Bao Xue Chen, Hiromi Hamanaka, and Kokuya Iwamura, "Recovery of refractive-index profiles of planar graded-index waveguides from measured mode indices: an iteration method," J. Opt. Soc. Am. A 9, 1301-1305 (1992)
An iteration method for calculating refractive-index profiles of graded-index waveguides from measured effective indices is presented. The refractive-index profiles recovered from the effective indices for the exponential function indicate that the accuracy of this method is better than ±10−4. We apply this method to Ti in-diffused Fe2O3–LiNbO3 and two-step ion-exchanged glass waveguides and obtain a good agreement between the theoretical and the experimental values.
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Inverse Solutions of Δn, d, and Turning Points Derived from a Pair of Exact Effective Indices for the Exponential-Index Profile
Modes
Δn
d
x0
x1
x2
x3
x4
0–1
0.0149
5.056
2.365
4.603
0–2
0.0149
5.040
2.362
6.852
0–3
0.0149
5.025
2.360
9.342
0–4
0.0149
5.029
2.360
12.301
1–2
0.0149
5.022
4.591
6.842
1–3
0.0157
4.880
4.696
9.317
1–4
0.0149
5.019
4.590
12.288
2–3
0.0150
4.993
6.824
9.314
2–4
0.0149
5.018
6.839
12.286
3–4
0.0148
5.043
9.358
12.317
Average
0.0150
5.013
2.362
4.620
6.839
9.333
12.298
Initial
0.0150
5.000
Table 2
Exact Effective Indices and Effective Indices Calculated from Average Values in Table 1
Effective Index
Mode
Exact
Present Study
0
2.21701
2.21706
1
2.21368
2.21367
2
2.21152
2.21153
3
2.21002
2.21003
4
2.20899
2.20899
Table 3
Measured and Calculated Effective Indices for Ti: Fe2O3–LiNbO3 Waveguidesa
Mode
Nmeas
Ncal
0
2.2174
2.2173
1
2.2097
2.2098
2
2.2053
2.2052
Average values in Table 4 were used for the calculations. Nmeas, experimental values; Ncal, calculated values.
Table 4
Inverse Solutions of Δn, d, and Turning Points for Gaussian Function Derived from Measured TM Effective Indices for Ti: Fe2O3–LiNbO3 Waveguides
Modes
Δn
d
x0
x1
x2
0–1
0.0200
2.547
1.716
2.998
0–2
0.0197
2.649
1.753
4.949
1–2
0.0185
2.744
3.140
5.082
Average
0.0194
2.647
1.735
3.069
5.016
Table 5
Effective Indices of Single-Step Ion-Exchanged Soda-Lime Glass Waveguidesa
Mode
Nmeas
Ncal
0
1.5810
1.5809
1
1.5639
1.5639
2
1.5468
1.5467
3
1.5296
1.5292
Average values in Table 7 were used for the calculations. Nmeas, experimental values; Ncal, calculated values.
Table 6
Measured Effective Indices in the Two-Step Ion-Exchanged Soda-Lime Glass Waveguides and Calculated Effective Indicesa
Mode
Nmeas
Ncal
0
1.5394
1.5394
1
1.5356
1.5356
2
1.5320
1.5322
3
1.5294
1.5291
Average values in Table 8 for the TE modes were used for the calculations. Nmeas, experimental values; Ncal, calculated values.
