Amit Hochman and Yehuda Leviatan, "Calculation of confinement losses in photonic crystal fibers by use of a source-model technique," J. Opt. Soc. Am. B 22, 474-480 (2005)

We extend our previous work on photonic-crystal fibers (PCFs) using the source-model technique to include leaky modes of fibers having a finite-sized photonic bandgap crystal (PBC) cladding. We concentrate on a hollow-core PCF and calculate the confinement losses by means of two different methods. The first method is more general but also more computationally expensive; we use sources that have a complex propagation constant and seek a transverse resonance in the complex plane. The second method, applicable only to modes with small confinement losses, uses sources with a real propagation constant to approximate leaky modes that have a propagation constant that is close to the real axis. We then apply Poynting’s theorem to calculate the attenuation constant in a manner akin to the perturbation methods used to calculate the losses in finite-conductivity metal waveguides. This first approximation can be improved through iterative application of the algorithm, i.e., by use of sources with the attenuation constant found in the first approximation. The two methods are shown to be in good agreement with each other and with previously published results for solid-core PCFs. Numerical results show that, for the hollow-core PCF analyzed, many layers of PBC cladding are needed to attain confinement losses that are acceptable for telecommunications.

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Complex Propagation Constants for Ten Modes of a Solid-Core PCF. Comparison with the Results Obtained with the Multipole Method, Given in Ref. 5a

SMT

MM

${\mathrm{\Delta}}_{r}$ (%)

${\mathrm{\Delta}}_{i}$ (%)

Class (p)

${\beta}_{\mathrm{r}}/{k}_{0}$

$-{\beta}_{\mathrm{i}}/{k}_{0}$

${\beta}_{\mathrm{r}}/{k}_{0}$

$-{\beta}_{\mathrm{i}}/{k}_{0}$

1.445395235

$3.197\times {10}^{-8}$

1.445395345

$3.15\times {10}^{-8}$

$1.9\times {10}^{-6}$

0.37

3, 4

1.438583574

$5.319\times {10}^{-7}$

1.438585801

$4.986\times {10}^{-7}$

$3.9\times {10}^{-5}$

1.62

2

1.438444750

$9.900\times {10}^{-7}$

1.438445842

$9.929\times {10}^{-7}$

$1.9\times {10}^{-5}$

0.07

5, 6

1.438364905

$1.375\times {10}^{-6}$

1.438366726

$1.374\times {10}^{-6}$

$3.2\times {10}^{-5}$

0.02

1

1.430408204

$2.162\times {10}^{-5}$

1.430175

$2.22\times {10}^{-5}$

$4.1\times {10}^{-3}$

0.67

8

1.429956727

$1.570\times {10}^{-5}$

1.4299694

$1.577\times {10}^{-5}$

$2.2\times {10}^{-4}$

0.11

3, 4

1.429247915

$8.739\times {10}^{-6}$

1.429255296

$9.337\times {10}^{-6}$

$1.3\times {10}^{-4}$

1.65

7

Six-hole Solid-core PCF. The calculations were made with the following parameters: $\mathrm{\Lambda}=6.75\mathit{\mu}\mathrm{m},$$\mathrm{\lambda}=1.45\mathit{\mu}\mathrm{m},$${R}_{v}=2.5\mathit{\mu}\mathrm{m},$
and ${\u220a}_{\mathrm{r}}={1.45}^{2}.$
The number of testing points per vein is 54, and the number of sources per vein is 36. The relative differences between the ${\beta}_{\mathrm{r}}$
and ${\beta}_{\mathrm{i}},$
found with the two methods, are given (in percents) in columns 5 and 6, respectively.

Table 2

Propagation Constant of the Fundamental Mode as Found by Different Studiesa

Parameters for this calculation are the same as in Table 1.
Adjustable boundary condition Fourier decomposition method.
Finite-difference frequency-domain method.

