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Near-field enhancement of the nanostructure on the fused silica with rigorous method HU WANG,1,2 HONGJI QI,1,* BIN WANG,1,2 YANYAN CUI,1,2 YINGJIE CHAI,1,2 YUNXIA JIN,1 KUI YI,1 AND JIANDA SHAO1 1
Key Laboratory of Materials for High Power Laser, Shanghai Institute of Optics and Fine Mechanics, No. 390 Qinghe Road, Jiading District, Shanghai 201800, China 2 University of Chinese Academy of Sciences, Beijing 100049, China *Corresponding author: [email protected] Received 26 January 2015; revised 18 March 2015; accepted 14 April 2015; posted 14 April 2015 (Doc. ID 233192); published 4 May 2015
A rigorous electromagnetic method is developed to analyze the resonance effect of near field caused by nanoscale subsurface defects, which play a key role in describing absorption enhancement during laser–matter interaction for transparent dielectric materials. The total electric field calculated with this new method is consistent with the result of finite-difference time-domain simulation. The concept of mode amplitude density spectrum is developed to analyze the specific modes of the total field. A new mode parameter is proposed to demarcate the contribution of the resonance. The frequency space is divided into four parts and the resonance effect is analyzed as well as the contributions of different modes to the total field. The influence of the structure parameters on the near-field modulation and energy transference is also discussed. It is found that the enhancement mechanism of the nearfield and local absorption is the resonance effect caused by the total internal reflection on the sidewall of the nanostructure. In addition, the surrounding energy is mainly guided into the structure by the root of the structure via the energy flow analysis. © 2015 Optical Society of America OCIS codes: (260.6970) Total internal reflection; (050.1755) Computational electromagnetic methods; (140.3330) Laser damage; (260.5740) Resonance. http://dx.doi.org/10.1364/AO.54.004318
1. INTRODUCTION Laser–matter interactions are of great importance in various applications such as laser ablation [1,2] and laser-induced damage of optical components [3–5]. To better describe interaction mechanisms in the nanosecond regime for transparent dielectric materials, we must first understand how the irradiation is affected and absorbed by the nominal transparent dielectric materials. Resonance effect caused by subsurface nanostructure on the substrate has been recognized as a key factor of absorption enhancement in the solar cells [6] or describing field enhancement during laser–matter interaction for dielectric materials [7]. Electromagnetic resonances in the dielectric object can be classified into morphology-dependent resonances (MDRs) or whispering gallery modes, dependent on the structure’s morphology and incident frequency [8,9]. The MDR can greatly enhance the near field by confining the rays around the structure via a total internal reflection (TIR). Near-field enhancement around the defects embedded in optical components irradiated by high-power laser, has been widely investigated based on the electromagnetic field theory over the last two decades. For example, Bloembergen used the exact solution to quantify the light intensity enhancement 1559-128X/15/144318-09$15/0$15.00 © 2015 Optical Society of America
by assuming that the dimensions of the defects were much smaller than wavelength and ignored the reflection, refraction, diffraction, and interference at cracks and surfaces [10]. Deford assumed rotational symmetry to model the near-field enhancement around nodule defects in dielectric mirror coatings with AMOS computer code by using the finite-different timedomain (FDTD) technique [11]. Gruzdev and Libenson discussed the instability of high-power electromagnetic field resulted from resonances in nonabsorbing microinclusions embedded in the transparent dielectric [12]. Genin et al. modeled the light intensity enhancement factor caused by microscale planar and conical cracks with the FDTD method [7]. Their results indicated that the largest field enhancement could be reached when TIR occurs at both the crack and surface. Stolz and Roger investigated the near-field modulation by nodular defects existing in high reflective coatings with various geometry dimensions [13]. They also discussed the impacts of mitigation pits in dielectric multilayers by using an FDTD algorithm [14]. Unfortunately, problems in semi-infinite space commonly demand a large enough domain with periodic boundary conditions to approximate the infinite dimension. Algorithms such as FDTD or the finite element method require
Research Article a tedious preparation to test the convergence of both grid size and period length to find the optimized configuration to decrease computation time without significant loss in the accuracy of solution. Recently Chen et al. came up with a rigorous method called the Fourier modal method [15], which is very popular in surface relief grating analysis [16,17], to calculate near-field modulation caused by the subsurface defects and spherical mitigation pits on KDP crystal [18]. Their results indicated that the occurrence of TIR should be responsible for the largest light intensification inside the crystal. However, the enhancement mechanism by TIR was not determined and quantified. In this paper, we extend the existing state-of-the-art rigorous coupled wave analysis method commonly used in grating structure to solve the nanostructure embedded in semi-infinite space problems and analyze the local field enhancement and energy localization mechanism by taking advantage of grasping the information of all harmonics via this rigorous method. To the best of our knowledge, it is the first time to quantify the contribution of TIR to the near-field distribution around the nanostructure on the substrate. With the concept of the mode amplitude density spectrum mentioned in Subsection 2.D and a new model parameter, the near-field modulation mechanism caused by the nanostructure is discussed. The rigorous analysis method developed by us would be beneficial to analyze the local absorption and near-field enhancement in various applications, such as laser-induced damage of optical components [19], searching for optimal mitigation geometry for damage pits [14], near-field investigation [20], and light trapping in the solar cells [21,22].
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wave transmitted at the substrate–air interface. Here only the transverse electric polarization is considered, because the results for transverse magnetic polarization can be easily obtained by exchanging the role of the electric field and magnetic field in Maxwell equations. The grating structure with a large enough period Λ is used to approximate the semi-infinite space problem, as shown in Fig. 1. However, convergence test for period Λ and harmonic number M should be completed before calculating the near field. The normalized electric amplitude of the ith reflected wave in region I and transmitted wave in region II are denoted as r i and t i , respectively. The detail procedure to solve the values of r i and t i has been widely developed over the years [16,17,23]. In this paper, they are taken as a starting point for the near field and energy flow analysis. The assumed harmonic time dependence in our paper is expiωt. B. Near-Field Distribution
The normalized incident electric field with an incident angle θ is given by E in;y exp−jk x0 x − k I;z0 z;
(1) 2
1∕2
where k x0 k 0 nI sinθ and kI;z0 k0 nI 1 − sin θ . The ith tangential wave-vector component scattering from the nanostructure is given by kxi k 0 nI sin θ iλ∕Λ;
i 0; 1; 2; …
(2)
Because ε0 and μ0 differ by several orders of magnitude, E and H will differ by several orders of magnitude. It can be avoided by making the following change of variables: H η0 H;
(3)
1∕2
2. BASIC MODEL AND THEORY A. Basic Definition and Theory Background
To lay down the general framework for the main idea, we focus on a simple rectangle protuberance on an infinite fused silica substrate, as shown in Fig. 1. However, the rectangle is a natural brick for constructing arbitrary structure morphology. Both the rectangle and substrate have the same refractive index n 1.46. The irradiation wavelength λ is set as 355 nm, which is mostly concerned in the high-power laser system. The wavevector magnitude in free space is denoted as k 0 . The width and height of the rectangle are denoted as w0 and h0 , respectively, and they are initially equal to the incident wavelength. Nearfield distribution is produced with normally incident plane
is the vacuum admittance. where η0 ε0 ∕μ0 According to Maxwell equations, the magnetic field can be expressed as 1 H− ∇ × E: (4) jk0 Then the near field in region I can be represented as the sum of the incident field and all reflected harmonics [16]: X r i exp−jk xi x kI;zi z; (5) E I;y E in;y i
and H I;x
1 ∂E I;y jk 0 ∂z
X kI;zi kI;z0 E in;y − r exp−jk xi x kI;zi z; k0 k0 i i
H I;z −
1 ∂E I;y jk0 ∂x
X k xi kx0 E in;y r exp−jk xi x kI;zi z: k0 k0 i i
Also, the near field in region II can be given as X t i expf−jkxi x − kII;zi z h0 g; E II;y Fig. 1. Geometry for the rectangle nanostructure on substrate surface.
