A computational model for label-free detection of non-fluorescent biochromophores by stimulated emission

Ádám Fekete; Árpád I. Csurgay

doi:10.1364/BOE.6.001021

1. Introduction

Observation of biomolecules (detected molecules) is usually done by fluorescent microscopy, and in most cases the detected molecules have to be labelled by binding a large fluorescent chromophore molecule. In this case only the response of the composite system of the detected and the chromophore molecule can be observed [1]. Frequently the labelling molecule is much larger than the detected one, thus the observation could become misleading.

A technique for label-free observation of biomolecules has been suggested and experimentally demonstrated as a “stimulated emission microscope” by Min et al. [2], which is one of the technique within the transient absorption microscopy family [3]. The molecules become excited by photon absorption as a result of the first pulse then they are forced back to ground state by stimulated coherent photon emission due to the second pulse (Fig. 1.).

Fig. 1 Possible transitions between molecular energy levels. An electron can absorb a photon exciting electronically and vibrationally the molecule; vibrational relaxation happens in the excited state; besides the spontaneous emission the excited molecules can be relaxed by stimulated emission induced by another photon.

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During the photon absorption both the electronic and vibrational states are excited. Since lower-energy photons are used for the second pulse, the measurements obtained by different time delay between the pulses highly depend on the relaxation processes of the molecules. Measuring the increase in the number of photons induced by the second pulse, we can estimate the presence of the molecule to be detected.

The number of molecules in the focal volume is small, thus a high-frequency, phase-sensitive detection technique was implemented by modulating the excitation signal. They measured the stimulated emission with and without excitation multiple times and amplified the difference. Increasing the number of repeated measurements, a detectable output signal have been achieved that’s amplitude is one order-of-magnitude higher against the laser noise [4]. Motivated by the proposal of Min et al. [5], we pursued a theoretical study with the aim to construct a new computational model of label-free detection of non-fluorescent molecules by stimulated emission.

First, a quantum optics model was used for the interaction of the electromagnetic field and matter. In order to understand the dominant physical processes during label-free detection and in the same time to insure a balance between the necessary computational requirements and accuracy, we modelled the interaction of matter and the electromagnetic field in the focus area. We constructed a master equation describing the interaction of quantized matter and a classical electromagnetic field.

Using density matrix formalism we simulated the time evolution of the master equation that contains both the electronic and the vibrational processes [6]. Capturing both the electronic and vibrational modes of a molecule in our model, we could determine the quantitative behavior of detection of a molecule by stimulated emission.

In order to check the validity of our model, we simulated the measured data of Min et al. published in reference [5], and good agreement of calculated and measured data were found. Our simulations provide an opportunity to assess the potential and also the limits of label-free detection by stimulated emission. Additional to calculate the measured data the model enables us to asses the role of parameter dependence as well as the role of effects of individual vibrational modes.

2. A new computational model and master equation

Our computational model (Fig. 2.) makes assumptions on the molecule (2.1), on the type of transitions (2.2), on the laser excitations (2.3), on the interactions between molecules and excitations (2.4), and on the environment (2.5).

Fig. 2 Block diagram of the main parts of model and the corresponding variables.

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2.1. The model of the molecule

The Born-Oppenheimer approximation is assumed, and the electronic configuration of the molecule to be detected is considered to be a two-state quantum system. The vibrational states of the molecule are modeled by M independent harmonic oscillators. We neglect the rotational states of the molecule, because they only cause spectrum broadening and thus have no significant role in the process of our interest.

The density matrix and the Hamiltonian operator of a molecule are given by:

ρ_{s y s} = ρ_{a} \otimes ρ_{v_{1}} \otimes \dots \otimes ρ_{v_{k}},

H_{s y s} = \frac{1}{2} ℏ ω_{a} σ_{z} + \sum_{k}^{M} ℏ ω_{v_{k}} a_{v_{k}}^{†} a_{v_{k}},

where ω_a = (E_e − E_g)/ħ is the frequency of electronic transition, σ_z is the Pauli operator,

ω_{v_{k}}

is the frequency and finally

a_{v_{k}}^{†}

and

a_{v_{k}}

are the creation and annihilation operators of vibrational states.