Table 3

Complex Propagation Constant. Comparison between Scanning the Complex β Plane (Subsection 4.A) and the Use of the Conservation of Energy (Subsection 4.B)a

${N}_{\mathrm{r}}$

${\beta}_{\mathrm{r}}/{k}_{0}$

Section 4.A

Section 4.B

$-{\beta}_{\mathrm{i}}/{k}_{0}$

Iterations

$-{\beta}_{\mathrm{i}}/{k}_{0}$

Iterations

2

0.9953

$1.4114\times {10}^{-2}$

33

$1.4049\times {10}^{-2}$

11

3

0.9870

$4.1730\times {10}^{-3}$

34

$4.1746\times {10}^{-3}$

6

4

0.9857

$1.7651\times {10}^{-3}$

32

$1.7752\times {10}^{-3}$

4

5

0.9855

$8.3782\times {10}^{-4}$

34

$8.3692\times {10}^{-4}$

3

Hollow-core PCF of Fig. 1, for $\mathrm{\Lambda}/\mathrm{\lambda}=1.2732,$
which is close to the bandgap center. The mode is of symmetry class $p=1.$
The number of testing points per vein is 54, and the number of sources per vein is 36.

Table 4

Dependence of Confinement Losses on the Number of Rings of Veinsa

$\mathcal{L}$ (dB/m), $p=3,$ 4

$\mathcal{L}$ (dB/m), $p=5,$ 6

${N}_{\mathrm{r}}$

First Mode

Second Mode

First Mode

Second Mode

2

$7.3\times {10}^{5}$

$7.3\times {10}^{5}$

$2.9\times {10}^{6}$

$2.9\times {10}^{6}$

3

$1.8\times {10}^{5}$

$1.5\times {10}^{5}$

$1.8\times {10}^{5}$

$1.8\times {10}^{5}$

4

$5.2\times {10}^{4}$

$6.8\times {10}^{4}$

$8.4\times {10}^{4}$

$7.0\times {10}^{4}$

5

$2.9\times {10}^{4}$

$3.3\times {10}^{4}$

$4.2\times {10}^{4}$

$2.7\times {10}^{4}$

Confinement losses for four modes at $\mathrm{\lambda}=1$µm and $\mathrm{\Lambda}=1.2732$µm. The real part of the propagation constant, ${\beta}_{\mathrm{r}},$
varies slightly with the number of rings of veins, ${N}_{\mathrm{r}}.$
For ${N}_{\mathrm{r}}=4,$
the values of ${\beta}_{\mathrm{r}}/{k}_{0}$
are 0.9927, 0.9975, 0.9859, 0.9947 for the modes in the second, third, fourth, and fifth columns, respectively.

Tables (4)

Table 1

Complex Propagation Constants for Ten Modes of a Solid-Core PCF. Comparison with the Results Obtained with the Multipole Method, Given in Ref. 5a

SMT

MM

${\mathrm{\Delta}}_{r}$ (%)

${\mathrm{\Delta}}_{i}$ (%)

Class (p)

