Effect of light beam on measurements of reflectance and transmittance of turbid media with integrating sphere: Monte Carlo simulation

Abstract

Integrating sphere technique is widely used to measure the total reflectance and transmittance of turbid sample, but the unavoidable light loss induces some measuring error. It has never been reported whether the error depends on the shape and size of light beam. In this paper, a convolution for computing the responses to rectangular incident light beam based on the Monte Carlo method was presented. The effects of light beam shape and size, and optical properties of sample on the measurements were addressed. The results show that the light loss with rectangular incident light beam is larger than that with circular one with the same area. The more the area of light beam, the more the light loss. And the light loss induced by the optical properties of sample is much more significant than that by the shape and size of the incident light beam.

Keywords

integrating spheres / optical properties / Monte Carlo

1 Introduction

Random lasers in various strongly and weakly scattering disordered medium with optical gain, were first theoretically predicted by Letokhov in the late 1960s [1] and were further experimentally observed by Lawandy et al. in 1994 [2], random lasers are well illustrated with a time dependent theory to perform a lasing numerical simulation in localized modes [1–11]. By this theory, many properties of random lasers have been investigated. Previous works had mainly focused on random lasers with a fixed pumping rate, in which all random lasers were pumped with a lasing pulse, while pumping rate was usually considered as a fixed value in the whole process of numerical simulations. Because the duration of the simulating time was usually a few picoseconds (pss), a fixed pumping rate may be available for the pumping pulse emitted from a nanosecond (ns) or ps laser, but it could be not adequate for the pumping pulse emitted from a femtosecond (fs) laser. Much information for random lasing could be lost in the simulations, especially in the case of fs pumping. Moreover, threshold gain behavior is very improtant subject for conventional lasers. Therefore, threshold gain properties of lasing modes in one-dimensional (ID) disordered media optically pumped by fs-lasing pulse are investigated here.

In this work, threshold gain behavior of random lasers is calculated in ID random medium pumped by an 80 fs pulse. In order to be compared with fixed pumping, the spectral intensity varying with the peak value of fs pulse pumping is calculated at the same disordered medium structure, and the simulation time of the system is 6 ps. Therefore, the fixed pumping denotes ps pulse in our simulation. The calculated results show that the excited modes between the fs pulse and fixed pumping (ps pulse) are only slightly different, which suggest that excited modes can not be determined by the duration of pumping. To further explore the physical nature of excited modes for random lasers, the modes’ lifetimes have been calculated. The results indicate that there exists a certain correspondence between the mode’s threshold pumping rate and its lifetime, and the lower the threshold is with the longer lifetime. Our works enrich the knowledge in case of short pulse pumping, as well as offer more guidance for relevant experiments.

2 Theoretical model

Binary layers of the system are consisted of two dielectric materials, as shown in Fig. 1. White layer simulates the air, which is characterized by a random variable thickness a_n and a dielectric constant ${ϵ}_{\mathrm{1}}={ϵ}_{\mathrm{0}}$, while black layer with a fixed thickness b = 300 nm and a dielectric constant ${ϵ}_{\mathrm{2}}=\mathrm{4}{ϵ}_{\mathrm{0}}$ simulates the scatters that are also a gain media with a four-level atomic system. The random variable a_n is described as $a_n=a(\mathrm{1}+w\gamma )$, where a =180 nm, w is the strength of randomness, and $\gamma$ is a random value in the range [-0.5, 0.5].

Fig.1 Schematic illustration of 1D random medium

Full size|PPT slide

For the 1D time-dependent Maxwell equations and an active and non-magnetic medium, we have

(1a)$${\displaystyle \frac{\partial H_y}{\partial x}}={ϵ}_{\mathrm{0}}{ϵ}_i{\displaystyle \frac{\partial E_z}{\partial t}}+{\displaystyle \frac{\partial P_z}{\partial t}},(i=\mathrm{1},\mathrm{2}),$$

(1b)${\displaystyle \frac{\partial E_z}{\partial x}}={\mu }_{\mathrm{0}}{\displaystyle \frac{\partial H_y}{\partial t}},$

where P_z is a polarization density component in z direction, ${ϵ}_{\mathrm{0}}$ and ${\mu }_{\mathrm{0}}$ are the electric permittivity and the magnetic permeability of vacuum, respectively.