2.2. Determination of allowed/possible transitions

In our model the photon transitions between the vibrational modes is neglected. We restrict ourselves to allow only the transitions |g,n⟩ → |e,n ± 1⟩ and |e,n⟩ → |g,n ± 1⟩, n = 0,1,2,… as shown in Fig. 3.

Fig. 3 Energy diagram of the molecule and its possible transitions induced by the excitations for vibrational mode v_k, k = 1,…,M.

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2.3. The model of the laser excitations

Due to the large number of photons we can use classical electromagnetic model for the description of excitation pulses as a zero order Gaussian beam [7]. In the focal volume the electromagnetic field is given by a plane-wave according to the near field approximation. The time evolution of the excitation pulses is an amplitude-modulated Gaussian pulse.

The excitation (Ẽ_e) and the stimulating (Ẽ_s) pulses can be described as:

\begin{array}{l} {\tilde{E}}_{e} = Re [A_{e} E_{0, e} e^{- j ω_{e} t}] = A_{e} E_{0, e} \frac{1}{2} (e^{- j ω_{e} t} + e^{j ω_{e} t}), \\ {\tilde{E}}_{s} = Re [A_{s} E_{0, s} e^{- j ω_{s} t}] = A_{s} E_{0, s} \frac{1}{2} (e^{- j ω_{s} t} + e^{j ω_{s} t}), \end{array}

where E₀_,e and E₀_,s are the peak intensities of the electric field, ω_e and ω_s are the frequencies of the excitation and the stimulating signal, A_s and A_e are the envelope of the laser pulses, that can be expressed as follows:

A_{e} = e^{- 4 \ln 2 {\frac{(t - t_{0, e})}{2 τ_{e}^{2}}}^{2}}, A_{s} = e^{- 4 \ln 2 {\frac{(t - t_{0, s})}{2 τ_{s}^{2}}}^{2}},

where t₀_,e and t₀_,s are the time delays and τ_e and τ_s are the full width at half maximum of the pulses.

2.4. Interaction of the molecule and excitations

To model the interaction between the field and molecule we use the electric dipole approximation because the wavelength of the excitation is much larger than the size of the molecule. We use Rabi frequencies and rotating wave approximation (RWA). Since the optical frequencies (∼ 10¹⁵ Hz) are a few order of magnitude larger than the vibrational frequencies (~ 10¹² Hz) and much larger than the rotational frequencies (∼ 10⁹ Hz) we can suppose that the excitation processes are occurring instantaneously. We can also suppose that the orientation of the molecules is static during the transition but arbitrary at each repeated measurement. This means that the orientation of molecules and the polarization of electromagnetic field cause only amplitude reduction in the measured signal.

Introducing the Rabi frequencies (Ω = d · E₀/ħ) according to the possible transitions the interaction (H_int = −d·Ẽ) can be expressed as follows:

\begin{array}{l} H_{e} = - ℏ A_{e} Ω_{e} S \frac{1}{2} (e^{- j ω_{e} t} + e^{j ω_{e} t}), \\ H_{s} = - ℏ A_{s} Ω_{s} S \frac{1}{2} (e^{- j ω_{s} t} + e^{j ω_{s} t}), \\ S = (σ_{+} + σ_{-}) \sum_{k} (a_{v_{k}}^{†} + a_{v_{k}}), \end{array}

where σ₊ and σ₋ are the Pauli operators of the electric transition.

Avoiding rapid oscillation and using the chosen excitation frequencies and rotating wave approximation the interaction operators can be written as follows:

\begin{array}{l} H_{e} = - ℏ \frac{1}{2} A_{e} Ω_{e} \sum_{k} (σ_{+} a_{v_{k}}^{†} e^{- j ω_{e} t} + σ_{-} a_{v_{k}} e^{j ω_{e} t}), \\ H_{s} = - ℏ \frac{1}{2} A_{s} Ω_{s} \sum_{k} (σ_{+} a_{v_{k}} e^{- j ω_{e} t} + σ_{-} a_{v_{k}}^{†} e^{j ω_{e} t}) . \end{array}

