Atomic Force Microscopy (AFM) allows for a highly sensitive detection of spectroscopic signals. This has been first demonstrated for NMR of a single molecule and recently extended to stimulated Raman in the optical regime. We theoretically investigate the use of optical forces to detect time and frequency domain nonlinear optical signals. We show that, with proper phase matching, the AFM-detected signals closely resemble coherent heterodyne-detected signals. Applications are made to AFM-detected and heterodyne-detected vibrational resonances in Coherent Anti-Stokes Raman Spectroscopy (χ(3)) and sum or difference frequency generation (χ(2)).

Atomic force microscopy of single molecules has seen many advancements over the past two decades since its theoretical and experimental advents in early 1990s.1,2 Optical forces have been used to create optical lattices for trapping cold atoms and ions.3 Magnetic and optical tweezers are versatile tools for measurements of forces and corresponding single molecule displacements in biological molecules such as DNA and proteins. AFM measurements provide high resolution and conformational control of bio-macromolecules. Noncontact AFM has been used to visualize intermolecular bonds such as hydrogen bonds.4 Magnetic tweezers are often used for analysing DNA topology, and like optical tweezers, can perform 3-dimensional manipulations. These techniques generally measure the displacement of molecular position caused by the forces applied by the probe.5–11 The optical potential acting on the single molecule by the fields can trap cold atoms in optical lattices.3 Measuring its gradient (the force) constitutes a new type of signal.

Here we theoretically investigate how optical forces may be used to detect nonlinear optical signals. Instead of measuring the mechanical displacement of single molecule induced by the applied force, one can look at the gradient force applied by the AFM tip to the molecule.12,13 In such measurements, the spectroscopic information is contained in the pulse parameters used to create the force. Stimulated Raman resonances related to χ(3) have been recently reported experimentally by Rajapksa et al.22 The mechanical force associated with different NonLinear Optical (NLO) techniques like FRET-force in force imaging has been investigated.14 A plethora of methods have been employed for detecting nonlinear optical signals. These include homo and hetrodyne detection of fields,15,16 incoherent fluorescence detection,17,18 photoaccoustic detection,19,20 photoelectron, and current detection.21 

We derive general expressions for ultrafast nonlinear optical spectroscopy with gradient force detection. A single molecule on a glass cover slide is subjected to several collinear laser beams and the variation of the gradient force with selected parameters such as delays between pulses, phases, or frequencies is measured by AFM tip using vibrational tapping mode,22 thus generating multidimensional signals. We show that by manipulating the phases between fields, it is possible in some cases to make the optical potential (and gradient force) to coincide with the heterodyne signal.23 

The conservative force is given by the gradient of the optical potential V(r).3,12,14

\begin{equation}{\bf F(r)}_{\bf gr}=-{\bigtriangledown}_{\bf r} {\bf V(r).}\end{equation}
F(r)gr=rV(r).
(1)

We first calculate the potential felt by the molecule due to its nonlinear coupling to electromagnetic fields in continuous wave (cw) frequency-domain experiments.24,n classical monochromatic fields induce a polarization P(ω) in the molecule, which then creates an optical potential (V),23 

\begin{eqnarray}& {\bf V}({\bf r}) = -\Re \big[\int d\omega {\bf P}(\omega ).{\bf E}^{\ast }({\bf r},\omega )\big]=- \Re \big[\int dt {\bf P}(t).{\bf E}^{\ast }({\bf r},t)\big].\nonumber \\\end{eqnarray}
V(r)=dωP(ω).E*(r,ω)=dtP(t).E*(r,t).
(2)

The classical electric field can be expanded in modes:

\begin{eqnarray}{\bf E}_i({\bf r},t)&=&\sum\nolimits _{\xi _i=\pm }\epsilon _i \int \frac{d\omega _i}{2 \pi } \mathcal {E}_i^{\xi _i}({\bf r},\omega _i) e^{i \xi _i (\omega _i t +\phi _i)}, \nonumber \\[-6pt]\\{\bf E}_i({\bf r},\omega ) &=& \sum\nolimits _{\xi _i=\pm } 2\pi . \epsilon _i \mathcal {E}_i^{\xi _i}({\bf r},\omega _i) e^{i\xi _i\phi _i} \delta (\omega ^\prime -\xi _i\omega ),\nonumber\end{eqnarray}
Ei(r,t)=ξi=±εidωi2πEiξi(r,ωi)eiξi(ωit+ϕi),Ei(r,ω)=ξi=±2π.εiEiξi(r,ωi)eiξiϕiδ(ωξiω),
(3)

where

$\mathcal {E}_i^{\xi _i}({\bf r},\omega )$
Eiξi(r,ω) is the position dependent envelope of the ith field and ξi = ± is the hermiticity (negative for positive frequency and positive for negative frequency component) of the field. ϕi is the phase angle between the incident classical field modes and εi is the respective polarization vector. The optical potential is given by

\begin{equation}{\bf V}({\bf r})= -\Re \int _0^\infty {d\omega }{\bf P}(\omega ).{\bf E}^\ast ({\bf r},\omega ),\\\end{equation}
V(r)=0dωP(ω).E*(r,ω),
(4)

where ω > 0. For comparison, the heterodyne detected signal measured at ω is given by25 

\begin{equation}S({\bf r};\omega ^\prime )= -\Im \int _0^\infty {d\omega } {\bf P}(\omega ).{\bf E}^\ast ({\bf r},\omega ).\\\end{equation}
S(r;ω)=0dωP(ω).E*(r,ω).
(5)

