Complex structured beam direct generation by coherent superposition of a complete set of degenerate eigenmodes

Zilong Zhang; Zilong Zhang; Zilong Zhang; Suyi Zhao; Suyi Zhao; Suyi Zhao; Xin Wang; Xin Wang; Xin Wang; Wei He; Wei He; Wei He; Yuqi Wang; Yuqi Wang; Yuqi Wang; Changming Zhao; Changming Zhao; Changming Zhao

doi:10.1364/OE.488812

1. Introduction

Structured laser beams with unique spatial characteristics have attracted great interests [1]. The rich beam structures have brought new breakthroughs to many fields, including imaging [2,3], metrology [4,5], optical communications [6–8], optical trapping [9,10], and quantum information processing [11,12]. The exploration of the formation mechanism and physical properties of optical transverse modes will further promote the related applications of structured laser beams.

In order to describe the formation dynamics of laser patterns under time average, the Helmholtz equation without time term was derived based on Maxwell's electromagnetic field theory [13]. It is the basic wave equation that the electric vector of the optical-frequency electromagnetic field should satisfy under the scalar approximation. A series of transverse laser modes are solved by simplifying the equation under certain conditions. The Hermite-Gaussian (HG) beam, Laguerre-Gaussian (LG) beam, and Ince-Gaussian (IG) beam are a set of complete solutions using Cartesian coordinate, cylindrical coordinate, and elliptical coordinate under the condition of paraxial approximation [14–17]. The diffraction-free beams, including the Bessel beam, Mathieu beam, and Weber beam, are solved in free space based on the cylindrical, elliptic, and parabolic cylindrical coordinates respectively [18–21]. Although these modes obtained under the ideal single-mode condition have extended the cognition of laser spatial characteristics, more complex structured laser patterns still can be observed from real laser cavities. The eigenmode coherent superposition regime can not only explain the conversion relationship between eigenmodes, but also fill in the blank transition region of light fields characterized by the Bloch sphere or Poincare sphere for structured light, greatly enriching the understanding of actual complex light field [22–24]. The Hermite-Laguerre-Gaussian (HLG) modes are superimposed to obtain complex geometric modes with high dimensional coherent states [25–27]. The coherent superposition of even and odd IG modes creates the Helical-Ince-Gaussian (HIG) mode [28]. SU(2) modes superposed by prior frequency-degenerate eigenmodes not only have the transverse singular distribution, but also have the controllable longitudinal divergence characteristic [29]. Although the optical field structure obtained by the coherent superposition of two or more eigenmodes can match the partial mathematical expressions derived from physical models, any optical field can be decomposed by a complete eigenmode set in theory. And based on the self-reproduction physical mechanism of laser transverse mode formation, the degenerate relation of eigenmode components can be taken as a constraint condition to simplify the mode spectrum model of propagation-invariant beams.

In this article, the degenerate eigenmodes spectrum model of complex structured beams is investigated in both theoretical analysis and experiments. The light field of propagation-invariant structured beams can be described accurately and concisely by the coherent superposition of degenerate eigenmodes in a complete set. The degenerate eigenmode spectrum considering both amplitude and phase relations is displayed visually in three-dimensional form. The formation of complex structured beam patterns based on this regime is well explained, and a good agreement with the experimental results of both significantly and slightly different beam patterns within the same order is also authenticated. The directly generated structured laser beams from a cavity in previous works were usually not quite complex. Here, the ‘complex’ is not only about the beam pattern structure itself, but also about the composition of the beam. And it has not been clearly and quantitatively investigated that a cavity mode can be decomposed to a complete set of degenerate eigenmodes, previously. This work offers a more universal method to analyze the eigenmode composition of the propagation-invariant complex structured beams from laser cavity.