i
and
(6)
(7)
(8)
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H II;x
H II;z
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1 ∂E II;y X k II;zi t expf−jkxi x − k II;zi z h0 g; k0 i jk0 ∂z i
(9) X ∂E 1 k xi II;y − t expf−jk xi x − kII;zi z h0 g: k0 i jk 0 ∂x i
mode amplitude density spectrum (RMADS) and transmitted mode amplitude density spectrum (TMADS), respectively, which can be expressed as RMADS
jr i j ; Δk∕k 0
(20)
TMADS
jt i j ; Δk∕k 0
(21)
(10) The variable k l;zi can be determined by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k0pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2l − k xi ∕k0 2ffi; k0 nl > k xi kl;zi ; l I; II: (11) −jk0 kxi ∕k0 2 − n2l ; k0 nl < k xi The electric field and magnetic field in the structure region can be written as X S yi z exp−jkxi x; (12) E gy i
and H gx −j
X U xi z exp−jk xi x;
(13)
i
H gz −
1 ∂E gy X kxi S z exp−jkxi x; k0 yi jk 0 ∂x i
(14)
in which Sy WX0 c Xd c− ;
(15)
Ux VX0 c − Xd c− ;
(16)
where W and V are electromagnetic eigen vectors, X0 and Xd are diagonal matrices with the diagonal elements exp−k0 qm z and expk0 qm z − d , respectively, qm are the electromagnetic eigen values, c and c− are column vectors with unknown constants to be determined from the boundary conditions. The detail deduction and description of these parameters can be found in [16].
where Δk∕k0 λ∕Λ. In Eqs. (20) and (21), the 0th harmonic is extracted and replaced by the interpolation of the 1th harmonics to avoid the different order of the magnitude. With the concepts of RMADS and TMADS, the electric field and energy flow density distribution formed by the modes in the selected region can be intentionally analyzed. We analyze the formation mechanism of TIR around the rectangle, which is also known as MDR, and quantify the contribution of TIR among all the modes to the total field. The formation mechanism of the near field in the rectangle is initially analyzed with the ray-tracing method. It is well known that harmonics satisfied the condition of jkx ∕k0 j ≥ 1 will be total reflected by the horizontal interface, and harmonics in the region of jkx ∕k0 j ≤ 1.0638 will be total reflected by the vertical interface (fused silica). As we can see in Fig. 2(a), the normal incident plane waves are scattered into various high orders by the nonuniform parts of the structure. Considering the harmonics in the region of jk x ∕k 0 j < 1, whether the harmonics can be total reflected by the sidewall is dependent on the possibility of reaching the sidewalls.
C. Energy Flow Density
The time-averaged energy flow density is determined from Poynting’s theorem and is given by 1 hSi η0 ReE × H: (17) 2 With the definition of normalized incident electric amplitude, the average incident energy flow density is η0 ∕2. Thus it is convenient to set η0 ∕2 as the basic unit of energy flow density. The components of hSi can be expressed as S x ReE y H z ;
(18)
S z −ReE y H x :
(19)
D. Mode Amplitude Density Spectrum
As mentioned in Subsection 2.A, the distribution of the reflected and transmitted mode amplitude at reflected interface or transmitted interface can be obtained. However, we find that it is better to import the concept of density spectrum to eliminate the effect of the choice of convergence period and harmonic number. The new concepts are called reflected
Fig. 2. Resonance analysis with ray-tracing method: (a) incident wave scattered by the nonuniform structure, (b) harmonics directly go through the reflected and transmitted interfaces or are total reflected by the sidewalls, (c) harmonics are total reflected by both horizontal wall and sidewalls, (d) harmonics directly go through the sidewalls or are total reflected by horizontal wall.
Research Article To illustrate the schematic, a new model parameter kc is defined, which is the boundary for the harmonics to reach the sidewall, and dependent on the aspect ratio of the structure. Harmonics in the region of jk x ∕k 0 j ≤ k c ∕k 0 will directly go through the incident and transmitted interfaces and the resonance effect caused by the sidewalls will not happen, as the magenta arrows shown in Fig. 2(b). The near field is weak modulated by the diffraction effect. However, harmonics in the region of kc ∕k0 ≤ jkx ∕k0 j < 1 will be total reflected by the sidewall and run away from the incident and transmitted interfaces, which is corresponding a strong resonance enhancement effect for both field and energy flow, as the orange arrows shown in Fig. 2(b). The near field will be strong modulated by the modes in this region and the resonance effect under this mechanism is dependent on the height of the structure. If the harmonics satisfy the condition of 1 ≤ jkx ∕k0 j < 1.0638, they will be total reflected by both horizontal wall and vertical walls and finally run away through the transmitted interface, as the orange color shown in Fig. 2(c). However, the orders in this region are too high and the intensity of these modes is initially weak. In addition, the width of this region is so small that the modulation to the total field is relatively weak. For the extreme high orders in the region of jkx ∕k0 j > 1.0638, the harmonics will directly go through the sidewalls or are total reflected by the horizontal wall, as shown in Fig. 2(d). Therefore, the value of mode parameter kc is very important to investigate the resonance effect of the near field, which is used to delimit the condition for activating the TIR on the sidewalls. Based on the analysis with the ray-tracing method, k c ∕k0 with a value close to 1 means that most of the harmonics directly go through the reflected interface and transmitted interface, and the resonance effect caused by the sidewalls is very weak as well as the near-field modulation. The field enhancement mechanism is attributed to the normal diffraction effect and weak total reflection by the horizontal interface. In contrast, kc ∕k 0 with a value close to 0 means that most of the harmonics are total reflected by the sidewalls, and the resonance effect caused by the sidewalls is very strong as well as the near-field modulation. Although the resonance effect is initially analyzed with the phenomenological method, the possible results are further discussed with the rigorous method in the next section. 3. RESULTS AND DISCUSSION A. Convergence Analysis
To approximate the semi-infinite space problem with a large enough period Λ, the convergence dependence on the period is implemented under a large enough harmonic number of 301. From Fig. 3, it is observed that the specular reflectance of the equivalent grating converge to a finite value when the period reaches 30 times the length of the incident wavelength, which is about 10.65 μm. By choosing the period as 30λ, 40λ, 50λ, 60λ, 70λ, and 80λ, the convergence test of the harmonic number is implemented, as shown in Fig. 4. It is obvious that the specular reflectance converge to a finite value when the harmonic level is large enough. In addition, the stable value is more accurate and the difference between the curves is less, while the period becomes
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Fig. 3. Reflectance dependence on the period.
