Electromagnetic power provided by sources within multilayer optics: free-space and modal patterns

Claude Amra and Sophie Maure

Author Affiliations

Claude Amra and Sophie Maure

^{}Laboratoire d'Optique des Surfaces et des Couches Minces, Unité
Propre de la Recherche et de l'Enseignement Supérieur Associée
au Centre National de la Recherche Scientifique 6080, Ecole Nationale Supérieure
de Physique de Marseille, Domaine Universitaire de St Jérôme,
Avenue Escadrille Normandie-Niemen, 13397 Marseille Cedex 20, France

Claude Amra and Sophie Maure, "Electromagnetic power provided by sources within multilayer optics: free-space and modal patterns," J. Opt. Soc. Am. A 14, 3102-3113 (1997)

An electromagnetic theory is presented that makes possible the development
of a complete energy balance within an arbitrary multilayer microcavity that
supports different kinds of classical optical sources. The theory is based
on a single Fourier spectrum of waves and is valid for transparent or dissipative
stacks, with no use of modal methods. We show how the power provided by the
cavity is converted into Poynting flux and absorption. Free-space and guided-mode
patterns are calculated for single layers, mirrors, and narrow-band filters.
The modal pattern is shown to be strongly dependent on the cavity poles. Discretization
of the high-frequency energy into a set of guided modes is introduced as an
asymptotic limit of the problem when absorption vanishes to zero. The applications
concern defect-induced absorption in optical multilayers or guided-mode coupling
through microirregularities in a stack, as well as spontaneous emission in
classical microcavities.

R. LaComb and J. P. Casey J. Opt. Soc. Am. A 32(10) 1780-1790 (2015)

References

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Ratios $\mu ({n}_{L}^{\u2033})$ of Embedded $(1.52<{\nu}^{*}<4)$ to Radiative $(0<{\nu}^{*}<1)$ Power, Calculated for Different Absorption Values (${n}_{L}^{\u2033}=0,$${10}^{-2},$${10}^{-4},$ and 1) and Two Polarizations ($\mathit{SS}$ and $\mathit{PP}$), Where the Cavity is the Low-Index Layer
of Fig. 5(a)

${n}^{\prime \prime}$

$\mathit{SS}$ Polarization

$\mathit{PP}$ Polarization

0

0

0

${10}^{-4}$

$1.85\times {10}^{-4}$

$4.23\times {10}^{-3}$

${10}^{-2}$

$1.85\times {10}^{-2}$

$4.22\times {10}^{-1}$

1

$9.77\times {10}^{-1}$

34.51

Table 2

Ratios ${\mu}_{i}$ of Modal Power Carried by Each Guided Mode, Normalized to
Radiative Power, Where the Calculation Is Performed for $\mathit{SS}$
(TE-Mode) and $\mathit{PP}$ (TM-Mode) Polarizations and the Cavity
Is the High-Index Layer of Fig. 7(a)

${f}_{i,max}$
is the maxima of normalized spectral densities at modal frequencies in the
case of slight absorption $({n}_{L}^{\u2033}={10}^{-4}).$
Σ is the sum of all ${\mu}_{i}$
ratios and gives the contribution of total embedded power.

Table 3

Ratios ${\mu}_{i}$ of Modal Power Carried by Each Guided Mode, Normalized to
Radiative Power, Where the Calculation Is Performed for $\mathit{SS}$
(TE-Mode) and $\mathit{PP}$ (TM-Mode) Polarizations and the Cavity
Is That of Fig. 9(a)

${f}_{i,max}$
is the maxima of spectral densities at modal frequencies in the case of slight
absorption $({n}_{H}^{\u2033}={10}^{-4}).$
Σ is the sum of all ${\mu}_{i}$
ratios and gives the contribution of total embedded power.

