Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group

Luminescence material characterizations on laser-phosphor lighting techniques

Open Access Open Access

Abstract

This study examines three different types of laser carriers, including the phosphors in silicone (PiS), the phosphor in glass (PiG), and alumina-based ceramic binders (CP), for laser-phosphor lighting characterizations via a laser projector light source module. The thermal influence of heat spreading on these phosphor materials and their luminescence performance is also investigated. The conversion efficiency of PiS, PiG and CP was found to be 29.7%, 34.6% and 31.8%, with their corresponding maximum laser power operations of 3.9 W, 7.9 W, and 17.2 W, respectively. This work further correlates the maximum laser operation power with the thermal conductivity of luminescence material. From the optical engine perspective, it was found that CP exhibits the superior thermal conductivity of 17.0 W/m⋅K for slight hot-spot IR observation and higher laser power operation. This work finally delivers a CP device for 50.2W maximum laser operation with the operating temperature below 100 °C. The simulation is also carried out in order to examine spreading resistance’s influence, when subject to convective boundary condition. CP’s response sensitivity to heat transfer coefficient was found to be rather small.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Corrections

25 March 2019: A typographical correction was made to the author listing.

1. Introduction

Solid-state lighting (SSL) has revolutionized the lighting circumstance with comparatively high power efficiency and is applicable for various versatile applications [1,2]. Recently, Laser-phosphor illumination has attracted massive attentions to special lightings, such as laser projectors, headlamp, spot illuminator, and endoscopic light source. For thermal issues associated with LED packaging processes, many fluid flow problems exist, such as phosphor coating, silicone injection, chip bonding, solder reflow, etc. while phosphor coating is the most important process which is essential for LED performance [3,4]. Different from LED chip emission, laser diode (LD) junction is constructed by an oscillator that delivers the coherent light beam with a high power density having small Etendue. This is why LD is mostly applied in field of high-performance collimated illumination instead of general diffused lighting. In this decade, the major LD manufacturer, NICHIA Corporation, had demonstrated a 445~460 nm-LD featuring power efficiencies form 1.6 W to 4.7 W with reliability up to 30000 hours at an affordable price. Since then, LD lighting had been penetrating into the market of traditional lamps, i.e., halogen, Xeon, and mercury, in projector light source or other collimated lighting. The laser-phosphor light source based on 80~300 W laser excitation with phosphor luminescence are now extensively implemented into professional projections, home-theater displays as well as projecting light [5,6].

Relative to LD, phosphor-luminescent component is another key factor in laser-phosphor lighting. A blue laser module comprised of tens- or hundreds- of LDs contains focused lens onto the phosphor surface in a confined spot of 2~4 mm2 for luminescence excitation. This implies phosphor suffers from severely blue photon bombardment in energetic density of 40~75 W/mm2, nearly one-sixth of solar surface thermal irradiation (~250 W/mm2). Consequently, the laser-phosphor power transaction is only 25~50% under such quantitative excitation when compared to the above 90% QE efficiency for LED lighting. In this regard, appreciable heat is generated from the residue laser power and phosphor transaction that eventually accumulates onto the irradiation spot during laser-photon bombardment, thereby resulting in high temperature and impairing luminance or damaging the material [7]. In general, phosphors are dispersed into silicone during coating process. Luo et al. [8] simulated that a 680 mW laser excited with 1 mm2-remote PiS could yield phosphor temperature as high as 549 °C for silicone carbonization. Cheng et al. [9,10] adopted glass-based phosphor wheel by employing 850 °C sintered-glass (Tg ~570 °C) design instead of silicone (Tg ~150 °C) for 30 W laser operation and depicted a superior thermal reliability. However, binders of silicone and glass are not only correlated with their usage temperature limit; and the thermal conductivities, k (W/m⋅K), would dominate the luminance characteristics of phosphors in binders. Zhang et al. [11] modulated the PiG borosilicate glass binder composition with a higher thermal conductivity (2.8 W/m⋅K at 80 °C, Tg = 711 °C), and yields a superior phosphor efficacy ~205 lm/W. Their design offered a very small luminescence drop of 0.5% after 100 hours operation at 300 °C. Lenef et al. [5] adopted ceramic phosphor to improve thermal conductivity subject to a remote 25 W laser-phosphor demonstration. In laser-phosphor configuration, illuminating phosphors are also the heat source to induce thermal phosphorous decay. These binder matrixes of silicone, glass, and ceramics exhibit different k as ~0.2, 1~2, and 10~20 W/m⋅K, respectively, which certainly imposes some effect on laser-phosphor lighting characteristics.

