Thermal lens effect with light&#x2019;s orbital angular momentum

Yuan Li; Wuhong Zhang; Wuhong Zhang; Lixiang Chen; Lixiang Chen

doi:10.1364/OE.465600

1. Introduction

When a Gaussian beam is incident on an absorbing medium, the temperature will rise due to the absorption of small amounts of energy from the beam. The localized temperature change brings about a transverse or along optical axis gradient in the index of refraction, which can be characterized by the thermal-optical coefficient $dn/dT$ of the medium. This phenomenon is named as thermal lens effect (TLE). It was first discovered by Gordon et al. in 1965, who built the theoretical basis for practical application of thermal lens effect [1]. The thermal lens spectrometry [2] is one of the important applications proposed to measure the absorption spectrum of the medium precisely [3]. Then, the next few decades has witnessed a growing interest in the TLE including the theoretical developments and new applications, which can be found in those good review papers and books [4–10]. Besides, with the emergence and development of high power laser, TLE can lead to significant laser beam quality degradation, depolarization losses, laser ceasing due to the cavity instability, or even fracture of the laser crystal [11–15]. Many techniques have been proposed to minimize these drawbacks induced by TLE [16–19]. TLE has been a standard tool to study high power laser [16], environmental, chemical and biomedical analysis, as well as thermal characterization and imaging, particularly working in micro-space [10]. However, all of these previous efforts are focused on using the fundamental mode-Gaussian beam induced TLE.

It is noted that the type of thermal lens relies on the sign of thermal-optical coefficient, which negative coefficient material can only provide concave lens, and positive coefficient material can only provide convex lens [20,21]. For a certain material, how to produce the desired lens effect is important for practical application. By changing the spatial distribution of the heating laser, it is possible to produce a different distribution of the refractive index. In 1992, Allen et al. recognized that the Laguerre-Gaussian (LG) laser beams with a helical phase front of $\exp (i\ell \phi )$ carrying a well-defined Orbital Angular Momentum (OAM) of $\ell \hbar$ per photon [22], where $\phi$ is azimuthal angle and $\ell$ is the OAM quantum number. Since then, OAM has found a vast variety of applications in both the classical and quantum realm which can be found in those excellent review papers [23–26]. By controlling the gain and spherical thermal lensing effect in a diode-side-pumped bounce amplifier laser, Chard et al. produced a high-power LG modes with high circularity and low astigmatism [27]. More recently, Pritam et al. proposed a temporally switchable mode converter that can realize mode conversion between the LG and the Hermite-Gaussian (HG) beam based on the TLE [28]. Last year, Rahman et al. presented a theoretical model to analyse the LG induced temperature and refractive index profiles in TLE [29]. Although these previous efforts have studied some part of the relation between the LG modes and the TLE, little attention has been paid to experimentally study the OAM induced TLE.

In this paper, theoretically and experimentally, we demonstrate an interesting TLE induced by OAM. We build a dual beam setup that uses LG beam carrying $\ell \hbar$ OAM with a wavelength $\lambda _1$ as the heating source to illuminate a certain absorption medium, and another laser beam with Gaussian distribution in $\lambda _2$ is utilized to measure the OAM induced TLE. We find that the formation of the OAM-thermal lens has a rapid change towards the evolution direction but falls back slowly and then approaches to the steady state. We also demonstrate the relationship among the power, radius and the focal length of the formed thermal lens induced by the OAM. Our work opens the door that uses the structured light beam to study the thermal-optical effect, which may be used to improve the sensitivity of the absorption spectrum for chemical and biomedical analysis.