Table 4

Same as Table 3, but the Cavity Is That of Fig. 10(a)

Mode

${\mu}_{i}(\mathit{SS})$

${f}_{i,max}(\mathit{SS})$

Mode

${\mu}_{i}(\mathit{PP})$

${f}_{i,max}(\mathit{PP})$

TE0

$5.7\times {10}^{-4}$

8

TM0

$2.0\times {10}^{-3}$

2131

TE1

$5.8\times {10}^{-3}$

45

TM1

$1.9\times {10}^{-3}$

8549

TE2

$8.7\times {10}^{-2}$

1897

TM2

$1.5\times {10}^{-1}$

12,461

$\sum =9.3\times {10}^{-2}$

$\sum =1.53\times {10}^{-1}$

Table 5

Same as Table
3, but the Cavity Is That of Fig. 11(a)

Mode

${\mu}_{i}(\mathit{SS})$

${f}_{i,max}(\mathit{SS})$

Mode

${\mu}_{i}(\mathit{PP})$

${f}_{i,max}(\mathit{PP})$

TE0

0.25

1586

TM0

$1.3\times {10}^{-2}$

47

TE1

0.94

6128

TM1

$4.5\times {10}^{-2}$

185

TE2

1.94

13,752

TM2

$8.2\times {10}^{-2}$

370

TE3

3.02

26,867

TM3

$7.9\times {10}^{-2}$

508

TE4

3.76

35,974

$\sum =9.91$

$\sum =0.22$

Table 6

Same as Table
3, but the Cavity Is the Fabry–Perot Cavity of Fig. 12(a), Whose Design is ${\mathrm{M}}_{5}6\mathrm{L}{\mathrm{M}}_{5}$

Mode

${\mu}_{i}(\mathit{SS})$

${f}_{i,max}(\mathit{SS})$

Mode

${\mu}_{i}(\mathit{PP})$

${f}_{i,max}(\mathit{PP})$

TE0

$2.1\times {10}^{-2}$

115

TM0

$6.1\times {10}^{-3}$

84

TE1

$2.9\times {10}^{-2}$

158

TM1

$1.1\times {10}^{-2}$

149

TE2

$6.8\times {10}^{-2}$

366

TM2

$5.0\times {10}^{-3}$

55

TE3

$7.6\times {10}^{-2}$

368

$\sum =0.19$

$\sum =2.2\times {10}^{-2}$

Table 7

Same as Table
3, but with a Fabry–Perot Cavity of Design ${\mathrm{M}}_{5}2\mathrm{H}{\mathrm{M}}_{6}$

Mode

${\mu}_{i}(\mathit{SS})$

${f}_{i,max}(\mathit{SS})$

Mode

${\mu}_{i}(\mathit{PP})$

${f}_{i,max}(\mathit{PP})$

TE0

$1.6\times {10}^{-2}$

218

TM0

0.4

2182

TE1

$3.2\times {10}^{-2}$

268

TM1

$8.9\times {10}^{-3}$

38

TE2

$9.6\times {10}^{-3}$

68

TM2

$7.7\times {10}^{-2}$

328

TE3

0.1

1053

TM3

$3.5\times {10}^{-2}$

321

TE4

$4.1\times {10}^{-2}$

358

$\sum =0.20$

$\sum =0.5$

Tables (7)

Table 1

Ratios $\mu ({n}_{L}^{\u2033})$ of Embedded $(1.52<{\nu}^{*}<4)$ to Radiative $(0<{\nu}^{*}<1)$ Power, Calculated for Different Absorption Values (${n}_{L}^{\u2033}=0,$${10}^{-2},$${10}^{-4},$ and 1) and Two Polarizations ($\mathit{SS}$ and $\mathit{PP}$), Where the Cavity is the Low-Index Layer
of Fig. 5(a)

${n}^{\prime \prime}$

$\mathit{SS}$ Polarization

$\mathit{PP}$ Polarization

0

0

0

${10}^{-4}$

$1.85\times {10}^{-4}$

$4.23\times {10}^{-3}$

${10}^{-2}$

$1.85\times {10}^{-2}$

$4.22\times {10}^{-1}$

1

$9.77\times {10}^{-1}$

34.51

Table 2

Ratios ${\mu}_{i}$ of Modal Power Carried by Each Guided Mode, Normalized to
Radiative Power, Where the Calculation Is Performed for $\mathit{SS}$
(TE-Mode) and $\mathit{PP}$ (TM-Mode) Polarizations and the Cavity
Is the High-Index Layer of Fig. 7(a)

${f}_{i,max}$
is the maxima of normalized spectral densities at modal frequencies in the
case of slight absorption $({n}_{L}^{\u2033}={10}^{-4}).$
Σ is the sum of all ${\mu}_{i}$
ratios and gives the contribution of total embedded power.

Table 3

Ratios ${\mu}_{i}$ of Modal Power Carried by Each Guided Mode, Normalized to
Radiative Power, Where the Calculation Is Performed for $\mathit{SS}$
(TE-Mode) and $\mathit{PP}$ (TM-Mode) Polarizations and the Cavity
Is That of Fig. 9(a)

${f}_{i,max}$
is the maxima of spectral densities at modal frequencies in the case of slight
absorption $({n}_{H}^{\u2033}={10}^{-4}).$
Σ is the sum of all ${\mu}_{i}$
ratios and gives the contribution of total embedded power.

Table 4

Same as Table 3, but the Cavity Is That of Fig. 10(a)

Mode

${\mu}_{i}(\mathit{SS})$

${f}_{i,max}(\mathit{SS})$

Mode

${\mu}_{i}(\mathit{PP})$

${f}_{i,max}(\mathit{PP})$

TE0

$5.7\times {10}^{-4}$

8

TM0

$2.0\times {10}^{-3}$

2131

TE1

$5.8\times {10}^{-3}$

45

TM1

$1.9\times {10}^{-3}$

8549

TE2

$8.7\times {10}^{-2}$

1897

TM2

$1.5\times {10}^{-1}$

12,461

$\sum =9.3\times {10}^{-2}$

$\sum =1.53\times {10}^{-1}$

Table 5

Same as Table
3, but the Cavity Is That of Fig. 11(a)

Mode

${\mu}_{i}(\mathit{SS})$

${f}_{i,max}(\mathit{SS})$

Mode

${\mu}_{i}(\mathit{PP})$

${f}_{i,max}(\mathit{PP})$

TE0

0.25

1586

TM0

$1.3\times {10}^{-2}$

47

TE1

0.94

6128

TM1

$4.5\times {10}^{-2}$

185

TE2

1.94

13,752

TM2

$8.2\times {10}^{-2}$

370

TE3

3.02

26,867

TM3

$7.9\times {10}^{-2}$

508

TE4

3.76

35,974

$\sum =9.91$

$\sum =0.22$

Table 6

Same as Table
3, but the Cavity Is the Fabry–Perot Cavity of Fig. 12(a), Whose Design is ${\mathrm{M}}_{5}6\mathrm{L}{\mathrm{M}}_{5}$

Mode

${\mu}_{i}(\mathit{SS})$

${f}_{i,max}(\mathit{SS})$

Mode

${\mu}_{i}(\mathit{PP})$

${f}_{i,max}(\mathit{PP})$

TE0

$2.1\times {10}^{-2}$

115

TM0

$6.1\times {10}^{-3}$

84

TE1

$2.9\times {10}^{-2}$

158

TM1

$1.1\times {10}^{-2}$

149

TE2

$6.8\times {10}^{-2}$

366

TM2

$5.0\times {10}^{-3}$

55

TE3

$7.6\times {10}^{-2}$

368

$\sum =0.19$

$\sum =2.2\times {10}^{-2}$

Table 7

Same as Table
3, but with a Fabry–Perot Cavity of Design ${\mathrm{M}}_{5}2\mathrm{H}{\mathrm{M}}_{6}$