From this work, in order to comprehend the thermal effect of phosphor material on laser transaction, a practical optical engine in commercial laser projector is employed to characterize these different phosphor materials in silicone, glass, and alumina-based binders. It benchmarks the luminescence performance with optical effects, thermal issues, and material properties from the viewpoint of light engine module. This attempts to provide some phosphor material characterizations to laser-lighting application.

2. Material and experimental setup

2.1 Materials

The luminescence materials used for testing include the phosphor in silicone (PiS), the phosphor in glass (PiG), and the ceramic phosphor (CP), respectively. The PiS is composed of 40 wt% of Ce-YAG phosphor powders (INTEMATIX, NYAG-4354, 553 ± 2nm, 13 um-sized) in silicone (Dow-Corning OE-6351) for coating. For comparisons, the PiG is also made of 40 wt% of the aforesaid phosphor powders dispersed in SiO2-Na2O-K2O-Al2O3 glass matrix. The recipe with 1~2 μm-sized glass frits (ASAHI Glass Co., Ltd) were mixed with phosphors, followed by an isocratic press into 4̎ ingot with 15~16 mm height. After 780 °C treatment with 30 minutes sinter-forming, the PiG ingot was wire-sawed into around 200 μm-thick discs, followed by grinding or polishing treatment. Thereafter, the PiG disc was diced into 5 × 5 mm2 chip for usage. CP is a pure material based on polycrystalline Ce-YAG solid material. Oxide precursors of Aluminum, Yttrium and Cerium were employed; after mixed and isocratic press molding, CP green body ingot was reacted at 1600~1700 °C under controlled atmosphere for YAG synthesis. 1.0 mol% of Cerium is proposed in this work for 550 nm emission. Same as PiG process, the back-end machining procedures were sawing, polishing, and then dicing into chips for further usage.

2.2 Phosphor device preparation

Experimental devices are performed by these PiS, PiG and CP materials onto a commercial 95% Al-reflective aluminum substrates (0.7 mm-thick). Figure 1(a) depicts PiS slurry was imprinted with a 200 μm-thick 5 × 5 mm2 area on Al substrate, followed by 150 °C -of 3 minutes curing process for characterizations. And the PiG- and CP-chips have two structures for comparisons. One is silicone-glued phosphor chips onto Al substrates as shown in Fig. 1(b). The 10 μm-silicone medium accessed the interface adhesion between phosphor chips and substrates. The other device structure is without silicone-glue, directly mounting phosphor chips onto Al substrates for comparisons. For easier distinguishing, they are denoted as PiG/CP(Silicone) and PiG/CP(Air) for discussion. Figure 2 shows the cross-sectional images of PiS, PiG, and CP materials. Figures 2(a) and 2(b) present the morphologies of PiS and PiG show their binary material properties of phosphors dispersed into the silicone and glass matrix. PiG further exhibited micro-pores inside with about 10% porosity. Figure 2(c) indicates that CP is a one-component property of YAG-based material. The morphology shows CP is densely made with 1~2 μm YAG grains, implying a higher thermal conductivity.

 figure: Fig. 1

Fig. 1 Phosphor device specifications (a) and appearance (b).

Download Full Size | PDF

 figure: Fig. 2

Fig. 2 Cross-sectional SEM images for (a) PiS, (b) PiG, and (c) CP.

Download Full Size | PDF

2.3 Luminescence characterization setup

A commercial NEC 6500-lumen laser-phosphor projector (NP-PA653UL) was employed in this work. Figure 3 illustrates the laser-phosphor optical engine architecture inside. The laser module was used by NICHIA 455 nm laser banks (28 W455nm-op / 2.5 A). As shown, the laser beam was condensed and focused by a relay lens into 2 × 2 mm2 spot irradiating on phosphor device. The maximum laser output on phosphor surface in this work is measured as about 50.2 Wop, which constructed as 12.6 W/mm2 of excitation energy density. The measured phosphor devices were attached onto a thermoelectric (TE) cooling module with a uniform temperature of 25 °C. While laser excitation, the relay lens collected the phosphorous irradiation into the integrating sphere (IS) via dichroic mirror reflection. The luminescence analysis was performed by the spectrometer (Ocean Optics USB2000 + , 250~850 nm with 1 nm accuracy). Simultaneously, an infrared thermal imaging camera (CHCT, P384-20 with 200 um-resolution) was used to capture the phosphor surface temperature profile. All experimental measurements were performed in continuous 10 minutes after steady-state was reached.

 figure: Fig. 3

Fig. 3 Schematic of the experimental setup for luminescence and thermal characterizations.

