Abnormal spectral bands in broadband sum frequency generation induced by bulk absorption and refraction

Yuhan He; Yongyan Zhang; He Ren; Jingjing Wang; Wei Guo; Shi-Gang Sun; Zhaohui Wang

doi:10.1364/OE.27.028564

1. Introduction

Sum frequency generation spectroscopy (SFG) with intrinsic surface selectivity is a powerful tool to investigate the chemical construction and molecular orientation on surfaces and interfaces.[1-3]. In SFG, an infrared pulse (IR) at ω_IR spatially and temporally overlaps with a visible pulse (VIS) at ω_VIS to generate the signal at ω_SFG = ω_VIS + ω_IR. With a femtosecond IR, broadband SFG (BB-SFG) can simultaneously obtain a SFG spectrum broader than 400 cm^-1 [4]. In BB-SFG, the broadband IR pulse excites the targeted vibrations and the VIS up-converts the induced polarization into the second order SFG signals. The measured BB-SFG spectrum represents a convolution of the surface response function (the “true spectrum”) with the incident VIS [5]. The duration of the VIS pulses in most BB-SFG setups is several picoseconds, which provides a proper time resolution to capture the surface dynamics, such as energy transfer and molecular motions [6-9]. However, the VIS, with a duration on the same order of the vibrational dephasing time of the surface molecules, would introduce extra distortions to the measured BB-SFG spectra [5]. With time-resolved BB-SFG, the distortion can be alleviated by the mathematical data retrieval based on the consistency between the temporal and frequency domain signals. J. E. Patterson et al. simultaneously fitted the BB-SFG spectra at different time delays to improve the fitting accuracy [10]. Frequency domain nonlinear regression (FDNLR) can restore the phase and intensity of the resonance without the pre-set fitting parameters [11]. Similar to FDNLR, some excellent algorithms, such as principal component generalized projection (PCGP), have mostly been used to retrieve the spectral properties of frequency-resolved optical gating (FROG), but not been used in SFG spectroscopy [12-16]. The distortion can also be removed with experiment methods. High resolution BB-SFG (HR-BB-SFG) provides an experimental way to treat the distortion. The discriminable probe can be removed with a much longer VIS pulse with the trade-off temporal resolution against frequency resolution, and the fine spectral features can be determined unambiguously [17].

Different from dielectric surfaces, the metal substrate can create strong nonresonant SFG signal, which coherently interference with the narrow molecular signals [4, 18, 19]. The spectral line-shape strongly depends on the relative phase and intensity of the resonant and non-resonant components [18-21]. The non-resonant signal mainly originates from the instant response of the metal substrate surface, and dephases much faster than the resonant signals (the vibrational response of the surface molecules, typically dephases in several ps) [22]. Suppression of the non-resonant background can be achieved using a time-asymmetric visible pulse [5, 23].

The buried interfaces have been intensively investigated with SFG in catalysis [24-26], electrochemistry [27, 28] and surfaces of functional materials [29-31]. Although the SFG is insensitive to the homogeneous bulk, the bulk layer, especially those containing the surface species, could still impact on the SFG spectra in several possible ways: (1) the bulk layer may significantly reduce the intensity of the incident laser pulses, and add extra dispersion to the ultra-short pulses [32, 33]; (2) the SFG signal from different surfaces/interfaces can lead to complicated interferences [34-36]; (3) the bulk layer may also contribute to the SFG signal in some exceptional conditions [37, 38]; (4) nonlinear scattering from the bulk, such as anti-Stokes Raman and sum frequency scattering, may interfere with the SFG signal as well [39, 40]. Although the spectral fitting and background suppression can be very helpful to identify the origin of the SFG bands, it is difficult to distinguish the bulk contributions with these approaches.

