Spectrum Broadening Due to Nonselective Linear Absorption

Abstract

The position and linewidth of an emission spectrum reflect the physical properties of the luminophor. So, keeping the spectrum from distortion is very important in its measurement. However, we find that the spectrum linewidth will be broadened when the near-infrared radiation from a sodium lamp passes through a nonselective linear absorbing filter. This counterintuitive linewidth-broadening phenomenon is obvious when the residual light power after the filter is low enough, typically lower than $2.48\times {10}^{-4} \mathsf{\mu }\mathrm{W}$. This novel linewidth-broadening effect is different from the well-known Lorentzian, Doppler, and Voigt broadening, and is likely to be more independent evidence of the discrete wavelet structure of classical plane light waves. The effect is significant in high-sensitivity spectroscopy measurements, for example streak camera spectroscopy and Raman spectroscopy experiments. In addition, this effect may also be significant for cosmological research.

1. Introduction

A spectrum is like a fingerprint that can be used to identify atoms, elements, and molecules [1] located in laboratory or in a faraway location such as stars, the galaxy, and a cloud of gas [2]. So, spectroscopy finds widespread application ranging from physics [3], analytical chemistry [4], astronomy [5,6], and micro/nano-processing technology to [7] biopharmaceuticals [8]. When a particle transits from an excited state to its lower energy state, it emits a photon with the same energy needed to raise the particle to its excited state. Since the lifetime of the excited state is finite, the emission spectrum has a certain spectrum linewidth. The spectral line is usually described by the Lorentzian line profile [3]. The Doppler effect caused by molecular thermal motion can broaden the spectrum linewidth and change the line shape to a Gaussian profile [3,9]. The spectral line may also be affected by particle collisions, lattice vibrations, defects, and other factors, resulting in a complex Voigt line-broadening profile [10]. In addition, the Hubble redshift of spectral lines originating from the Doppler effect has been found in cosmological studies [11]. In 1986, Wolf proposed that the normalized spectrum line will remain unchanged when a light wave propagates in free space if the spatial coherence of the light source satisfies a certain scaling law [12]. If the scaling law is broken, for example, if the spatial coherence of the light source is frequency-dependent [13] or the fluctuation in the scattering kernel is random in space and time, the normalized spectrum line will be red-shifted when viewed from a long distance [14,15].

Since the position and linewidth of an emission spectrum reflect the physical properties of the luminophor, keeping them from distortion is very important in spectrum measurement. However, the forms of emission spectra are not always stable. Recently, Zhang et al. found in the Michelson interference experiment that the coherence length of a He-Ne laser beam will be shortened when the beam passes through a nonselective linear absorbing medium with low enough optical power left [16]. This indicates that the nonselective linear absorption in the optical path of a spectrum measurement system may change the linewidth of the spectrum, since the shortening of the coherence length may lead to the broadening of the spectrum. Here, we report the linewidth-broadening phenomenon observed by letting the near-infrared radiation from a sodium lamp pass through a nonselective linear absorbing filter, which is likely to be more independent evidence of the discrete wavelet structure of classical plane light waves [17]. The effect is significant in high-sensitivity spectral measurements such as in streak camera spectroscopy experiments, where a nonselective linear absorption filter is often used to attenuate the light signal to avoid saturation or damage of the streak camera, although the original light signal is weak. And the spectrum-broadening effect caused by the filter would make the recorded physical information distorted. Another example is the experiment of Raman spectroscopy [8,18], whose signal is often very weak. And the use of a filter for signal attenuation or noise reduction would lead to spectral line broadening. In addition, this effect may also be significant for cosmological research. Because the light from distant stars is weak when it reaches Earth and the light has traveled a long space path, most likely through a thick absorbing medium of galaxies, the spectral broadening effect described here may occur.