Download Full Size | PDF

3. Results and discussion

Figure 4 depicts the luminescence emission spectra for PiS, PiG(Air), and CP(Air) devices at ~3.0 W455nm excitation. The inset indicates that those phosphor materials exhibit the similar spectra with color coordinate (x, y) = (0.415, 0.566) at 545 nm where main peak emission is optically benchmarked. By integrating the luminescence intensity from 460 nm to 700 nm, the PiS, PiG, and CP would deliver their power efficiencies (PE) (convert ratio of luminescence power to laser power) respectively as 29.7%, 34.6% and 31.8% under 3.0 W455nm in Table 1. Note that PiG exhibits the superior luminescence efficiency, which coincides with Lenef’s report [10]. Micro-pores in Fig. 2(b) of PiG would benefit the laser scattering for higher absorption usage and superior luminescence output. Hence, PiG prevails PiS and CP at low laser power excitation. Further, IR camera showed the thermal images of these laser-irradiated phosphor materials’ surfaces in Fig. 5. Under 2 × 2 mm2-3.0 W455nm laser bombardment, these phosphor materials demonstrate the thermal responses as Gaussian thermal distributions. PiS shows that-the appreciable heat dissipation with hot-spot profile. Denoting T50 as the temperatures at FWHM (full-width at half-maximum) of phosphor temperature distributions, Table 1 shows that the highest T50 is 50.5 °C for PiS and the lowest T50 is 34.1 °C for CP. The highest ratio of T80/T50 (T80 noted as the temperatures at 80% maximum) for PiS is 1.28, revealing severe temperature rise in PiS. This result coincides with the trend of the thermal conductivity measurements (k, Hot Disc, TPS-2500S) as 0.2, 1.3, and 17.0 W/m⋅K for PiS, PiG, and CP, respectively. It is believed that, in high power laser operation, high k value of CP would provide superior thermal conduction to substrate for heat dissipation, thereby yielding a lower temperature for preferable luminescence output. Another aspect, for silicone and glass binders, the inferior property of thermal conductivity k may constrain the phosphor to be operated in high temperature surroundings for low luminescence output and even results in material damage.

 figure: Fig. 4

Fig. 4 Luminescence emission spectra for PiS(Air), PiG(Air), and CP(Air)

Download Full Size | PDF

Tables Icon

Table 1. Some characteristics for different luminescence materials under ~3W laser excitation.

 figure: Fig. 5

Fig. 5 Temperature distributions for different phosphor materials under ~3W laser excitations.

Download Full Size | PDF

By increasing the laser excitation power, Fig. 6(a) and 6(b) show the luminescence and temperature characteristics for various phosphor materials. The phosphor output increases efficiently with increasing laser power. With further laser power rise, a thermal decay with output saturation is observed, and soon sequence by thermal quenching with abrupt luminescence drop and temperature rise. The maximum laser power operation is defined as the phosphor started to thermal decay, closed but not quenching. Accordingly, PiS, PiG(Air) and CP(Air) deliver the approximate efficiencies of 29.9%, 30.9%, and 29.1% with their maximum laser power operations at 3.9 W, 7.9 W, and 17.2 W, respectively. And the operation temperature is limited to around 150 °C due to the phosphor thermal decay. CP(Air) exhibits the mild temperature-rise response for withstanding higher laser power excitation. These similar PE values imply phosphors dominate the luminescence characteristics, and the maximum laser operation power is positively correlated to the thermal conductivity of binder materials. This result also coincides with Fig. 4 for the same emission spectra in these three different phosphor materials; hence the device properties might be determined by binder’s thermal conductivity or device structure, etc. Inset of Fig. 6(a) further reveals the maximum laser power operation is roughly logarithm function to k3.03, and no dependence on efficacy. This appears that CP luminescence material having higher thermal conductivity would contribute a higher laser power operation.

 figure: Fig. 6

Fig. 6 (a) Luminescence and (b) Temperature characteristics for PiS (Glue), PiG (Air), and CP (Air) at different laser power excitations.