For the four-level atomic system, the rate equations read

(2a)$${\displaystyle \frac{d{N}_{\mathrm{1}}}{dt}}={\displaystyle \frac{N_{\mathrm{2}}}{{\tau }_{\mathrm{21}}}}-W_p(t)N_{\mathrm{1}},$$

(2b)$${\displaystyle \frac{d{N}_{\mathrm{2}}}{dt}}={\displaystyle \frac{N_{\mathrm{3}}}{{\tau }_{\mathrm{32}}}}-{\displaystyle \frac{N_{\mathrm{2}}}{{\tau }_{\mathrm{21}}}}-{\displaystyle \frac{E_z}{\hslash {\omega }_l}}·{\displaystyle \frac{dP}{dt}},$$

(2c)$${\displaystyle \frac{d{N}_{\mathrm{3}}}{dt}}={\displaystyle \frac{N_{\mathrm{4}}}{{\tau }_{\mathrm{43}}}}-{\displaystyle \frac{N_{\mathrm{3}}}{{\tau }_{\mathrm{32}}}}+{\displaystyle \frac{E_z}{\hslash {\omega }_l}}·{\displaystyle \frac{dP}{dt}},$$

(2d)$${\displaystyle \frac{d{N}_{\mathrm{4}}}{dt}}=-{\displaystyle \frac{N_{\mathrm{4}}}{{\tau }_{\mathrm{43}}}}+W_p(t)N_{\mathrm{1}}.$$

Set particle population in unit volume of each level is N₄, N₃, N₂ and N₁ individually. The pumping rate from E₁ to E₄ is W_p; the particles arrive and E₄ transfer to E₃ quickly in the form of radiationless transition, the factor of probability is $\mathrm{1}/{\tau }_{\mathrm{43}}$. Before the population inverse, E₃ transfer to E₂ quickly in the form of spontaneous activity emission, the factor of probability is $\mathrm{1}/{\tau }_{\mathrm{32}}$. E₂ transfer to E₁ mostly in the form of spontaneous activity emission, the factor of probability is $\mathrm{1}/{\tau }_{\mathrm{21}}$.

The polarization obeys the following equation:

(3)$${\displaystyle \frac{d^{\mathrm{2}}P}{d{t}^{\mathrm{2}}}}+\Delta {\omega }_l{\displaystyle \frac{dP}{dt}}+{\omega }_l^{\mathrm{2}}P=\kappa \Delta N{E}_z.$$

This equation links Maxwell’s equations with rate equations. $\Delta N=N_{\mathrm{2}}-N_{\mathrm{3}}$ is the population difference density between the populations in the lower and upper levels of the atomic transition. Amplification takes place when external pumping mechanism produces population inversion $\Delta N<\mathrm{0}$. The linewidth of atomic transition is $\Delta {\omega }_l=\mathrm{1}/{\tau }_{\mathrm{32}}+\mathrm{2}/T_{\mathrm{2}}$ where collision time T₂ is usually much smaller than lifetime ${\tau }_{\mathrm{32}}$. The constant κ is given by $\kappa =\mathrm{6}\pi {ϵ}_{\mathrm{0}}{c}^{\mathrm{3}}/({\omega }_l^{\mathrm{2}}{\tau }_{\mathrm{32}})$.

The pumping rate W_p(t) is

(4)$$W_p(t)=W_{peak}\exp \left(-{\displaystyle \frac{\mathrm{4}{(t-t_{\mathrm{0}})}^{\mathrm{2}}}{{\tau }^{\mathrm{2}}}}\right),$$

where W_peak is the peak value of the pumping, τ is the width of the pumping, t₀ is the time corresponding to the peak value, as shown in Fig. 2. In our simulations, τ and t₀ are set to 80 fs and 200 fs for all conditions, respectively.