2. Theoretical analysis

In fact, any orthogonal basis set as a solution to the paraxial wave equation can be used to describe the optical field of any complex transverse laser mode based on the mode superposition theory. However, the careful selection of the basis and parameters of the eigenmode components will optimize the mathematical expression of the whole optical field [30]. Considering the two-dimensional characteristics of the transverse plane, we choose the axisymmetric Hermite-Gaussian (HG) mode as the basic mode. HG beam is the general analytic solution of Helmholtz equation in Cartesian coordinate system, and the complex amplitude distribution of its light field is [14],

(1)$$H{G_{mn}}({x,y,\textrm{z}} )= \frac{{C_{m,n}^{HG}}}{{{\omega ^2}}}\exp ( - \frac{{{x^2} + {y^2}}}{{{\omega ^2}}}){H_m}(\frac{{\sqrt 2 x}}{\omega }){H_n}(\frac{{\sqrt 2 y}}{\omega })\exp \left[ {ikz + ik\frac{{{x^2} + {y^2}}}{{2R(z)}} - i(m + n + 1)\psi (z)} \right],$$

where $\textrm{C}_{\textrm{m,n}}^{\textrm{HG}}$ is the normalized constant of HG modes, H_m and H_n are the m-th and n-th order Hermite polynomial. $\mathrm{\omega }$ is the half-width of the beam at the position z, ${\mathrm{\omega }^\textrm{2}}\mathrm{\ =\ \omega }_\textrm{0}^\textrm{2}\textrm{(z}_\textrm{R}^\textrm{2}\textrm{ + }{\textrm{z}^\textrm{2}}\textrm{)/z}_\textrm{R}^\textrm{2}$, ${\mathrm{\omega }_\textrm{0}}$ is the waist radius of the fundamental mode beam, and z_R is the Rayleigh length. R(z) is the radius of curvature associated with z and the Gouy phase varying with the transmission distance is $\mathrm{\psi }(\textrm{z} )\textrm{ = arctan(z/}{\textrm{z}_\textrm{R}}\textrm{)}$.

Considering the coupling relationship between the eigenmode components, the complex transverse modes in the laser cavity can be regarded as the direct superposition of multiple complex amplitudes of HG modes, and their optical field can be expressed as,

(2)$$E({x,y,z} )= \sum\nolimits_{m,n} {{a_{m,n}}H{G_{m,n}}({\cdot} )\exp [i{\phi _{m,n}} + ikz + ik\frac{{{x^2} + {y^2}}}{{R(z)}} - i({N + 1} )\psi (z)]},$$

where HG_m,n(·) is the pure strength item of HG modes, a_m,n is scale coefficient of each mode. The modal weightings are in general a set of complex number whose phase plays no less role than the modulus. ${\phi _{\textrm{m,n}}}$ are initial phase differences between the basic modes. For higher-order transverse modes, different degenerate modes are involved in the same mode order N, and the change rate of Gouy phase is proportional to the mode order N. For HG modes, N = m + n (m,n = 0,1,2…).

Degenerate eigenmodes with the same order have consistent diverging properties. Due to the coupling effect triggered by spontaneous or self-induced processes, combined with the inherent nonlinearity of the laser cavity, complex transverse modes can be formed in the cavity [22–23]. The transverse modes have a common frequency, that is, the average of the eigenmodes participating in the oscillation, while having a stable and self-similar transverse pattern independent of propagation [31]. Therefore, the intracavity output complex mode can also be considered as the contributions of all degenerate eigenmodes under the same order, and Eq. (2) can be rewritten as,

(3)$$E(x,y,z) = \exp \left( {i\overline \omega t + i\frac{{\overline \omega }}{c}\frac{{{x^2} + {y^2}}}{{R(z)}} - i({N + 1} )\psi (z)} \right) \cdot \sum\limits_{m = 0}^{m = N} {{a_{m,N - m}}H{G_{m,N - m}}({\cdot} )\exp (i{\varphi _{m,N - m}})} .$$

Here, $\mathrm{\bar{\omega }}$ is the averaged optical frequency of all degenerate eigenmodes under the same order. As for a real laser beam, the optical frequency of the degenerate eigenmodes may be slightly different due to the cavity nonuniformity. Furthermore, the light field is simplified to the form of coherent superposition of whole degenerate eigenmodes with corresponding complex scale coefficients, which can be calculated according to the orthogonal mode decomposition theory [32,33].