The total induced polarization is the sum of positive and negative frequency components, Ptotal(ω) = P+(ω) + P(ω). One can identify P+(ω) with P(ω) and P(ω) with P*(ω). Similarly, we can identify E+(ω) with E(ω) and E(ω) with E*(ω). The superscripts + and − denote the positive or negative frequency components. For a cw measurement involving discrete set of modes, we have

\begin{eqnarray}{\bf V}_d({\bf r};\omega _j)=-\Re \sum\nolimits _j {\bf P}(\omega _j).{\bf E}^\ast ({\bf r},\omega _j),\end{eqnarray}
Vd(r;ωj)=jP(ωj).E*(r,ωj),
(6)
\begin{eqnarray}S_d({\bf r};\omega _j^\prime )=-\Im \sum\nolimits _j {\bf P}(\omega _j).{\bf E}^\ast ({\bf r},\omega _j).\end{eqnarray}
Sd(r;ωj)=jP(ωj).E*(r,ωj).
(7)

For the linear response, we get the optical potential and heterodyne signal,

\begin{eqnarray}{\bf V}^{(1)}({\bf r};\omega _j)= - \sum\nolimits _j \chi ^{(1)^ \prime }(\omega _j).|{\bf E}({\bf r},\omega _j)|^2,\end{eqnarray}
V(1)(r;ωj)=jχ(1)(ωj).|E(r,ωj)|2,
(8)
\begin{eqnarray}S^{(1)}_{het}({\bf r};\omega _j)= -2 \sum\nolimits _j \chi ^{(1)^{ \prime \prime} }(\omega _j).|{\bf E}({\bf r},\omega _j)|^2,\end{eqnarray}
Shet(1)(r;ωj)=2jχ(1)(ωj).|E(r,ωj)|2,
(9)

where, linear susceptibility, χ(1)(ω) = χ(1)(ω) + iχ(1)″(ω). Superscripts and denote the real and imaginary part. For a multilevel system, we have

\begin{eqnarray}\chi ^{(1)}(\omega )=\frac{1}{\hbar }\sum _{a,c}P(a)\frac{|\mu _{ac}|^2\omega _{ac}}{(\omega _{ac})^2-(\omega +i\Gamma _{ac})^2},\end{eqnarray}
χ(1)(ω)=1a,cP(a)|μac|2ωac(ωac)2(ω+iΓac)2,
(10)

where,

$P(a)=e^{-\beta \varepsilon _a}/\sum _ae^{-\beta \varepsilon _a}$
P(a)=eβɛa/aeβɛa with ɛa as the eigen energy of level a, and β = 1/kbT with temperature T. The linear optical potential and heterodyne signal have dispersive and absorptive profiles at ω = ωac as depicted in Fig. 1.

FIG. 1.

Optical potential (solid red) and heterodyne signal (dashed blue) induced on single chromophore at ℏωac = 2.5 eV using real and imaginary part of Eq. (10)

FIG. 1.

Optical potential (solid red) and heterodyne signal (dashed blue) induced on single chromophore at ℏωac = 2.5 eV using real and imaginary part of Eq. (10)

Close modal

We now turn to a second order process (Fig. 2) where ω1 in the IR regime, ω2 and ω3 are in visible region , and ω3 = ω1 + ω2, the optical potential is then given by

\begin{eqnarray}{\bf V}^{(2)}({\bf r};\omega _1,\omega _2)&=& -\Re \big[\sum\nolimits _j (2\pi)^3 \mathcal {E}_1({\bf r},\omega _j)\mathcal {E}_2({\bf r},\omega _j)\nonumber \\&&\mathcal {E}_3^\ast ({\bf r},\omega _j)\sum\nolimits _p\lbrace \chi ^{(2)}(-\omega _3;\omega _1,\omega _2)\rbrace \big],\nonumber\\\end{eqnarray}
V(2)(r;ω1,ω2)=[j(2π)3E1(r,ωj)E2(r,ωj)E3*(r,ωj)p{χ(2)(ω3;ω1,ω2)}],
(11)

where, ∑p represents the sum over all permutations of frequencies {ω1, ω2, −ω3}.χ(2)( − ω3; ω1, ω2) represents Sum Frequency Generation (SFG) whereas the other two permutations, when ω2 or ω1 are detected, are Difference Frequency Generation (DFG). All possible processes contribute to the optical potential. The total heterodyne-detected signals is:

\begin{eqnarray}&&S^{(2)}_{total}({\bf r},\omega _3;\omega _1,\omega _2)\nonumber\\&&\quad = -\Im \big[\sum\nolimits _j (2\pi )^3 \mathcal {E}_1({\bf r},\omega _j)\nonumber \\&&\qquad \mathcal {E}_2({\bf r},\omega _j)\mathcal {E}_3^\ast ({\bf r},\omega _j)\sum\nolimits _p\chi ^{(2)}(-\omega _3;\omega _1,\omega _2)\big].\qquad\end{eqnarray}
Stotal(2)(r,ω3;ω1,ω2)=[j(2π)3E1(r,ωj)E2(r,ωj)E3*(r,ωj)pχ(2)(ω3;ω1,ω2)].
(12)