In this paper, BB-SFG spectra of a bare Au surface, obtained with and without a model bulk layer in the IR path, were used to investigate the bulk effects (frequency-dependent absorption and refraction) on the SFG spectra with the help of PCGP algorithm. To avoid the nonlinear scattering of the bulk due to the overlap of the incident IR and VIS ultrashort pulses, a ∼10 µm poly-ethylene thin film (PE) was inserted only into the IR light path to model the bulk effects. Then, the SFG spectra from an air/Au surface at different time delays were recorded, and analyzed with PCGP algorithm to retrieve the time and frequency domain properties of the Au surface SFG response. With the retrieved information, phase, intensity distributions in both the time and frequency domains, the bulk distortion on the incident IR pulse can be accessed.

2. Principle of SFG and PCGP algorithm

The principle of SFG has been substantially discussed previously [5, 41, 42]. Briefly, an IR pulse at frequency ω_IR creates coherent vibrational polarization P⁽¹⁾ in the vibrational excited state. Then, P⁽¹⁾ is probed by the following VIS pulse at ω_VIS. The SFG signal at ω_SFG= ω_IR + ω_VIS can be expressed as

(1)$${E_{SFG}}(t )\propto {P^{(1 )}}(t ){E_{VIS}}({t,\tau } )= ({R(t )\otimes {E_{IR}}(t )} ){E_{VIS}}({t,\tau } )$$

Where τ is the temporal interval between the IR and VIS pulses (delay time). τ is positive when the IR arrives ahead of the VIS and vice versa. With Fourier transform, E_SFG in the frequency domain can be expressed as

(2)$$\begin{array}{l} {E_{SFG}}({{\omega_{SFG}}} )\propto {P^{(1 )}}({{\omega_{IR}}} )\otimes {E_{VIS}}({{\omega_{VIS}},\tau } )\\ = ({{\chi^{(2 )}}{E_{IR}}({{\omega_{IR}}} )} )\otimes {E_{VIS}}({{\omega_{VIS}},0} ){e^{ - i\omega \tau }} \end{array}$$

P⁽¹⁾ can be divided into resonant component, P_R, from the surface molecules, and non-resonant component, P_NR, from the metal substrate surface. P_NR dephases much faster than P_R, thus the spectra with only the narrow resonant bands can be obtained at suitable delays [5, 23].

To model the bulk effects, a 10 µm PE film was inserted into the IR path. The BB-SFG spectra from a bare Au surface were used to diagnose the distortion of the IR pulse. According to Eqs. (1) and (2), P⁽¹⁾ and E_VIS can be calculated from the deconvolution of the measured E_SFG with a known phase. However, there is no direct phase information in a single intensity SFG spectrum. Fortunately, with a series of time-resolved BB-SFG spectra at different delay time τ, and proper mathematical retrieval algorithm, the profile of the IR both in the frequency and time domains can be retrieved. χ⁽²⁾ of the bare Au surface is nearly constant in our measurement window (2400∼3400 cm^-1), therefore, the recovered P⁽¹⁾ can be used as a probe of the IR pulse.

PCGP algorithm has been widely used in FROG. The temporal electronic field E_FROG(t, τ) and frequency field E_FROG(ω, τ) can be expressed as Eq. (3) and Eq. (4) respectively.

(3)$${E_{FROG}}({t,\tau } )\propto {E_P}(t )G({t,\tau } )$$

(4)$${E_{FROG}}({\omega ,\tau } )\propto {E_P}(\omega )\otimes G({\omega ,\tau } )$$

Where E_P is the electric field of the pulse, and G is the gate function. With a series of time-resolved spectra, namely |E_FROG(ω,τ)|² at different τ, E_P and G can be retrieved simultaneously. The similarity between the SFG Eqs. (1) and (2) and the FROG Eqs. (3) and (4) indicates that P⁽¹⁾ and E_VIS can also be retrieved with PCGP algorithm in the same way. Detailed description of PCGP can be found in [12] and the appendices. Home-made PCGP algorithm with an example run was coded with Matlab and included in Code 1 in [43].

3. Experimental method

The BB-SFG setup is shown in Fig. 1. A portion of the fs amplifier (Astrella, Coherent, 800 nm, 35 fs, 1 KHz) was used to pump a near IR optical parameter amplifier (TOPAS-C, Light Conversion) followed with a non-collinear difference frequency generator to produce the mid IR pulse. The narrowband VIS (805 nm) was created by filtering the fs amplifier output with an étalon. The IR and VIS were temporally and spatially overlapped at the sample surface with incident angles both around 60° (respect to the surface normal, and crossed vertically with a 5° angle). The SFG spectra were recorded by a spectrograph with a CCD detector (Horiba).