2. Theoretical analysis

We start from the original work by Gordon et al. [1]. Instead of using a Gaussian laser mode, here we use a OAM mode as the heating source in the medium. The LG beam, which is the most commonly used light beam that carrying OAM, can be described as:

(1)$$\begin{aligned} LG^\ell_p(r,\phi,z)&=A^\ell_p({-}i)^{(2p+\ell+1)}\frac{w_0}{w}(\frac{\sqrt{2}r}{w})^{\ell}L_p^\ell(\frac{2r^2}{w^2})exp(-\frac{r^2}{w^2})exp({-}i\ell\phi)\\ & exp(i[kz-\frac{kr^2}{2z}-(2p+\ell+1)tan^{{-}1}(\frac{z}{Z_R})]), \end{aligned}$$

where the radius of the beam at the position of $Z$ is $w=w_0\sqrt {1+({z}/{Z_R})^2}$ , the Rayleigh Length is $Z_R={kw_0^2}/{2}$, and the phase term $e^{i\ell \phi }$ indicates the existence of OAM. According to Beer-Lambert law, the intensity of light coming into a medium decreases with the entering depth, which is denoted as: $I(r,\phi,z)=I_0(r,\phi, z=0) exp(-\alpha z)$, where $I_0(r,\phi,z=0)$ denotes the intensity at the surface of the medium, $\alpha$ is the absorption coefficient of the medium. Under the low absorption approximation, the heat generated per unit time in a small region ($dV$) of the medium by the laser can be written as: $q(r,\phi,z)=\alpha I(r,\phi,z)dV$. Then, a non-steady heat transfer equation can be described as:

(2)$$c\rho\frac{\partial T(r,\phi,z, t)}{\partial t}-\kappa\nabla^2[T(r,\phi,z, t)]-q(r,\phi,z)=0,$$

where $c$ is the specific heat, $\rho$ is the density, $\kappa$ is the thermal conductivity [30]. It is noted that the different boundary condition will give the different solution of the Eq. (2). Considering in real experimental conditions, the boundary condition is:

i. far from the center of light in radial distance, temperature change is zero, i.e., at $T=T_0,(r\rightarrow \infty )$;
ii. initially, the temperature of the medium is the same, there is no temperature change, i.e., at $T=T_0,(t=0)$;
iii. a thin medium is used in our experiment, which is satisfied $h\ll D$, $h$ is the thickness and $D$ is the diameter of the medium;
iv. the two surfaces along the $z$ direction of the medium will lose heat when their temperature is higher than environment, the rate of losing heat at a small area $ds$ is proportional to the temperature differences between the surface of medium and environment, which is denoted as $d\Phi =\eta (T-T_0)ds$. $\eta$ is the heat transfer coefficient ($W/m^2\cdot K$), and $\Phi$ is the heat flux ($W$).

Under the boundary condition iii, the temperature along the $z$ direction can be approximated as a constant. It is noted that without the boundary condition iii, for example, a thick medium is used, then the temperature distribution of the medium will become complex and cannot act as a lens any longer, which deserves our further studies in the near future. Therefore, we can neglect the propagation of the heating laser and just consider the structure of the transverse plane. So, by considering $z=0$ in Eq. (1), the intensity of the heating source on the medium can be denoted as:

(3)$$\begin{aligned} I^\ell_p(r,\phi)=LG^\ell_p(r,\phi)LG^{\ell*}_p(r,\phi)=\frac{2p!}{(1+p)\pi}\frac{P_0}{w_0^2}(\frac{2r^2}{w_0^2})^{\lvert \ell \rvert}exp(\frac{-2r^2}{w_0^2})(L^{|\ell |}_p(\frac{-2r^2}{w_0^2}))^2. \end{aligned}$$

According to the term $({2r^2}/{w_0^2})^{|\ell |}$, at the center of light, the intensity is zero, showing an axial symmetry. And the radius of maximum intensity is $R=r(I_{MAX})=w_0\sqrt {|\ell |/2}$ [31], which is used to describe the radius of OAM beam in our work.