Using the level scheme of Fig. 2 and invoking the Rotating Wave Approximation (RWA), we obtain,

\begin{eqnarray}&&{\bf V}^{(2)}({\bf r};\omega _1,\omega _2)\nonumber\\&&\quad = -\Re \Bigg[\frac{-i}{2\hbar^{2} }\sum\nolimits _j (2\pi )^3 \mathcal {E}_1({\bf r},\omega _j)\nonumber \\&&\qquad \mathcal {E}_2({\bf r},\omega _j)\mathcal {E}_3^\ast ({\bf r},\omega _j)\mu _{ac}\mu _{cb}\mu _{ba}\sum\nolimits _a P(a)\nonumber \\&&\qquad\Bigg\lbrace \frac{1}{(\omega _{IR}-\omega _{ca}+i\Gamma _{ca})\big(\omega _{IR}+\omega _{vis}^{(2)}-\omega _{ab}+i\Gamma _{ab}\big)}\nonumber \\&&\qquad +\,\frac{1}{\big(\omega _{vis}^{(3)}-\omega _{ab}+i\Gamma _{ab}\big)}\Bigg(\frac{-1}{\big(\omega _{vis}^{(3)}-\omega _{vis}^{(2)}-\omega _{ac}+i\Gamma _{ac}\big)}\nonumber \\&&\qquad +\,\frac{1}{\big(\omega _{vis}^{(3)}-\omega _{IR}^{(1)}-\omega _{ca}+i\Gamma _{ca}\big)}\Bigg)\Bigg\rbrace \Bigg].\end{eqnarray}
V(2)(r;ω1,ω2)=[i22j(2π)3E1(r,ωj)E2(r,ωj)E3*(r,ωj)μacμcbμbaaP(a){1(ωIRωca+iΓca)ωIR+ωvis(2)ωab+iΓab+1ωvis(3)ωab+iΓab(1ωvis(3)ωvis(2)ωac+iΓac+1ωvis(3)ωIR(1)ωca+iΓca)}].
(13)

The sum of the three heterodyne signals (Eq. (12)) is given by replacing ℜ with ℑ in Eq. (13) as shown in Fig. 3.

FIG. 2.

(a) The three level model scheme used to calculate the χ(2) optical potential and heterodyne signal. (b) Loop diagram for SFG and DFG associated to the level diagram in Fig. 2(a) 

FIG. 2.

(a) The three level model scheme used to calculate the χ(2) optical potential and heterodyne signal. (b) Loop diagram for SFG and DFG associated to the level diagram in Fig. 2(a) 

Close modal
FIG. 3.

(a) Optical potential Eq. (13) for cw χ(2) with μac = 6D, μbc = 1D, ωac = 2.3 eV, ωba = 10 eV, ωbc = 10 eV, Γac = 5 cm−1, Γbc = 5 cm−1, and Γab = 5 cm−1. (b) Total heterodyne signal (SFG+DFG) for the same parameters.

FIG. 3.

(a) Optical potential Eq. (13) for cw χ(2) with μac = 6D, μbc = 1D, ωac = 2.3 eV, ωba = 10 eV, ωbc = 10 eV, Γac = 5 cm−1, Γbc = 5 cm−1, and Γab = 5 cm−1. (b) Total heterodyne signal (SFG+DFG) for the same parameters.

Close modal

We next turn to the third order optical potential and heterodyne signal (measured at ω4) for a four wave mixing of configuration ω4 = ω1 − ω2 + ω3 as shown in Fig. 4, at resonance frequency ω1 − ω2 = ωac and coherent fields (−sgnj)∑i ϕi = 0) are given by