Fig. 1. The schematic of BB-SFG setup.

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To model the bulk impacts on the IR, a 10 µm poly-ethylene film (PE, art. NO. 3335, Chahua Modern Housewares Co. Ltd.) was inserted into the IR light path. The pulse energies of the IR (∼3450 cm^-1, ∼40 fs) before and after the PE film were 12 and 8 µJ, and the VIS (805 nm) was 2µJ . The Au surface (100 nm Au coated on a borosilicate glass window) was treated for 30 min with a UV lamp (253 and 184 nm, HWOTECT) to remove the organic contaminants before the SFG measurement. The polarization combination of the SFG measurement was PPP (SFG, VIS and IR at P polarization).

4. Result and discussion

To understand the role of the bulk in the BB-SFG spectroscopy, a 10 µm PE film was inserted into the IR light path, and a series of BB-SFG spectra from a bare Au surface were measured with a time interval of 0.167 ps. The BB-SFG spectra without and with the PE film, namely with normal and distorted IR pulses, were shown in Fig. 2. With normal IR pulse (Fig. 2(a)), a broadband spectral feature with ∼200 cm^-1 FWHM, the SFG response of the Au surface, was observed at delay times τ ≤ 0.5 ps, and almost vanished when τ ≥ 1 ps. With the distorted IR (Fig. 2(b)), there were extra narrow band spectral structures, peaks/valleys, superposed on the broad Au BB-SFG band. The narrow valleys around 2853 and 2927 cm^-1 (τ ≤ 0.5 ps) evolved into positive peaks around 2848 cm^-1 and 2960 cm^-1 at τ ≥ 1 ps.

Fig. 2. The BB-SFG spectra of the Au with (a) normal and (b) distorted IR pulse. Dash blue lines are the measured spectra, and the red solid lines are the reconstructed spectra from PCGP. Offset for comparison.

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With the distorted IR, the evolution of the BB-SFG spectral feature of the bare Au surface was very similar to that of the Au surface with adsorbed molecules, where the resonant molecular signals presented as narrow peaks at positive delay times when the nonresonant vanished [5, 11, 23]. However, all the spectra shown in Fig. 2 were measured from the same bare Au surface. Therefore, the narrowband structures in Fig. 2(b), which survived even after the suppression of the broadband non-resonant at τ ≥ 1 ps, should be attributed to the distortion of the IR pulse by the PE film. This result implies that SFG spectroscopy is not entirely ‘bulk-free’ for the buried interfaces because of the pulse distortion, and close attention should be paid to the bulk effects in the SFG experiments for buried interfaces.

Apparently, the BB-SFG spectra around time zero could be understood as the absorption of the PE film. When the IR passed through the PE film, part of the IR was absorbed (2853 cm^-1 and 2927 cm^-1, the symmetric and asymmetric stretching of the -CH₂, respectively). Then the distorted IR interacted with the VIS at the Au surface, and a BB-SFG spectrum with reduced intensity around the PE absorptions was generated. Thus, 2 valleys around 2853 cm^-1 and 2927 cm^-1 were observed in the BB-SFG spectra as shown in Fig. 2(b). However, at τ ≥ 1 ps, the valleys evolved into peaks, and shifted to 2848 and 2960 cm^-1, respectively. The BB-SFG spectra at τ ≥ 1 ps cannot be simply interpreted by the absorption of the IR, or any kind of interference of the SFG from the substrate and surface molecules (there is no surface molecule at all). The only source of SFG response here is the Au surface, thus, the surprising long survived narrow bands at positive delays should be attributed to the Au surface.

Group velocity dispersion is a common phenomenon in the interaction between ultrashort pulsed lasers with dispersive medium. In our BB-SFG measurement, the frequency and time dependent dispersion could be introduced by the interaction of the fs IR and the PE film.

