Applied and Adaptive Optics Lab, Department of Physics, Indian Institute of Space Science and Technology, Valiamala P.O., Thiruvananthapuram 695547, India
Swaliha B. H., S. Asokan, and J. Solomon Ivan, "Estimation of dislocated phases and tunable orbital angular momentum using two cylindrical lenses," Appl. Opt. 62, 3083-3092 (2023)
A first-order optical system consisting of two cylindrical lenses separated by a distance is considered. It is found to be non-conserving of orbital angular momentum of the incoming paraxial light field. The first-order optical system is effectively demonstrated to estimate phases with dislocations using a Gerchberg–Saxton-type phase retrieval algorithm by making use of measured intensities. Tunable orbital angular momentum in the outgoing light field is experimentally demonstrated using the considered first-order optical system by varying the distance of separation between the two cylindrical lenses.
Long Li, Guodong Xie, Yongxiong Ren, Nisar Ahmed, Hao Huang, Zhe Zhao, Peicheng Liao, Martin P. J. Lavery, Yan Yan, ChangJing Bao, Zhe Wang, Asher J. Willner, Nima Ashrafi, Solyman Ashrafi, Moshe Tur, and Alan E. Willner Appl. Opt. 55(8) 2098-2103 (2016)
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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Orbital Angular Momentum Values at Various Iteration Numbersa
Charge on SPP
Charge on SPP
Iteration Number ()
100
1.9599
1.9540
4.0433
−3.9395
−3.9195
−8.9036
200
1.9633
1.9575
4.0694
−3.9324
−3.9054
−8.1545
300
1.9652
1.9588
4.0960
−3.9317
−3.9043
−8.1265
400
1.9661
1.9594
4.1098
−3.9310
−3.9037
−8.1233
500
1.9665
1.9596
4.1158
−3.9305
−3.9032
−8.1211
600
1.9667
1.9597
4.1187
−3.9300
−3.9028
−8.1191
700
1.9668
1.9598
4.1199
−3.9297
−3.9024
−8.1175
Column 1 shows the iteration number n. Columns 2–4 show the OAM values at planes ${\text{P}_1}$, ${\text{P}_2}$, and ${\text{P}_3}$, estimated from the phases extracted through the algorithm by making use of the measured intensities, for an input singular light field of topological charge 2, for $d = 10\;\text{cm} $ and ${d^\prime} = 9\;\text{cm} $. Columns 5–7 repeat the exercise for an input singular light field of topological charge ${-}4$ for $d = 10\;\text{cm} $ and ${d^\prime} = 9\;\text{cm} $.
Table 2.
Converged Orbital Angular Momentum Values and Intensity Correlation Valuesa
Converged OAM
Correlation
Charge on SPP
Transverse Plane
−3.93
−4.05
−3.89
0.95
0.94
0.97
−3.90
−3.99
−3.88
0.96
0.95
0.98
−8.12
−8.05
−8.03
0.98
0.98
0.97
−2.92
−2.92
−2.95
0.99
0.98
0.99
−2.93
−2.94
−2.94
0.99
0.99
0.99
−6.12
−6.03
−6.09
0.99
0.99
0.99
−1.98
−1.97
−2.00
0.98
0.98
0.99
−1.98
−1.96
−1.99
0.99
0.99
0.99
−4.07
−4.03
−4.06
0.98
0.98
0.98
−0.96
−0.98
−1.02
0.97
0.98
0.97
−0.97
−0.96
−1.04
0.98
0.98
0.98
−2.12
−2.02
−2.02
0.97
0.98
0.97
1
0.99
0.99
1.01
0.97
0.97
0.98
0.97
0.97
1.00
0.98
0.98
0.99
2.16
2.16
2.01
0.99
0.99
0.99
2
1.97
1.97
1.97
0.98
0.99
0.98
1.96
1.96
1.96
0.99
0.99
0.99
4.12
4.03
4.07
0.98
0.98
0.98
3
2.91
2.90
2.92
0.98
0.99
0.98
2.92
2.91
2.95
0.99
0.99
0.99
6.06
6.01
6.10
0.97
0.98
0.97
4
3.95
3.93
3.94
0.98
0.97
0.97
3.98
3.93
3.93
0.98
0.98
0.99
7.97
7.99
7.99
0.98
0.97
0.97
Column 1 shows the charge on SPP, and column 2 shows the transverse plane. Columns 3–5 show the converged OAM values at planes ${\text{P}_1}$, ${\text{P}_2}$, and ${\text{P}_3}$ (estimated from the phases extracted through the algorithm by making use of the measured intensities), i.e., ${\tau _{\text{in}}}(500;{z_1})$, ${\tau _{\text{in}}}(500;{z_2})$, and ${\tau _{\text{out}}}(500;{z_3})$, for ${d^\prime} = 9,10$, and 11 cm. Columns 6–8 show the saturated correlation values at planes ${\text{P}_1}$, ${\text{P}_2}$, and ${\text{P}_3}$, i.e., $C(500;{z_1})$, $C(500;{z_2})$, and $C(500;{z_3})$, for ${d^\prime} = 9,10$, and 11 cm. Here, $d = 10\;\text{cm} $ throughout.