\begin{eqnarray}&&{\bf V}^{(3)}({\bf r};\omega _1,-\omega _2,\omega _3)\nonumber\\&&\quad = -\Re \big[\sum\nolimits _j (2\pi )^4 \mathcal {E}_1({\bf r},\omega _j)\mathcal {E}_2^\ast ({\bf r},\omega _j)\nonumber \\&&\qquad \mathcal {E}_3({\bf r},\omega _j)\mathcal {E}_4^\ast ({\bf r},\omega _j)\sum\nolimits _p\lbrace \chi ^{(3)}(-\omega _4;\omega _1,-\omega _2,\omega _3)\rbrace \big],\nonumber\\\end{eqnarray}
V(3)(r;ω1,ω2,ω3)=[j(2π)4E1(r,ωj)E2*(r,ωj)E3(r,ωj)E4*(r,ωj)p{χ(3)(ω4;ω1,ω2,ω3)}],
(14)
\begin{eqnarray}&&S^{(3)}_{het}({\bf r},\omega _4;\omega _1, -\omega _2,\omega _3)\nonumber\\&&\quad = -\Im \big[\sum\nolimits _j (2\pi )^4 \mathcal {E}_1({\bf r},\omega _j)\mathcal {E}_2^\ast ({\bf r},\omega _j)\nonumber \\&&\qquad \mathcal {E}_3({\bf r},\omega _j)\mathcal {E}_4^\ast ({\bf r},\omega _j)\chi ^{(3)}(-\omega _4;\omega _1,-\omega _2,\omega _3)\big].\end{eqnarray}
Shet(3)(r,ω4;ω1,ω2,ω3)=[j(2π)4E1(r,ωj)E2*(r,ωj)E3(r,ωj)E4*(r,ωj)χ(3)(ω4;ω1,ω2,ω3)].
(15)

For the three level system shown in Fig. 4, with ground state (a), vibrational excited state (c) and electronic excited state (b), the CARS resonances of χ(3) for (ω4 = ω1 − ω2 + ω3) is25 

\begin{eqnarray}&&\chi _{CARS}^{(3)}(-\omega _4;\omega _1,-\omega _2,\omega _3)\nonumber\\&&\quad =\displaystyle\frac{P(a)|\mu _{ba}|^2|\mu _{ac}|^2}{(\omega _1-\omega _{ba}+i\eta _e)}\nonumber \\&&\qquad \frac{1}{(\omega _1-\omega _2+\omega _3-\omega _{bc}+i\eta _e)(\omega _1-\omega _2-\omega _{ac}+i\eta _R)}.\nonumber \\\end{eqnarray}
χCARS(3)(ω4;ω1,ω2,ω3)=P(a)|μba|2|μac|2(ω1ωba+iηe)1(ω1ω2+ω3ωbc+iηe)(ω1ω2ωac+iηR).
(16)

Unlike the heterodyne signal which is measured at a selected frequency, the optical potential Eq. (14) is given by a sum over all possible permutations of signal frequency for χ(3). The two signals coincide if we change the overall phase of

$\mathcal {E}_1({\bf r},\omega _j)\mathcal {E}_2^\ast ({\bf r},\omega _j)\mathcal {E}_3({\bf r},\omega _j)\mathcal {E}_4^\ast ({\bf r},\omega _j)$
E1(r,ωj)E2*(r,ωj)E3(r,ωj)E4*(r,ωj) by
$\frac{\pi }{2}$
π2
and assume the Rotating Wave Approximation (RWA) as was done in deriving Eq. (13) for χ(2).

FIG. 4.

The three level model system used to calculate the optical potential and heterodyne χ(3) signal and the contributing Unrestricted loop diagram.

FIG. 4.

The three level model system used to calculate the optical potential and heterodyne χ(3) signal and the contributing Unrestricted loop diagram.

Close modal

Consider a special case involving only two fields with frequencies ω1 and ω2,

$\mathcal {E}_3({\bf r},\omega _j)=\mathcal {E}_1({\bf r},\omega _j)$
E3(r,ωj)=E1(r,ωj) and
$\mathcal {E}_2({\bf r},\omega _j)\break=\mathcal {E}_4({\bf r},\omega _j)$
E2(r,ωj)=E4(r,ωj)
, we can rewrite Eqs. (14) and (15) as

\begin{eqnarray}&&{\bf V}^{(3)}({\bf r};\omega _1,-\omega _2)\nonumber\\&&\quad = -\sum\nolimits _j (2\pi )^4 |\mathcal {E}_1({\bf r},\omega _j)|^2\nonumber \\&&\qquad |\mathcal {E}_2({\bf r},\omega _j)|^2\sum\nolimits _p\lbrace \chi ^{(3)^ \prime }(-\omega _2;\omega _1,-\omega _2,\omega _1)\rbrace ,\end{eqnarray}
V(3)(r;ω1,ω2)=j(2π)4|E1(r,ωj)|2|E2(r,ωj)|2p{χ(3)(ω2;ω1,ω2,ω1)},
(17)
\begin{eqnarray}&&S_{total}^{(3)}({\bf r};\omega _1,-\omega _2)\nonumber\\&&\quad = -\sum\nolimits _j (2\pi )^4 |\mathcal {E}_1({\bf r},\omega _j)|^2\nonumber \\&&\qquad |\mathcal {E}_2({\bf r},\omega _j)|^2\sum\nolimits _p\lbrace \chi ^{(3)^{ \prime \prime} }(-\omega _2;\omega _1,-\omega _2,\omega _1)\rbrace .\end{eqnarray}
Stotal(3)(r;ω1,ω2)=j(2π)4|E1(r,ωj)|2|E2(r,ωj)|2p{χ(3)(ω2;ω1,ω2,ω1)}.
(18)

AFM provides the same information as heterodyne detection but with a much higher sensitivity (∼pN force) that allows to detect single molecules [SI from Ref. 22]. If the relevant field factors are real, for example, in linear regime or in Eqs. (17) and (18), then the heterodyne signal is given by imaginary part of susceptibility (χ) and the optical potential (and gradient force) is given by the real part of susceptibility (χ). In some cases, it is possible to vary the phases and make the optical potential (or gradient force) coincide with the heterodyne signal.