Fig.2 Illustration of pumping process with time

Full size|PPT slide

The values of those parameters in the above equations that will be used in simulating the active part in the following numerical calculations, and they are respectively taken as: T₂ = 2.18 × 10^-14 s, τ₃₂ = 1 × 10^-8 s, τ₂₁ = 5 × 10^-12 s, τ₄₃ = 1 × 10^-13 s, and $N_T={\displaystyle \sum N_i=3.313\times {10}^{24}{m}^{-3}}$.

When pumping is provided over the whole system, the electromagnetic fields can be calculated. In order to model such an open system, a Liao absorbing layer [11] is used to absorb outward wave. The space and time increments have been chosen to be $\Delta x$=1×10^-8 m and $\Delta t<\Delta x/(\mathrm{2}c)$, where $\Delta t$ is taken to be 1.67×10^-17 s, respectively. The pulse response is recorded during a time window of length T_w = 6 ps at all nodes in the system and Fourier transform is used to obtain an intensity spectrum.

3 Calculation and discussion

Analysis with calculating the spectral intensity varying with the peak value of fs pulse pumping was carried out. Three long-life modes indicated by their central wavelengths λ₀,λ₁ and λ₂ are marked respectively, while their spectral peak intensities are tracked and their threshold gain properties under different peak values of fs pumping pulse are analyzed. The duration of the pumping pulse is set to τ =80 fs for all conditions. As seen from Fig. 3(a), when the peak value of the pumping rate is quite low (W_p = 1×10¹⁰ s^-1), there are many discrete peaks, peak intensity of each peak is weak and the ordering of peak intensity is as same as that of spectral intensity. Note that each peak corresponds to a lasing mode supported by the disordered medium. With the peak value of the pumping rate increasing to a special value (W_p = 1×10¹¹ s^-1), the spectral intensity of mode λ₀ in Fig. 3(b) is stronger than that in Fig. 3(a) about two orders of magnitude, indicating that only the mode λ₀ is effectively amplified and dominates the whole spectrum. It is worth noticing that the spectral width of the mode λ₀ becomes quite larger at W_p = 1×10¹¹ s^-1 than that at lower W_p (W_p = 1×10¹⁰ s^-1). When the pumping rate further increases, accompanied by the increase of the peak intensity and the decrease of the spectral width for the mode λ₀, but the spectral widths of the modes λ₁ and λ₂ become larger than those at lower W_p, as shown in Figs. 3(c) and 3(d). As the pumping rate increases farther, due to mode competition more modes are excited, as shown in Figs. 3(e) and 3(f). The intensities and widths of the three modes finally reach to their stable values as the pump rate increases greatly as well as another mode also is excited, as shown in Fig. 3(f).

Fig.3 Intensity spectrum in arbitrary units versus the wavelength for 1D disordered medium shown in Fig. 1 at (a) Wp = 1×1010 s-1; (b) Wp = 1×1011 s-1; (c) Wp = 1×1012 s-1; (d) Wp = 1×1013 s-1; (e) Wp = 1×1014 s-1;(f) Wp = 1×1015 s-1

Full size|PPT slide

To obtain more information about the threshold gain behavior for the modes, numerous calculations are performed at different pumping rates, and curves of the peak intensity and the spectral width versus the pumping rates for the three modes can be obtained, as shown in Fig. 4. According to traditional method, pumping thresholds for the three modes have been shown in Fig. 4(a) as W_I0=5×10^-11 s^-1, W_I1=3×10^-12 s^-1 and W_I2=1.2×10^-12 s^-1 respectively, in which I denotes the threshold determined by intensity. For the mode λ₀, a jump for the spectral width within a pump regime near the pumping threshold has been observed. The peak value of the jump appears at the point that is close to the threshold of the mode (Fig. 4(b)).

Fig.4 Plots of the peak intensity and spectral width of the lasing modes versus the pump rate Wp under fs pulse pumping. (a) Peak intensities for the four indicated modes, and the lasing threshold measured from the plots are WI0 = 5×10-11 s-1, WI1=3×10-12 s-1 and WI2=1.2×10-12 s-1; (b) peak intensity and spectral width for the mode λ0

Full size|PPT slide

For comparison, we calculated the spectral intensity varying with the fixed pumping rate using the same medium structure under 80 fs pulse pumping, as shown in Fig. 5. The calculated results show that there are only slightly differences under the fs pulse pumping among the excited modes. The sequence of the excited modes using the fixed ps pulse pumping is identical to that of using fs pulse pumping.