(4)$$E(x,y,z) = \sum\limits_{m = 0}^{m = N} {{c_{m,N - m}}H{G_{m,N - m}}(x,y,z)} ,$$

(5)$${c_{m,N - m}} = \frac{{\int\!\!\!\int {E(x,y)HG_{m,N - m}^\ast (x,y)dxdy} }}{{\sqrt {\left( {\int\!\!\!\int {{{|{E(x,y)} |}^2}dxdy} } \right)\left( {\int\!\!\!\int {{{|{H{G_{m,N - m}}(x,y)} |}^2}dxdy} } \right)} }}.$$

Since the eigenmode components are coupled with certain initial phases, the scale coefficients are complex, c_m,N-m= p_m,N-m+ iq_m,N-m, The intensity and phase information of eigenmode components can be separated by calculating the modulus and phase angle. The relative scale coefficient (normalization based on the maximum value) and the initial phase of each eigenmode component can be calculated as,

(6)$${{a_{m,N - m}} = |{{c_{m,N - m}}} |= \frac{{\sqrt {{p_{m,N - m}}^2 + {q_{m,N - m}}^2} }}{{\max \left( {\sqrt {{p_{j,N - j}}^2 + {q_{j,N - j}}^2} } \right)}}}\;\;\;{j = 0,1, \cdots ,N}$$

(7)$${\varphi _{m,N - m}} = \arg ({c_{m,N - m}}) = \arctan \left( {\frac{{{q_{m,N - m}}}}{{{p_{m,N - m}}}}} \right).$$

Based on the above theoretical analysis, transverse modes of any cavity output can be accurately represented by a finite number of degenerate eigenmodes. All degenerate eigenmodes with the same order are the analytical function related to the transmission distance z, and their transmission behaviors are similar. The spatio-temporal bistable state formed by the transverse mode has consistent intensity distribution in the near-field and far-field, and the mode spectral distribution remains unchanged at different transmission distances. The intensity patterns of the complex transverse mode and the corresponding mode spectrum distribution are shown in Fig. 1.

Fig. 1. The light intensity patterns and mode spectrum distributions of structured beams generated by coherent superposition of complete set of degenerate eigenmodes. The mode orders of degenerate eigenmodes are N = 3, 4, 5 due to the different pumping conditions in the laser cavity. The X, Y and Z axes of the mode spectrum represent the eigenmode index, initial phases and relative scale coefficients (normalization based on the maximum value).

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Structured beams can be generated by coherent superposition of whole degenerate eigenmodes under a certain mode order according to the conditions of the laser cavity. The eigenmode index, initial phases, and relative scale coefficients can be characterized by the three-dimensional mode spectrum. The three parameters controlling the coherent superposition relationship of degenerate eigenmodes in a complete set determinate the generated structured beams. And their complicated combination changes can describe the light field of any complex structured beams from the cavity. Considering the transformations between different coordinate systems, Eq. (4) can also give the light field under other eigenmode basis set such as LG and IG modes. And one can discover that it’s not too complicated for the degenerate HG eigenmodes to express these modes. It’s should be noted that laser beams with simple structures also belong to Eq. (4), just with most of the scale coefficients being zero.

One more actual condition should be clarified here. It’s that for a real laser cavity with a large Fresnel number and high power output, patterns of the laser beam are so complicated that no obvious structures can be observed. These patterns are the results of complex combinations of different eigenmodes of multiple orders. Some of the modes may be coherently coupled, while others may be in incoherent oscillations. The eigenmodes in different orders are hardly spatially coherent due to the relatively large frequency differences. While the eigenmodes in the same order are easy to be coherently coupled due to their quite close frequencies. In general, the eigenmodes in the same order will coherently couple into a complex mode, while these modes in different orders will incoherently superimpose a beam pattern without apparent structures. If the modes’ orders in the cavity are high enough, a top-hat liked beam profile can be obtained. Thus, we could simulate almost any laser beam by a sum of several equations similar to Eq. (4) but with different orders N. Conversely, any laser beam can be analyzed by the eigenmode spectrum distributions shown in Fig. 1. The intensity ratio and initial phase of each eigenmode are determined by comprehensive effect of the net gain distribution and the cavity parameters.