We now apply the results to the three-level model system shown in Fig. 4, with ground state (a), vibrational excited state (c), and electronic excited state (b),χ(3) for stimulated CARS for two beam (ω4 = 2ω1 − ω2) within the RWA is25 

\begin{eqnarray}&&\chi _{CARS}^{(3)}(-\omega _2;\omega _1)\nonumber\\&&\quad =\frac{P(a)|\mu _{ba}|^2|\mu _{ac}|^2}{(2\omega _1-\omega _2-\omega _{ab^\prime }+i\eta _{ab^\prime })}\nonumber \\&&\qquad \frac{1}{(\omega _1-\omega _2-\omega _{ac}+i\eta _{ac})(\omega _1-\omega _{ba}+i\eta _{ba})},\qquad\end{eqnarray}
χCARS(3)(ω2;ω1)=P(a)|μba|2|μac|2(2ω1ω2ωab+iηab)1(ω1ω2ωac+iηac)(ω1ωba+iηba),
(19)
\begin{eqnarray}&&\chi _{CARS}^{(3)}(-\omega _1;\omega _2)\nonumber\\&&\quad =\frac{P(c)|\mu _{ba}|^2|\mu _{ac}|^2}{(2\omega _2-\omega _1-\omega _{ac}+i\eta _{ac})}\nonumber \\&&\qquad \frac{1}{(\omega _2-\omega _1-\omega _{bb^{\prime }}+i\eta _{bb^{\prime }})(\omega _2-\omega _{cb}+i\eta _{cb})}.\qquad\end{eqnarray}
χCARS(3)(ω1;ω2)=P(c)|μba|2|μac|2(2ω2ω1ωac+iηac)1(ω2ω1ωbb+iηbb)(ω2ωcb+iηcb).
(20)

Since the electronic transition frequencies ωbc and ωab are much higher than the vibrational frequency ωac, we can rewrite Eq. (16) for optical potential as, Eq. (21), where

$\chi ^{(3)}_{\Sigma }=\Re [\sum _{p}\chi _{CARS}^{(3)}(-\omega _2;\omega _1)]$
χΣ(3)=[pχCARS(3)(ω2;ω1)]⁠. We get the total heterodyne signal by simply replacing ℜ in Eq. (21) with ℑ,

\begin{eqnarray}{\bf V}^{(3)}({\bf r};\omega _1,-\omega _2){=} -\big[\!\sum\nolimits _j\! (2\pi )^4 |\mathcal {E}_1({\bf r},\omega _j)|^2|\mathcal {E}_2({\bf r},\omega _j)|^2\chi ^{(3)}_{\Sigma }\big].\!\!\!\!\!\!\nonumber \\[-4pt]\end{eqnarray}
V(3)(r;ω1,ω2)=j(2π)4|E1(r,ωj)|2|E2(r,ωj)|2χΣ(3).
(21)

Raman resonances are observed when ω1 − ω2 = ±ωac where ωac is the difference between two ground vibrational states. The calculated optical potential and heterodyne signals are shown in Fig. 5. The gradient force due to this Raman resonance was calculated in Ref. 22 using phenomenological nonlinear polarizability. Our expression is recast in terms of the general third order polarizability χ(3) and allows to calculate the optical potential (or gradient force) for other techniques as well.

FIG. 5.

(a) Optical potential (Eq. (21)) for the model system of Fig. 4, at Raman Resonances for four wave mixing with two frequencies ω1 and ω21 = ω1, Ω2 = ω1 − ω2; Ω3 = 2ω1 − ω2) with,

$\mu _{ac} = 6 D,\mu _{bc}=1 D,\omega _{ac} = 0.2014\, eV, \omega _{bc}= 20\, eV, \omega _{ba}\approx 20\, eV,\break\omega _{bb^\prime }\approx 0 eV;\eta _{ab}= 0.1\, \text{cm}^{-1}\,\text{and} \, \eta _{ac}= 0.2\, \text{cm}^{-1}.$
μac=6D,μbc=1D,ωac=0.2014eV,ωbc=20eV,ωba20eV,ωbb0eV;ηab=0.1cm1andηac=0.2cm1. (b) Total Heterodyne signal for the same parameters by changing ℜ to ℑ in Eq. (21).

FIG. 5.

