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Introduction
where the ratio of the peak amplitude b to the base amplitude a, b/a = m is termed
the modulation of the incident light. As described above, the phase and modulation
of the emission will be different but at the same frequency. The population of the
excited state is:
N(t) = A + B sin(ωt − φ) (1.21)
The demodulation is given by the ratio of the amplitudes B/A and phase shift
φ. Recall the single exponential intensity decay (Equation 1.13). Differentiating the
single exponential intensity decay gives an expression for the time-dependent excited
state population:
dN(t)
dt = −1/τN(t) + I0(t) (1.22)
Using d/dx(sin u) = cos u du/dx, the derivative of Equation 1.21 gives:
dN(t)
dt
Substitution of equation 1.23 into equation 1.22 yields:
= ωB cos(ωt − φ) (1.23)
ωB cos(ωt − φ) = −1/τ[A + B sin(ωt − φ)] + a + b sin ωt (1.24)
If the sin and cos functions are expanded using the standard trigonometric identities
i.e.:
Cosine Expansion : cos(α − β) = cos α cos β + sin α sin β
Sine Expansion : sin(α − β) = sin α cos β − sin β cos α
then the following relationships can be found by equating the constant terms and
those terms in φ and ωt.
a − (1/τ)A = 0 (1.25)
This base amplitude is commonly called the Direct Component (DC) offset. This term comes
from the function for a sinusoidal wave as opposed to the b term or Alternate Component (AC)
part.
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