2.5. Model of the environment

The environment is taken into account as a thermal bath that is capable of determining the excited state and vibrational relaxations characterized by γ, $κ_{v_{k}}$ . The excitation of the molecule by the thermal bath is negligible. The Liouville-von Neumann master equation for the reduced density operator is the following:

\begin{array}{l} \frac{d ρ}{d t} = - \frac{j}{ℏ} [H_{s y s} + H_{e} + H_{s,} ρ] \\ + γ (σ_{-} ρ σ_{+} - \frac{1}{2} σ_{+} σ_{-} ρ - \frac{1}{2} ρ σ_{+} σ_{-}) \\ + \sum_{k} κ_{v_{k}} (1 + n_{v_{k}}^{(th)}) (a_{v_{k}} ρ a_{v_{k}}^{†} - \frac{1}{2} a_{v_{k}}^{†} a_{v_{k}} ρ - \frac{1}{2} ρ a_{v_{k}}^{†} a_{v_{k}}) \\ + \sum_{k} κ_{v_{k}} n_{v_{k}}^{(th)} (a_{v_{k}}^{†} ρ a_{v_{k}} - \frac{1}{2} a_{v_{k}} a_{v_{k}}^{†} ρ - \frac{1}{2} ρ a_{v_{k}} a_{v_{k}}^{†}) \end{array}

where γ is the dissipation rate of the excited state,

κ_{v_{k}}

are the dissipation rates of the vibrational relaxations and

n_{v_{k}}^{(th)}

are the average photon numbers of thermal bath excitation corresponding to the v_k vibrational mode:

n_{v_{k}}^{(th)} = {(e^{\frac{ℏ ω_{v_{k}}}{k_{B} T}} - 1)}^{- 1},

where k_B is the Boltzmann constant and T is the absolute temperature. For the excited state relaxation we suppose that all of the vibrational modes have the same dissipation rate. In that case the relaxation rate of the actual vibrational mode is negligible compared to the whole molecule relaxation.

For numerical evaluation the rotating frame approximation can be used:

H_{s y s} = \sum_{k} ℏ ω_{v_{k}} a_{v_{k}}^{†} a_{v_{k}},

H_{e} = - ℏ \frac{1}{2} A_{e} Ω_{e} \sum_{k} (σ_{+} a_{v_{k}}^{†} e^{- j (ω_{e} - ω_{a}) t} + σ_{-} a_{v_{k}} e^{j (ω_{e} - ω_{a}) t}),

H_{s} = - ℏ \frac{1}{2} A_{s} Ω_{s} \sum_{k} (σ_{+} a_{v_{k}} e^{- j (ω_{s} - ω_{a}) t} + σ_{-} a_{v_{k}}^{†} e^{j (ω_{s} - ω_{a}) t}) .

2.6. Determination of relative excitation

The expected number of additional photons produced by stimulated emission (N_re) can be determined as a time integral of relative excitation (n_re):

N_{r e} = \int n_{r e} d t,

where n_re can be calculated as the expectation value from the time evolution of master equation:

n_{r e} = tr [- \frac{j}{ℏ} [H_{s}, ρ] (σ_{-}^{†} σ_{-})] .

3. Computational evaluation of the model

Because of computational complexities the new model has been evaluated following heuristic approximations as follows:

A molecule in a given environment is specified by ω_a, $ω_{v_{1}}$ , …, $ω_{v_{M}}$ , γ and $κ_{v_{k}}$ .
One of the vibrational modes is selected – e.g. the mode belonging to absorption maximum – and denoted by $ω_{v_{1}}$ .
First, excite the system at $ω_{e x c} = ω_{a} + ω_{v_{1}}$ . Than we generate stimulated emission by stimulating signal at $ω_{s t i} = ω_{a} - ω_{v_{1}}$ . We solve a reduced master equation for $ρ_{s y s} = ρ_{a} \otimes ρ_{v_{1}}$ , and calculate the relative excitation $N_{r e_{1}}$ as a function of the the product of E_eE_s (Fig. 4(a)). We calculate the relative excitation as a function of time delay between excitation and stimulation for different E_eE_s (Fig. 4(b)), and for different γ and κ (Fig. 4(c) and (d)).
In order to calculate the spectrum of the relative excitation N_re as a function of the frequency of the stimulating signal we have to take into account the contribution of every vibrational mode v₁,…,v_M. In this case the relative excitation N_re(ω) will be the sum of M relative excitations as: $N_{r e} (ω) = N_{r e_{1}} (ω) + \sum_{k = 2}^{M} N_{r e_{k}} (ω),$ where $N_{r e_{k}} (ω)$ is calculated from the solution of reduced master equation for the $ρ_{s y s_{1}} = ρ_{a} \otimes ρ_{v_{1}}$ at $ω_{s t i} = ω_{a} - ω_{v_{1}}$ and $ρ_{s y s_{k}} = ρ_{a} \otimes ρ_{v_{1}} \otimes ρ_{v_{k}}$ at $ω_{s t i} = ω_{a} - ω_{v_{k}}$ where k = 2, …, M.