Then the heat transfer of Eq. (2) under the above boundary condition can be specified as:

(4)$$c\rho\frac{\partial T(r,\phi,t)}{\partial t}-\kappa\nabla^2 T(r,\phi,t)-\alpha I^\ell_p(r,\phi)h{\it ds}+\eta(T-T_0){\it ds}=0.$$

The term $\eta (T-T_0){\it ds}$ in Eq. (4) increases the complexity of the heat transfer function to get an analytical solution. We note that without this term, the equation has been analytically solved recently [29]. However, the term induced by the boundary condition iv is considered for real environment condition and important for practical application. Here, instead of providing the analytical solution, we use the finite difference method to solve the problem and then calculate the temperature field.

In order to study the formation process of the temperature field with time, we choose a normal medium polymethyl methacrylate (PMMA) as our sample medium. PMMA has been widely used in the wind shield of aircraft because of its excellent transparency and strength. The thermal conductivity of PMMA is $k=0.19W/m\cdot K$, $\alpha =26.3m^{-1}$ of $532nm$ and nearly zero of $633nm$, the specific heat is $c=1464J/Kg\cdot K$ and the density is $\rho =1160Kg/m^3$, the diameter $D=25.4mm$, the thickness is $h=0.40mm$. Besides, the heating beam is a $LG^2_0$ mode with radius $w=0.6mm$ and power $P=10mW$. Under the normal temperature and pressure, the convective heat transfer coefficient is $\eta =5W/m^2K$. Our simulation results are shown in Fig. 1(a). One can find that only a circular temperature field raised at the beginning ($t=1s$), just like the intensity distribution of the heating source $I^2_0(r,\phi )$. Due to the thermal diffusion, the temperature field spreads to the center area of the sample, as the graphs shown from $t=2s$ to $t=4s$. With the temperature rising of the sample, the rates of heat loss at the both sides increase and gradually achieve balance with heating and thermal diffusing. This process is shown from $t=5s$ to $t=\infty$.

Fig. 1. (a) the time evolution process of the sample’s temperature field under a heating beam with OAM $\ell =2$. (b) the steady state of the sample’s temperature field under various heating beams with OAM $\ell =0, \ell =1, \ell =2, \ell =3$. (c) the plot of the transverse temperature fields with different heating power and OAM values.

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To further study the steady-state of the temperature field, we show our simulation results in Fig. 1(b). From left to right, we use $\ell =0,1,2,3$ as the heating source. Under the same input power, one can see the temperature fields induced by OAM modes are much lower than the Gaussian mode, and there is a temperature hollow at the center of the sample instead of a peak. To clearly show the differences of the temperature fields, we plot the distribution along the center of the transverse plane under different input power $P$, as shown in Fig. 1(c). The first line shows the temperature distribution induced by Gaussian mode while the second and third line denote the temperature distribution induced by OAM. One can see that there is a clear temperature drop in the center for the beam containing OAM and the drop is deeper with increasing the OAM value $\ell$ and the input power P. And according to the relationships between refractive index and temperature field, $n=n_0+(T-T_0){dn}/{dT}$, we can also calculate the reflective index distribution. Here, $dn/dT=-8.5 \times 10^{-5} (1/K)$ is the thermal-optical coefficient of the PMMA, and $n_0\approx 1.5$ is the initial refractive index of the PMMA under the normal temperature of the environment. This phenomenon shows that if we consider the center of the temperature field induced by OAM, it will produce a convex lens TLE effect which is just opposite to the concave lens produced by Gaussian mode due to the reverse temperature fields under paraxial conditions. Thus it may provide an interesting application that changes the beam waist of a probe light just though changing the heating beam’s OAM value instead of changing the medium like previous study. Besides, in order to calculate the focal length of the TLE induced by OAM, we can use a similar way like the previous study [1].