(a) Optical potential (Eq. (21)) for the model system of Fig. 4, at Raman Resonances for four wave mixing with two frequencies ω1 and ω21 = ω1, Ω2 = ω1 − ω2; Ω3 = 2ω1 − ω2) with,

$\mu _{ac} = 6 D,\mu _{bc}=1 D,\omega _{ac} = 0.2014\, eV, \omega _{bc}= 20\, eV, \omega _{ba}\approx 20\, eV,\break\omega _{bb^\prime }\approx 0 eV;\eta _{ab}= 0.1\, \text{cm}^{-1}\,\text{and} \, \eta _{ac}= 0.2\, \text{cm}^{-1}.$
μac=6D,μbc=1D,ωac=0.2014eV,ωbc=20eV,ωba20eV,ωbb0eV;ηab=0.1cm1andηac=0.2cm1. (b) Total Heterodyne signal for the same parameters by changing ℜ to ℑ in Eq. (21).

Close modal

We next turn to time-domain measurements which use temporally well separated impulsive pulses. The induced nth order optical potential V(r; tj) and heterodyne signal S(r; tj) are given by

\begin{eqnarray}{\bf V}^{(n)}({\bf r};t_j)&=& -\Re \hbar^{(-n)} \left[\int d t {\bf P}^{(n)}(t).{\bf E}^\ast ({\bf r},t)\right]\nonumber \\[-6pt]\\[-6pt]S^{(n)}_{het}({\bf r};t_j)&=& -\frac{2^n}{\hbar ^n}\Im \left[\int { dt} {\bf P}^{(n)}(t).{\bf E}^\ast ({\bf r},t)\right].\nonumber\end{eqnarray}
V(n)(r;tj)=(n)dtP(n)(t).E*(r,t)Shet(n)(r;tj)=2nndtP(n)(t).E*(r,t).
(22)

Signals are now parameterized by the time delays between pulses t1, t2, t3 rather than their frequencies. We can write the optical potential and corresponding heterodyne signals, to first order,

\begin{eqnarray}{\bf V}^{(1)}({\bf r};t_1)&=& -\Re \left[\int {d t}\int _0^\infty {d\tau _1}S^{(1)}(\tau _1){\bf E}^\ast (t){\bf E}(\tau _1)\right]\nonumber \\[-6pt]\\[-6pt]S_{het}^{(1)}({\bf r};t_1)&=& -\Im \left[\int {d t}\int _0^\infty {d\tau _1}S^{(1)}(\tau _1){\bf E}^\ast (t){\bf E}(\tau _1)\right],\nonumber\end{eqnarray}
V(1)(r;t1)=dt0dτ1S(1)(τ1)E*(t)E(τ1)Shet(1)(r;t1)=dt0dτ1S(1)(τ1)E*(t)E(τ1),
(23)

in second order,

\begin{eqnarray}{\bf V}^{(2)}({\bf r};t_1,t_2)&=& -\Re \Bigg[\int {d t}\int _0^\infty {d\tau _1}\int _0^\infty {d\tau _2}S^{(2)}(\tau _1,\tau _2)\nonumber \\&&{\bf E}_1(\tau _1) {\bf E}_2(\tau _2){\bf E}_3^\ast (t)\Bigg]\nonumber \\S_{het}^{(2)}({\bf r};t_1,t_2)&=& -\Im \Bigg[\int {d t}\int _0^\infty {d\tau _1}\int _0^\infty {d\tau _2}S^{(2)}(\tau _1,\tau _2)\nonumber \\&&{\bf E}_1(\tau _1){\bf E}_2(\tau _2){\bf E}_3^\ast (t)\Bigg],\end{eqnarray}
V(2)(r;t1,t2)=[dt0dτ10dτ2S(2)(τ1,τ2)E1(τ1)E2(τ2)E3*(t)]Shet(2)(r;t1,t2)=[dt0dτ10dτ2S(2)(τ1,τ2)E1(τ1)E2(τ2)E3*(t)],
(24)

finally, in third order,

\begin{eqnarray}&&{\bf V}^{(3)}({\bf r};t_1,t_2,t_3)\nonumber\\&&\quad = -\Re \Bigg[\int {d t}\int _0^\infty {d\tau _1}\int _0^\infty {d\tau _2}\int _0^\infty {d\tau _3}\nonumber \\&&\qquad S^{(3)}(\tau _3,\tau _2,\tau _1){\bf E}_4^\ast (t){\bf E}_1(\tau _1){\bf E}_2^\ast (\tau _2){\bf E}_3(\tau _3)\Bigg]\end{eqnarray}
V(3)(r;t1,t2,t3)=[dt0dτ10dτ20dτ3S(3)(τ3,τ2,τ1)E4*(t)E1(τ1)E2*(τ2)E3(τ3)]
(25)
\begin{eqnarray}&&S_{het}^{(3)}({\bf r};t_1,t_2,t_3)\nonumber\\&&\quad = -\Im \Bigg[\int {d t}\int _0^\infty {d\tau _1}\int _0^\infty {d\tau _2}\int _0^\infty {d\tau _3}\nonumber \\&&\qquad S^{(3)}(\tau _3,\tau _2,\tau _1){\bf E}_4^\ast (t){\bf E}_1(\tau _1){\bf E}_2^\ast (\tau _2){\bf E}_3(\tau _3)\Bigg].\end{eqnarray}
Shet(3)(r;t1,t2,t3)=[dt0dτ10dτ20dτ3S(3)(τ3,τ2,τ1)E4*(t)E1(τ1)E2*(τ2)E3(τ3)].
(26)