Fig. 4. Harmonic convergence test under the different periods.
larger. However, larger period means smaller mode interval and larger required harmonic number, which will significantly increase the amount of calculation. Thus it is advisable to choose the appropriate period value and harmonic number to decrease computation time without significant loss in the accuracy of the solution. B. Near-Field Calculation
Near-field calculations with the convergence condition of Λ 30λ, Λ 60λ, Λ 80λ, and M 117 are implemented with the rigorous method and then compared with that calculated by the FDTD method, as shown in Fig. 5. The maxima values of the electric field in Figs. 5(a)–5(c) are 1.7447, 1.7479, and 1.7476, respectively. Compared with that in Fig. 5(d), which is about 1.7493, the error between the calculation with Λ 30λ and the FDTD method is about 0.0046. The negligible difference among Figs. 5(a)–5(d) is attributed to the different simulation dimension and mesh accuracy in the FDTD method. Nevertheless, calculation under the condition of Λ 30λ and M 117 can obtain enough
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Fig. 5. Near-field jEj distribution calculated with rigorous method: (a) Λ 30λ, (b) Λ 60λ, (c) Λ 80λ, and (d) FDTD method.
accuracy with little consumption amount. It can be seen that the results obtained with the rigorous method under this condition agree well with that calculated by the FDTD method. From Fig. 5(a) it is found that the maximal electric value is about 1.745 and locates beneath the rectangle, which is similar to the focusing effect. An equation of P abs ωε 0 0 jEj2 ∕2 has been formulated to quantify the local absorption distribution inside the material [23,24], where ε 0 0 is the imaginary part of the permittivity and ω is the incident angle frequency. Comparing with the electric value (about 0.813) in the perfect substrate, the electric field enhancement factor in the structure is about 2.146. It means that the enhancement of local absorption beneath the rectangle is about 4.6, which will greatly increase the damage probability and decrease the damage threshold of the substrate. Moreover, the local absorption will be further enhanced in case that there are some absorptive defects beneath the rectangle. To further investigate the energy transport around the structure, energy flow density distribution is calculated according to Eqs. (6)–(19). The x-component and z-component of energy flow density are presented in Fig. 6, and the sign of energy flow
value is consistent with the positive direction of the coordinate axis. As seen in Fig. 6(a), the energy around the nanostructure is guided into the rectangle by the sidewall. From Fig. 6(b) we can see that energy inside the rectangle is amplified and focused beneath the rectangle. Therefore the nanoscale rectangle on the substrate behaves as an optical lens. However, the enhancement mechanism of the field and the focused effect should be discussed. C. Near-Field and Energy Flow Decomposition with RMADS
The mode amplitude distribution r i at reflected interface with different periods is analyzed. From Fig. 7(a) we can see that different configurations show similar variation but different intensity distribution. However, with the concept of RMADS, the difference among these periods is eliminated, as shown in Fig. 7(b). Thus RMADS can reflect the intrinsic properties as long as the convergence conditions are satisfied. As seen in Fig. 7(b), the modes that satisfied the condition of jkx ∕k0 j ≥ 1 are evanescent waves at the horizontal interface. The model parameter k c mentioned in Subsection 2.D is about 0.3333k 0 ,
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Fig. 6. Energy flow density distribution: (a) S x , (b) S z .
which is the first minimum point near the origin in Fig. 7(b). There is no resonance effect in the region of jkx ∕k0 j ≤ kc ∕k0 . Therefore, the contribution of the modes can be divided into four parts: jkx ∕k0 j ≤ kc ∕k0 , kc ∕k 0 ≤ jk x ∕k 0 j < 1, 1 ≤ jkx ∕k0 j < 1.0638, and jkx ∕k0 j > 1.0638, as mentioned in Subsection 2.D. The near field is decomposed according to Eqs. (5), (8), and (12) by selectively adding the modes in different parts respectively. The contribution of the modes in the region of jkx ∕k0 j ≤ k c ∕k0 to the near-field |E| and the component of energy flow is calculated and given in Fig. 8. In this region, the incident field and the corresponding 0th harmonic are
Fig. 7. Mode amplitude distribution at reflected interface: (a) reflected amplitude, (b) reflected amplitude density.
Fig. 8. The contribution of the modes in the region of jk x ∕k 0 j ≤ k c ∕k 0 to the (a) near-field jEj, (b) energy flow S x , and (c) S z .
considered. For the plane substrate, the maximum field in the air and substrate should be 1.187 and 0.813, and the z-component of energy flow should be −0.965 without the x-component. Figure 8(a) shows that the near field is weakly modulated by the modes in this region produced by the structure, and only the basic diffraction effect can be observed beneath the substrate. As seen from Fig. 8(b), it is obvious that the energy outside the structure near the root of the rectangle is transferred away. However, energy flow focus phenomenon is not observed according to the z-component of energy flow, as shown in Fig. 8(c). Therefore, energy can only be transferred to the higher orders in other parts according to the energy law. To further investigate the resonance effect and the energy transference, the contribution of the modes in the region of kc ∕k0 ≤ jkx ∕k0 j < 1 is also studied. The incident field is not considered in this region and it is found that near-field enhancement is very obvious, as shown in Fig. 9(a). The maximum value of the electric field is comparable to the incident field. The resonance effect caused by the TIR from the sidewalls is so strong that the maximum field is concentrated in the middle of the rectangle. Thus the modes in this region are the major source for the total field enhancement. With the analysis in Fig. 8, it is very obvious that the energy transferred from the modes in the region of jkx ∕k0 j ≤ kc ∕k 0 is guided into the rectangle by the root of the structure, as shown in Fig. 9(b), and then focused beneath the rectangle with the contribution of the modes in the region of kc ∕k 0 ≤ jkx ∕k0 j < 1, as we can see in Fig. 9(c). Therefore, it provides a possible direction to enhance or suppress the local field by optimizing the profile of the root of the structure. Since the near-field modulation and energy flow distribution are very weak for the modes in the region of
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Fig. 11. Modulation dependence on the height of structure: (a) mode parameter k c , (b) maximum squared electric field jEmax j2 .
