Jia-Sheng Ye,
Bi-Zhen Dong,
Ben-Yuan Gu,
and Shu-Tian Liu
J.-S. Ye, B.-Z. Dong (bzdong@aphy.iphy.ac.cn), and B.-Y. Gu are with the Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100080, China.
J.-S. Ye and S.-T. Liu are with the Department of Physics, Harbin Institute of Technology, Harbin 150001, China.
Jia-Sheng Ye, Bi-Zhen Dong, Ben-Yuan Gu, and Shu-Tian Liu, "Analysis of a cylindrical microlens array with long focal depth by a rigorous boundary-element method and scalar approximations," Appl. Opt. 43, 5183-5192 (2004)
We investigated the focal characteristics of open-regional cylindrical microlens arrays with long focal depth by using a rigorous boundary-element method (BEM) and three scalar methods, i.e., a Kirchhoff and two Rayleigh-Sommerfeld diffraction integral forms. Numerical analysis clearly shows that the model cylindrical microlens arrays with different f-numbers can generate focusing beams with both long focal depth and high transverse resolution. The performance of the cylindrical microlens arrays, such as extended focal depth, relative extended focal depth, diffraction efficiency, and focal spot size, is appraised and analyzed. From a comparison of the results obtained by the rigorous BEM and by scalar approximations, we found that the results are quite similar when the f-number equals f/1.6; however, they are quite different for f/0.8. We conclude that the BEM should be adopted to analyze the performance of a microlens array system whose f-number is less than f/1.0.
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Extended Focal Depths Δf or Relative Extended Focal Depths Δf/Δf
0
(in Micrometers) of LFD DCMAs for Several f-Numbers Calculated by the BEM and by Three Scalar Methods
The data in parentheses denote the relative extended focal depth Δf/Δf0, where Δf0 corresponds to the extended focal depth of the conventional DCMA, i.e., δf = 0 μm.
Table 2
Diffraction Efficiencies η (%) of LFD DCMAs for Several f-Numbers Calculated by the BEM and by Three Scalar Methods
Microlens Array
Measurement Method
f/#
Preset Focal Depth δf (μm)
BEM
Rayleigh-Sommerfeld
Kirchhoff (KIM)
TE
TM
RSM1
RSM2
f/0.8
0
56.05
59.99
79.97
86.59
83.23
10
73.31
76.62
67.59
70.42
69.00
20
61.72
63.34
57.98
59.35
58.66
f/1.0
0
68.64
72.23
82.30
87.12
84.69
10
77.34
79.73
75.31
77.70
76.51
20
70.12
71.51
68.19
69.58
68.88
f/1.2
0
77.32
80.88
84.00
87.55
85.77
10
80.30
82.31
79.48
81.49
80.48
20
75.06
76.24
74.17
75.42
74.79
f/1.6
0
81.35
83.39
84.67
86.71
85.70
10
83.23
84.55
83.43
84.83
84.13
20
80.64
81.51
80.51
81.48
81.00
Table 3
Focal Spot Sizes d (in Micrometers) of LFD DCMAs for Several f-Numbers Calculated by the BEM and by Three Scalar Methods
Microlens Array
Measurement Method
Diffraction-Limited Spot Size d0 (μm)
f/#
Preset Focal Depth δf (μm)
BEM
Rayleigh-Sommerfeld
Kirchhoff (KIM)
TE
TM
RSMI
RSM2
f/0.8
0
1.93
1.91
1.79
1.71
1.74
1.02
10
2.38
2.36
2.50
2.46
2.58
1.02
20
3.11
3.11
3.21
3.18
3.18
1.02
f/1.0
0
2.05
1.98
2.15
2.10
2.14
1.27
10
2.75
2.72
2.89
2.85
2.89
1.27
20
3.50
3.50
3.59
3.59
3.59
1.27
f/1.2
0
2.48
2.44
2.55
2.52
2.55
1.53
10
3.14
3.14
3.25
3.21
3.21
1.53
20
3.88
3.85
3.96
3.95
3.95
1.53
f/1.6
0
3.13
3.11
3.23
3.20
3.23
2.04
10
3.93
3.90
4.00
3.96
3.98
2.04
20
4.58
4.58
4.65
4.65
4.65
2.04
Table 4
Comparison of Focusing Performances of the Single, Dual, and Triple LFD Cylindrical Microlens Systems for the TE Polarization
Microlens numbers 1, 2, and 3 represent single, dual, and triple LFD cylindrical microlens systems, respectively.
Tables (4)
Table 1
Extended Focal Depths Δf or Relative Extended Focal Depths Δf/Δf
0
(in Micrometers) of LFD DCMAs for Several f-Numbers Calculated by the BEM and by Three Scalar Methods
The data in parentheses denote the relative extended focal depth Δf/Δf0, where Δf0 corresponds to the extended focal depth of the conventional DCMA, i.e., δf = 0 μm.
Table 2
Diffraction Efficiencies η (%) of LFD DCMAs for Several f-Numbers Calculated by the BEM and by Three Scalar Methods
Microlens Array
Measurement Method
f/#
Preset Focal Depth δf (μm)
BEM
Rayleigh-Sommerfeld
Kirchhoff (KIM)
TE
TM
RSM1
RSM2
f/0.8
0
56.05
59.99
79.97
86.59
83.23
10
73.31
76.62
67.59
70.42
69.00
20
61.72
63.34
57.98
59.35
58.66
f/1.0
0
68.64
72.23
82.30
87.12
84.69
10
77.34
79.73
75.31
77.70
76.51
20
70.12
71.51
68.19
69.58
68.88
f/1.2
0
77.32
80.88
84.00
87.55
85.77
10
80.30
82.31
79.48
81.49
80.48
20
75.06
76.24
74.17
75.42
74.79
f/1.6
0
81.35
83.39
84.67
86.71
85.70
10
83.23
84.55
83.43
84.83
84.13
20
80.64
81.51
80.51
81.48
81.00
Table 3
Focal Spot Sizes d (in Micrometers) of LFD DCMAs for Several f-Numbers Calculated by the BEM and by Three Scalar Methods
Microlens Array
Measurement Method
Diffraction-Limited Spot Size d0 (μm)
f/#
Preset Focal Depth δf (μm)
BEM
Rayleigh-Sommerfeld
Kirchhoff (KIM)
TE
TM
RSMI
RSM2
f/0.8
0
1.93
1.91
1.79
1.71
1.74
1.02
10
2.38
2.36
2.50
2.46
2.58
1.02
20
3.11
3.11
3.21
3.18
3.18
1.02
f/1.0
0
2.05
1.98
2.15
2.10
2.14
1.27
10
2.75
2.72
2.89
2.85
2.89
1.27
20
3.50
3.50
3.59
3.59
3.59
1.27
f/1.2
0
2.48
2.44
2.55
2.52
2.55
1.53
10
3.14
3.14
3.25
3.21
3.21
1.53
20
3.88
3.85
3.96
3.95
3.95
1.53
f/1.6
0
3.13
3.11
3.23
3.20
3.23
2.04
10
3.93
3.90
4.00
3.96
3.98
2.04
20
4.58
4.58
4.65
4.65
4.65
2.04
Table 4
Comparison of Focusing Performances of the Single, Dual, and Triple LFD Cylindrical Microlens Systems for the TE Polarization