Table 3.
Theoretically Expected Values as Well as the Converged Values of Orbital Angular Momentum at the Output, for a
Charge on SPP
−4
−7.24
−7.25
−8.00
−8.05
−8.84
−8.84
−3
−5.43
−5.46
−6.00
−6.03
−6.63
−6.66
−2
−3.62
−3.71
−4.00
−4.03
−4.42
−4.47
−1
−1.81
−1.88
−2.00
−2.02
−2.21
−2.23
1
1.81
1.79
2.00
2.16
2.21
2.19
2
3.62
3.61
4.00
4.03
4.42
4.41
3
5.43
5.44
6.00
6.01
6.63
6.63
4
7.24
7.24
8.00
7.99
8.84
8.84
Column 1 shows the charge on SPP. Columns 2, 4, and 6 show the theoretically expected values of OAM at the output, i.e., ${\tau _{\text{out}}}$ of Eq. (14) for the choice of ${f_x = 5\;\text{cm}} $, ${f_y} = 10\;\text{cm} $, and ${d^\prime} = 10\;\text{cm} $ for $d = 9,10$, and 11 cm, respectively. Columns 3, 5, and 7 show the converged values of OAM at plane ${\text{P}_3}$ (estimated from the phases extracted through the algorithm by making use of the measured intensities), i.e., ${\tau _{\text{out}}}(500;{z_3})$, for $d = 9,10$, and 11 cm. See Fig. 5.
Tables (3)
Table 1.
Orbital Angular Momentum Values at Various Iteration Numbersa
Charge on SPP
Charge on SPP
Iteration Number ()
100
1.9599
1.9540
4.0433
−3.9395
−3.9195
−8.9036
200
1.9633
1.9575
4.0694
−3.9324
−3.9054
−8.1545
300
1.9652
1.9588
4.0960
−3.9317
−3.9043
−8.1265
400
1.9661
1.9594
4.1098
−3.9310
−3.9037
−8.1233
500
1.9665
1.9596
4.1158
−3.9305
−3.9032
−8.1211
600
1.9667
1.9597
4.1187
−3.9300
−3.9028
−8.1191
700
1.9668
1.9598
4.1199
−3.9297
−3.9024
−8.1175
Column 1 shows the iteration number n. Columns 2–4 show the OAM values at planes ${\text{P}_1}$, ${\text{P}_2}$, and ${\text{P}_3}$, estimated from the phases extracted through the algorithm by making use of the measured intensities, for an input singular light field of topological charge 2, for $d = 10\;\text{cm} $ and ${d^\prime} = 9\;\text{cm} $. Columns 5–7 repeat the exercise for an input singular light field of topological charge ${-}4$ for $d = 10\;\text{cm} $ and ${d^\prime} = 9\;\text{cm} $.
Table 2.