Fig. 9. The contribution of the modes in the region of k c ∕k 0 ≤ jk x ∕k 0 j < 1 to the (a) near-field jEj, (b) energy flow S x , and (c) S z .
1 ≤ jkx ∕k0 j < 1.0638 and jkx ∕k0 j > 1.0638, we only give the near-field distribution caused by the modes in these regions, as shown in Figs. 10(a) and 10(b). The modes in these regions caused a weak modulation just below the horizontal interface of the rectangle. D. Influence of the Height on the Near-Field Modulation
The resonance effect and near-field modulation dependent on the height of the structure are studied and the results are given in Fig. 11. It can be seen that the value of kc ∕k 0 obtained from the calculation is close to 1.0 with a small height, which is consistent with the analysis in Subsection 2.D. In this case, the squared field intensity is about 0.86, which is close to the ideal flatten substrate. As the height of the structure increases, the value of k c ∕k0 decreases rapidly and then tends to be steady at a critical point, which is corresponding to a transition of physical mechanism. When the height of the structure exceeds the critical height, large amount of modes can reach the sidewalls and the resonance effect becomes stronger and stronger as well as the maximum squared electric field.
Fig. 10. The contribution of the modes in the region of (a) 1 ≤ jk x ∕k 0 j < 1.0638 and (b) jk x ∕k 0 j > 1.0638 to the near field.
The RMADS for different height is also investigated, as shown in Fig. 12. For the condition of h0 ∕λ 0.1, the resonance effect in the guided region is not observed and the model parameter k c is about 0.9333k 0 . When the height reaches about λ, a resonance appeared in the guided region. The resonance phenomenon becomes more complex and the model parameter kc is smaller. To further investigate the influence of the height, the near field for h0 ∕λ 0.1, h0 ∕λ 5, and h0 ∕λ 9 are calculated respectively. As a special case, the total field and energy flow for h0 ∕λ 0.1 are shown in Fig. 13. It can be seen that the near-field modulation is very weak due to lack of the resonance resulting from the sidewalls. All the components of the field are close to that of the ideal flatten substrate. Other extreme cases for h0 ∕λ 5 and h0 ∕λ 9 are also analyzed and shown in Fig. 14. The resonance effect resulting from the sidewalls is the dominated mechanism for the enhancement of the near field. It is obvious that the energy around the structure is guided in to the structure and trapped by the sidewall, then interfering and coupling with each other. From Figs. 14(c) and 14(f), we can see that the z-component of energy flow is trapped in the local point. The localization point
Fig. 12. RMADS for different height.
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14(d), and 14(f), which indicates a formation of the standingwave electric field with the waveguide effect inside the nanostructure. By further analyzing the x-component of the energy flow in Figs. 14(b) and 14(e), it can be found that the incident energy trapped by the rectangle is reallocated into two similar waveguide modes, which are corresponding to the two sidewalls. Both modes are total reflected twice in Fig. 14(b) and thrice in Fig. 14(e), respectively, by the sidewalls and propagate along the vertical direction. The resonance peak in Fig. 12 can be related to the intrinsic excitation of the waveguide modes inside the rectangle. Although the investigation in this paper is concentrated on the rectangle structure on the fused silica used for the highpower laser system, the analysis method is very convenient to be extended to other nanostructures with arbitrary shape and different materials. In addition, the concept of RMADS can reflect the resonance enhancement of the near field, which can be simply calculated with the rigorous method developed here. 4. CONCLUSION Fig. 13. (a) Near-field jEj distribution, energy flow density (b) S x and (c) S z calculated with rigorous method for h0 ∕λ 0.1.
of energy is more when the height increases, as well as the near field. It is very interesting that the localization points are equally spaced inside the rectangle, as shown in Figs. 14(a), 14(c),
Rigorous near-field and energy flow calculations are implemented for the structure with different height, and the results are compared with that obtained from FDTD simulation. The concept of mode amplitude density spectrum is developed to analyze the resonance effect caused by the nanostructure. With the new proposed model parameter the total field is decomposed into four regions, and the contribution of different regions to the near field is also studied. It is found that the field enhancement caused by the resonance effect is strongly dependent on the sidewalls of the structure. The energy trapping is originated from the root of the structure according to the energy flow analysis, although the local field enhancement is not obvious at that point. The analysis schematic here provides a possible direction to evaluate and optimize the profile of the most important section of the structure. The influence of the height on the near-field modulation and energy flow distribution is also discussed. The more and more complex resonance effect will result in more energy and field localization. REFERENCES
Fig. 14. (a) Near-field jEj distribution, energy flow density (b) S x and (c) S z calculated with rigorous method for h0 ∕λ 5. (d) Nearfield jEj distribution, energy flow density (e) S x and (f) S z calculated with rigorous method for h0 ∕λ 9.
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7. F. Y. Genin, A. Salleo, T. V. Pistor, and L. L. Chase, “Role of light intensification by cracks in optical breakdown on surfaces,” J. Opt. Soc. Am. A 18, 2607–2616 (2001). 8. J. R. Arias-Gonzalez and M. Nieto-Vesperinas, “Near-field distributions of resonant modes in small dielectric objects on flat surfaces,” Opt. Lett. 25, 782–784 (2000). 9. J. R. Arias-Gonzalez, P. C. Chaumet, and M. Nieto-Vesperinas, “Nanoparticles on surfaces: resonances and optical forces,” Proceedings of the International School of Physics "Enrico Fermi" 144, 405–424 (2001). 10. N. Bloember, “Role of cracks, pores, and absorbing inclusions on laser-induced damage threshold at surfaces of transparent dielectrics,” Appl. Opt. 12, 661–664 (1973). 11. J. F. Deford and M. R. Kozlowski, “Modeling of electric-field enhancement at nodular defects in dielectric mirror coatings,” Proc. SPIE 1848, 455–472 (1993). 12. V. E. Gruzdev and M. N. Libenson, “Field instability in high-quality dielectric sphere and disk: implication to the problem of laser induced damage of microsphere and microdisk resonators,” Proc. SPIE 3578, 784–791 (1999). 13. C. J. Stolz, M. D. Feit, and T. V. Pistor, “Laser intensification by spherical inclusions embedded within multilayer coatings,” Appl. Opt. 45, 1594–1601 (2006). 14. S. R. Qiu, J. E. Wolfe, A. M. Monterrosa, M. D. Feit, T. V. Pistor, and C. J. Stolz, “Searching for optimal mitigation geometries for laserresistant multilayer high-reflector coatings,” Appl. Opt. 50, C373– C381 (2011). 15. M. Chen, M. Li, W. Jiang, and Q. Xu, “Influence of period and amplitude of microwaviness on KH2PO4 crystal’s laser damage threshold,” J. Appl. Phys. 108, 043109 (2010).