4. Numerical results

We have examined the case when the lifetime of vibrational state is in the same order as the width of the pulse used for excitation and the lifetime of excited state. We suppose that the excited state relaxation occurs through the thermal environment. This means that spontaneous emission with detectable photon emission is negligible.

The parameter values used for the simulations are summarized in Table 1. The initial values of state probabilities have been chosen according to the thermal equilibrium. The occupancy of the smallest states of vibrational modes with frequency larger than 400cm⁻¹ is greater than 90% at room temperature. We can observe enough accuracy by using 10 states for vibrational mode because the occupation probability of any higher energy state is negligible.

Table 1. Simulation parameters.

View Table

4.1. Simulation of relative excitation as a function of delay between the excitation and stimulating signal

The simulation results of the relative excitation as a function of the intensity and the pulse delay are shown in Fig. 4. We suppose that the density matrix formalism describes precisely the time average of multiple pulse excitations.

Fig. 4 Numerical results of the model: (a) shows the relative excitation as a function of the two excitation signal amplitude’s product using a fixed (t₀_,s = 0.15ps) delay and the linear dependence of signal intensities at weak excitation. The effect of the two excitation signal’s amplitude (b), the rate of the vibrational relaxation (c) and the rate of excited state relaxation (d) are shown as a function of time delay.

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In each simulation we took into account only a single vibrational mode; moreover, processes in the frequency domain partially overlap. The rate of vibrational $(κ_{v_{1}})$ and excited state (γ) relaxations can be highly molecular specific. Nevertheless it suggests that changing the delay but keeping the same parameter set will result in a molecule specific shape of the simulation result.

Since the density matrix formalism describes only the population probabilities we can suppose that the measured value depends linearly on the concentration of the molecules in the solvent.

The Rabi frequencies are determined by the light intensities and the transition probabilities, thus the measured value depends on the two laser pulses following each other, and the response of the system depends on the multiplication of the pulse intensities at weak excitation. This quadratic behavior in the space domain is well known from other nonlinear techniques like two photon or stimulated emission depletion (STED) microscopy [8, 9].

4.2. Spectrum of relative excitation

To determine the spectrum of relative excitation we need to consider all vibrational modes of frequencies $(ω_{v_{k}})$ and their relaxation rates $(κ_{v_{k}})$ . The excitation signal frequency is fixed, the stimulating frequency is changed. In order to compare calculated and measured data, we ran our simulation for crystal violet molecule, the same as in the work of Min et al. The numerical results are shown in Fig. 5 estimated according to Eq. (14) using:

N_{r e_{k}} (ω) = N_{r e_{k}} \cdot g_{σ} (ω)

where

N_{r e_{k}}

is the relative excitation of single vibration mode and g_σ is a Gaussian function with σ variance describing the spectral broadening caused by molecular rotations.

Fig. 5 Numerical results of spectrum calculation: (a) shows the relative excitation as a function of wavelength in the simulation model (σ = 50) compared to the experimental data presented by Min at al. [5] and (b) shows the relative excitation values of each vibrational modes calculated according to the model.

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5. Conclusions

A new model in form of master equation (Eq. (7)) is proposed to describe photonic-electronic-molecular dynamics for label-free detection by stimulated emission. Our model represents a two-state electronic quantum system and finite number of vibrational modes. Based on this model algorithms are proposed to evaluate relative excitations.

To calculate the relative excitation the time-domain master equation has to be solved. The computational complexity of the numerical solution depends on the number of entangled vibration modes taken into account.

