Metal nanospheres under intense continuous-wave illumination: A unique case of nonperturbative nonlinear nanophotonics

Abstract

We show that the standard perturbative (i.e., cubic) description of the thermal nonlinear response of a single metal nanosphere to intense continuous-wave (CW) illumination is sufficient only for a temperature rise of up to 100 degrees above room temperature. Beyond this regime, the slowing down of the temperature rise requires a nonperturbative description of the nonlinear response, even though the permittivity is linearly dependent on the temperature and despite the deep subwavelength effective propagation distances involved. Using experimental data, we show that, generically, the increase of the imaginary part of the metal permittivity dominates the increase of the host permittivity as well as the resonance shift due to the joint changes to the real parts of the metal and host. Thus, the main nonlinear effect is a decrease of the quality factor of the resonance. We further analyze the relative importance of the various contributions to the temperature rise and thermal nonlinearity, compare the nonlinearity of Au and Ag, demonstrate the potential effect of the nanoparticle morphology, and show that although the thermo-optical nonlinearity of the host typically plays a minor role, its thermal conductivity and its temperature dependence is important. Finally, we discuss the differences between CW and ultrafast thermal nonlinearities.

The thermo-optic nonlinearity of single metal nanoparticles under intense continuous-wave illumination

“…As mentioned, this is an unusual large nonlinearity for the associated subwavelength scales involved. As noted already in [65], the greater sensitivity of the scattered intensity on the imaginary part of the permittivity is in accord with the experimental findings reported in [59].…”

“…< 0; this occurs because of the increase of ε m with the temperature causes a reduction of the resonance quality factor. On the other hand, 1 C abs ∂C abs ∂ε m q 1 =0, T NP =T h,0 → 0, so that B m has a negligible effect on the temperature rise, in agreement with previous report on the weak (second-order) effect of ε m on the temperature [65]. In this case, Eq.…”

“…Specifically, for a = 20 nm and 30 nm, the rate of the temperature increase is sub-linear (i.e., it slows down as the incoming intensity increases with respect to the low intensity response), in good agreement with the quasi-static calculations in Ref. [65]; for a = 50 nm, by contrast, the temperature increase rate is super-linear (i.e., it grows with the incoming intensity due to the increase of absorption with temperature). The latter effect is not captured by the quasistatic approximation used in Ref.…”

The thermo-optic nonlinearity of single metal nanoparticles under intense continuous-wave illumination

“…As mentioned, this is an unusual large nonlinearity for the associated subwavelength scales involved. As noted already in [65], the greater sensitivity of the scattered intensity on the imaginary part of the permittivity is in accord with the experimental findings reported in [59].…”

“…< 0; this occurs because of the increase of ε m with the temperature causes a reduction of the resonance quality factor. On the other hand, 1 C abs ∂C abs ∂ε m q 1 =0, T NP =T h,0 → 0, so that B m has a negligible effect on the temperature rise, in agreement with previous report on the weak (second-order) effect of ε m on the temperature [65]. In this case, Eq.…”

“…Specifically, for a = 20 nm and 30 nm, the rate of the temperature increase is sub-linear (i.e., it slows down as the incoming intensity increases with respect to the low intensity response), in good agreement with the quasi-static calculations in Ref. [65]; for a = 50 nm, by contrast, the temperature increase rate is super-linear (i.e., it grows with the incoming intensity due to the increase of absorption with temperature). The latter effect is not captured by the quasistatic approximation used in Ref.…”

Size-dependence of the photothermal response of a single metal nanosphere

“…(A8) can also be used to explain the material-dependence of the particle temperature rise. For small particle sizes, the absorption cross-section (A8) is inversely proportional to ε ′′ m , so the temperature rise is higher for low loss metal NP, in agreement with previous results [11,34]. However, when the particle size increases and becomes large enough so that the radiation damping term (x 3 term in the denominator of (A8)) dominates, the absorption cross-section becomes directly proportional to ε ′′ m , so the temperature rise is higher for lossy metal NPs.…”