Figure 3 displayed the delay time dependent intensity at 2853 cm^-1 and 3020 cm^-1 from our time-resolved BB-SFG spectra. The 2853 cm^-1 is the peak absorption of the PE, and 3020 cm^-1 is far away from any of the PE absorptions. The profiles of the 2853 cm^-1 without the PE film and that of the 3020 cm^-1 both with and without PE are almost identical within our measurement accuracy, which represent the temporal profile of the VIS pulse (shaped from the laser with an étalon). In Fig. 3(a), three local maximums at -0.16 ps, 0.83 ps and 1.67 ps can be observed in the intensity evolution curve with the PE film. These results clearly indicated that frequency and time dependent dispersion of the IR had been induced by the PE film.

Fig. 3. The intensity of BB-SFG at (a) 2853 cm^-1 (the resonant frequency of PE) and (b) 3020 cm^-1 (out of the resonant range of PE).

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The frequency-dependent dispersion can be modeled by the electric field E of the incident IR propagating in the PE with Eq. (5)

(5)$$E({k,\omega } )= \textrm{A}\exp ({in{k_0}x - i\omega t} )$$

Where A, k₀ and n are the amplitude, wave vector in vacuum, and the apparent refraction consisting of both the linear and non-linear effects, respectively. The real and imaginary parts of the frequency-dependent n are associated with the refraction and absorption, respectively. Inserting the PE film reduced the SFG intensity at 2853 cm^-1 by 93% because of the absorption as shown in Fig. 3(a). The frequency dependent refraction stretched the IR pulse, and led to the delay-dependent spectral features (valleys/peaks) in the BB-SFG spectra (Fig. 2(b)). The SFG intensity at 2853 cm^-1 approached its global maximum at 0.83 ps with the 10 µm PE film, namely the real part of n was greater than 20 (real(n) = cτ/l + 1, where c is the speed of light; τ is the time delay, 0.83 ps; and l is the length of the PE, 10 µm, then real(n) = 25.9). The intensity of BB-SFG survived at τ > 2 ps because of the stretching of the IR pulse. Besides the linear refraction and absorption, power dependent non-linear effects on n may also cause distortion of the IR pulse which led to the three local maximums in the temporal evolution at 2853 cm^-1.

According to Eq. (2), both the VIS and IR could influence the line-shapes of the BB-SFG spectra. To remove the VIS effects, the induced P⁽¹⁾ was restored through PCGP algorithm as shown in Fig. 4. χ⁽²⁾ of the Au surface can be approximated as constant, so that P⁽¹⁾(ω)=χ⁽²⁾(ω)E_IR(ω) represented the spectral line-shape of the IR pulse. As shown in Fig. 4(a), the pulse duration of the IR was stretched from ∼40 fs to more than 500 fs with the PE film. The P⁽¹⁾ sections at t > 100 fs and t < -100 fs (marked with the black and green rectangles in Fig. 4(a)) were transferred into the frequency domain, and displayed along with the retrieved P⁽¹⁾(ω) in Fig. 4(b). The section parts of the P⁽¹⁾ at t > 100 fs and t < -100 fs were centered at ∼2853 and ∼2930 cm^-1, respectively. The strong time dependent section properties suggested significant changes of the refraction index n near the PE absorption resonance. The phase of the P⁽¹⁾(ω) with and without the PE film were shown in Fig. 4(c). The phase of the P⁽¹⁾(ω) with the PE changed dramatically around the absorption resonances. The phase lag and advance indicated the frequency dependent refractions of the PE film. The baseline or the profile of the P⁽¹⁾(ω) phase (if smooth out the rapid changes around the absorption resonances) was close to the profile of the E_IR (the red curve in Fig. 4(b)), they both were close to a Gaussian lineshape. In the nonlinear refraction theory, the nonlinear refraction should be proportional to the incident light intensity (the IR intensity herein). And, the red curve in Fig. 4(b) was the distribution of the incident IR pulse. Therefore, this similarity suggested that the nonlinear refraction had additional effects on the IR pulse besides the linear refraction.