The time delay dependent relative excitation can be evaluated by taking into account only a single vibrational mode $(ω_{v_{1}})$ . The excitation frequency is $ω_{a} + ω_{v_{1}}$ and the stimulation frequency is $ω_{a} - ω_{v_{1}}$ . Linear dependence in concentration and quadratic dependence in space resolution are obtained at weak excitation that is a consequence of using two pulses.

To calculate the spectrum of relative excitation two entangled vibrational modes are necessary. The dominant and the other vibrational modes has to be entangled one-by-one. The proposed algorithm overcomes the problem of computational complexity and enables to evaluate the spectrum on a high-end computer.

High correlation between calculated and measured time delay and frequency dependent relative excitation confirm the validity of the proposed model.

Acknowledgments

The research has been supported by the Hungarian Research Faculty grant. We are grateful to the Multidisciplinary Doctoral School of Sciences and Technology, Pázmány Péter Catholic University (Budapest, Hungary), Professor Tamás Roska and Professor Péter Szolgay for providing the conditions of the research work. We thank also our colleague, Imre Juhász for inspiring discussions.

References and links

1. S. W. Hell and E. Rittweger, “Microscopy: Light from the dark,” Nature (London) 461(7267), 1069–1070 (2009). [CrossRef]

2. W. Min, C. W. Freudiger, S. Lu, and X. S. Xie, “Coherent nonlinear optical imaging: beyond fluorescence microscopy,” Annu. Rev. Phys. Chem. 62, 507–530 (2011). [CrossRef] [PubMed]

3. P. Wang, M. N. Slipchenko, J. Mitchell, C. Yang, E. O. Potma, X. Xu, and J.-X. Cheng, “Far-field imaging of non-fluorescent species with subdiffraction resolution,” Nature Photon. 7(6), 449–453 (2013). [CrossRef]

4. R. Paschotta, “Noise of mode-locked lasers (Part I): numerical model,” Appl. Phys. B 79(2), 153–162 (2004). [CrossRef]

5. W. Min, S. Lu, S. Chong, R. Roy, G. R. Holtom, and X. S. Xie, “Imaging chromophores with undetectable fluorescence by stimulated emission microscopy,” Nature (London) 461(7267), 1105–1109 (2009). [CrossRef]

6. H. Carmichael, An Open Systems Approach to Quantum Optics (Springer, 1993).

7. Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed gaussian beam,” IEEE J. Quantum Electron. 33(4), 566–573 (1997). [CrossRef]

8. T. Ye, D. Fu, and W. S. Warren, “Nonlinear absorption microscopy,” Photochem. Photobiol. 85(3), 631–645 (2009). [CrossRef] [PubMed]

9. J. N. Farahani, M. J. Schibler, and L. A. Bentolila, “Stimulated emission depletion (STED) microscopy: from theory to practice,” Microscopy: Science, Technology, Applications and Education 2, 1539–1547 (2010).

Parameter	Value	Description
T	298	temperature [K]
ω_a	618.5	frequency of two-state model [nm]
γ	1/0.4	dissipation rate of excited state [1/ps]
$ω_{v_{1}}$	782	frequency of vibrational mode [cm⁻¹]
$κ_{v_{1}}$	1/0.1	dissipation rate of vibrational mode [1/ps]
λ_e	590	wavelength of excitation signal [nm]
λ_s	650	wavelength of stimulating signal [nm]
Ω_e	7	Rabi frequency of excitation signal
Ω_s	7	Rabi frequency of stimulating signal
τ_e	0.2	pulse width of excitation signal [ps]
τ_s	0.2	pulse width of stimulating signal [ps]

A computational model for label-free detection of non-fluorescent biochromophores by stimulated emission

Abstract

1. Introduction

2. A new computational model and master equation

2.1. The model of the molecule

2.2. Determination of allowed/possible transitions

2.3. The model of the laser excitations

2.4. Interaction of the molecule and excitations

2.5. Model of the environment

2.6. Determination of relative excitation

3. Computational evaluation of the model

4. Numerical results

4.1. Simulation of relative excitation as a function of delay between the excitation and stimulating signal

4.2. Spectrum of relative excitation

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Tables (1)

Equations (15)

Biomedical Optics Express