Fig.5 Spectral intensity in arbitrary units versus the wavelength for 1D disordered medium pumped by a fixed pumping rate shown in Fig. 1 at (a) Wp = 1×108 s-1; (b) Wp = 1×109 s-1; (c) Wp = 1×1010 s-1; (d) Wp = 1×1011 s-1

Full size|PPT slide

In order to further investigate intrinsic reason that makes the mode λ₀ be excited firstly, the rates of the energy decays for the modes λ₀, λ₁ and λ₂ at the same disordered structure described as above. Three models are separately excited by a monochromatic source. The monochromatic source has a Gaussian envelope with same amplitudes for the three modes. For each mode, the energy is recorded at all nodes and the total energy is obtained by summing the energy at all nodes. Figure 6 shows the evolution of the normalized total energy decay with time, from which we can obtain the mode’s lifetime τ for each mode if we define τ to be the time that the total energy decreases from its maximal value E_max to E_max/e, in which e is approximately equal to 2.71828. The computed results show that the ordering of the three modes’ lifetime τ is τ₁<τ₂<τ₀. This indicates that there is a certain correspondence between the mode’s threshold and its lifetime. Because the quality factor of a mode is directly proportional to τ, i.e., Q=2πcτ/λ, there exists a certain correspondence between the mode’s threshold pumping rate and its Q-factor and a localized mode with a larger Q-factor has a lower threshold.

Fig.6 Normalized total energy decays with time for each marked mode. Three lifetimes are τ0=2.84 ps, τ1=0.92 ps, and τ2=1.85 ps

Full size|PPT slide

4 Conclusions

In conclusion, threshold gain behavior of random lasers is investigated in 1D random medium pumped by an 80 fs pulse. The results show that the excited modes are only slightly different between the fs pulses and fixed pumping (6 ps). This suggests the excited modes strongly depend on the medium structure instead of the duration of pumping, at lease for both fs and ps case.

References

[1] Noda H M, Motohka T, Murakami K, Muraoka H, Nasahara K N. Accurate measurement of optical properties of narrow leaves and conifer needles with a typical integrating sphere and spectroradiometer. Plant, Cell & Environment, 2013, 36(10): 1903–1909

[2] van Staveren H J, Moes C J M, van Marie J, Prahl S A, van Gemert M J C. Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm. Applied Optics, 1991, 30(31): 4507–4514

[3] Aernouts B, Zamora-Rojas E, Van Beers R, Watté R, Wang L, Tsuta M, Lammertyn J, Saeys W. Supercontinuum laser based optical characterization of Intralipid® phantoms in the 500-2250 nm range. Optics Express, 2013, 21(26): 32450–32467

[4] Zamora-Rojas E, Aernouts B, Garrido-Varo A, Saeys W, Perez-Marin D, Guerrero-Ginel J E. Optical properties of pig skin epidermis and dermis estimated with double integrating spheres measurements. Innovative Food Science & Emerging Technologies, 2013, 20: 343–349

[5] Zhong X, Wen X, Zhu D. Lookup-table-based inverse model for human skin reflectance spectroscopy: two-layered Monte Carlo simulations and experiments. Optics Express, 2014, 22(2): 1852–1864

[6] Kim S, Jeong S. Effects of temperature-dependent optical properties on the fluence rate and temperature of biological tissue during low-level laser therapy. Lasers in Medical Science, 2014, 29(2): 637–644

[7] Välisuo P, Kaartinen I, Tuchin V, Alander J. New closed-form approximation for skin chromophore mapping. Journal of Biomedical Optics, 2011, 16(4): 046012

[8] Adams M T, Wang Q, Cleveland R O, Roy R A. Thermal dose dependent optical property changes of ex vivo chicken breast tissues between 500 and 1100 nm. Physics in Medicine and Biology, 2014, 59(13): 3249–3260

