Unformatted text preview: ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain Blackbody Radiation, Gain and Broadening
ECE 455 Optical Electronics Summary Tom Galvin
Gary Eden
ECE Illinois Introduction
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary In this section, we will learn how to do the following:
Understand the properties of blackbody (thermal) light
Describe absorption, spontaneous emission and stimulated
emission with rate equations
Predict the line width of atomic and molecular transitions Blackbody Radiation
ECE 455
Lecture 3 Blackbody
Radiation Blackbody radiation is emitted by objects because they are hot.
The optical energy density for a blackbody radiator at
temperature T between the frequencies of ν and ν + d ν is Lineshape
Gain
Summary ρ(ν )d ν = 8π h ν 3
c 3 exp 1
hν
kB T dν (1) −1 where ν is the frequency, h is Planck’s constant, c is the speed
of light, and kB is Boltzmann’s constant. The peak wavelength
of this distribution can be found with the following equation
λmax = 2.898 × 106 K·nm
T (2) Blackbody Radiation Picture
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary Example: Human Blackbody
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary Problem: The temperature of a healthy person is about 37◦ C.
If a human could be considered an ideal blackbody, at what
wavelength would the blackbody spectrum peak? What are the
energy and photon densities of visible light emitted by people
immediately above the surface of the skin? What are the
energy and photon densities of photons in ±1% bandwidth
around the peak wavelength?
Solution: The peak wavelength can be found with the aid of
Equation 2
λp = 2.898 × 106 K·nm
= 9344 nm
(37 + 273.15) K (3) Example: Human Blackbody
ECE 455
Lecture 3 To ﬁnd the energy density of visible light, numerically integrate
Equation 1 over the appropriate limits: Blackbody
Radiation
Lineshape ρvis = c
400 nm
c
700 nm Gain 8π h ν 3
c 3 exp 1
hν
kB T dν (4) −1 = 5.44 × 10−30 Jcm−3 Summary To convert energy density to photon density, divide by the
energy per photon, hν . Numerical integration is once again
required
ρvis
¯ = c
400 nm
c
700 nm 8πν 2
c 3 exp 1
hν
kB T dν
−1 = 1.87 × 10−11 photonscm−3 (5) Example: Human Blackbody
ECE 455
Lecture 3 For a ±1% bandwidth around the peak, the energy density is
ρIR Blackbody
Radiation = 1.01·c
9344 nm
0.99·c
9344 nm 8π h ν 3
c 3 exp Lineshape 1
hν
kB T dν (6) −1 = 9.21 × 10−8 Jcm−3 Gain
Summary Because the bandwidth of this integration region is much
narrower, there is no need to perform another numerical
integration; we can simply divide the above answer by the
hc
energy per photon λp = 2.13 × 10−20 J.
9.21 × 10−8 Jcm−3
= 4.33 × 1012 photonscm−3
2.13 × 10−20 J/photon
(7)
The lesson: If you are looking for somebody in the dark, don’t
use your eyes. Use an IR camera.