2. Experiment Results and Discussion

To observe the linewidth-broadening phenomenon, a sodium lamp and an optical fiber spectrometer (AmSpec-ULS3648-USB2) with a nominal resolution of 0.21–0.25 nm are used. The experimental setup is shown in Figure 1, where the near-infrared light (radiation) from the sodium lamp is collected by a convex lens (made of K9 glass) with a diameter of 35 mm and a focal length of 40 mm. The lens is placed at a distance of 23 cm from the exit of the sodium lamp to focus the sodium lamp light onto the free end of the lead-in fiber of the spectrometer. A nonselective linear absorption filter (composed of multiple pieces) is placed close to the fiber entrance to attenuate the signal light for the experiment. There are six types of nonselective linear absorption filters, whose transmittances vs. the wavelengths are shown in Figure 2 (one type involves several pieces of filters). A black tube is put between the convex lens and filter to prevent the environmental light from entering into the fiber. By adjusting the free end of the fiber, sufficient strong optical power can enter into the spectrometer to produce an observable signal.

There are many near-infrared spectral lines of the radiation of the sodium lamp. To calibrate the optical powers of each spectral line, we measure the total optical power of the radiation by using a power meter, which is placed behind the convex lens. In the measurement, the nonselective linear absorption filter is removed, and a band-pass filter is added to restrict the wavelength range to 700–820 nm. With an optical fiber spectrometer, the initial spectrum intensity distribution of the sodium lamp behind the convex lens is measured. From the distribution, the spectral lines with single-line structures can be identified and the optical power of each spectral line can be calculated.

Table 1. Wavelengths and powers of near-infrared spectral lines of sodium lamp with single-line structure in range of 700–820 nm.

Wavelength (nm)	706.62	727.17	738.27	763.25	772.24	794.54
$\mathrm{power} \left(P_0\right)$ (μW)	9	3	17	52	19	15

gives the powers of several spectral lines with a single-line structure.

It can be seen from Table 1 that the power of the 763.25 nm line is much higher than that of the others. According to ref. [17], this line will undergo a weaker broadening effect than the others. Therefore, three typical spectral lines of 706.62, 727.17, and 738.27 nm are chosen to investigate the linewidth-broadening effect. To this end, for a chosen spectral line, three nonselective linear absorption filters with different transmittances are inserted before the optical fiber spectrometer sequentially. The integration time for each measurement and the average number are fixed as 9 s and 13, respectively. In order to minimize the influence of the environment and measurement system noises, the dark background signal is recorded and deducted from each measurement. For each absorption filter, we record 18 sets of measurement data. The average of these 18 sets of data gives the final experiment results, which are shown in Figure 3, Figure 4 and Figure 5.

Spectrum broadening at 706.62 nm is shown in Figure 3, where the points denote the experimentally measured spectra (the same in Figure 4 and Figure 5). Figure 3a is the initial spectrum without passing through the absorption filter (the same for subfigure (a) in Figure 4 and Figure 5 below), while Figure 3b–d give the spectra for the absorption filters with transmittances T of $0.146\times {10}^{-4}$, $0.076\times {10}^{-4}$, and $0.048\times {10}^{-4}$, respectively. The corresponding optical powers of this spectral line after passing through the absorption filters can be estimated by $P_{0}T$ and are $1.31\times {10}^{-4} \mathsf{\mu }\mathrm{W}$, $0.68\times {10}^{-4} \mathsf{\mu }\mathrm{W}$, and $0.43\times {10}^{-4} \mathsf{\mu }\mathrm{W}$, respectively. Compared to Figure 3a, there is slight broadening in the spectrum of Figure 3b. The broadening becomes evident in Figure 3c,d. Spectrum broadening starts from the lower part of the spectral line. And the lower the transmittance of the filter, the more evident the broadening.

Spectrum broadening at 727.17 nm is shown in Figure 4. The transmittances of the absorption filters for Figure 4b–d are measured as $1.228\times {10}^{-4}$, $0.262\times {10}^{-4}$, and $0.131\times {10}^{-4}$, respectively. And the corresponding optical powers $P_{0}T$ of this spectral line are $2.44\times {10}^{-4} \mathsf{\mu }\mathrm{W}$, $0.77\times {10}^{-4} \mathsf{\mu }\mathrm{W}$, and $0.39\times {10}^{-4} \mathsf{\mu }\mathrm{W}$, respectively. Similar to Figure 3, the spectrum line undergoes slight linewidth broadening in Figure 4b, while it shows evident broadening phenomena in Figure 4c,d. And the broadening also increases with the decrease in transmittance of the filter. It should be noticed that the optical powers $P_{0}T$ of the spectral line for Figure 4b,c are, respectively, larger than those in Figure 3b,c, and the broadening effects are correspondingly weaker.