We again focus on vibrational resonances and assume electronically off-resonant frequencies. For χ(2) (Fig. 2), if we have two temporally resolved impulsive electric fields with delay t1 and total phase

$\sum _i^n \phi _i=0$
inϕi=0⁠, we can then write the P(2)(r, t1) as26–28 

\begin{eqnarray}&{\bf P}^{(2)}({\bf r},t_1)= \mathcal {E}_{1} \mathcal {E}_{2} \sum\nolimits _{a} P(a)\mu _{ab}|\mu _{ca}|^2 I_{ac}(t_1)e^{i \omega _1 t_1},\end{eqnarray}
P(2)(r,t1)=E1E2aP(a)μab|μca|2Iac(t1)eiω1t1,
(27)

where we define,

$P(a)=e^{-\beta \varepsilon _a}/\sum _a e^{-\beta \varepsilon _a}$
P(a)=eβɛa/aeβɛa and
$I_{\nu \nu ^\prime }(t)\break =exp[-i \omega _{\nu \nu ^\prime }t_1 - \gamma _{\nu \nu ^\prime } t_1]$
Iνν(t)=exp[iωννt1γννt1]
. For simplicity, the r dependence is engraved in the envelope
$\mathcal {E}$
E
and has not been explicitly written. Now we can write the second order optical potential measured as a function of t at t = t1 with t2 = 0 as,

\begin{eqnarray}{\bf V}^{(2)}({\bf r},t_1) \!=\! \Re\hbar^{(-2)} \left[i\mathcal {E}_1\mathcal {E}_2 \mathcal {E}_3^{\ast } \sum\nolimits _{a}\! P(a)\mu _{ac}\mu _{cb}\mu _{bc} |I_{ac}(t_1)|^2 e^{i \omega _1 t_1}\right].\!\!\!\!\!\!\nonumber \\\end{eqnarray}
V(2)(r,t1)=(2)iE1E2E3*aP(a)μacμcbμbc|Iac(t1)|2eiω1t1.
(28)

Heterodyne signals are given by replacing ℜ in Eqs. (28) and (29) with ℑ. One could observe the Raman resonance at conjugate frequency (Ω1) by Fourier transforming Eq. (28) with respect to t1 as

\begin{eqnarray}{\bf V}^{(2)}({\bf r},\Omega _1) = \int _0^{\infty }{d t_1 }{\bf V}^{(2)}({\bf r},t_1)e^{i \Omega _1 t_1}.\end{eqnarray}
V(2)(r,Ω1)=0dt1V(2)(r,t1)eiΩ1t1.
(29)

For three temporally separated electric fields (Fig. 4), we assume that the field envelopes

$\mathcal {E}_i(t)$
Ei(t) does not allow matter to evolve during the pulse and the fields are long enough so that their spectral bandwidth is narrow enough to show vibrational Raman resonances. Using coherent
$(\sum _i^n \phi _i=0)$
(inϕi=0)
and impulsive
$(\mathcal {E}_i(t)=\mathcal {E}_i\delta (t))$
(Ei(t)=Eiδ(t))
fields, we can immediately write the Raman resonance term for optical potential using loop diagrams in Fig. 4 and Eq. (25), as Eq. (30),

\begin{eqnarray}{\bf V}^{(3)}({\bf r},t_2)&=&\Re \Bigg[\frac{-1}{\hbar ^3}|\mathcal {E}_1|^2|\mathcal {E}_2^\ast |^2 \sum\nolimits _{a} P(a)|\mu _{ab}|^2|\mu _{ca}|^2 \nonumber \\&& |I_{ca}(t_2)|^2 e^{i(\omega _1-\omega _2)t_2}\Bigg].\end{eqnarray}
V(3)(r,t2)=[13|E1|2|E2*|2aP(a)|μab|2|μca|2|Ica(t2)|2ei(ω1ω2)t2].
(30)

The heterodyne-detected signal corresponding to Eq. (30) is obtained by simply replacing ℜ in Eq. (30) with ℑ. If we Fourier transform Eq. (30), similar to Eq. (29), but with respect to t2 using conjugate frequency (Ω2) instead, we can observe the vibrational Raman resonance at Ω2 = ωac. In our case, this will give a similar optical potential and heterodyne-detected signal as shown in Fig. 6 for χ(2) processes but with a different prefactor. Since we considered electronically off-resonant processes, second and third order optical potentials can simply be parameterized by conjugate frequencies Ω1 or Ω2 respectively, as in Eqs. (29) and (30), giving one dimensional spectra. When extended to electronically resonant system, one would be able to get multidimensional time-domain spectra with variable pulse delays.

FIG. 6.

Time resolved χ(2) (Eqs. (28) and (29)) with similar parameters as Fig. 3.

FIG. 6.

Time resolved χ(2) (Eqs. (28) and (29)) with similar parameters as Fig. 3.