ρIR =
¯ Absorption
ECE 455
Lecture 3 Blackbody
Radiation 2 Lineshape
Gain
Summary A twolevel system
absorbs a photon
and is left in an
excited state
1 Spontaneous Emission
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary Twolevel system
spontaneously
decays from a higher
energy level to a
lower energy level
and emits a photon. 2 The decay rate is
characterized by the
spontaneous
emission lifetime τ
Emission is uniform
in all 4π steridians
Spontaneous
emission can only be
explained by QED 1 Stimulated Emission
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary Photon interacts
with twolevel
system in an excited
state
The system goes to
a lower energy level
and a photon with
the corresponding
energy is emitted
Emitted photon has
the same direction,
polarization, phase
and energy as the
incident photon 2 hγ 2hγ 1 Three Fundamental Process
ECE 455
Lecture 3 Blackbody
Radiation Absorption
dN2
dt = +B12 ρ(ν21 )N1 (8) Absorption Lineshape
Gain
Summary Spontaneous Emission
dN2
dt = −A21 N2 (9) SpontaneousEmission Stimulated Emission
dN2
dt = −B21 ρ(ν21 )N2 (10) StimulatedEmission Total population transfer
dN2
= −A21 N2 − B21 ρ(ν21 )N2 + B12 ρ(ν21 )N1
dt (11) Relationship Between Einstein Coeﬃcients I
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary Assume atoms are in equilibrium with a blackbody radiation
d
ﬁeld. In steady state, dt () = 0. Solving Equation 11 for ρ, we
ﬁnd:
A21 N2
ρ(ν ) =
(12)
B12 N1 − B21 N2
In thermal equillibrium, the populations of energy levels can be
assumed to follow a Boltzmann distiribution. Thus:
N2
g2
hν21
= exp −
N1
g1
kB T (13) Relationship Between Einstein Coeﬃcients II
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary Substitution of Equation 13 into 12 gives:
ρ(ν21 ) = A21
B21 1
B12 g1 hν21 /kt
B21 g2 e −1 (14) When the above equation is compared with the blackbody
spectrum (Equation 1),
ρ(ν ) = 8π h ν 3
c 3 exp 1
hν
kB T the following relationships are discovered:
g1
B21 = B12
g2
A21
8π hν 3
=
B21
c3 (1)
−1 (15)
(16) Example: Relative Strength of Stimulated and
Spontaneous Emission
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary Problem: Some sodium atoms contaminated the ﬁlament of a
incandescent lightbulb ﬁlament. When the bulb is operating at
T = 5XXX ◦ C, assume the sodium atoms and the optical ﬁeld
are in thermal equilibrium with the . Find the ratio of
spontaneous emission to stimulated emission on the D1 line.
Solution: A signiﬁcant optical energy density is necessary to
make stimulated emission as strong as spontaneous emission. Introduction to Lineshape
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary Atomic and molecular transition lines have a ﬁnite
bandwidth; They are not perfectly sharp.
Various physical processes broaden the spectrum lines.
Homogenous line broadening occurs when all emitters are
equally eﬀected by broadening mechanisms.
Inhomogenous line broadening occurs when all emitters are
unequally eﬀected by broadening mechanisms.
Line broadening reduces the eﬀective gain because not all
atoms are capable of interacting with the radiation ﬁeld. The Lineshape Function
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary For example, if there are N2 atoms in the upper energy level,
then the number capable of emitting a photon with d ν of the
frequency ν is:
N (ν )d ν = g (ν )N2 d ν
(17)
If this equation is integrated over all frequencies, we must get
N2 . Thus g (ν ) must be a probability distribution. In other
words:
∞
g (ν )d ν = 1
(18)
0 Lifetime Broadening
ECE 455
Lecture 3 Blackbody
Radiation g (ν ) = Lineshape
Gain
Summary 1
∆ν
2π (ν − ν0 )2 + where
∆ν = 1
2π 1
1
+
τ1 τ2 ∆ν 2
2 (19) (20) The peak of the lineshape function is:
g (ν0 ) = 2
π ∆ν (21) Collisional Broadening
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape Siegman 128 It can be shown from the principles of that the
collision rate is: Gain
Summary νcol = Nm σ 8kT
π 1
1
−
Mm M2 1/2 (22) An more phenominalogical description
∆ν = A + BP (23) Total Homogenous Broadening
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain The total linewidth from all homogenous broadening
mechanisms is: Summary ∆νtotal = 1
[(A1 + k1 ) + (A2 + k2 ) + 2vcol ]
2π where the Ai ’s and ki ’s are the total radiative and
nonradiative relaxation rates for level i . (24) Doppler Broadening I
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape Doppler broadening occurs in room temperature gases. In their