Research Article 16. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995). 17. P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45, 1357–1374 (1998). 18. J. Cheng, M. Chen, W. Liao, H. Wang, Y. Xiao, and M. Li, “Fabrication of spherical mitigation pit on KH2PO4 crystal by micro-milling and modeling of its induced light intensification,” Opt. Express 21, 16799–16813 (2013). 19. J. Liu, W. L. Zhang, H. Cui, J. Sun, H. Li, K. Yi, and M. P. Zhu, “Study on high-reflective coatings of different designs at 532 nm,” Chin. Opt. Lett. 12, 083101 (2014). 20. J. F. Zhang, W. Liu, Z. H. Zhu, X. D. Yuan, and S. Q. Qin, “Strong field enhancement and light-matter interactions with all-dielectric metamaterials based on split bar resonators,” Opt. Express 22, 30889–30898 (2014). 21. Z. Xia, Y. Wu, R. Liu, P. Tang, and Z. Liang, “Light trapping enhancement with combined front metal nanoparticles and back diffraction gratings,” Chin. Opt. Lett. 11, S10503 (2013). 22. Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light-trapping in solar cells,” Proc. SPIE 7772, 77720Z (2010). 23. K. H. Brenner, “Aspects for calculating local absorption with the rigorous coupled-wave method,” Opt. Express 18, 10369–10376 (2010). 24. F. Ding, L. Mo, J. Zhu, and S. He, “Lithography-free, broadband, omnidirectional, and polarization-insensitive thin optical absorber,” Appl. Phys. Lett. 106, 061108 (2015).
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Research Article
Near-field enhancement of the nanostructure on the fused silica with rigorous method HU WANG,1,2 HONGJI QI,1,* BIN WANG,1,2 YANYAN CUI,1,2 YINGJIE CHAI,1,2 YUNXIA JIN,1 KUI YI,1 AND JIANDA SHAO1 1
Key Laboratory of Materials for High Power Laser, Shanghai Institute of Optics and Fine Mechanics, No. 390 Qinghe Road, Jiading District, Shanghai 201800, China 2 University of Chinese Academy of Sciences, Beijing 100049, China *Corresponding author: [email protected] Received 26 January 2015; revised 18 March 2015; accepted 14 April 2015; posted 14 April 2015 (Doc. ID 233192); published 4 May 2015
A rigorous electromagnetic method is developed to analyze the resonance effect of near field caused by nanoscale subsurface defects, which play a key role in describing absorption enhancement during laser–matter interaction for transparent dielectric materials. The total electric field calculated with this new method is consistent with the result of finite-difference time-domain simulation. The concept of mode amplitude density spectrum is developed to analyze the specific modes of the total field. A new mode parameter is proposed to demarcate the contribution of the resonance. The frequency space is divided into four parts and the resonance effect is analyzed as well as the contributions of different modes to the total field. The influence of the structure parameters on the near-field modulation and energy transference is also discussed. It is found that the enhancement mechanism of the nearfield and local absorption is the resonance effect caused by the total internal reflection on the sidewall of the nanostructure. In addition, the surrounding energy is mainly guided into the structure by the root of the structure via the energy flow analysis. © 2015 Optical Society of America OCIS codes: (260.6970) Total internal reflection; (050.1755) Computational electromagnetic methods; (140.3330) Laser damage; (260.5740) Resonance. http://dx.doi.org/10.1364/AO.54.004318
1. INTRODUCTION Laser–matter interactions are of great importance in various applications such as laser ablation [1,2] and laser-induced damage of optical components [3–5]. To better describe interaction mechanisms in the nanosecond regime for transparent dielectric materials, we must first understand how the irradiation is affected and absorbed by the nominal transparent dielectric materials. Resonance effect caused by subsurface nanostructure on the substrate has been recognized as a key factor of absorption enhancement in the solar cells [6] or describing field enhancement during laser–matter interaction for dielectric materials [7]. Electromagnetic resonances in the dielectric object can be classified into morphology-dependent resonances (MDRs) or whispering gallery modes, dependent on the structure’s morphology and incident frequency [8,9]. The MDR can greatly enhance the near field by confining the rays around the structure via a total internal reflection (TIR). Near-field enhancement around the defects embedded in optical components irradiated by high-power laser, has been widely investigated based on the electromagnetic field theory over the last two decades. For example, Bloembergen used the exact solution to quantify the light intensity enhancement 1559-128X/15/144318-09$15/0$15.00 © 2015 Optical Society of America
by assuming that the dimensions of the defects were much smaller than wavelength and ignored the reflection, refraction, diffraction, and interference at cracks and surfaces [10]. Deford assumed rotational symmetry to model the near-field enhancement around nodule defects in dielectric mirror coatings with AMOS computer code by using the finite-different timedomain (FDTD) technique [11]. Gruzdev and Libenson discussed the instability of high-power electromagnetic field resulted from resonances in nonabsorbing microinclusions embedded in the transparent dielectric [12]. Genin et al. modeled the light intensity enhancement factor caused by microscale planar and conical cracks with the FDTD method [7]. Their results indicated that the largest field enhancement could be reached when TIR occurs at both the crack and surface. Stolz and Roger investigated the near-field modulation by nodular defects existing in high reflective coatings with various geometry dimensions [13]. They also discussed the impacts of mitigation pits in dielectric multilayers by using an FDTD algorithm [14]. Unfortunately, problems in semi-infinite space commonly demand a large enough domain with periodic boundary conditions to approximate the infinite dimension. Algorithms such as FDTD or the finite element method require
Research Article a tedious preparation to test the convergence of both grid size and period length to find the optimized configuration to decrease computation time without significant loss in the accuracy of solution. Recently Chen et al. came up with a rigorous method called the Fourier modal method [15], which is very popular in surface relief grating analysis [16,17], to calculate near-field modulation caused by the subsurface defects and spherical mitigation pits on KDP crystal [18]. Their results indicated that the occurrence of TIR should be responsible for the largest light intensification inside the crystal. However, the enhancement mechanism by TIR was not determined and quantified. In this paper, we extend the existing state-of-the-art rigorous coupled wave analysis method commonly used in grating structure to solve the nanostructure embedded in semi-infinite space problems and analyze the local field enhancement and energy localization mechanism by taking advantage of grasping the information of all harmonics via this rigorous method. To the best of our knowledge, it is the first time to quantify the contribution of TIR to the near-field distribution around the nanostructure on the substrate. With the concept of the mode amplitude density spectrum mentioned in Subsection 2.D and a new model parameter, the near-field modulation mechanism caused by the nanostructure is discussed. The rigorous analysis method developed by us would be beneficial to analyze the local absorption and near-field enhancement in various applications, such as laser-induced damage of optical components [19], searching for optimal mitigation geometry for damage pits [14], near-field investigation [20], and light trapping in the solar cells [21,22].