3. Experimental results

To experimentally observe the OAM induced TLE, we build a dual beam experimental setup, as shown in Fig. 2. The probe beam, is a $633nm$ He-Ne laser which is modulated to a standard Gaussian distribution by the following spatial pinhole filtering system (L1-P1-L2). The heating source is a $532nm$ laser expanded onto a spatial light modulator (SLM, FSLM-2K55-P, CAS Microstar) to produce the light fields contained OAM. The resolution of the SLM is $1920 \times 1080$ pixels and each pixel size is 6.4$\mu$m. The light fields are filtered to get the desired OAM beam by a 4f filtering system (L3-P2-L4). Then the heating source is directed into the sample. Since the beam radius influences the TLE significantly, we have put the sample on the image plane of the SLM to precisely control the radius of the heating beam. A board band beam spliter (BS) is used to make sure the co-linearity between the probe beam and the heating source. In order to increase the absorption coefficient for the heating laser but decrease the influences of probing beam, we add the red dye into the PMMA as our sample, which is shown as the insert of Fig. 2. After the sample, a long-pass filter is used to block the heating laser. We use two different setups to test the steady state (a) and dynamics process (b) of the sample’s temperature field. Figure 2(a) contains a attenuator to reduce the intensity in the CCD camera. Figure 2(b) shows that the probe beam is collected to record the power by a fiber coupling system (FCS).

Fig. 2. The dual beam experimental setup. L: Lens, $L1=25mm, L2=50mm,L3=L4=300mm$, P: pinhole, SLM: spatial light modulator, MMF: multi-mode fiber, FCS: fiber coupling system. (a) The recording setup is used to analyse the thermal lens effect. (b) An optical fiber coupling system is used to analyse the time evolution of the thermal lens effect.

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In our first experiment, we directly show the opposite effect of the TLE between the Gaussian and OAM beam as the heating source. The power of the probe beam is just $20\mu W$ to reduce the possible thermal effect on the sample. Our experimental results are shown in Fig. 3. One can see that the probe beam has expanded significantly (b) but reduced to smaller (f) compared with non-heating source (e). This effect can be seen more clearer in Fig. 3(g)-(i). The beam waist of the corresponding probe beam is obtained by using Gaussian surface fitting of the recorded intensity pattern, which is $597\mu m,372\mu m,287\mu m$, respectively. The hollow temperature fields of the sample give a convex lens that focus the collimated probe beam while the Gaussian temperature fields give a concave lens that expands the beam waist, which are well consistent with our theoretical analysis.

Fig. 3. Thermal lens effect induced by Gaussian beam and OAM beam. The intensity distribution of (a) Gaussian and (c) OAM beam with $\ell =3$. (b) denote no heating source. The probe beam’s intensity distribution under Gaussian heating (d), without heating (e) and under OAM beam heating (f). (g)-(i) are the corresponding fitting curve of the intensity pattern (d)-(f).

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In order to study the formation process of the TLE induced by OAM, we design a FCS to detect the power of the probe beam. As we have shown in Fig. 3, the thermal lens shapes the beam diameter of the probe beam clearly. Note that the coupling efficiency of the FCS is influenced by the beam diameter due to the limited numerical aperture (NA). So the detected power is decided by the thermal lens. By recording the power changes over time, one can analyze the formation process of TLE induced by OAM. Under the condition of no heating source and the probe beam’s diameter before the FCS, we choose a $f=4.43 mm$ coupling lens connected with a multi-mode fiber (MMF) with a core diameter of $105\mu m$ as our FCS. The MMF is connected with a high time resolution fiber laser power meter to record the power in time. Our experimental results are shown in Fig. 4(a). All of the data points are collected under $50ms$ and averaged by 3 times. Compared with Gaussian heating beam, the detected power of OAM heating beam gets rapidly enhanced at the first step due to the low temperature of the center so that a short focal length of the convex lens is formed. Then, as the heat spreading to the center, the focal length increases to a constant value, so the detected power dropped slowly until to a steady state. A similar reverse process can be seen in the cooling region. To explain this effect more clearly, we simulate the time evolution of focal length as the red dot line (upper) shown in Fig. 4(b). The focal length drops quickly to the lowest value and then go to a steady one slowly. Note that the short convex focal length focus the beam diameters better. As for the cooling process, at the very beginning of turning off the heating beam, the temperatures of the circular drop quickly and then get to a balance with the center area. So, the focal length increases at first but has a singular point due to the plane lens formed, and then the temperatures spread from the center which are just like the Gaussian heating beam. An inverse phenomenon in heating and cooling process can be observed for the Gaussian heating beam, as predicted by the numerical simulation blue dot line (lower) shown in Fig. 4(b). These interesting effects may provide a potential slow thermal-optical gate that can control of light passing through or blocking by changing the heating beam from OAM beam to a Gaussian one.