rest frame all atoms will emit light with the same homogenous
proﬁle, gH (ν ). However, light is observed in the lab frame,
where the proﬁle will be doppler shifted. Gain
Summary The doppler shift of a photon emitted from a nonrelativistic
particle moving toward the observer with velocity vz is
ν =ν 1+ vz
c (25) The observed lineshape proﬁle in the lab frame is then
gH (ν ) = gH ν 1 + vz
c (26) Doppler Broadening II
ECE 455
Lecture 3 It can be shown from thermodynamics that the distribution of
particle velocities in the z direction is Blackbody
Radiation M
2π kT Lineshape
Gain 1/2 exp − 2
Mvz
2kT (27) Summary The total lineshape proﬁle in the lab frame from all particles
moving with all velocities is
g (ν ) = M
2π kT 1/2 ∞ gH (ν ) · exp −
−∞ 2
Mvz
2kT dvz (28) For light molecules at room temperature, doppler broadening
dominates lifetime broadening, we may therefore make the
approximation gH (ν ) ≈ δ (ν − ν0 ). Doppler Broadening Picture
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary Doppler Broadening III
ECE 455
Lecture 3 The lineshape function of a Doppler broadened transition is:
Blackbody
Radiation
Lineshape g (ν ) = Gain
Summary 2
∆νD ln2
π 1/2 where
∆νD = ν − ν0
∆ν exp −4ln(2) 8kTln2
Mc 2 2 (29) 1/2 ν0 (30) The peak of the lineshape function is:
g (ν0 ) = 2
∆νD ln2
π 1/2 (31) Doppler Broadening and Laser Cooling
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary Tune a laser beam to
slightly below an atomic
resonance
Atoms moving toward the
laser see the light doppler
shifted into resoannce
The atom is free to reemit
the photon in any direction
Six counterpropogating
beams must be used to
ensure zero net
momentum transfer Laser COOLING PICTURE Example: Doppler Broadening in the Copper Vapor
Laser
ECE 455
Lecture 3 Blackbody
Radiation Problem: The copper vapor laser (CVL) operates at
T = 1750 K. Its wavelength is λ = 510.6 nm and the
spontaneous emission lifetime is τsp = 5×10−7 s. Determine
the form of the lineshape function and its FWHM. Lineshape
Gain
Summary Solution: The homogenous linewidth is
∆νH = 1
= 3.18 × 105 Hz
2πτsp (32) The doppler linewidth is
∆νD = ν0 8kTln2
= 2.20 × 109 Hz
Mc 2 (33) Doppler broadening dominates this transition. Therefore the
lineshape will be gaussian with a FHWM ∆νD . Energy Density and Intensity
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary If all radiation is travelling in a
given direction, the following
relationship exisists between
the energy density and the
intensity:
I (ν ) = c
ρ(ν )
n (34) Figure Demonstrating
Relationship Stimulated Emission in Broadband Light
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary Equation 11 If the transition linewidth is much narrower than
optical bandwidth, not all photons will be able to interact with
the transition:
dN2
dt ∞ (B12 N1 g (ν )ρ(ν ) − B21 N2 g (ν )ρ(ν )) d ν (35) =
0 ∞ = (B12 N1 − B21 N2 ) g (ν )ρ(ν )d ν
0 (36) ∞ ≈ (B12 N1 − B21 N2 ) ρ(ν21 ) g (ν )d ν (37) 0 = (B12 N1 − B21 N2 ) ρ(ν21 )
Here, ν21 is the center wavelength of the transition. (38) Stimulated Emission in Narrowband Light
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain If the bandwith of the optical ﬁeld is much narrower than the
transition linewidth, then not all atoms will be able to interact
with the ﬁeld.
dN2
dt ∞ (B12 N1 g (ν )ρ(ν ) − B21 N2 g (ν )ρ(ν )) d ν (39) =
0 ∞ Summary = (B12 N1 − B21 N2 ) g (ν )ρ(ν )d ν
0 (40) ∞ ≈ (B12 N1 − B21 N2 ) g (νp ) ρ(ν )d ν (41) 0 = (B12 N1 − B21 N2 ) g (νp )ρp (42) Here νp is the wavelength of the stimulating optical ﬁeld.
Generally, lasers will oscillator at or very near the peak of the
gain medium. It is this case we’ll be interested in for the rest of
the semester. The Possibility of Gain
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary Consider the interaction of a gain medium with narrowband
light
N2
dN2
= −B21 N2 ρν g (ν ) + B12 N1 ρν g (ν ) −
(43)
dt
τ2
g2
N2
= −B21 N2 ρν g (ν ) + B21 N1 ρν g (ν ) −
(44)
g1
τ2
N2
g2
= −B21 ρν g (ν ) N2 − N1 −
(45)
g1
τ2
λ2
N2
ρν c 1
g2
= −A21
(46)
g (ν )
N 2 − N1 −
2
8π n
n hν
g1
τ2
σse (ν )Iν
g2
N2
=−
N 2 − N1 −
(47)
hν
g1
τ2
where:
σse (ν ) ≡ A21 λ2
g (ν )
8π n 2 (48) The Possibility of Gain II
ECE 455
Lecture 3 Notice that if
Blackbody
Radiation
Lineshape
Gain
Summary N2 > g2
N1
g1 (49) The population of the upper state will decrease. Where does
the energy go? Into the optical ﬁeld! The term N2 − g2 N1 will appear repeatedly in our equations.