Close modal

In summary, we showed that AFM detection closely resembles coherent heterodyne signals with appropriate phase matching. This is in contrast with other incoherent detection techniques like Florescence,15 photo-acoustic,19,20 and photo-electron detection.21 

The authors wish to thank H. K. Wickramasinghe and E. Potma for useful discussions. They gratefully acknowledge the support of the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy. In addition, they also thank the National Science Foundation (Grant No. CHE-1058791) and the National Institute of Health (Grant No. GM-59230) for their support.

1.
T.
Plakhotnik
,
E. A.
Donley
, and
U. P.
Wild
,
Annu. Rev. Phys. Chem.
48
,
181
(
1997
).
2.
S.
Mukamel
,
D.
Healion
,
Y.
Zhang
, and
J. D.
Biggs
,
Annu. Rev. Phys. Chem.
64
,
101
(
2013
).
3.
M. B.
Dahan
,
E.
Peik
,
J.
Reichel
,
Y.
Castin
, and
C.
Salomon
,
Phys. Rev. Lett.
76
,
4508
(
1996
).
4.
J.
Zhang
,
P.
Chen
,
B.
Yuan
,
W.
Ji
,
Z.
Cheng
, and
X.
Qiu
,
Science
342
,
611
(
2013
).
5.
K. C.
Neuman
and
A.
Nagy
,
Nat. Methods
5
,
491
(
2008
).
6.
S.
Kasas
,
N.
Thomson
,
B.
Smith
,
P.
Hansma
,
J.
Miklossy
, and
H.
Hansma
,
Int. J. Imaging Syst. Technol.
8
,
151
(
1997
).
7.
J.
Yan
,
D.
Skoko
, and
J. F.
Marko
,
Phys. Rev. E
70
,
011905
(
2004
).
8.
S. C.
Kuo
and
M. P.
Sheetz
,
Science
260
,
232
(
1993
).
9.
S.
Hohng
,
S.
Lee
,
J.
Lee
, and
M. H.
Jo
,
Chem. Soc. Rev.
43
,
1007
(
2014
).
10.
M.
Orrit
,
T.
Ha
, and
V.
Sandoghdar
,
Chem. Soc. Rev.
43
,
973
(
2014
).
11.
D.
Kilinc
and
G. U.
Lee
,
Integr. Biol.
6
,
27
(
2014
).
12.
T.
Kudo
and
H.
Ishihara
,
Phys. Chem. Chem. Phys.
15
,
14595
(
2013
).
13.
I.
Rajapaksa
,
K.
Uenal
, and
H. K.
Wickramasinghe
,
Appl. Phys. Lett.
97
,
073121
(
2010
).
14.
A. E.
Cohen
and
S.
Mukamel
,
J. Phys. Chem. A
107
,
3633
(
2003
).
15.
S.
Mukamel
and
M.
Richter
,
Phys. Rev. A: At., Mol., Opt. Phys.
83
,
013815
(
2011
).
16.
G.
Roumpos
and
S. T.
Cundiff
,
JOSA B
30
,
1303
(
2013
).
17.
D.
Brinks
,
F. D.
Stefani
,
F.
Kulzer
,
R.
Hildner
,
T. H.
Taminiau
,
Y.
Avlasevich
,
K.
Müllen
, and
N. F.
Van Hulst
,
Nature (London)
465
,
905
(
2010
).
18.
D.
Brinks
,
R.
Hildner
,
E. M.
van Dijk
,
F. D.
Stefani
,
J. B.
Nieder
,
J.
Hernando
, and
N. F.
van Hulst
,
Chem. Soc. Rev.
43
,
2476
(
2014
).
19.
G. A.
West
,
J. J.
Barrett
,
D. R.
Siebert
, and
K. V.
Reddy
,
Rev. Sci. Instrum.
54
,
797
(
1983
).
20.
S. Y.
Emelianov
,
P.-C.
Li
, and
M.
O’Donnell
,
Phys. Today
62
,
34
(
2009
).
21.
H.
Cohen
,
Appl. Phys. Lett.
85
,
1271
(
2004
).
22.
I.
Rajapaksa
and
H. K.
Wickramasinghe
,
Appl. Phys. Lett.
99
,
161103
(
2011
).
23.
T.
Iida
and
H.
Ishihara
,
Phys. Rev. Lett.
97
,
117402
(
2006
).
24.
S.
Mukamel
,
Principles of Nonlinear Optical Spectroscopy
(
Oxford University Press New York
,
1995
), Vol.
29
.
25.
S.
Rahav
and
S.
Mukamel
,
Proc. Natl. Acad. Sci. U.S.A.
107
,
4825
(
2010
).
26.
J. Y.
Huang
and
Y.
Shen
,
Phys. Rev. A
49
,
3973
(
1994
).
27.
S.
Roke
,
A. W.
Kleyn
, and
M.
Bonn
,
Chem. Phys. Lett.
370
,
227
(
2003
).
28.
I. V.
Stiopkin
,
H. D.
Jayathilake
,
A. N.
Bordenyuk
, and
A. V.
Benderskii
,
J. Am. Chem. Soc.
130
,
2271
(
2008
).