Fig. 4. (a) Experimental results of the time evolution of thermal lens. The red diamond (upper) denotes OAM heating beam while the blue dot (lower) is the Gaussian heating beam. (b) denotes the numerical simulation of the time evolution of the focal length.

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In order to further explore the influences of the different parameters on the OAM beam induced TLE in a steady state, we conduct a series of experimental tests by changing the heating power $P$, the radius of maximum intensity $R$ and the value of OAM beam, as shown in Fig. 5. The trend of the three lines are similar. The focal length is mainly determined by the power. And the radius of the beam influence the focal length a little bit. While the different OAM value is not the main factor to influence the focal length. The error bar of each data is obtained by averaging three times of the independent experimental runs. All of the data points are obtained at 120s after turning on the heating beam to ensure the steady state. Note that the focal length of the formatted lens increases inversely with the $P$, and the large focal length is hard to measure experimentally. Therefore, the error bar for lower power is relatively high. Considering the external invariant temperature, the possible non-uniformity in materials, the slight inhomogeneity of the produced OAM beam, and the measurement error caused by the deviation between the probe beam and heating beam, some of the experimental points deviate a little bit of the theory. In spit of these effects, it is clearly shown that our experimental results are consistent well with the theoretical predictions.

Fig. 5. Experimental results (data point with error bar) and numerical simulation (point line) for the relation of heating power $P$, the radius of maximum intensity $R$ and the OAM value $\ell$ in thermal lens effect.

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4. Conclusion

In conclusion, we have formulated the theoretical TLE with OAM beams and found the inverse effect compared with the traditional Gaussian heating beam under the condition of surface heat exchanging. Our experimental results have clearly demonstrated the inverse effect and also shown an interesting thermal optical gate which is controlled by the values of OAM. Note that by replacing the single OAM beam with the superposed OAM modes, Eq. (4) is also applicable for the calculation of the temperature and refractive index fields in the medium, which is expected to have richer effects but not limited to TLE. In this regard, our work opens the door that uses the structured light beam to study the thermal-optical effect and may improve the sensitivity of the absorption spectrum for chemical and biomedical analysis [8–10]. Besides, the dynamic process of the OAM induced TLE shows a rapid change at the beginning but slowly changes to steady state, and it even creates a peak overshoot phenomenon. The spiral energy flow of OAM beam is supposed to increase the speed of thermal exchange from the beam to the sample, which still remains to be studied further. From the experimental point, one may need an ultrafast temperature sensor to analyse the sample’s spiral energy flow induced by OAM beam. What’s more, the vector beams have raised much more attention recently [32–34], we only use linear polarization beam and isotropic material here. It is supposed to have more interesting phenomena by utilizing the vector beam with anisotropic materials such as the radially polarized light beam which we are devoting our efforts on currently.

Funding

National Natural Science Foundation of China (11904303, 12034016, 61975169); Fundamental Research Funds for the Central Universities (20720200074, 20720220030); Youth Innovation Fund of Xiamen (3502Z20206045); Science Fund for Distinguished Young Scholars of Fujian Province (2015J06002); Program for New Century Excellent Talents in University (NCET-13-0495).

Acknowledgments

We thank the prof. Shanhe Su for the useful discussion about calculating the heat function. We also thank Shaochen Fang for discussing the experimental setup.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Thermal lens effect with light’s orbital angular momentum

Abstract

1. Introduction

2. Theoretical analysis

3. Experimental results

4. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (4)

Optics Express