Fig. 4. The retrieved results of the induced polarization (a) P⁽¹⁾(t), (b) P⁽¹⁾(ω), and (c) the phase of P⁽¹⁾(ω).

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From our BB-SFG measurement and PCGP retrieval, IR absorption spectrum of the PE film could be calculated from the ratio of the retrieved P⁽¹⁾ (R_P= |P_PE⁽¹⁾(ω)/P_NPE⁽¹⁾(ω)|² = |E_IR,PE(ω)/E_IR,NPE(ω)|², red dashed line in Fig. 5, rescaled for comparison) and the ratio of the measured BB-SFG spectra R_SFG (green dotted line in Fig. 5) with and without the PE at time zero. Both R_P and R_SFG could approximately represent the major profile of the regular IR absorption spectrum (blue solid line in Fig. 5). However, the retrieved data, Rp, matched better to the subtle features, such as shoulders and weak absorptions. According to our results, PCGP algorithm can remove the distortion from the VIS pulse and provide more accurate spectral profiles with extra phase information as FDNLR algorithm [11], which is significant for the analysis of BB-SFG spectra with congested bands. Additionally, from the intensity ratio of the SFG spectra with and without the absorber (can be many kind of targets of interest), the main features of an IR absorption spectrum can be derived, therefore, SFG up-conversion of the IR could provide more efficient IR detection (in the visible spectral range) especially for ultrashort IR pulses used in time-resolved IR spectroscopy.

Fig. 5. Comparison of calculated IR absorption from the retrieved |P_NPE(ω)/P_PE(ω)|² (red solid line), the ratio of the measured spectra at τ = 0 ps (green dashed line), and FTIR spectrum of the PE film (blue solid line).

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5. Conclusions

BB-SFG is a powerful tool to explore the buried interfaces because of its intrinsic surface selectivity. A PE film was inserted into the IR light path to model the bulk distortion of the fs IR pulse. The results suggest that the frequency and time dependent bulk distortion on the IR leads to the complicated SFG spectral profile even though there are no surface molecules in our measurements. This distortion adds uncertainty to the spectral assignment of the SFG bands from a buried interface, and is hard to diminish even with background suppression or subtraction especially on the metal surface with strong non-resonant signals. Extra caution should be paid when analyzing the in situ SFG spectrum from a buried interface because SFG spectroscopy cannot be guaranteed as ‘bulk-free’. Additionally, our results also imply that the spectral resolution and acquisition efficiency can be improved through the SFG up-conversion and the PCGP algorithm in time-resolved IR spectroscopy.

A. Appendices

A.1 Principle of PCGP

Detailed discussion about the PCGP method can be found in [12].

As defined by Eq. (1), E_SFG(t, τ) is proportional to P⁽¹⁾(t)E_VIS(t, τ). E_SFG(t, τ), P⁽¹⁾(t) and E_VIS(t, τ) can be discretized into vectors E_SFG(N, τ), P⁽¹⁾(N) and E_VIS(N, τ) with a fixed interval Δt, where N is the vector index. Equation (1) can be re-written as

(6)$${E_{\textrm{SFG}}}({N,\tau } )= {P^{(1 )}}(N ){E_{\textrm{VIS}}}({N,\tau } )$$

If the interval of delay time, Δτ, is also set to Δt, then E_VIS(N, τ) can be expressed as

(7)$${E_{\textrm{VIS}}}({N,\tau } )= {E_{\textrm{VIS}}}({N,M\Delta t} )= {E_{\textrm{VIS}}}({N - M,0} )$$

The matrix form of the SFG electric field, E_SFG(t), can be constructed by arranging all E_SFG(N, τ) elements under each τ into one column.