Figure 5 shows spectrum broadening at 738.27 nm. The transmittances of the filter corresponding to Figure 3b–d are $0.146\times {10}^{-4}$, $0.097\times {10}^{-4}$, and $0.057\times {10}^{-4}$, respectively. And the optical powers $P_{0}T$ of the spectral line are $2.48\times {10}^{-4} \mathsf{\mu }\mathrm{W}$, $1.65\times {10}^{-4} \mathsf{\mu }\mathrm{W}$, and $0.97\times {10}^{-4} \mathsf{\mu }\mathrm{W}$, respectively. One can see that in Figure 5, the broadening of the spectral line in Figure 5b–d is slower than that in Figure 3b–d and Figure 4b–d. The broadening can only be seen evidently in Figure 5c,d. The optical powers $P_{0}T$ of the spectral line corresponding to Figure 5b–d are, respectively, larger than those corresponding to Figure 3b–d or to Figure 4b–d. Comparing these three groups of experimental results, it is easy to see that the smaller the optical power $P_{0}T$ of the spectral line, the easier it is to observe the broadening effect of the spectral line. This is because, according to ref. [16], the smaller the optical power of the spectral line, the more evident the shortening of the coherence length.

Here, we try to provide an explanation for the above experiments using the discrete wavelet structure theory of classic plane light waves proposed by Zhang and She [17]. According to the theory, a plane light wave of finite length can be described by the following wave train with a discrete wavelet structure:

$$E_k(z-ct)={\displaystyle \frac{E_{k0}}{2}}{e}^{ik(z-ct)}{\displaystyle \sum _{r=-n}^{n+1}{e}^{\frac{-{[(z-ct)-r{\lambda }_0+{\lambda }_0/2]}^2}{2s}}},$$

(1)

where t and z are the time and the coordinate along the direction of traveling light, respectively; E_k0 and c are the amplitude of the wave and the speed of light in the vacuum, respectively. n is the discrete wavelet structure parameter, which is an integer. The wave vector is $k={\omega }_0/c$, with ${\omega }_0$ being the idler frequency. ${\lambda }_0=2\pi /k$; $s={({\lambda }_0/c_1)}^2/2$, with $c_1=0.886231921$. We regard the near-infrared radiation of each spectral line of the sodium lamp as the superposition of many independent basic wave trains with amplitudes of E_k0min [17]. According to [16], the coherence length of the radiation will be shortened when the radiation passes through a nonselective absorption filter. And the formula for the variation in the coherence length is

$$L={\displaystyle \frac{(1+C{P}_0)T}{1+C{P}_{0}T}}{L}_0,$$

(2)

where $P_0$ and $T$ are the incident light power before the filter and the transmittance of filter, respectively; C is a parameter, which is $C=1.76\times 10^9 {\mathrm{W}}^{-1}$ [16]; $L_0$ and L are the initial coherence length of the incident radiation and the coherence length of the radiation after the filter, respectively. However, Equation (2) only describes the average shortening of the coherence length of basic wave trains. In fact, the shortening of the coherence lengths should be uneven when the wave trains are absorbed segment by segment. The coherence lengths of basic wave trains after the filter should be in the range from $L$ to $L_0$. The numerical results show that if we assume that the amplitude distribution of the basic wave trains (with idler wavelength ${\lambda }_0$ and parameter n) after passing through the absorption filter obeys the law $e^{-n/2m}/e^{-1/2}$ ($n=m,m+1,\dots ,n_0=[L_0/2{\lambda }_0]$, $m=[L/2{\lambda }_0]$ (where $[x]$ denotes the smallest integer greater than or equal to x) and that the phases ${\phi }_n$ of basic wave trains are random, the radiation of the idler wavelength ${\lambda }_0$ after the filter can be described by

$$E(z-ct)={\displaystyle \frac{E_{k0\min }}{2}}{\displaystyle \sum _{n=m}^{n_0}{e}^{1/2(1-n/m)}{e}^{i[k(z-ct)+{\phi }_n]}{\displaystyle \sum _{r=-n}^{n+1}{e}^{\frac{-{[(z-ct)-r{\lambda }_0+{\lambda }_0/2]}^2}{2s}}}},$$

