2 Surface Plasmons for Chiral Sensing
43
2.4 Complete Measurement of Chirality
So far we have examined CHISPR signals assuming only a real-valued chirality
parameter κ. Under realistic experimental conditions, this is a valid approximation
for interpreting the results when one performs measurements at optical frequencies far detuned from any molecular resonances where chiral-dependent differential
absorption (i.e. circular dichroism), which is proportional to Im(κ), is negligible
[11]. However, when the optical frequency of an SPR instrument is near a molecular
transition, both circular birefringence and circular dichroism [proportional to Re(κ)
and Im(κ), respectively] become substantial and the proposed measurements should
be interpreted with care.
To examine circular dichroism effects we introduce a nonzero imaginary part in
the chirality parameter κ and demonstrate its effect on the CHISPR signals. However,
to appropriately examine the case of circular dichroism (by introducing an imaginary
part in the chirality parameter κ) without violating the passivity, we must ensure that
Im(n c ± κ) > 0. For this reason, we introduce artificial loss in the average refractive
index of the chiral layer, i.e. n c = 1.33 + 0.01 i.
We start by demonstrating how each CHISPR signal (θ and DR signals) change
under the influence of an imaginary-valued chirality parameter. In Fig. 2.12a we
present the chiral-dependent angular split θ as a function of Re(κ) for Im(κ) =
−10
−3
, 0, 10
−3 . We see that in the presence of absorption [Im(κ) = 0] the angular
split θ obtains a linear chiral-dependent offset, and the effects of Re(κ) and Im(κ)
appear as linear superpositions in the total θ . Moreover, in Fig. 2.12b we present
the change in the differential signals, ρ DR and φ DR , in the presence of absorption.
In particular, we calculate ρ DR and φ DR for κ = 10
−5 , κ = 10
−5 i, and the sum of
the two individual signals. We also present the same signals for the case of κ =
10
−5
+ 10
−5 i, which we show to coincide with the sums of the individual signals
[Fig. 2.12b]. In overall, we see that in the presence of absorption and birefringence,
the effects of the real and imaginary parts of the chirality parameter appear as linear
superpositions in the final CHISPR signals (θ and DR signals).
In Fig. 2.13 we examine the resulting DR signals for the cases of both chiraldependent refraction, κ = ±10
−5 , and absorption, κ = ±10
−5 i. From the individual
simulations we observe a clear distinction in the DR signals between the four cases,
that is, CHISPR enables the detection of the magnitude and sign of both the real
and imaginary part of the chirality parameter, and discrimination of their contribution through the distinct CHISPR signals. In combination with the results shown in
Fig. 2.12, it becomes apparent that in the case where the SPR operational wavelength
is near the vicinity of a molecular resonance, where circular dichroism is accompanied by an dispersive circular birefringence, i.e Re(κ) & Im(κ) = 0 (Cotton effect
[11]), the resulting DR signals will be the result of a linear superposition of the individual signals for the real and the imaginary part of the total chirality parameter (as
these are presented in Fig. 2.13).
43
2.4 Complete Measurement of Chirality
So far we have examined CHISPR signals assuming only a real-valued chirality
parameter κ. Under realistic experimental conditions, this is a valid approximation
for interpreting the results when one performs measurements at optical frequencies far detuned from any molecular resonances where chiral-dependent differential
absorption (i.e. circular dichroism), which is proportional to Im(κ), is negligible
[11]. However, when the optical frequency of an SPR instrument is near a molecular
transition, both circular birefringence and circular dichroism [proportional to Re(κ)
and Im(κ), respectively] become substantial and the proposed measurements should
be interpreted with care.
To examine circular dichroism effects we introduce a nonzero imaginary part in
the chirality parameter κ and demonstrate its effect on the CHISPR signals. However,
to appropriately examine the case of circular dichroism (by introducing an imaginary
part in the chirality parameter κ) without violating the passivity, we must ensure that
Im(n c ± κ) > 0. For this reason, we introduce artificial loss in the average refractive
index of the chiral layer, i.e. n c = 1.33 + 0.01 i.
We start by demonstrating how each CHISPR signal (θ and DR signals) change
under the influence of an imaginary-valued chirality parameter. In Fig. 2.12a we
present the chiral-dependent angular split θ as a function of Re(κ) for Im(κ) =
−10
−3
, 0, 10
−3 . We see that in the presence of absorption [Im(κ) = 0] the angular
split θ obtains a linear chiral-dependent offset, and the effects of Re(κ) and Im(κ)
appear as linear superpositions in the total θ . Moreover, in Fig. 2.12b we present
the change in the differential signals, ρ DR and φ DR , in the presence of absorption.
In particular, we calculate ρ DR and φ DR for κ = 10
−5 , κ = 10
−5 i, and the sum of
the two individual signals. We also present the same signals for the case of κ =
10
−5
+ 10
−5 i, which we show to coincide with the sums of the individual signals
[Fig. 2.12b]. In overall, we see that in the presence of absorption and birefringence,
the effects of the real and imaginary parts of the chirality parameter appear as linear
superpositions in the final CHISPR signals (θ and DR signals).
In Fig. 2.13 we examine the resulting DR signals for the cases of both chiraldependent refraction, κ = ±10
−5 , and absorption, κ = ±10
−5 i. From the individual
simulations we observe a clear distinction in the DR signals between the four cases,
that is, CHISPR enables the detection of the magnitude and sign of both the real
and imaginary part of the chirality parameter, and discrimination of their contribution through the distinct CHISPR signals. In combination with the results shown in
Fig. 2.12, it becomes apparent that in the case where the SPR operational wavelength
is near the vicinity of a molecular resonance, where circular dichroism is accompanied by an dispersive circular birefringence, i.e Re(κ) & Im(κ) = 0 (Cotton effect
[11]), the resulting DR signals will be the result of a linear superposition of the individual signals for the real and the imaginary part of the total chirality parameter (as
these are presented in Fig. 2.13).