Download Full Size | PDF

Moreover, since the thermal conduction is crucial to phosphor device, the air interfaces at present PiG(Air) and CP (Air) might cause considerable thermal resistance to restrict their maximum laser power operation. In order to reduce the contact resistances, a 10 μm silicone-adhesion was further introduced into PiG(Air) and CP(Air) for luminescence and thermal comparisons. As illustrated in Fig. (7), with the silicone-adhesion interval, the light and heat would be easily spreading onto Al substrate. Figure 7(a) shows the insert of silicone medium delivers less luminescence improvement, but largely benefit from the thermal response and laser operation limit power. Figure 7(b) shows the PiG(Silicone) and CP(Silicone) deliver lower temperature responses as compared to PiG(Air) and CP(Air). This result again confirms the maximum laser operation not only relates to material thermal conductivity but also correlates with device structure which all correspond to the thermal conduction. Without the air-gap thermal conduction obstruction, CP(Silicone) relies on its advantage of good thermal conductivity for extending the maximum laser operation power to 50.2 W within only 100 °C temperature response and 13.5W luminescence output.

 figure: Fig. 7

Fig. 7 (a) Luminescence and (b) Temperature characteristics for PiG (Air, Glue) and CP (Air, Glue) at different laser power excitations.

Download Full Size | PDF

In order to comprehend the influence of material thermal conductivities at laser-phosphor lighting, this work also constructs some thermal resistance analysis to describe these luminescence materials, performances. Figure 8 depicts a schematic diagram of phosphor luminescence device in a phosphor plate with a dimensions of 5 mm (2c) × 5 mm (2d) and a thickness of 0.2 mm (T1) for 2 mm (2a) × 2 mm (2b) laser spot irradiation. Silicone glue layer (T2) of 2 μm was employed to impose the heat flux onto underlying reflective aluminum. Although the aluminum substrate was cooled at 25 °C by a thermoelectric cooler (TEC), an effective forced convection heat transfer coefficient h was introduced as this boundary condition for further simulation [12]. The heat source (Qh) on spot, coming from the difference between laser power input (Qlaser) and phosphor luminescence output (Qphosphor), is derived from Eq. (1) with the temperature difference (ΔT) and its corresponding device thermal resistance (Rt). Rt is actually comprised of material thermal resistance in z-direction, Rz, and thermal spreading resistance, Rs, as

Qh=ΔT/Rt
Where
Rt=Rz+Rs
and Rz is given by
Rz=t1/k1A+t2/k2A
with A as the phosphor plate surfave area of 4cd. Furthermore, Rs is estimated from Muzychka et al. [12] who derived the analytical solution for thermal spreading resistance of rectangular two-layer substrates with given geometric and thermal parameters as
Rs,Muzychka=12a2cdk1/m=1sin2(aδm)/δm3+12b2cdk1m=1sin2(bλn)λn3φ(λn)φ(δm)+12a2b2cdk1m=1n=1sin2(aδm)sin2(bλn)δm2λm2βm,nφ(βm,n)
where
φ(ζ)=(αe4ζt1+αe2ζt1)+ρ(e2ζ(2t1+t2)+αe2ζ(2t1+t2))(αe4ζt1αe2ζt1)+ρ(e2ζ(2t1+t2)αe2ζ(2t1+t2))
with
ρ=ζ+h/k2ζh/k2,α=1+k2/k11k2/k1,δm=mπ/c,λn=nπ/d,andβm,n=(δm2+λn2)1/2
and k1 and k2 are the thermal conductivities for layers 1 and 2 as phosphor plate and silicone glue, respectively. Based on these simulations, Fig. 9 depicts the sensitivity of Rt to the heat transfer coefficient of TEC boundary for different phosphor luminescence materials. As clearly shown in the figure, PiS still presents the highest Rt value in 1000~200 K/W. By increasing the convection coefficient from 100 to 3500 W/m2⋅K, PiS and PiG have apparent drops in their Rt tendency within 2000 W/m2⋅K, and subsequently approach to their minimal values as 200 and 80 K/W. This simulation result also implies that the used TEC in this work acts heat transport as apparent convective coefficient <2000 W/m2⋅K due to previous temperature responses is quadratic polynomial to the laser power. However, from the viewpoint of a commercial optical engine, such convective coefficient more than 2000 W/m2⋅K is hardly performed by air-cooling. Water-cooling, high power TEC or other high-performance cooling technique is essentially adopted for PiS and PiG to get rid of thermal influence, which would increase the burden onto system cost, volume and complexity. That explains why most present laser-phosphor lighting products suffer severe thermal issues. In contrast, CP shows much less sensitive response to the heat transfer coefficient. These numerical solutions for CP’s superior experimental consequences are attributed to its comparatively small spreading resistances ~20-30 K/W with slight hot-spot IR observation and higher laser power operation. This explains that ceramic luminescence materials are now penetrating into the laser-phosphor lighting market. Application, the thermal conductivities of luminescence phosphor materials would play an essential role in laser transaction performances.