Table 7
Inverse Solutions of Δn, d, and Turning Points for the Parabolic Function Derived from Measured TE Effective Indices for the Single-Step Ion-Exchanged Soda-Lime Glass Waveguides
Modes
Δn
d
x0
x1
x2
x3
0–1
0.0812
3.686
1.430
2.215
0–2
0.0811
3.711
1.436
2.804
0–3
0.0811
3.724
1.439
3.295
1–2
0.0808
3.730
2.231
2.814
1–3
0.0808
3.736
2.234
3.304
2–3
0.0806
3.742
2.821
3.308
Average
0.0809
3.722
1.435
2.227
2.813
3.302
Table 8
Inverse Solutions of Δn, d, and Turning Points for Fermi Function (α = 5.954) Derived from Measured TE Effective Indices for the Two-Step Ion-Exchanged Soda-Lime Glass Waveguides
Modes
Δn
d
x0
x1
x2
x3
0–1
0.0320
7.657
2.805
4.910
0–2
0.0324
6.868
2.696
6.456
0–3
0.0321
7.341
2.761
8.105
1–2
0.0336
5.935
4.548
6.216
1–3
0.0324
7.131
4.797
8.041
2–3
0.0303
8.667
6.944
8.517
Average
0.0321
7.267
2.754
4.752
6.539
8.221
Tables (8)
Table 1
Inverse Solutions of Δn, d, and Turning Points Derived from a Pair of Exact Effective Indices for the Exponential-Index Profile
Modes
Δn
d
x0
x1
x2
x3
x4
0–1
0.0149
5.056
2.365
4.603
0–2
0.0149
5.040
2.362
6.852
0–3
0.0149
5.025
2.360
9.342
0–4
0.0149
5.029
2.360
12.301
1–2
0.0149
5.022
4.591
6.842
1–3
0.0157
4.880
4.696
9.317
1–4
0.0149
5.019
4.590
12.288
2–3
0.0150
4.993
6.824
9.314
2–4
0.0149
5.018
6.839
12.286
3–4
0.0148
5.043
9.358
12.317
Average
0.0150
5.013
2.362
4.620
6.839
9.333
12.298
Initial
0.0150
5.000
Table 2
Exact Effective Indices and Effective Indices Calculated from Average Values in Table 1
Effective Index
Mode
Exact
Present Study
0
2.21701
2.21706
1
2.21368
2.21367
2
2.21152
2.21153
3
2.21002
2.21003
4
2.20899
2.20899
Table 3
Measured and Calculated Effective Indices for Ti: Fe2O3–LiNbO3 Waveguidesa
Mode
Nmeas
Ncal
0
2.2174
2.2173
1
2.2097
2.2098
2
2.2053
2.2052
Average values in Table 4 were used for the calculations. Nmeas, experimental values; Ncal, calculated values.
Table 4
Inverse Solutions of Δn, d, and Turning Points for Gaussian Function Derived from Measured TM Effective Indices for Ti: Fe2O3–LiNbO3 Waveguides
Modes
Δn
d
x0
x1
x2
0–1
0.0200
2.547
1.716
2.998
0–2
0.0197
2.649
1.753
4.949
1–2
0.0185
2.744
3.140
5.082
Average
0.0194
2.647
1.735
3.069
5.016
Table 5
Effective Indices of Single-Step Ion-Exchanged Soda-Lime Glass Waveguidesa
Mode
Nmeas
Ncal
0
1.5810
1.5809
1
1.5639
1.5639
2
1.5468
1.5467
3
1.5296
1.5292
Average values in Table 7 were used for the calculations. Nmeas, experimental values; Ncal, calculated values.
Table 6
Measured Effective Indices in the Two-Step Ion-Exchanged Soda-Lime Glass Waveguides and Calculated Effective Indicesa
Mode
Nmeas
Ncal
0
1.5394
1.5394
1
1.5356
1.5356
2
1.5320
1.5322
3
1.5294
1.5291
Average values in Table 8 for the TE modes were used for the calculations. Nmeas, experimental values; Ncal, calculated values.
Table 7
Inverse Solutions of Δn, d, and Turning Points for the Parabolic Function Derived from Measured TE Effective Indices for the Single-Step Ion-Exchanged Soda-Lime Glass Waveguides
Modes
Δn
d
x0
x1
x2
x3
0–1
0.0812
3.686
1.430
2.215
0–2
0.0811
3.711
1.436
2.804
0–3
0.0811
3.724
1.439
3.295
1–2
0.0808
3.730
2.231
2.814
1–3
0.0808
3.736
2.234
3.304
2–3
0.0806
3.742
2.821
3.308
Average
0.0809
3.722
1.435
2.227
2.813
3.302
Table 8
Inverse Solutions of Δn, d, and Turning Points for Fermi Function (α = 5.954) Derived from Measured TE Effective Indices for the Two-Step Ion-Exchanged Soda-Lime Glass Waveguides