3. Experimental setup and results

To achieve the large Fresnel number pumping, a microchip laser cavity is used in the experiment. The microchip is composed of a gain crystal and a birefringent crystal chip. The gain crystal is Nd:YAG chip with 1.0% concentration for Nd atom. The birefringent crystal is LiTaO₃ (LTO) with n_e= 2.14 and n_o= 2.136 at 1064 nm. The sizes of the two chips are both 0.7mm × 5mm × 5 mm. The cavity is a Fabry-Perot cavity with two plane surfaces of the chips. The pump laser is an 808 nm laser diode with a coupled fiber whose core diameter is 100µm. And a coupled lens system with 1:1 magnification is used to transmit the pump beam into the microchip cavity. The oscillated transverse modes in the cavity have two perpendicular polarizations. As the refractive indexes of the two polarizations are different, the cavity lengths for two polarized modes are also different, making the two perpendicularly polarized transverse modes in different states. Under low pump power, both polarization states have clear structures, which are the results of coherent superpositions of HG or LG eigenmodes. The orders of the two polarized modes are usually different due to the difference between Fresnel numbers, gain coefficients and modes competitions. Once the pump power increases, one of the two polarized modes will capture most of the power in the cavity and the remaining one is still in a low gain condition. Then the pattern of this high-power mode gets into a quite fuzzy state, which is the result of a complex superposition of lots of basic eigenmodes. While the other polarized mode can maintain a purely coherent state at a relatively large pump power range. Thus, we can use this dual-medium microchip laser to obtain pure structured laser beams easily. The experimental setup and principle are shown in Fig. 2.

Fig. 2. Schematic diagram of experimental device for generating complex structured beams.

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With the microchip cavity shown in Fig. 2, we can obtain various of structured modes. Lots of these modes are shown in our earlier works [34–37]. We select the order of transverse modes as 4 to prove the theory in section 2 of this work. By carefully adjusting the pump spot size incident the cavity and pump power, we can obtain the stable state of the 4^th order eigenmode oscillation. Since the pump spot is incident to the cavity in a circular shape and its intensity is close to the flat-top distribution. The output beam has a good symmetry, and it’s easy to obtain a complex mode which is composed by all eigenmodes with the same order. As the short Fabry-Perot cavity has no obvious selectivity on certain eigenmode, most of the eigenmodes under a certain order can oscillate successfully. The switch of transverse modes comes from the variation of gain distribution and related thermal effects, which can be achieved by slightly adjusting the pump beam incident angle under a certain pump power. Around the normal incident of the pump beam, the laser output has a structured pattern, which is a coherent combination of all the eigenmodes under the same order. The phase and intensity of each mode component may be different according to the whole beam profile modulation in the cavity. Various of structured beam patterns have been obtained experimentally, which almost each of them can be analyzed by the degenerate eigenmodes spectrum. Partial experimental patterns are shown in Fig. 3 and Fig. 4. The structured beam patterns shown in Fig. 3 have obvious variations from each other. These patterns all can be decomposed into the form of HG-mode spectrums with the complete 4^th order eigenmodes, which are the five modes of HG₀₄, HG₁₃, HG₂₂, HG₃₁, and HG₄₀ in the series. The intensity ratios and the relative phases of these five modes are characterized by the 3-dimentional charts in Fig. 3(c). The X-axis presents the indexes of HG eigenmodes, Y-axis presents the initial phases (from 0 to π) of each eigenmode, and Z-axis presents the normalized intensity coefficients of each eigenmodes. By comparing row (a) and (b) of Fig. 3, one can discover a good similarity between them. As the phase and the intensity of each eigenmode used for simulation is discretized with an interval of 1/4π and 0.25 respectively, the simulated patterns may have little differences with the experimentally measured ones in some detail positions. We believe that quite high similarity can be achieved by setting parameters accurately.

