J. Eur. Opt. Society-Rapid Publ. 21, 35( 2025) 47
Using equation( 5) and the integrals of
Z 2p
0 exp½iðm � nÞuŠdu ¼ 2pd m; n, the following expression can be obtained:
Uðq; h; zÞ ¼ � 4p2 k 2 zf
X1
m¼�1
Z 1
J m
0
ik exp ðikzÞexp 2f v2 0
exp
ik
2f q2
ð�1Þ m c m exp ðimhÞv 0 ð13Þ k v f q0 0
J m
k q z q0 exp
ik
2 q 0 dq 0:
2z q0
In accordance with Euler’ s formula, equation( 13) can be written as:
Uðq; h; zÞ ¼ � 4p2 ik exp ðikzÞexp k 2 zf 2f v2 0 exp ik
2f q2
2
6 4 þi
P1 m¼�1
Z 1
J m
0
Z 1
J m
0 ð�1Þ m c m exp ðimhÞv 0 k k k
2 v f q0 0 J m q exp z q0 2z q0 k k f q0 v 0 J m z q0 q exp k
2z q0 2
q 0 dq 0
3
7 5: q 0 dq 0 ð14Þ
Z 1
�
The Z integrals of x sin ax 2 J v ðbxÞJ v ðcxÞdx and
1
� 0 x cos ax 2 J v ðbxÞJ v ðcxÞdx [ 18 ], and the formula
0
J �m( x)=( �1) m J m( x) are used in deriving equation( 14).
Z 1
0
� x sin ax 2
J v ðbxÞJ v ðcxÞdx
¼ 1 2a cos b 2 þ c 2
� v 4a 2 p
Z 1
0
� x cos ax 2
J v
bc 2a
½a > 0; b > 0; c > 0; Rev > �2Š J v ðbxÞJ v ðcxÞdx
¼ 1 2a sin b2 þ c 2
� v 4a 2 p
J v
bc 2a
½a > 0; b > 0; c > 0; Rev > �1Š
See the Equation( 17) bottom of the page ð15Þ
ð16Þ
To generate nondiffracting radial carpet-lattice beams, we choose a radial amplitude grating with a sinusoidal transmission function [ 1 ]: tðuÞ ¼ AðuÞ ¼ 1 1 þ cos mu 2 ½ ð ÞŠ ¼ 1
2 þ 1 ð
4 eimu þ e �imu Þ; ð18Þ where m is the number of spokes of the grating.
We can obtain c 0 = 1 / 2, c m = c �m = 1 / 4 and c n6¼m = 0 by comparing equations( 2) and( 17). We then substitute the coefficients into equation( 17). The distribution of the resulting light field is given as:
Uðq; h; zÞ ¼� ik exp ik z þ v 2
0
2f
2f � zv
2 0
� q2 v f 2 0
2z
k
J 0 f qv 0
þð�iÞ m k cos ðmhÞJ m f qv 0
: ð19Þ
The intensity distribution of the light field at distance z can be calculated as I( q, h, z)= U( q, h, z) U *( q, h, z), where * denotes a complex conjugate. Hence, we can obtain:
I ðq; h; zÞ ¼ k2 2
v 0
4f 2
þJ 0
2
½ ð�iÞ m þ ðÞ i m k f qv 0
k
Šcos ðmhÞJ 0 f qv 0 J m
þ cos 2 2 k ðmhÞJ m f qv 0
k f qv 0
: ð20Þ
Equation( 17) shows that our constructed radial carpetlattice beams are propagation-invariant because the propagating distance z does not appear in the argument of the Bessel function J 0 and J m. The optical distribution of radial carpet-lattice beams is always the same for different propagating distances z. On the basis of equation( 20), we typically simulate several types of nondiffracting radial carpet-lattice beams with different parameters in MATLAB, as shown in Figure 1. Noted that all simulated subgraphs in Figure 1 share the same color bars. Figure 1 shows that the lattice characteristics of nondiffracting radial carpet-lattice beams are different when the number of spokes m of the radial grating is odd and even. From equation( 20), [( �i) m +( i) m ] isalways0whenm is odd. When m is even, there is [( �i) m +( i) m ] = ± 2. These results
indicate that the optical structures of odd and even order nondiffracting radial carpet-lattice beams are fundamentally different. As shown in Figure 1, the m-order radial carpet-lattice beams have m spokes for the radial grating with even number of spokes. Figure 1 shows that as the spokes of m increases, the carpet beam becomes increasingly close to the lattice structure, which is the reason for naming it as the carpet-lattice beam. For the radial grating with odd number of spokes, the m-order radial carpet beams have 2m spokes, thereby exhibiting fundamentally different from even order radial carpet-lattice beams. These different lattice properties may lead to new different uses. U ð q; h; z Þ ¼ � ik f
exp ik z þ v2 0 2f
� zv2 0 f 2
� q2
2z
( k v 0 c 0 J 0 f qv 0
þ) X1 ð�iÞ m k
½ c m exp ðimhÞþc �m exp ð�imhÞŠJ m f qv 0: m¼�1 ð17Þ