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Accepted: May. 27, 2020

Posted: Jul. 24, 2020

Published Online: Jul. 27, 2020

The Author Email: Y. F. Chen (yfchen@cc.nctu.edu.tw)

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Y. F. Chen, C. C. Lee, C. H. Wang, M. X. Hsieh. Laser transverse modes of spherical resonators: a review [Invited][J]. Chinese Optics Letters, 2020, 18(9): 091404

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Fig. 1. Calculated patterns of HLG modes $ψn1,n2(α,β)(x˜,y˜)$ with $n1=3$ and $n2=8$ for various values of $α$ and $β$ on the Poincaré sphere.

Fig. 2. Calculated patterns for the standing waves given by $Re[ψn1,n2(α,β)(x˜,y˜)]$ (golden color) as well as $Im[ψn1,n2(α,β)(x˜,y˜)]$ (green color), corresponding to the results in Fig. 1.

Fig. 3. Calculated results for $ψN1,N2(HG)(x˜,y˜)$ and $ΦN1,N2(x˜,y˜,ϕn)$ by using Eqs. (30) and (34) with $(N1,N2)=(3,4)$.

Fig. 4. Calculated results for $ψN1,N2(HG)(x˜,y˜)$ and $ΦN1,N2(x˜,y˜,ϕn)$ by using Eqs. (30) and (34) with $(N1,N2)=(7,8)$.

Fig. 5. Calculated results for $ψN1,N2(HG)(x˜,y˜)$ and $ΦN1,N2(x˜,y˜,ϕn)$ by using Eqs. (30) and (34) with $(N1,N2)=(3,12)$.

Fig. 6. Calculated results for $ψN1,N2(LG)(x˜,y˜)$ and $ΦN1,N2(α,β)(x˜,y˜,ϕn)$ by using Eqs. (43) and (45) with $(α,β)=(π/2,π/2)$ and $(N1,N2)=(3,12)$.

Fig. 7. Calculated results for the standing waves of $Re[ψN1,N2(LG)(x˜,y˜)]$ and $Re[ΦN1,N2(α,β)(x˜,y˜,ϕn)]$ corresponding to the traveling wave shown in Fig. 6.

Fig. 8. Calculated results for $ψN1,N2(α,β)(x˜,y˜)$ and $ΦN1,N2(α,β)(x˜,y˜,ϕn)$ by using Eqs. (43) and (45) with $(α,β)=(2π/5,2π/5)$ and $(N1,N2)=(4,11)$.

Fig. 9. (a) Configuration of the single lens mode converter. Two vertical lines show the positions of the beam waists produced by the spherical matching lens and by the active axis of the cylindrical lens with focal length $f$. (b) Relationship between the $xy$-Cartesian coordinate system and the $x′y′$-Cartesian coordinate system. The $x′$ and $y′$ axes are the active and inactive components of the cylindrical lens.

Fig. 10. Experimental results (first column), numerical wave patterns (second column), and phase structures (third column) for the propagation evolution of the converted beam $Ψ9,0(x,y,z;ζ)$ with $ζ=π/4$. The number in the right side denotes the size of the pattern with the unit $ωo/2$.

Fig. 11. Experimental results (first column), numerical wave patterns (second column), and phase structures (third column) for the propagation evolution of the converted beam $Ψ9,4(x,y,z;ζ)$ with $ζ=π/4$. The number in the right side denotes the size of the pattern with the unit $ωo/2$.

Fig. 12. Experimental results (first column), numerical wave patterns (second column), and phase structures (third coulmn) for the propagation evolution of the converted beam $Ψ8,8(x,y,z;ζ)$ with $ζ=π/4$. The number in the right side denotes the size of the pattern with the unit $ωo/2$.
Fig. 13. Experimental results (first column), numerical wave patterns (second column), and phase structures (third column) for the propagation evolution of the converted beam $Ψ9,4(x,y,z;ζ)$ with $ζ=−5π/36$. The number in the right side denotes the size of the pattern with the unit $ωo/2$.