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wave transmitted at the substrate–air interface. Here only the transverse electric polarization is considered, because the results for transverse magnetic polarization can be easily obtained by exchanging the role of the electric field and magnetic field in Maxwell equations. The grating structure with a large enough period Λ is used to approximate the semi-infinite space problem, as shown in Fig. 1. However, convergence test for period Λ and harmonic number M should be completed before calculating the near field. The normalized electric amplitude of the ith reflected wave in region I and transmitted wave in region II are denoted as r i and t i , respectively. The detail procedure to solve the values of r i and t i has been widely developed over the years [16,17,23]. In this paper, they are taken as a starting point for the near field and energy flow analysis. The assumed harmonic time dependence in our paper is expiωt. B. Near-Field Distribution
The normalized incident electric field with an incident angle θ is given by E in;y exp−jk x0 x − k I;z0 z;
(1) 2
1∕2
where k x0 k 0 nI sinθ and kI;z0 k0 nI 1 − sin θ . The ith tangential wave-vector component scattering from the nanostructure is given by kxi k 0 nI sin θ iλ∕Λ;
i 0; 1; 2; …
(2)
Because ε0 and μ0 differ by several orders of magnitude, E and H will differ by several orders of magnitude. It can be avoided by making the following change of variables: H η0 H;
(3)
1∕2
2. BASIC MODEL AND THEORY A. Basic Definition and Theory Background
To lay down the general framework for the main idea, we focus on a simple rectangle protuberance on an infinite fused silica substrate, as shown in Fig. 1. However, the rectangle is a natural brick for constructing arbitrary structure morphology. Both the rectangle and substrate have the same refractive index n 1.46. The irradiation wavelength λ is set as 355 nm, which is mostly concerned in the high-power laser system. The wavevector magnitude in free space is denoted as k 0 . The width and height of the rectangle are denoted as w0 and h0 , respectively, and they are initially equal to the incident wavelength. Nearfield distribution is produced with normally incident plane
is the vacuum admittance. where η0 ε0 ∕μ0 According to Maxwell equations, the magnetic field can be expressed as 1 H− ∇ × E: (4) jk0 Then the near field in region I can be represented as the sum of the incident field and all reflected harmonics [16]: X r i exp−jk xi x kI;zi z; (5) E I;y E in;y i
and H I;x
1 ∂E I;y jk 0 ∂z
X kI;zi kI;z0 E in;y − r exp−jk xi x kI;zi z; k0 k0 i i
H I;z −
1 ∂E I;y jk0 ∂x
X k xi kx0 E in;y r exp−jk xi x kI;zi z: k0 k0 i i
Also, the near field in region II can be given as X t i expf−jkxi x − kII;zi z h0 g; E II;y Fig. 1. Geometry for the rectangle nanostructure on substrate surface.
i
and
(6)
(7)
(8)
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H II;x
H II;z
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1 ∂E II;y X k II;zi t expf−jkxi x − k II;zi z h0 g; k0 i jk0 ∂z i
(9) X ∂E 1 k xi II;y − t expf−jk xi x − kII;zi z h0 g: k0 i jk 0 ∂x i
mode amplitude density spectrum (RMADS) and transmitted mode amplitude density spectrum (TMADS), respectively, which can be expressed as RMADS
jr i j ; Δk∕k 0
(20)
TMADS
jt i j ; Δk∕k 0
(21)
(10) The variable k l;zi can be determined by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k0pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2l − k xi ∕k0 2ffi; k0 nl > k xi kl;zi ; l I; II: (11) −jk0 kxi ∕k0 2 − n2l ; k0 nl < k xi The electric field and magnetic field in the structure region can be written as X S yi z exp−jkxi x; (12) E gy i
and H gx −j
X U xi z exp−jk xi x;
(13)
i
H gz −
1 ∂E gy X kxi S z exp−jkxi x; k0 yi jk 0 ∂x i
(14)
in which Sy WX0 c Xd c− ;
(15)
Ux VX0 c − Xd c− ;
(16)
where W and V are electromagnetic eigen vectors, X0 and Xd are diagonal matrices with the diagonal elements exp−k0 qm z and expk0 qm z − d , respectively, qm are the electromagnetic eigen values, c and c− are column vectors with unknown constants to be determined from the boundary conditions. The detail deduction and description of these parameters can be found in [16].
where Δk∕k0 λ∕Λ. In Eqs. (20) and (21), the 0th harmonic is extracted and replaced by the interpolation of the 1th harmonics to avoid the different order of the magnitude. With the concepts of RMADS and TMADS, the electric field and energy flow density distribution formed by the modes in the selected region can be intentionally analyzed. We analyze the formation mechanism of TIR around the rectangle, which is also known as MDR, and quantify the contribution of TIR among all the modes to the total field. The formation mechanism of the near field in the rectangle is initially analyzed with the ray-tracing method. It is well known that harmonics satisfied the condition of jkx ∕k0 j ≥ 1 will be total reflected by the horizontal interface, and harmonics in the region of jkx ∕k0 j ≤ 1.0638 will be total reflected by the vertical interface (fused silica). As we can see in Fig. 2(a), the normal incident plane waves are scattered into various high orders by the nonuniform parts of the structure. Considering the harmonics in the region of jk x ∕k 0 j < 1, whether the harmonics can be total reflected by the sidewall is dependent on the possibility of reaching the sidewalls.
C. Energy Flow Density
The time-averaged energy flow density is determined from Poynting’s theorem and is given by 1 hSi η0 ReE × H: (17) 2 With the definition of normalized incident electric amplitude, the average incident energy flow density is η0 ∕2. Thus it is convenient to set η0 ∕2 as the basic unit of energy flow density. The components of hSi can be expressed as S x ReE y H z ;
(18)
S z −ReE y H x :
(19)
D. Mode Amplitude Density Spectrum
As mentioned in Subsection 2.A, the distribution of the reflected and transmitted mode amplitude at reflected interface or transmitted interface can be obtained. However, we find that it is better to import the concept of density spectrum to eliminate the effect of the choice of convergence period and harmonic number. The new concepts are called reflected
Fig. 2. Resonance analysis with ray-tracing method: (a) incident wave scattered by the nonuniform structure, (b) harmonics directly go through the reflected and transmitted interfaces or are total reflected by the sidewalls, (c) harmonics are total reflected by both horizontal wall and sidewalls, (d) harmonics directly go through the sidewalls or are total reflected by horizontal wall.