g1
To simplify equations, we deﬁne the population inverison
∆N ≡ N 2 − g2
N1
g1 (50) The Stimulated Emission Cross Section
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary σse is not just a random
collection of constants
The stimulated emission
cross section can be
viewed as
’Optical thickness’
1/(σse N ) Cross Section Picture Typical Values of the Stimulated Emisson Cross
Section
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary Laser
Argon Ion
Nitrogen
HeNe
HeCd
Copper Vapor
GreeNe
Dye (Rh6G)
KrF
CO2
Ti:Al2 O3
Nd:YAG
Nd:Glass
Ruby λ (nm)
488.0
337.1
632.8
441.6
510.5
543.5
577.0
248.0
10600
800.0
1064
1062.3
694.3 σse (m2 )
2.6 × 10−16
4.0 × 10−17
3.0 × 10−17
9.0 × 10−18
8.6 × 10−18
2.0 × 10−18
2.9 × 10−20
2.6 × 10−20
3.0 × 10−22
3.4 × 10−23
2.8 × 10−23
2.9 × 10−24
2.5 × 10−24 Optical Ampliﬁcation
ECE 455
Lecture 3 Blackbody
Radiation Consider a plane wave propagating in the +z direction through
an atomic medium. Then the change in the density of photons
Np after propagating through this medium is Lineshape
Gain
Summary dNp = N2 n
σse (ν )Iν
∆N − η
dz
hν
τ2 c (51) where η is an eﬃciency factor. For now, assume the beam is
intense enough so that the second term on the right hand side
is neglible.
c
Iν = Np hν
(52)
n
dI
= I σse ∆N
(53)
dz
These equations are nonlinear. Degeneracy and Gain I
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary Why does degeneracy aﬀect the gain?
An optical ﬁeld will cause both stimulated emission and
absorption; It is the net diﬀerence between these two
processes which determines whether a beam is ampliﬁed or
attenuated.
Fermi’s Golden Rule (FGR) predicts a constant transition
probability, W10 , per unit time between each individual
pair of upper and lower states
FGR also predicts that the absorption and stimulated rates
are equal, so W01 = W10 .
If there are multiple states at a given energy, the
population at that energy will be evenly distributed among
the states. Degeneracy and Gain II
ECE 455
Lecture 3 Blackbody
Radiation N1 Lineshape
Gain
Summary W10 W01 N0 The net stimulated emission is:
dN1
dt = N1 W10 − N0 W01
= W10 (N1 − N0 ) (54) Degeneracy and Gain III
ECE 455
Lecture 3 ½N1 Blackbody
Radiation ½N1 Lineshape
Gain W10 Summary W01 W10 W01 N0 The net stimulated emission is:
dN1
dt N1
N1
W10 − N0 W01 +
W10 − N0 W01
2
2
= W10 (N1 − 2N0 )
= (55) Degeneracy and Gain IV
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary The argument on the previous slide can be extended to any
number of upper and lower states. We will always ﬁnd
g1
dN1
= W10 N1 − N0
dt
g0 (56) where g1 and g2 are the upper and lower state degeneracies.
We happen to know
W10 = I σse
hν10 Warning!! In some situations with very intense ﬁelds, the
population will not have time to equilibrate among the
degenerate energy levels. This analysis will then fail. (57) The Gain Coeﬃcient
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary The
γ (ν ) ≡ σ (ν )∆N (58) Summary
ECE 455
Lecture 3 Blackbody
Radiation
Lineshape
Gain
Summary Blackbodies
There are three primary mechanisms of broadening in gas:
lifetime, doppler, and collisional.
Broadening mechanisms in solids are more diﬁcult to
predict.
A system will absorb light if ∆N < 0 and amplify it if
∆N > 0 ...
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 Fall '08
 Eden,J
 Photon, Lineshape

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