 figure: Fig. 8

Fig. 8 Schematic of the two-layer model for thermal spreading resistance analysis.

Download Full Size | PDF

 figure: Fig. 9

Fig. 9 Effect of convective coefficient on Rt for PiS, PiG and CP.

Download Full Size | PDF

4. Conclusions

In this study, three different types of phosphor materials, including the phosphors in silicone (PiS), the phosphor in glass (PiG), and alumina-based ceramic binders (CP), are characterized by a commercial laser projector optical engine. These laser carries are directly transacted for phosphorous luminescence comprehended with their optical effects, thermal issues, and material properties. It is found that PiS, PiG and CP deliver their thermal conductivities as 0.2, 1.3, and 17.0 W/m⋅K. For same phosphor emission property, these three laser carriers exhibit similar conversion efficiency ~30.0%, but offer different maximum laser power operations are 3.9 W, 7.9 W, and 17.2 W, respectively. Experimental results explain the maximum laser operation power is determined by the thermal conductivity of binder materials. The maximum laser power is roughly logarithm function to thermal conductivity with the power of 3.03. By using ceramic binder as the contact medium for the phosphor powder and reflector can ease the heat transport effectively and hinder the temperature raise to higher laser power operation. This work also greatly delivers a CP luminescence device with 50.2W laser power operation under response temperature of 100 °C. A simulation is also carried out to examine the influence of spreading resistance subject to convective boundary condition. These numerical solutions explained the PiS and PiG in present market suffer much thermal issue. It is also found that the response sensitivity of CP to heat transfer coefficient is rather small.

Funding

Ministry of Science and Technology (MOST, Taiwan) (107-2622-E-009-002-CC2 and 107-2221-E-009-142). Department of Industrial Technology, Ministry of Economic Affairs Taiwan

References

1. C. Basu, M. Meinhardt-Wollweber, and B. Roth, “Lighting with laser diodes,” Adv. Opt. Technol. 2, 313–321 (2013).

2. J. Y. Tsao, J. J. Wierer, L. E. S. Rohwer, M. E. Coltrin, M. H. Crawford, J. A. Simmons, P.-C. Hung, H. Saunders, D. S. Sizov, R. Bhat, and C.-E. Zah, “Introduction Part B. ultra-efficient solid-state lighting: likely characteristics, economic benefits,” Technological Approaches 126, 11–26 (2013). [CrossRef]  

3. X. Luo, R. Hu, S. Liu, and K. Wang, “Heat and fluid flow in high-power LED packaging and applications,” Pror. Energy Combust. Sci. 56, 1–32 (2016). [CrossRef]  

4. X. Luo, X. Fu, F. Chen, and H. Zheng, “Phosphor self-heating in phosphor converted light emitting diode packaging,” Int. J. Heat Mass Transfer 58(1-2), 276–281 (2013). [CrossRef]  

5. M. H. Kane, J. Jiao, N. Dietz, J.-J. Huang, A. Lenef, J. Kelso, M. Tchoul, O. Mehl, J. Sorg, and Y. Zheng, “Laser-activated remote phosphor conversion with ceramic phosphors,” 9190, 91900C (2014).