(8)$${\textbf{M}} = \left[ \begin{array}{ccccc} P(1 ){E_{VIS}}(1 )&P(1 ){E_{VIS}}(2 )&P(1 ){E_{VIS}}(3 )&\ldots &P(1 ){E_{VIS}}(m )\\ P(2 ){E_{VIS}}(2 )&P(2 ){E_{VIS}}(3 )&P(2 ){E_{VIS}}(4 )&\ldots &P(2 ){E_{VIS}}({m + 1} )\\ P(3 ){E_{VIS}}(3 )&P(3 ){E_{VIS}}(4 )&P(3 ){E_{VIS}}(5 )&\ldots &P(3 ){E_{VIS}}({m + 2} )\\ \ldots &\ldots &\ldots &\ldots & \\ P(n ){E_{VIS}}(n )&P(n ){E_{VIS}}({n + 1} )&P(n ){E_{VIS}}({n + 2} )&\ldots &P(n ){E_{VIS}}({m + n - 1} )\end{array} \right]$$

The values of τ for each column in the matrix are set as τ₀, τ₀ - Δt, τ₀ - 2Δt,…τ₀ - mΔt. And P(N) and E_VIS(N) are the short forms of P⁽¹⁾(N) and E_VIS(N,τ₀), respectively. Then Eq. 8 can be rearranged into the SFG trace M (analogy to the FROG trace) which is the outer product of vector P(t) and E_VIS(t) as in Eq. 9.

(9)$${\textbf{M}} = \left[ \begin{array}{ccccc} P(1 ){E_{VIS}}(1 )&P(1 ){E_{VIS}}(2 )&P(1 ){E_{VIS}}(3 )&\ldots &P(1 ){E_{VIS}}(m )\\ P(2 ){E_{VIS}}(1 )&P(2 ){E_{VIS}}(2 )&P(2 ){E_{VIS}}(3 )&\ldots &P(2 ){E_{VIS}}(m )\\ P(3 ){E_{VIS}}(1 )&P(3 ){E_{VIS}}(2 )&P(3 ){E_{VIS}}(3 )&\ldots &P(3 ){E_{VIS}}(m )\\ \ldots &\ldots &\ldots &\ldots &\\ P(n ){E_{VIS}}(1 )&P(n ){E_{VIS}}(2 )&P(n ){E_{VIS}}(3 )&\ldots &P(n ){E_{VIS}}(m )\end{array} \right]$$

With singular value decomposition (SVD), singular values V and the unitary matrix S and D can be calculated (M = SVD^T, where D^T is the transposition of D). The first row of S and D are the largest singular value V(1,1), namely S(:,1) and D(:,1) correspond to the normalized P(t) and E_VIS(t).

In the BB-SFG spectroscopy, P⁽¹⁾ contains the non-resonant part P_NR and the resonant part P_R (commonly more than one resonance), and P_NR almost vanished instantly, and PR will last for few ps (on the same order of the VIS pulse duration). For example, with the ∼40 fs IR pulse in our BB-SFG measurement, a spectral range of ∼1000 cm^-1 will be needed to cover the whole P_NR signal, while the typical band width of the P_R is only several wavenumbers. For the PCGP algorithm, if we set the frequency interval as 1 cm^-1, to cover the 1000 cm^-1 range, 1000×1000 matrix has to be used for both ${{\boldsymbol{E}}_{{\boldsymbol{SFG}}}}\left( {\boldsymbol{t}} \right)$ and M if m = n. However, a large portion of the element values will be zero. To save the computing time, those columns containing only zero value elements can be removed.

In BB-SFG experiments, we measure the SFG intensity in the frequency domain, namely I_SFG = |E_SFG(ω, τ)|², where E_SFG(ω, τ) is the Fourier transform of E_SFG(t, τ). I_SFG is a real quantity, and carries no direct phase information. The retrieval of (P⁽¹⁾, E_VIS) and the phase of E_SFG should be conducted alternatively and iteratively.

A.2 PCGP algorithm

PCGP algorithm is processed in seven steps:

