β ¼ k sin θ ¼ nk 0 sin θ:
ð8:153Þ
As the phase refractive index n has a dispersion, the propagation constant β has a
dispersion as well. In optics, n sin θ is often referred to as an effective index. We
denote it by
n eff n sin θ or β ¼ k 0 n eff :
ð9:104Þ
As 0 θ π/2, we have n sin θ n. In general, it is going to be difficult to obtain
analytical description or solution for the dispersion of both the phase refractive index
and effective index. As in Example 9.1, we usually obtain the relevant data by the
numerical calculations.
3
4
5
6
7
500
600
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
Wavelength (nm)
Group index
n
g
BP1T
AC5
AC'7
AC'7
BP1T
AC5
Refractive index
n
(a)
(b)
Fig. 9.9 Examples of the
wavelength dispersion of (a)
group indices and (b)
refractive indices for several
organic semiconductor
crystals. Reproduced from
Yamao T, Okuda Y,
Makino Y, Hotta S (2011)
Dispersion of the refractive
indices of thiophene/
phenylene co-oligomer
single crystals. J Appl Phys
110(5): 053113/7 pages [6],
with the permission of AIP
Publishing. https://doi.org/
10.1063/1.3634117
Table 9.1 Optimized constants of A, B, and C for Sellmeier’s dispersion formula (9.101) with
several organic semiconductor crystals
a
Material
A
B
C (nm)
BP1T
5.7
1.04
397
AC5
3.9
1.44
402
AC’7
6.0
1.06
452
a Reproduced from Yamao T, Okuda Y, Makino Y, Hotta S (2011) Dispersion of the refractive
indices of thiophene/phenylene co-oligomer single crystals. J Appl Phys 110(5): 053113/7 pages,
with the permission of AIP Publishing. https://doi.org/10.1063/1.3634117
362
9 Light Quanta: Radiation and Absorption