Research Article To illustrate the schematic, a new model parameter kc is defined, which is the boundary for the harmonics to reach the sidewall, and dependent on the aspect ratio of the structure. Harmonics in the region of jk x ∕k 0 j ≤ k c ∕k 0 will directly go through the incident and transmitted interfaces and the resonance effect caused by the sidewalls will not happen, as the magenta arrows shown in Fig. 2(b). The near field is weak modulated by the diffraction effect. However, harmonics in the region of kc ∕k0 ≤ jkx ∕k0 j < 1 will be total reflected by the sidewall and run away from the incident and transmitted interfaces, which is corresponding a strong resonance enhancement effect for both field and energy flow, as the orange arrows shown in Fig. 2(b). The near field will be strong modulated by the modes in this region and the resonance effect under this mechanism is dependent on the height of the structure. If the harmonics satisfy the condition of 1 ≤ jkx ∕k0 j < 1.0638, they will be total reflected by both horizontal wall and vertical walls and finally run away through the transmitted interface, as the orange color shown in Fig. 2(c). However, the orders in this region are too high and the intensity of these modes is initially weak. In addition, the width of this region is so small that the modulation to the total field is relatively weak. For the extreme high orders in the region of jkx ∕k0 j > 1.0638, the harmonics will directly go through the sidewalls or are total reflected by the horizontal wall, as shown in Fig. 2(d). Therefore, the value of mode parameter kc is very important to investigate the resonance effect of the near field, which is used to delimit the condition for activating the TIR on the sidewalls. Based on the analysis with the ray-tracing method, k c ∕k0 with a value close to 1 means that most of the harmonics directly go through the reflected interface and transmitted interface, and the resonance effect caused by the sidewalls is very weak as well as the near-field modulation. The field enhancement mechanism is attributed to the normal diffraction effect and weak total reflection by the horizontal interface. In contrast, kc ∕k 0 with a value close to 0 means that most of the harmonics are total reflected by the sidewalls, and the resonance effect caused by the sidewalls is very strong as well as the near-field modulation. Although the resonance effect is initially analyzed with the phenomenological method, the possible results are further discussed with the rigorous method in the next section. 3. RESULTS AND DISCUSSION A. Convergence Analysis
To approximate the semi-infinite space problem with a large enough period Λ, the convergence dependence on the period is implemented under a large enough harmonic number of 301. From Fig. 3, it is observed that the specular reflectance of the equivalent grating converge to a finite value when the period reaches 30 times the length of the incident wavelength, which is about 10.65 μm. By choosing the period as 30λ, 40λ, 50λ, 60λ, 70λ, and 80λ, the convergence test of the harmonic number is implemented, as shown in Fig. 4. It is obvious that the specular reflectance converge to a finite value when the harmonic level is large enough. In addition, the stable value is more accurate and the difference between the curves is less, while the period becomes
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Fig. 3. Reflectance dependence on the period.
Fig. 4. Harmonic convergence test under the different periods.
larger. However, larger period means smaller mode interval and larger required harmonic number, which will significantly increase the amount of calculation. Thus it is advisable to choose the appropriate period value and harmonic number to decrease computation time without significant loss in the accuracy of the solution. B. Near-Field Calculation
Near-field calculations with the convergence condition of Λ 30λ, Λ 60λ, Λ 80λ, and M 117 are implemented with the rigorous method and then compared with that calculated by the FDTD method, as shown in Fig. 5. The maxima values of the electric field in Figs. 5(a)–5(c) are 1.7447, 1.7479, and 1.7476, respectively. Compared with that in Fig. 5(d), which is about 1.7493, the error between the calculation with Λ 30λ and the FDTD method is about 0.0046. The negligible difference among Figs. 5(a)–5(d) is attributed to the different simulation dimension and mesh accuracy in the FDTD method. Nevertheless, calculation under the condition of Λ 30λ and M 117 can obtain enough
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Fig. 5. Near-field jEj distribution calculated with rigorous method: (a) Λ 30λ, (b) Λ 60λ, (c) Λ 80λ, and (d) FDTD method.
accuracy with little consumption amount. It can be seen that the results obtained with the rigorous method under this condition agree well with that calculated by the FDTD method. From Fig. 5(a) it is found that the maximal electric value is about 1.745 and locates beneath the rectangle, which is similar to the focusing effect. An equation of P abs ωε 0 0 jEj2 ∕2 has been formulated to quantify the local absorption distribution inside the material [23,24], where ε 0 0 is the imaginary part of the permittivity and ω is the incident angle frequency. Comparing with the electric value (about 0.813) in the perfect substrate, the electric field enhancement factor in the structure is about 2.146. It means that the enhancement of local absorption beneath the rectangle is about 4.6, which will greatly increase the damage probability and decrease the damage threshold of the substrate. Moreover, the local absorption will be further enhanced in case that there are some absorptive defects beneath the rectangle. To further investigate the energy transport around the structure, energy flow density distribution is calculated according to Eqs. (6)–(19). The x-component and z-component of energy flow density are presented in Fig. 6, and the sign of energy flow
value is consistent with the positive direction of the coordinate axis. As seen in Fig. 6(a), the energy around the nanostructure is guided into the rectangle by the sidewall. From Fig. 6(b) we can see that energy inside the rectangle is amplified and focused beneath the rectangle. Therefore the nanoscale rectangle on the substrate behaves as an optical lens. However, the enhancement mechanism of the field and the focused effect should be discussed. C. Near-Field and Energy Flow Decomposition with RMADS
The mode amplitude distribution r i at reflected interface with different periods is analyzed. From Fig. 7(a) we can see that different configurations show similar variation but different intensity distribution. However, with the concept of RMADS, the difference among these periods is eliminated, as shown in Fig. 7(b). Thus RMADS can reflect the intrinsic properties as long as the convergence conditions are satisfied. As seen in Fig. 7(b), the modes that satisfied the condition of jkx ∕k0 j ≥ 1 are evanescent waves at the horizontal interface. The model parameter k c mentioned in Subsection 2.D is about 0.3333k 0 ,
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Fig. 6. Energy flow density distribution: (a) S x , (b) S z .
which is the first minimum point near the origin in Fig. 7(b). There is no resonance effect in the region of jkx ∕k0 j ≤ kc ∕k0 . Therefore, the contribution of the modes can be divided into four parts: jkx ∕k0 j ≤ kc ∕k0 , kc ∕k 0 ≤ jk x ∕k 0 j < 1, 1 ≤ jkx ∕k0 j < 1.0638, and jkx ∕k0 j > 1.0638, as mentioned in Subsection 2.D. The near field is decomposed according to Eqs. (5), (8), and (12) by selectively adding the modes in different parts respectively. The contribution of the modes in the region of jkx ∕k0 j ≤ k c ∕k0 to the near-field |E| and the component of energy flow is calculated and given in Fig. 8. In this region, the incident field and the corresponding 0th harmonic are
Fig. 7. Mode amplitude distribution at reflected interface: (a) reflected amplitude, (b) reflected amplitude density.