Converged Orbital Angular Momentum Values and Intensity Correlation Valuesa
Converged OAM
Correlation
Charge on SPP
Transverse Plane
−3.93
−4.05
−3.89
0.95
0.94
0.97
−3.90
−3.99
−3.88
0.96
0.95
0.98
−8.12
−8.05
−8.03
0.98
0.98
0.97
−2.92
−2.92
−2.95
0.99
0.98
0.99
−2.93
−2.94
−2.94
0.99
0.99
0.99
−6.12
−6.03
−6.09
0.99
0.99
0.99
−1.98
−1.97
−2.00
0.98
0.98
0.99
−1.98
−1.96
−1.99
0.99
0.99
0.99
−4.07
−4.03
−4.06
0.98
0.98
0.98
−0.96
−0.98
−1.02
0.97
0.98
0.97
−0.97
−0.96
−1.04
0.98
0.98
0.98
−2.12
−2.02
−2.02
0.97
0.98
0.97
1
0.99
0.99
1.01
0.97
0.97
0.98
0.97
0.97
1.00
0.98
0.98
0.99
2.16
2.16
2.01
0.99
0.99
0.99
2
1.97
1.97
1.97
0.98
0.99
0.98
1.96
1.96
1.96
0.99
0.99
0.99
4.12
4.03
4.07
0.98
0.98
0.98
3
2.91
2.90
2.92
0.98
0.99
0.98
2.92
2.91
2.95
0.99
0.99
0.99
6.06
6.01
6.10
0.97
0.98
0.97
4
3.95
3.93
3.94
0.98
0.97
0.97
3.98
3.93
3.93
0.98
0.98
0.99
7.97
7.99
7.99
0.98
0.97
0.97
Column 1 shows the charge on SPP, and column 2 shows the transverse plane. Columns 3–5 show the converged OAM values at planes ${\text{P}_1}$, ${\text{P}_2}$, and ${\text{P}_3}$ (estimated from the phases extracted through the algorithm by making use of the measured intensities), i.e., ${\tau _{\text{in}}}(500;{z_1})$, ${\tau _{\text{in}}}(500;{z_2})$, and ${\tau _{\text{out}}}(500;{z_3})$, for ${d^\prime} = 9,10$, and 11 cm. Columns 6–8 show the saturated correlation values at planes ${\text{P}_1}$, ${\text{P}_2}$, and ${\text{P}_3}$, i.e., $C(500;{z_1})$, $C(500;{z_2})$, and $C(500;{z_3})$, for ${d^\prime} = 9,10$, and 11 cm. Here, $d = 10\;\text{cm} $ throughout.
Table 3.
Theoretically Expected Values as Well as the Converged Values of Orbital Angular Momentum at the Output, for a
Charge on SPP
−4
−7.24
−7.25
−8.00
−8.05
−8.84
−8.84
−3
−5.43
−5.46
−6.00
−6.03
−6.63
−6.66
−2
−3.62
−3.71
−4.00
−4.03
−4.42
−4.47
−1
−1.81
−1.88
−2.00
−2.02
−2.21
−2.23
1
1.81
1.79
2.00
2.16
2.21
2.19
2
3.62
3.61
4.00
4.03
4.42
4.41
3
5.43
5.44
6.00
6.01
6.63
6.63
4
7.24
7.24
8.00
7.99
8.84
8.84
Column 1 shows the charge on SPP. Columns 2, 4, and 6 show the theoretically expected values of OAM at the output, i.e., ${\tau _{\text{out}}}$ of Eq. (14) for the choice of ${f_x = 5\;\text{cm}} $, ${f_y} = 10\;\text{cm} $, and ${d^\prime} = 10\;\text{cm} $ for $d = 9,10$, and 11 cm, respectively. Columns 3, 5, and 7 show the converged values of OAM at plane ${\text{P}_3}$ (estimated from the phases extracted through the algorithm by making use of the measured intensities), i.e., ${\tau _{\text{out}}}(500;{z_3})$, for $d = 9,10$, and 11 cm. See Fig. 5.
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