[9] Bashkatov A N, Genina E A, Kochubey V I, Gavrilova A A, Kapralov S V, Grishaev V A, Tuchin V V. Optical properties of human stomach mucosa in the spectral range from 400 to 2000 nm: prognosis for gastroenterology. Medical Laser Application, 2007, 22(2): 95–104

[10] Bashkatov A N, Genina E A, Kochubey V I, Rubtsov V S, Kolesnikova E A, Tuchin V V. Optical properties of human colon tissues in the 350-2500 nm spectral range. Quantum Electronics, 2014, 44(8): 779–784

[11] Bosschaart N, Edelman G J, Aalders M C, van Leeuwen T G, Faber D J. A literature review and novel theoretical approach on the optical properties of whole blood. Lasers in Medical Science, 2014, 29(2): 453–479

[12] Grabtchak S, Montgomery L G, Whelan W M. Optical absorption and scattering properties of bulk porcine muscle phantoms from interstitial radiance measurements in 650-900 nm range. Physics in Medicine and Biology, 2014, 59(10): 2431–2444

[13] Zamora-Rojas E, Aernouts B, Garrido-Varo A, Perez-Marin D, Guerrero-Ginel J, Saeys W. Double integrating sphere measurements for estimating optical properties of pig subcutaneous adipose tissue. Innovative Food Science & Emerging Technologies, 2013, 19(0): 218–226

[14] Zhang L, Shi A, Lu H. Determination of optical coefficients of biological tissue from a single integrating-sphere. Journal of Modern Optics, 2012, 59(2): 121–125

[15] Zhang Y, Chen Y, Yu Y, Xue X, Tuchin V V, Zhu D. Visible and near-infrared spectroscopy for distinguishing malignant tumor tissue from benign tumor and normal breast tissues in vitro. Journal of Biomedical Optics, 2013, 18(7): 077003

[16] Zhu D, Lu W, Zeng S, Luo Q. Effect of light losses of sample between two integrating spheres on optical properties estimation. Journal of Biomedical Optics, 2007, 12(6): 064004

[17] Pickering J W, Prahl S A, van Wieringen N, Beek J F, Sterenborg H J C M, van Gemert M J C. Double-integrating-sphere system for measuring the optical properties of tissue. Applied Optics, 1993, 32(4): 399–410

[18] Friebel M, Helfmann J, Meinke M C. Influence of osmolarity on the optical properties of human erythrocytes. Journal of Biomedical Optics, 2010, 15(5): 055005

[19] Morawiec S, Mendes M J, Filonovich S A, Mateus T, Mirabella S, Aguas H, Ferreira I, Simone F, Fortunato E, Martins R, Priolo F, Crupi I. Broadband photocurrent enhancement in a-Si:H solar cells with plasmonic back reflectors. Optics Express, 2014, 22(S4 Suppl 4): A1059–A1070

[20] Bruns S, Rademacher D, Vergöhl M, Weiss P. Properties of reactive sputtered alumina-silica mixtures. Applied Optics, 2014, 53(4): A334–A338

[21] Cong A X, Hofmann M C, Cong W, Xu Y, Wang G. Monte Carlo fluorescence microtomography. Journal of Biomedical Optics, 2011, 16(7): 070501

[22] Wang L, Jacques S L, Zheng L. MCML—Monte Carlo modeling of light transport in multi-layered tissues. Computer Methods and Programs in Biomedicine, 1995, 47(2): 131–146

[23] Wang L, Jacques S L, Zheng L. CONV—convolution for responses to a finite diameter photon beam incident on multi-layered tissues. Computer Methods and Programs in Biomedicine, 1997, 54(3): 141–150

[24] Vo-Dinh T. Biomedical Photonics Handbook. New York: CRC Press, 2003

[25] Prahl S A, van Gemert M J C, Welch A J. Determining the optical properties of turbid mediaby using the adding-doubling method. Applied Optics, 1993, 32(4): 559–568

Acknowledgements

This study was supported by the National Natural Science Foundation of China (Grant Nos. 81171376 and 91232710), the Research Fund for the Doctoral Program of Higher Education of China (No. 20110142110073), and the Science Fund for Creative Research Group of China (No. 6141064).

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg