectra measurements of hemoglobin adsorbed on the Ag−Fe₃O₄ nanocomposites under different external magnetic fields are also performed to explore its bio-applications. Finite element method (FEM) is used to simulate the local electromagnetic field distribution in Ag–Fe₃O₄ nanocomposites, revealing the SERS enhanced mechanism is determined mainly by the near field enhanced electromagnetic field. Due to its tunable and enhanced properties, Ag–Fe₃O₄ nanocomposites are expected to be promising SERS substrates for chemical and biological sensing applications.

In this paper, we report a widely tunable, narrow linewidth, low noise continuous-wave double-clad Er:Yb doped fiber ring laser. Tunability is demonstrated in wide range spanning from 1520 to almost 1620 nm covering the C and L spectral bands. The cavity design is optimized in order to achieve the largest tuning range with very high optical signal-to-noise ratio (SNR). The output coupling ratio greatly influences the tuning range of the laser while the position of the spectral filter determines the SNR. The obtained laser exhibits a tuning range over 98 nm with a nearly constant SNR of about 58.5 dB.

We study the effects of pump modulation in a cavity soliton laser consisting of a vertical cavity surface emitting laser with an intra-cavity saturable absorber. We show that a drifting soliton experiences enhanced mobility features by modulating the pump at the resonance frequency, and the effects are even larger below resonance. In particular, pump modulation reduces the rest time of the soliton in the initial stage of the motion and it increases its drift velocity in this regime. Moreover, pump modulation allows a decrease in the switching energy of the soliton to an amount equal to 36 photons. These results indicate that pump modulation is a promising way for the use of a cavity soliton laser as a fast optical buffer and an ultra low-energy optical switch.

Gain spectrum for a diffraction-free solution of the steady-state paraxial wave equation for the Stokes amplitude is shown to be frequency-asymmetric relative to an exact Raman resonance. The asymmetry is directly related to a light guiding originated due to Raman-induced change of the refractive index and imprinted in the Stokes far field. The index guiding with a zero-order Bessel beam pump is experimentally observed at the Stokes generation in high-pressure hydrogen and barium nitrate, and reasonable agreement with numerical data is demonstrated.

We investigate the scattering properties of the invisible lens in two ways. First, we describe the scattering of electromagnetic waves by the invisible lens realised by a purely dielectric, purely magnetic, and impedance-matched medium, respectively, using Debye potentials. Second, we employ the Wentzel–Kramers–Brillouin method to analyse the scattering of scalar waves by the lens. We show that in all cases the scattering is negligible for a discrete set of frequencies, while for other frequencies there is a phase slip at the boundary of the lens 'shadow'.

We report a series of simulation studies which extends pattern-illuminated Fourier ptychography microscopy by integrating with the nonlinearity arising from saturation of the fluorophore excited state for super-resolution fluorescence imaging. This extended technique, termed Saturated pattern-illuminated Fourier ptychography (SpiFP) microscopy, could achieve a resolution four times that of wide field when the illuminating light intensity approaches the saturation threshold in simulations. Increasing light intensity leads to further resolution enhancement. In order to demonstrate the performance of SpiFP, we make a comparison between SpiFP and saturated structure illumination microscopy in simulations, and prove that the SpiFP exhibits superior robustness to noise, aberration correcting ability, and pattern's flexibility. Introducing the saturation of the fluorescent emission brings in notable improvements in imaging performance, implying its potential in nanoscale-sized biological observations by wide-field microscopy.

The aim of the present work is to give a geometrical characterization of Durnin's beams. That is, we compute the wavefronts and caustic associated with the nondiffracting solutions to the scalar wave equation introduced by Durnin. To this end, first we show that in an isotropic optical medium $\psi ({\bf{r}},t)={{\rm{e}}}^{{\rm{i}}[{k}_{0}S({\bf{r}})-\omega t]}$ is an exact solution of the wave equation, if and only if, S is a solution of both the eikonal and Laplace equations, then from one and two-parameter families of this type of solution and the superposition principle we define new solutions of the wave equation, in particular we show that the ideal nondiffracting beams are one example of this type of construction in free space. Using this fact, the wavefronts and caustic associated with those beams are computed. We find that their caustic has only one branch, which is invariant under translations along the direction of evolution of the beam. Finally, the Bessel beam of order m is worked out explicitly and we find that it is characterized by wavefronts that are deformations of conical ones and the caustic is an infinite cylinder of radius proportional to m. In the case m = 0, the wavefronts are cones and the caustic degenerates into an infinite line.

A simple experimental method for determining the number of modes in planar dielectric multi-mode waveguides, and the effective index difference of these modes, is presented. Applying a thin, dye-doped polymer cladding, the fluorescence excited by multiple modes propagating in a silicon nitride slab waveguide is imaged to extract information. Interference between the modes produces a structured intensity profile along the waveguide which is constant in time. The spatial frequencies of this intensity profile are directly linked to the propagation constants of the underlying modes. Through a discrete Fourier transform, the modes' effective index differences are found and compare well with analytically calculated values. Furthermore, the amplitudes in the Fourier transform are directly related to the power in each mode. Comparing the amplitudes of the Fourier components as a function of propagation distance, an estimate of the propagation losses of the individual modes relative to one another is made. The method discussed could be applied to analysing mode behaviour in integrated photonic devices, most notably in mode-division multiplexing.

We propose a flexible design for implementation of the Luneburg lens with gradient photonic crystals. The full-wave simulation results demonstrate the excellent performance of omnidirectional focusing of the designed Luneburg lens over a broad frequency band, and firstly exhibit anisotropic focusing in the designed Luneburg lens with a specific frequency band. In this study, our effort is focused on figuring out the operating wavelength range where the effective medium approximation theory is applicable, and the mechanism for generating anisotropic and omnidirectional focusing in Luneburg lens structure.

The propagation of TE modes in a rectangular metal waveguide with an integrated structure of finite length containing a graphene monolayer was studied by a modal decomposition method. The modal decomposition method generates a number of linear algebraic equations related to waveguide modes taken into account. The system of linear algebraic equations for the amplitudes of all modes was derived taking into account the propagation as well as a number of local evanescent modes. Truncation errors in the function of the number of modes taken were quantified by using the discontinuities of the boundary conditions for the transverse electric and magnetic field components at the interface of the discontinuities of the structure. The transmitted and reflected TE₀₁ mode amplitudes were calculated versus the graphene layer length, its position inside the waveguide cavity and the graphene electrons chemical potential. It was shown that the symmetrical positioning of the graphene layer inside the waveguide cavity is the most suitable to achieve the largest amplitude modulation, whereas an asymmetrical positioning is suitable to achieve a large phase modulation of the transmitted mode. It was shown that the phase modulation increases with a transmission coefficient modulation increase and decreases with a transmission coefficient increase. It was found that the phase modulation firstly increases almost proportionally with the graphene layer length for relatively short structures; however, after passing through a maximum, the phase modulation becomes almost constant and hence only weakly dependent on the graphene layer length. Typical phase modulation values turn out to be about 15–20 degrees for minimal transmission coefficients of about 0.6–0.4; however, the phase modulation reaches a value of about 45 degrees if the transmission coefficient modulation equals 50%.

We studied the nonlinear parametric interaction of femtosecond fractionally-charged optical vortices in a Raman-active medium. Propagation of such beams was described using the Kirchhoff–Fresnel integrals by embedding a non-integer 2π phase step in a Gaussian beam profile. When using fractionally-charged pump or Stokes beams, we observed the production of new topological charge and phase discontinuities in the Raman field. These newly generated fractionally-charged Raman vortex beams were found to follow the same orbital angular momentum algebra derived by Strohaber et al (2012 Opt. Lett.37 3411) for integer vortex beams.

An algorithm for the direct unwrapped phase estimation from the linearly phase shifted interferograms is presented. The temporal fringe intensity along each pixel is represented as a function of fringe amplitude, phase step and the searched phase. These parameters are estimated in the nonlinear least squares sense using the Levenberg–Marquardt algorithm. The proposed method allows the masked interferograms to be handled using a pixel selection approach to provide the appropriate initial conditions at a given pixel utilizing the estimated parameters at one of its neighboring pixels, which results in direct unwrapped phase estimation. Simulation results are provided to evaluate the performance of the proposed method as a function of noise power, spatially varying phase step, number of interferograms and phase step detuning error. The experimental results are also provided in the case of a holographic interferometry setup.

We demonstrate for the first time a Bragg grating-based Fabry–Perot interferometer (FPI) fabricated in the polymer fiber with a core made of PMMA/PS copolymer and pure PMMA cladding. The FPI was formed by two gratings with the same Bragg wavelength, λ_B = 1312 nm, separated by a small gap. The FP cavity was created directly during the grating inscription process by placing a narrow blocking aperture, in the center of the UV beam. Good long-term stability was achieved by fabricating the gratings of type II with long irradiation time (8 min). By choosing an appropriate width of the blocking aperture, we could control the number and the width of interference fringes visible in the grating's reflection spectrum. Sharp fringes were obtained, of 3 dB width within the range of 50–100 pm, allowing for a significant increase in measurement resolution compared to direct interrogation of a single grating. The proposed interferometer was tested in measurements of strain and temperature in the range of 0–20 mstrain and 20 °C–70 °C, showing the sensitivity of 1.074 nm mstrain⁻¹ and −25.1 pm °C⁻¹, respectively.

By performing 1-norm operation for phase-shifting interferograms, a rapid phase shift extraction and phase retrieval algorithm is proposed in random phase-shifting interferometry (PSI). First, by performing the 1-norm operation for a sequence of phase-shifting interferograms, we can construct a group of new interference patterns, possessing less data information but the same phase shifts relative to the original interferograms. Second, based on the least-square iteration (LSI) algorithm, accurate phase shifts can be extracted from these construct interference patterns. Finally, the measured phase can be achieved from the above extracted phase-shifts using the LSI algorithm. The obtained results show that as long as the pixel number of the constructed interference pattern through performing 1-norm operation is about 0.4% pixel number of the original interferogram, the proposed algorithm can achieve accurate phase shifts and be measured. Compared with other iteration algorithms of PSI, in addition to maintaining the advantage of high accuracy, the proposed algorithm reveals outstanding performance in processing speed.

An all-optical tuning method of whispering gallery modes (WGMs) in a silica capillary based on lateral pumping scheme has been proposed and experimentally demonstrated by exploiting the photo-thermal effect of magnetic fluids (MFs) infiltrated into a micro-capillary. The WGMs are coupled from a tapered fiber perpendicular to the silica capillary. Experimental results indicate that the microresonator integrated with MFs shows a Q-factor of 1.2867 × 10⁴ and its optical tunability reaches 0.0382 nm/(mW · mm⁻²) under 532 nm laser illumination. Compared with the one infiltrated with pure water, our proposed capillary shows an optical tuning sensitivity increase of about 4270 times. The experimental results on the WGM resonance wavelength sensitivity for the capillaries with different diameters are in accordance with our simulation outcome using radially dependent heat conducting equation. Further experimental studies indicate that the proposed microresonator possesses a good spectral reversibility. Moreover, the dynamic response experiment shows that the rise and fall time of this microresonator is about 231 ms and 255 ms, respectively. The above intriguing features make the proposed WGM resonator a promising candidate for potential applications in optical filtering, microfluidic sensing, and signal processing as well as reconfigurable devices for future all-optical networks.

Transcranial photobiomodulation (PBM), also known as low level laser therapy (LLLT), relies on the use of red/NIR light to stimulate, preserve and regenerate cells and tissues. The mechanism of action involves photon absorption in the mitochondria (cytochrome c oxidase), and ion channels in cells leading to activation of signaling pathways, up-regulation of transcription factors and increased expression of protective genes. We have studied PBM for treating traumatic brain injury in mice using a NIR laser spot delivered to the head. Mice had improved memory and learning, increased neuroprogenitor cells in the dentate gyrus and subventricular zone, increased BDNF and more synaptogenesis in the cortex. These highly beneficial effects on the brain suggest that the applications of LLLT are much broader than first conceived. Other groups have studied stroke (animal models and clinical trials), Alzheimer's disease, Parkinson's disease, depression and cognitive enhancement in healthy subjects.

We derive exact solutions of Maxwell's equations based on superoscillatory superpositions of vectorial Bessel beams. These novel beams are diffraction-free and can support subwavelength features in their transverse electromagnetic fields, without the presence of any evanescent waves. These features can be propagated into the far field. Approximate solutions in closed form are also derived based on asymptotic expansions of Bessel functions for simple prescribed subwavelength patterns. The superoscillatory characteristics of both electric, magnetic field components (transverse and longitudinal), and the Poynting vector, as well as, the effect of nonparaxiality are systematically investigated.

In this paper we investigate how the dynamics of a two-level atom is affected by its interaction with the quantized near field of a plasmonic slab in relative motion. We demonstrate that for small separation distances and a relative velocity greater than a certain threshold, this interaction can lead to a population inversion, such that the probability of the excited state exceeds the probability of the ground state, corresponding to a negative spontaneous emission rate. It is shown that the developed theory is intimately related to a classical problem. The problem of quantum friction is analyzed and the differences with respect to the corresponding classical effect are highlighted.

We theoretically investigate the nonlinear optical response of a heterodimer comprising a semiconductor quantum dot strongly coupled to a metal nanoparticle. The quantum dot is considered as a three-level ladder system with ground, one-exciton, and bi-exction states. As compared to the case of a two-level quantum dot model, adding the third (bi-exciton) state produces fascinating effects in the optical response of the hybrid system. Specifically, we demonstrate that the system may exhibit picosecond and sub-picosecond self-oscillations and quasi-chaotic behaviour under single-frequency continuous wave excitation. An isolated semiconductor quantum dot does not show such features. The effects originate from competing one-exciton and bi-exciton transitions in the semiconductor quantum dot, triggered by the self-action of the quantum dot via the metal nanoparticle. The key parameter that governs the phenomena mentioned is the ratio of the self-action strength and the bi-exciton shift. The self-oscillation regime can be achieved in practice, in particular, in a heterodimer comprised of a closely spaced ZnS/ZnSe core-shell quantum dot and a spherical silver nanoparticle. The results may have applications in nanodevices for generating trains of ultrashort optical pulses.

It has been demonstrated that textured closed surfaces are good platforms to support spoof localized surface plasmons with great field confinement and enhancement. Here, we propose a non-concentric textured closed structure to significantly increase the level of field confinement and enhancement. By properly texturing the closed surface with gradient-depth grooves, we show that the incoming electromagnetic waves can be highly confined and focused at the deepest groove on the closed surface. Moreover, we find that the field enhancement factor of this structure is quite sensitive to the polarization angle of the incident wave, which can find potential applications in polarization sensors and energy harvesting devices at microwave and terahertz frequencies.

The optical Tamm states (OTSs) localized at the edges of a photonic crystal bounded by a nanoporous silver (NPS) film are investigated. NPS involves spherical vacuum nanopores dispersed in the metal matrix and is characterized by the effective resonance permittivity. The transmission, reflection, and absorption spectra of the structures under study at the normal incidence of light are calculated. It is shown that each Tamm state has its own frequency range where the real part of effective permittivity is negative. The light field localization at the high- and low-frequency OTSs is investigated. The specific features of spectral manifestation of the OTSs are studied in dependence on the nanopore concentration in the metal matrix and on the NPS film thickness.

Confocal laser scanning microscopy (CLSM) has wide applications in biological research and medical diagnosis. However, the spatial resolution and signal to noise ratio (SNR) of CLSM is reduced in the presence of an aberration. Here we improve the pupil-segmentation method to measure and compensate for aberrations in focus modulation CLSM (FM-CLSM), which uses Gaussian-type and doughnut-like foci to scan a sample in sequence. As a result, FM-CLSM can provide images with a high resolution and a high SNR for biomedical or industrial applications.

We present a theoretical study of the influence of the misfit strain on the transverse magneto-optic Kerr effect (TMOKE) and the transverse shift (Imbert–Fedorov effect) experienced by a light beam reflected from the surface of a magnetic/non-magnetic bilayer. The bilayer consists of a magnetic, gyrotropic yttrium-iron garnet film epitaxially grown on a non-magnetic dielectric gadolinium-gallium garnet slab. We use Green's function method to calculate the reflection matrix in the presence of strain. It is shown that the mechanical strain present in the vicinity of the geometrical interface between the constituents of the bilayer can induce a non-negligible contribution to the TMOKE and the Imbert–Fedorov shift (IFS) for incidence angles close to those satisfying the half-wave condition for both layers, at which neither the TMOKE nor the transverse shift would appear in the absence of strain. We analyze the dependence of the IFS on the state of polarization of the incident beam, the thickness of both layers, and the direction of magnetization in the magnetic layer.

We demonstrate, both theoretically and experimentally, a kind of fan-shaped optical beam propagating along the arbitrary trajectories (such as parabolic, hyperbolic and three-dimensional spiraling trajectories). With a controlled profile, this fan-shaped optical beam can be obtained from superposition of the Bessel-like beam and vortex Bessel-like beam. Also, the ability of guiding and transporting microparticles along its lobes is explored. These beams may find a variety of applications in optical trapping and manipulation.

In the laser heterodyne interferometry based on the microchip Nd:YAG dual-frequency laser, the amplitude of the beat note periodically fluctuates in time domain, which leads to the instability of the measurement. On the frequency spectrums of the two mono-frequency components of the laser and their beat note, several weak sideband signals are observed on both sides of the beat note. It is proved that the sideband frequencies are associated with the relaxation oscillation frequencies of the laser. The mechanism for the relaxation oscillations inducing the occurrence of the sideband signals is theoretically analyzed, and the quantitative relationship between the intensity ratio of the beat note to the sideband signal and the level of the amplitude fluctuation is simulated with the derived mathematical model. The results demonstrate that the periodical amplitude fluctuation of the beat note is actually induced by the relaxation oscillation. And the level of the amplitude fluctuation is lower than 10% when the intensity ratio is greater than 32 dB. These conclusions are beneficial to reduce the amplitude fluctuation of the microchip Nd:YAG dual-frequency laser and improve the stability of the heterodyne interferometry.