6. Y. C. Chi, D. H. Hsieh, C. Y. Lin, H. Y. Chen, C. Y. Huang, J. H. He, B. Ooi, S. P. DenBaars, S. Nakamura, H. C. Kuo, and G. R. Lin, “Phosphorous diffuser diverged blue laser diode for indoor lighting and communication,” Sci. Rep. 5(1), 18690 (2016). [CrossRef]   [PubMed]  

7. J. Xu, J. Wang, Y. Gong, X. Ruan, Z. Liu, B. Hu, B. Liu, H. Li, X. Wang, and B. Du, “Investigation of an LuAG:Ce translucent ceramic synthesized via spark plasma sintering: Towards a facile synthetic route, robust thermal performance, and high-power solid state laser lighting,” J. Eur. Ceram. Soc. 38(1), 343–347 (2018). [CrossRef]  

8. Y. Ma, W. Lan, B. Xie, R. Hu, and X. Luo, “An optical-thermal model for laser-excited remote phosphor with thermal quenching,” Int. J. Heat Mass Transfer 116, 694–702 (2018). [CrossRef]  

9. B. Zhou, W. Luo, S. Liu, S. Gu, M. Lu, Y. Zhang, Y. Fan, W. Jiang, and L. Wang, “Enhancing the performance of Ce:YAG phosphor-in-silica-glass by controlling interface reaction,” Acta Mater. 130, 289–296 (2017). [CrossRef]  

10. C.-C. Tsai, W.-C. Cheng, J.-K. Chang, L.-Y. Chen, J.-H. Chen, Y.-C. Hsu, and W.-H. Cheng, “Ultra-high thermal-stable glass phosphor layer for phosphor-converted white light-emitting diodes,” J. Disp. Technol. 9(6), 427–432 (2013). [CrossRef]  

11. Q. Zhang, R. Zheng, J. Ding, and W. Wei, “Excellent luminous efficiency and high thermal stability of glass-in-LuAG ceramic for laser-diode-pumped green-emitting phosphor,” Opt. Lett. 43(15), 3566–3569 (2018). [CrossRef]   [PubMed]  

12. Y. S. Muzychka, J. R. Culham, and M. M. Yovanovich, “Thermal spreading resistance of eccentric heat sources on rectangular flux channels,” J. Electron. Packag. 125(2), 178–185 (2003). [CrossRef]  

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1 Phosphor device specifications (a) and appearance (b).
Fig. 2
Fig. 2 Cross-sectional SEM images for (a) PiS, (b) PiG, and (c) CP.
Fig. 3
Fig. 3 Schematic of the experimental setup for luminescence and thermal characterizations.
Fig. 4
Fig. 4 Luminescence emission spectra for PiS(Air), PiG(Air), and CP(Air)
Fig. 5
Fig. 5 Temperature distributions for different phosphor materials under ~3W laser excitations.
Fig. 6
Fig. 6 (a) Luminescence and (b) Temperature characteristics for PiS (Glue), PiG (Air), and CP (Air) at different laser power excitations.
Fig. 7
Fig. 7 (a) Luminescence and (b) Temperature characteristics for PiG (Air, Glue) and CP (Air, Glue) at different laser power excitations.
Fig. 8
Fig. 8 Schematic of the two-layer model for thermal spreading resistance analysis.
Fig. 9
Fig. 9 Effect of convective coefficient on Rt for PiS, PiG and CP.

Tables (1)

Tables Icon

Table 1 Some characteristics for different luminescence materials under ~3W laser excitation.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

Q h = Δ T / R t
R t = R z + R s
R z = t 1 / k 1 A + t 2 / k 2 A
R s , M u z y c h k a = 1 2 a 2 c d k 1 / m = 1 sin 2 ( a δ m ) / δ m 3 + 1 2 b 2 c d k 1 m = 1 sin 2 ( b λ n ) λ n 3 φ ( λ n ) φ ( δ m ) + 1 2 a 2 b 2 c d k 1 m = 1 n = 1 sin 2 ( a δ m ) sin 2 ( b λ n ) δ m 2 λ m 2 β m , n φ ( β m , n )
φ ( ζ ) = ( α e 4 ζ t 1 + α e 2 ζ t 1 ) + ρ ( e 2 ζ ( 2 t 1 + t 2 ) + α e 2 ζ ( 2 t 1 + t 2 ) ) ( α e 4 ζ t 1 α e 2 ζ t 1 ) + ρ ( e 2 ζ ( 2 t 1 + t 2 ) α e 2 ζ ( 2 t 1 + t 2 ) )
ρ = ζ + h / k 2 ζ h / k 2 , α = 1 + k 2 / k 1 1 k 2 / k 1 , δ m = m π / c , λ n = n π / d , a n d β m , n = ( δ m 2 + λ n 2 ) 1 / 2
Select as filters


Select Topics Cancel
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.