Fig. 3. Complete set of eigenmodes spectrum analysis and comparison to experimentally obtained structured patterns under order 4. The experimental patterns, simulated patterns and eigenmodes spectrums are shown in (a), (b) and (c) respectively.

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To better investigate and reveal the variation of complete eigenmode components forming structured beams, we also select four patterns which are quite similar to each other. These patterns are obtained by a continuously slight adjusting of the incident angle (about 0.03°, 1/10 of the whole rotation) of pump beam in the horizontal direction. We can observe that these patterns all have an elliptical profile, while the internal structure having continuous changes. By analyzing the eigenmode spectrum in Fig. 4(c), we can discover that these patterns have regularly varied eigenmode spectrums.

Fig. 4. Beam patterns and mode spectrum of a slightly adjusted process.

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We make further analyzations by Fig. 5 for convenience. Figures 5(a) and (b) show the variation curves of intensity scale coefficients and relative phases of each mode component. By analyzing the mode spectrum, it can be seen that most eigenmodes’ distributions can keep basically similar for profile slowly varied patterns. The intensity coefficients of HG₀₄ and HG₄₀ eigenmodes maintain the same (to be 1.0) with the phase of HG₄₀ unchanged during the modulation. For HG₁₃ and HG₃₁ eigenmodes, both the intensity and the phase are not changed too much. While both the intensity and phase experience obvious variations for the HG₂₂ mode.

Fig. 5. Eigenmode spectrum analyzation of the similar beam patterns. The mode states (1)–(4) are in correspondence with four light field shown in Fig. 4.

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In the Fabry-Perot microchip cavity, the structured patterns will be decided by the most efficient energy output result covering the gain distribution and nonlinear couplings [22,38,39]. The relative intensity and phase of the eigenmodes are modulated under a comprehensive effect. Therefore, the mode spectrum has no apparent regularity during large tuning scales. While the patterns really experience a relatively continuous variation when the cavity parameter is slightly changed. For two quite similar patterns, the eigenmode spectrums might be strongly related. As there is a similarity between this work and the frequency-degenerate eigenmodes composed SU(2) modes, it’s necessary to make a distinguish of them. Different from eigenmodes achieved by making the ratio of transverse mode spacing to longitudinal mode spacing satisfy specific parameters, the degenerate eigenmodes referred in this article mainly focus on the same longitudinal mode. The degenerate eigenmodes with different transverse mode indexes are complete for a certain transverse mode order. Not only their optical frequency is degenerate, but also their divergence speed and Gouy phase change rate are the same, resulting in the propagation-invariant structures of their coherent superposed beams.

4. Conclusion and discussions

We proved that complex and propagation-invariant structured laser beams can be decomposed to a complete set of degenerate eigenmodes. The three-dimensional eigenmode spectrum can be used to well express these complex beams. Taking the HG beam as an example, the feasibility of resolving complex propagation-invariant modes with the complete eigenmode spectrum is illustrated. The spectral analysis of complete degenerate eigenmodes can analyze the eigenmodes’ combination characteristics of any propagation-invariant modes in theory, such as LG, IG, and BG beams, etc. When the exact complex amplitude of a light field is known, the detailed combination of eigenmodes can be obtained by solving the mode spectrum. In practice, it is relatively difficult to accurately detect the phase of beams with complex structures, while the light intensity collected by CCD is easy to obtain. With help of the deep learning method, we can match the structured beams with high accuracy by establishing a database of coherent superposition states of complete eigenmodes under a certain order, thus obtaining the accurate eigenmode spectral distribution.

Funding

National Natural Science Foundation of China (61805013).

Disclosures

The authors declare no conflict of interest.

Data availability

Data that support the findings of this study are available from the corresponding author upon reasonable request.

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Complex structured beam direct generation by coherent superposition of a complete set of degenerate eigenmodes

Abstract

1. Introduction

2. Theoretical analysis

3. Experimental setup and results

4. Conclusion and discussions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (7)

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