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Unformatted text preview: Group Velocity and Dispersion 1 The phase velocity index of refraction of a dielectric material as a function of wavelength may be accurately represented using a Sellmeier equation of the form GM2 G2)\2 6113A2 2—
" ‘A+A2—,\§+,\2—,\g A2Ag’ where A, M, and Gk are called the Sellmeier coefﬁcients. The quantities Ak and Gk rep
resent resonant wavelengths and oscillator strengths respectively. The phase velocity is
given by up = c/n. This corresponds to the velocity of a point of constant phase in the
electromagnetic wave. For example, the phase velocity represents the velocity of a wave
front traveling through an arrayed waveguide grating router. The group velocity is given
by 129 = c/[n — A (dn /d)\)]. This corresponds to the velocity of the wave packet. The
group velocity is the speed of a pulse of light traveling through the medium. It is the
speed of the energy flow. The sign of the derivative of the group velocity with respect to
wavelength determines within a wavelength channel whether the long wavelengths move
ahead of the short wavelengths (+ sign) or whether the long wavelengths fall behind the
short wavelengths (— sign). Silica is typically the transmission medium for optical ﬁber communications. the
transmission medium is silica. For this material, A = 1, A1 = 68.4 nm, 01 = 0.69617,
A2 = 116.2 nm, G2 = 0.40794, A3 = 9896.2 nm, G3 = 0.89748. For a silica optical ﬁber,
considering material dispersion as described above and ignoring waveguide dispersion,
calculate the phase velocity, the group velocity, the ﬁrst derivative of the group velocity
with respect to wavelength for A = 1.200 pm, = 1.300 ,um, = 1.400 pm. Express the
phase velocity and the group velocity in meters / sec accurately to six signiﬁcant ﬁgures.
Express the group velocity derivative in meters / 560/ nm accurately to four signiﬁcant ﬁgures. Put your ﬁnal answers in the spaces provided. Note that d/\ dn A (:le7 (mg 03A§
_ 2 2 2 + 2 2 2 + 2 2 2
n (A  A1) (A  A2) (A  A3) and that 123 _ l G1X‘1’(3)\2+A§) +G2A§(3)\2+/\§) +G3A§(3/\2+A§) _ 1 dn 2
d)\2 n (A2 — m3 (A2 — A93 (A2 — A§)3 72(5) °
/\ = 1.200,um Phase velocity = —_______ meters / sec
Group velocity = —_ meters / sec
First derivative of group velocity with respect to wavelength = —_ meters / 360/ nm ,\ = 1.300 pm Phase velocity = _____________— meters/ sec
Group velocity = _________________________ meters / sec
First derivative of group velocity with respect to wavelength = —— meters/sec/nm ,\ = 1.400 pm Phase velocity = __—___________ meters / sec
Group velocity = ____ meters/ sec
First derivative of group velocity with respect to wavelength = ___—______________. meters / sec/nm Of the three wavelengths, which gives the fastest pulse through the medium?
A = — pm
Of the three wavelengths, which gives the fastest phase speed through the medium? A: ___________um Group Velocity and Dispersion 1 The phase velocity is
11,, = c/n. The group velocity is
c v =———.
g [n—A%§ Differentiating the group velocity gives TL 211 n 2n
ﬂz‘c(%i_)‘g_,\7_g_x)=+ cAg—A;
d)\ (n—/\%)2 (n—Ai—KV
and so
A=1.200,um Phase velocity = 207,031,911 meters / sec
Group velocity = 205,098,393 meters/ sec
First derivative of group velocity with respect to wavelength = 331.88 meters/sec/nm
A = 1.300 pm Phase velocity = 207, 193, 950 meters / sec
Group velocity = 205, 108, 554 meters/ sec First derivative of group velocity with respect to wavelength = «111.95 meters/sec/nm
A = 1.400 pm
Phase velocity = 207 , 357, 030 meters/ sec Group velocity = 205,078, 719 meters/ sec
First derivative of group velocity with respect to wavelength = —473.79 meters / sec/ nm The wavelength of the fastest pulse through the medium is A = 1.300 pm The wavelength of the light with the fastest phase speed through the medium is /\ =
1.400 mm ...
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 Spring '08
 Gaylord

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