${\beta}_{\mathrm{r}}/{k}_{0}$

$-{\beta}_{\mathrm{i}}/{k}_{0}$

${\beta}_{\mathrm{r}}/{k}_{0}$

$-{\beta}_{\mathrm{i}}/{k}_{0}$

1.445395235

$3.197\times {10}^{-8}$

1.445395345

$3.15\times {10}^{-8}$

$1.9\times {10}^{-6}$

0.37

3, 4

1.438583574

$5.319\times {10}^{-7}$

1.438585801

$4.986\times {10}^{-7}$

$3.9\times {10}^{-5}$

1.62

2

1.438444750

$9.900\times {10}^{-7}$

1.438445842

$9.929\times {10}^{-7}$

$1.9\times {10}^{-5}$

0.07

5, 6

1.438364905

$1.375\times {10}^{-6}$

1.438366726

$1.374\times {10}^{-6}$

$3.2\times {10}^{-5}$

0.02

1

1.430408204

$2.162\times {10}^{-5}$

1.430175

$2.22\times {10}^{-5}$

$4.1\times {10}^{-3}$

0.67

8

1.429956727

$1.570\times {10}^{-5}$

1.4299694

$1.577\times {10}^{-5}$

$2.2\times {10}^{-4}$

0.11

3, 4

1.429247915

$8.739\times {10}^{-6}$

1.429255296

$9.337\times {10}^{-6}$

$1.3\times {10}^{-4}$

1.65

7

Six-hole Solid-core PCF. The calculations were made with the following parameters: $\mathrm{\Lambda}=6.75\mathit{\mu}\mathrm{m},$$\mathrm{\lambda}=1.45\mathit{\mu}\mathrm{m},$${R}_{v}=2.5\mathit{\mu}\mathrm{m},$
and ${\u220a}_{\mathrm{r}}={1.45}^{2}.$
The number of testing points per vein is 54, and the number of sources per vein is 36. The relative differences between the ${\beta}_{\mathrm{r}}$
and ${\beta}_{\mathrm{i}},$
found with the two methods, are given (in percents) in columns 5 and 6, respectively.

Table 2

Propagation Constant of the Fundamental Mode as Found by Different Studiesa

Parameters for this calculation are the same as in Table 1.
Adjustable boundary condition Fourier decomposition method.
Finite-difference frequency-domain method.

Table 3

Complex Propagation Constant. Comparison between Scanning the Complex β Plane (Subsection 4.A) and the Use of the Conservation of Energy (Subsection 4.B)a

${N}_{\mathrm{r}}$

${\beta}_{\mathrm{r}}/{k}_{0}$

Section 4.A

Section 4.B

$-{\beta}_{\mathrm{i}}/{k}_{0}$

Iterations

$-{\beta}_{\mathrm{i}}/{k}_{0}$

Iterations

2

0.9953

$1.4114\times {10}^{-2}$

33

$1.4049\times {10}^{-2}$

11

3

0.9870

$4.1730\times {10}^{-3}$

34

$4.1746\times {10}^{-3}$

6

4

0.9857

$1.7651\times {10}^{-3}$

32

$1.7752\times {10}^{-3}$

4

5

0.9855

$8.3782\times {10}^{-4}$

34

$8.3692\times {10}^{-4}$

3

Hollow-core PCF of Fig. 1, for $\mathrm{\Lambda}/\mathrm{\lambda}=1.2732,$
which is close to the bandgap center. The mode is of symmetry class $p=1.$
The number of testing points per vein is 54, and the number of sources per vein is 36.

Table 4

Dependence of Confinement Losses on the Number of Rings of Veinsa

$\mathcal{L}$ (dB/m), $p=3,$ 4

$\mathcal{L}$ (dB/m), $p=5,$ 6

${N}_{\mathrm{r}}$

First Mode

Second Mode

First Mode

Second Mode

2

$7.3\times {10}^{5}$

$7.3\times {10}^{5}$

$2.9\times {10}^{6}$

$2.9\times {10}^{6}$

3

$1.8\times {10}^{5}$

$1.5\times {10}^{5}$

$1.8\times {10}^{5}$

$1.8\times {10}^{5}$

4

$5.2\times {10}^{4}$

$6.8\times {10}^{4}$

$8.4\times {10}^{4}$

$7.0\times {10}^{4}$

5

$2.9\times {10}^{4}$

$3.3\times {10}^{4}$

$4.2\times {10}^{4}$

$2.7\times {10}^{4}$

Confinement losses for four modes at $\mathrm{\lambda}=1$µm and $\mathrm{\Lambda}=1.2732$µm. The real part of the propagation constant, ${\beta}_{\mathrm{r}},$
varies slightly with the number of rings of veins, ${N}_{\mathrm{r}}.$
For ${N}_{\mathrm{r}}=4,$
the values of ${\beta}_{\mathrm{r}}/{k}_{0}$
are 0.9927, 0.9975, 0.9859, 0.9947 for the modes in the second, third, fourth, and fifth columns, respectively.