(3)

then the overall coherence length after the filter is close to $L$. For simplicity, we set the constant fact $E_{k0\min }/2=1$. In the frequency domain, the spectrum of wave trains (3) is

$$F(\omega )=e^{-{\displaystyle \frac{s{(\omega -{\omega }_0)}^2}{2c^2}}}{\displaystyle \sum _{n=m}^{n_0}{e}^{(1-n/m)}}{\left|{\displaystyle \sum _{r=-n}^{n+1}{e}^{-i(\omega -{\omega }_0)[r{\lambda }_0/c-{\lambda }_0/(2c)]}}\right|}^2.$$

(4)

And in the wavelength domain, the spectrum becomes

$$F(\lambda )=e^{-2{\pi }^{2}s{(1/\lambda -1/{\lambda }_0)}^2}{\displaystyle \sum _{n=m}^{n_0}{e}^{1-n/m}}{\left|{\displaystyle \sum _{r=-n}^{n+1}{e}^{-i2\pi (1/\lambda -1/{\lambda }_0)[r{\lambda }_0-{\lambda }_0/2]}}\right|}^2$$

(5)

We find through the experiments (see Figure 3, Figure 4 and Figure 5) that the initial linewidths ($\Delta \lambda$) at 706.62, 727.17, and 738.27 nm are 0.375, 0.380, and 0.380 nm, respectively. For these three observed idler wavelengths (${\lambda }_0$), with the approximation $L_0\approx {\lambda }_0^2/\Delta \lambda$, we can use Equation (5) to calculate the spectra after the filter with different transmittances. The results are the red lines shown in Figure 3, Figure 4 and Figure 5, from which we can find the fitting linewidths and make detailed comparisons for each wavelength studied in terms of how different transmittances of the filters affect the FWHM of the spectral lines. For the 706.62 nm line, when without the filter, the initial linewidth is $\Delta \lambda =0.375 \mathrm{nm}$. And when with the filters of transmittances $T=0.146\times {10}^{-4}$, $0.076\times {10}^{-4}$, and $0.048\times {10}^{-4}$, the linewidth becomes $\Delta \lambda =0.68, 1.08,$ and $1.58 \mathrm{nm}$, respectively. For the 727.17 nm line, when without the filter, the initial linewidth is $\Delta \lambda =0.380 \mathrm{nm}$. And when with the filters of transmittances $T=1.228\times {10}^{-4},$ $0.262\times {10}^{-4}$ and $0.131\times {10}^{-4}$, the linewidth becomes $\Delta \lambda =0.52, 0.96,$ and $1.76 \mathrm{nm}$, respectively. For the 738.27 nm line, when without the filter, the initial linewidth is $\Delta \lambda =0.380 \mathrm{n}\mathrm{m}$. And when with the filters of transmittances $T=0.146\times {10}^{-4}$, $0.097\times {10}^{-4}$, and $0.057\times {10}^{-4}$, the linewidth becomes $\Delta \lambda =0.51, 0.60,$ and $0.79 \mathrm{nm}$, respectively. All of these show that the spectral lines after the filter are really broadened. And the lower the transmittance of the filter, the more evident the effect.

3. Conclusions

In summary, the attenuation of the near-infrared radiation of a sodium lamp is studied by using a nonselective linear absorption filter, and it is found that the spectral line of the emission spectrum will be broadened when the power after the filter becomes low enough (typically lower than $2.48\times {10}^{-4} \mathsf{\mu }\mathrm{W}$) due to linear absorption, and the lower the transmittance of the filter, the more evident the effect. This is different from the known spectrum-broadening effects. The effect is likely to be more independent evidence of the discrete wavelet structure of classical plane light waves [17]. We should pay attention to this effect in spectral experiments, especially in high-sensitivity spectral measurement such as in streak camera spectroscopy experiments and Raman spectroscopy experiments [8,18]. It may also be significant for cosmological research, since the spectral broadening effect described here may be superimposed on the Habble redshift effect [19] and the redshift effect predicted by Wolf [14]. So, the application of the effect in cosmological research would be an important direction in the future. However, the effect only occurs under very low light intensity, which makes it easily enshrouded by noise. To reduce the noise is a further experimental task.