Fig. 8. The contribution of the modes in the region of jk x ∕k 0 j ≤ k c ∕k 0 to the (a) near-field jEj, (b) energy flow S x , and (c) S z .
considered. For the plane substrate, the maximum field in the air and substrate should be 1.187 and 0.813, and the z-component of energy flow should be −0.965 without the x-component. Figure 8(a) shows that the near field is weakly modulated by the modes in this region produced by the structure, and only the basic diffraction effect can be observed beneath the substrate. As seen from Fig. 8(b), it is obvious that the energy outside the structure near the root of the rectangle is transferred away. However, energy flow focus phenomenon is not observed according to the z-component of energy flow, as shown in Fig. 8(c). Therefore, energy can only be transferred to the higher orders in other parts according to the energy law. To further investigate the resonance effect and the energy transference, the contribution of the modes in the region of kc ∕k0 ≤ jkx ∕k0 j < 1 is also studied. The incident field is not considered in this region and it is found that near-field enhancement is very obvious, as shown in Fig. 9(a). The maximum value of the electric field is comparable to the incident field. The resonance effect caused by the TIR from the sidewalls is so strong that the maximum field is concentrated in the middle of the rectangle. Thus the modes in this region are the major source for the total field enhancement. With the analysis in Fig. 8, it is very obvious that the energy transferred from the modes in the region of jkx ∕k0 j ≤ kc ∕k 0 is guided into the rectangle by the root of the structure, as shown in Fig. 9(b), and then focused beneath the rectangle with the contribution of the modes in the region of kc ∕k 0 ≤ jkx ∕k0 j < 1, as we can see in Fig. 9(c). Therefore, it provides a possible direction to enhance or suppress the local field by optimizing the profile of the root of the structure. Since the near-field modulation and energy flow distribution are very weak for the modes in the region of
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Fig. 11. Modulation dependence on the height of structure: (a) mode parameter k c , (b) maximum squared electric field jEmax j2 .
Fig. 9. The contribution of the modes in the region of k c ∕k 0 ≤ jk x ∕k 0 j < 1 to the (a) near-field jEj, (b) energy flow S x , and (c) S z .
1 ≤ jkx ∕k0 j < 1.0638 and jkx ∕k0 j > 1.0638, we only give the near-field distribution caused by the modes in these regions, as shown in Figs. 10(a) and 10(b). The modes in these regions caused a weak modulation just below the horizontal interface of the rectangle. D. Influence of the Height on the Near-Field Modulation
The resonance effect and near-field modulation dependent on the height of the structure are studied and the results are given in Fig. 11. It can be seen that the value of kc ∕k 0 obtained from the calculation is close to 1.0 with a small height, which is consistent with the analysis in Subsection 2.D. In this case, the squared field intensity is about 0.86, which is close to the ideal flatten substrate. As the height of the structure increases, the value of k c ∕k0 decreases rapidly and then tends to be steady at a critical point, which is corresponding to a transition of physical mechanism. When the height of the structure exceeds the critical height, large amount of modes can reach the sidewalls and the resonance effect becomes stronger and stronger as well as the maximum squared electric field.
Fig. 10. The contribution of the modes in the region of (a) 1 ≤ jk x ∕k 0 j < 1.0638 and (b) jk x ∕k 0 j > 1.0638 to the near field.
The RMADS for different height is also investigated, as shown in Fig. 12. For the condition of h0 ∕λ 0.1, the resonance effect in the guided region is not observed and the model parameter k c is about 0.9333k 0 . When the height reaches about λ, a resonance appeared in the guided region. The resonance phenomenon becomes more complex and the model parameter kc is smaller. To further investigate the influence of the height, the near field for h0 ∕λ 0.1, h0 ∕λ 5, and h0 ∕λ 9 are calculated respectively. As a special case, the total field and energy flow for h0 ∕λ 0.1 are shown in Fig. 13. It can be seen that the near-field modulation is very weak due to lack of the resonance resulting from the sidewalls. All the components of the field are close to that of the ideal flatten substrate. Other extreme cases for h0 ∕λ 5 and h0 ∕λ 9 are also analyzed and shown in Fig. 14. The resonance effect resulting from the sidewalls is the dominated mechanism for the enhancement of the near field. It is obvious that the energy around the structure is guided in to the structure and trapped by the sidewall, then interfering and coupling with each other. From Figs. 14(c) and 14(f), we can see that the z-component of energy flow is trapped in the local point. The localization point
Fig. 12. RMADS for different height.
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14(d), and 14(f), which indicates a formation of the standingwave electric field with the waveguide effect inside the nanostructure. By further analyzing the x-component of the energy flow in Figs. 14(b) and 14(e), it can be found that the incident energy trapped by the rectangle is reallocated into two similar waveguide modes, which are corresponding to the two sidewalls. Both modes are total reflected twice in Fig. 14(b) and thrice in Fig. 14(e), respectively, by the sidewalls and propagate along the vertical direction. The resonance peak in Fig. 12 can be related to the intrinsic excitation of the waveguide modes inside the rectangle. Although the investigation in this paper is concentrated on the rectangle structure on the fused silica used for the highpower laser system, the analysis method is very convenient to be extended to other nanostructures with arbitrary shape and different materials. In addition, the concept of RMADS can reflect the resonance enhancement of the near field, which can be simply calculated with the rigorous method developed here. 4. CONCLUSION Fig. 13. (a) Near-field jEj distribution, energy flow density (b) S x and (c) S z calculated with rigorous method for h0 ∕λ 0.1.
of energy is more when the height increases, as well as the near field. It is very interesting that the localization points are equally spaced inside the rectangle, as shown in Figs. 14(a), 14(c),
Rigorous near-field and energy flow calculations are implemented for the structure with different height, and the results are compared with that obtained from FDTD simulation. The concept of mode amplitude density spectrum is developed to analyze the resonance effect caused by the nanostructure. With the new proposed model parameter the total field is decomposed into four regions, and the contribution of different regions to the near field is also studied. It is found that the field enhancement caused by the resonance effect is strongly dependent on the sidewalls of the structure. The energy trapping is originated from the root of the structure according to the energy flow analysis, although the local field enhancement is not obvious at that point. The analysis schematic here provides a possible direction to evaluate and optimize the profile of the most important section of the structure. The influence of the height on the near-field modulation and energy flow distribution is also discussed. The more and more complex resonance effect will result in more energy and field localization. REFERENCES
Fig. 14. (a) Near-field jEj distribution, energy flow density (b) S x and (c) S z calculated with rigorous method for h0 ∕λ 5. (d) Nearfield jEj distribution, energy flow density (e) S x and (f) S z calculated with rigorous method for h0 ∕λ 9.
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