“…For sufficiently small particle sizes (a 10 nm), the Mieheat solutions show that ∆T NP increases quadratically with the radius, in agreement with the QS-heat solution; this verifies the validity of the results in Ref. [33,34] for such small NPs.…”

The thermo-optic nonlinearity of single metal nanoparticles under intense continuous-wave illumination

“…As mentioned, this is an unusual large nonlinearity for the associated subwavelength scales involved. As noted already in [65], the greater sensitivity of the scattered intensity on the imaginary part of the permittivity is in accord with the experimental findings reported in [59].…”

“…< 0; this occurs because of the increase of ε m with the temperature causes a reduction of the resonance quality factor. On the other hand, 1 C abs ∂C abs ∂ε m q 1 =0, T NP =T h,0 → 0, so that B m has a negligible effect on the temperature rise, in agreement with previous report on the weak (second-order) effect of ε m on the temperature [65]. In this case, Eq.…”

“…Specifically, for a = 20 nm and 30 nm, the rate of the temperature increase is sub-linear (i.e., it slows down as the incoming intensity increases with respect to the low intensity response), in good agreement with the quasi-static calculations in Ref. [65]; for a = 50 nm, by contrast, the temperature increase rate is super-linear (i.e., it grows with the incoming intensity due to the increase of absorption with temperature). The latter effect is not captured by the quasistatic approximation used in Ref.…”

Size-dependence of the photothermal response of a single metal nanosphere

“…(A8) can also be used to explain the material-dependence of the particle temperature rise. For small particle sizes, the absorption cross-section (A8) is inversely proportional to ε ′′ m , so the temperature rise is higher for low loss metal NP, in agreement with previous results [11,34]. However, when the particle size increases and becomes large enough so that the radiation damping term (x 3 term in the denominator of (A8)) dominates, the absorption cross-section becomes directly proportional to ε ′′ m , so the temperature rise is higher for lossy metal NPs.…”

“…For sufficiently small particle sizes (a 10 nm), the Mieheat solutions show that ∆T NP increases quadratically with the radius, in agreement with the QS-heat solution; this verifies the validity of the results in Ref. [33,34] for such small NPs.…”

The thermo-optic nonlinearity of single metal nanoparticles under intense continuous-wave illumination

“…As mentioned, this is an unusual large nonlinearity for the associated subwavelength scales involved. As noted already in [65], the greater sensitivity of the scattered intensity on the imaginary part of the permittivity is in accord with the experimental findings reported in [59].…”

“…< 0; this occurs because of the increase of ε m with the temperature causes a reduction of the resonance quality factor. On the other hand, 1 C abs ∂C abs ∂ε m q 1 =0, T NP =T h,0 → 0, so that B m has a negligible effect on the temperature rise, in agreement with previous report on the weak (second-order) effect of ε m on the temperature [65]. In this case, Eq.…”

“…Specifically, for a = 20 nm and 30 nm, the rate of the temperature increase is sub-linear (i.e., it slows down as the incoming intensity increases with respect to the low intensity response), in good agreement with the quasi-static calculations in Ref. [65]; for a = 50 nm, by contrast, the temperature increase rate is super-linear (i.e., it grows with the incoming intensity due to the increase of absorption with temperature). The latter effect is not captured by the quasistatic approximation used in Ref.…”

Size-dependence of the photothermal response of a single metal nanosphere

“…(A8) can also be used to explain the material-dependence of the particle temperature rise. For small particle sizes, the absorption cross-section (A8) is inversely proportional to ε ′′ m , so the temperature rise is higher for low loss metal NP, in agreement with previous results [11,34]. However, when the particle size increases and becomes large enough so that the radiation damping term (x 3 term in the denominator of (A8)) dominates, the absorption cross-section becomes directly proportional to ε ′′ m , so the temperature rise is higher for lossy metal NPs.…”

“…For sufficiently small particle sizes (a 10 nm), the Mieheat solutions show that ∆T NP increases quadratically with the radius, in agreement with the QS-heat solution; this verifies the validity of the results in Ref. [33,34] for such small NPs.…”