(1) Interpolation. Data used for PCGP should meet 2 requirements.
(i) the measured BB-SFG spectra should cover the whole range with non-vanishing SFG signal both in the frequency and time domain.
(ii) Keep Δτ (the delay time interval between each BB-SFG spectrum) and Δt (the time interval between data points in the evolution time t of E_SFG(t), P(t) and E_VIS(t)) equal. Δt should be determined by the BB-SFG spectral range through Eq. 10.
$(10)$$\Delta \tau = 100/3F.$$$ Where F is the width of spectral range in cm^-1 and Δτ is the interval of τ in ps. In our treatment, 1 cm^-1 interval for 2400∼3400 cm^-1 spectral range was adopted. Thus the measured spectra was interpolate with Δτ = 1/30 ps. The matrix sizes of I_SFG(ω, τ) from interpolation are 1001 × 551 for the spectra without PE and 1001 × 436 with PE, respectively.
(2) Use |E_SFG(ω, τ)| (square root of the measured BB-SFG spectra I_SFG(ω, τ)) and an initial guess value of the phase, calculate the temporal profile of the SFG, E_SFG(t), with inverse Fourier transform. The initial phase used in our program is 0 for all elements in E_SFG(ω, τ).
(3) Rearrange E_SFG(t) to M (Eqs. (8) and (9)), and set all the matrix elements in M (Eq. 9) which are not appearing in E_SFG(t) (Eq. 8) to zero.
(4) Decompose the matrix M with SVD to obtain singular values V, and unitary matrix S and D;
(5) Construct M₁ = S(:,1)V(1,1)D(:,1)^T, where S(:,1) and D(:,1) are the unitary vectors corresponding to the largest singular value V(1,1).
(6) Rearrange M₁ to E_SFG(t, τ) (reverse process as step 3) and calculate E_SFG(ω, τ) with Fourier transform.
(7) If the calculated |E_SFG(ω, τ)|² from M₁ is close enough to the measured spectra, stop the iteration, otherwise replace the guess value with the phase of the calculated E_SFG(ω, τ), and go to step (2). The ERROR between the calculated |E_SFG(ω, τ)|² and the measured spectra is calculated by Eq. 11, $(11)$$\textrm{ERROR} = \frac{{\textrm{norm}({{{\boldsymbol{I}}_{\textrm{SFG}}}({\omega ,\tau } )- {{|{{{\boldsymbol{E}}_{\textrm{SFG,R}}}({\omega ,\tau } )} |}^2}} )}}{{\textrm{norm}({{I_{\textrm{SFG}}}({\omega ,\tau } )} )}}$$$

Where norm(A) is the modulus of the matrix A. I_SFG and E_SFG,R are the measured BB-SFG spectra and the calculated SFG electric field.

S(:,1) and D(:,1) in PCGP algorithm are the normalized temporal profiles of the P⁽¹⁾ and E_VIS, and V(1,1) is the scale factor.

In this paper, the interpolated BB-SFG spectra used for the calculation were from 2400 to 3400 cm^-1 with Δω = 1 cm^-1 and Δτ = 1/30 ps. Homemade PCGP algorithm run on a personal computer with Intel(R) core(TM) i5-4210u and 4G DDR3 RAM. The ERRORs converged to 1.2% for spectra with normal IR and 4.3% for spectra with distorted IR within 500 iterations (<3 min computing time).

A.3 Matlab code of the PCGP algorithm

The interpolation (auto_pcgp.m, step (1) in A2) and the iteration of the PCGP algorithm (svdsfg.m, step (2)-(6) in A2) were coded with Matlab and included in Code 1 [43]. A sample run of the PCGP codes (command.m) for the measured BB-SFG spectra from an Au surface with the PE film (matlab_example.mat) were also included in Code 1 [43]. The units of delay time and frequency used in our programs should be ps and cm^-1, respectively. See the comments in the programs for more information.

Funding

National Natural Science Foundation of China (21327901); National Key Research and Development Program of China (2016YFA0200702, 2017YFA0206500).

Disclosures

The authors declare no conflicts of interest.

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43. Programm, example run and example data of PCGP algorithm. https://doi.org/10.6084/m9.figshare.9119666

Abnormal spectral bands in broadband sum frequency generation induced by bulk absorption and refraction

Abstract

1. Introduction

2. Principle of SFG and PCGP algorithm

3. Experimental method

4. Result and discussion

5. Conclusions

A. Appendices

A.1 Principle of PCGP

A.2 PCGP algorithm

A.3 Matlab code of the PCGP algorithm

Funding

Disclosures

References

Supplementary Material (1)

Cited By

Figures (5)

Equations (11)

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