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OPTICS LETTERS / Vol. 38, No. 23 / December 1, 2013
Increase of spatial coherence by subwavelength metallic gratings Toni Saastamoinen* and Hanna Lajunen Institute of Photonics, University of Eastern Finland, FI-80101 Joensuu, Finland *Corresponding author: [email protected] Received July 5, 2013; revised October 16, 2013; accepted October 22, 2013; posted October 23, 2013 (Doc. ID 193469); published November 21, 2013 We study the coherence changes in partially coherent beams transmitted through binary metallic gratings. The interaction of Gaussian Schell-model beams with grating structures supporting surface plasmons is rigorously modeled using the Fourier modal method and the coherent mode representation of partially coherent fields. Our numerical results show that, by choosing suitable parameters for the grating, the degree of coherence of the beam can be significantly increased. The studied approach offers new possibilities to alter the coherence properties of fields using nanophotonic components. © 2013 Optical Society of America OCIS codes: (030.1640) Coherence; (050.6624) Subwavelength structures; (240.6680) Surface plasmons. http://dx.doi.org/10.1364/OL.38.005000
The escalating ability to fabricate elaborate metallic nanostructures has opened up new ways to manipulate light based on plasmonic effects, and has led to a wide range of novel applications, such as nanoscopic field confinement, waveguiding, and detection [1,2]. One of the key discoveries in the field of plasmonics is the extraordinary transmission of light through metallic film with subwavelength holes [3], which has inspired numerous studies on the transmission resonances of electromagnetic fields incident on periodic slit and hole arrays [4]. On the whole, these works have shown that it is possible to tailor the spectral transmission properties of the metallic subwavelength structures by varying their geometrical parameters. While most studies on metallic subwavelength structures concentrate on the changes of the spectral intensity in transmission, there are fewer studies considering the effects of similar structures on the coherence properties of electromagnetic fields. However, it has been observed that surface plasmon polaritons generated in periodic microstructures can also enhance the coherence of light emitted by thermal sources [5]. Surface plasmons have also been experimentally shown to increase coherence between two independent optical sources [6]. Furthermore, the effect of plasmons on field correlations between specific points has been theoretically studied for systems consisting of two and three slits [7,8], and finite hole arrays [9] in metallic films. Previous theoretical studies have also considered also the effect of plasmon resonances in metallic nanocylinders on the coherence and polarization characteristics of the nearfield [10] and the influence of the state of coherence on plasmon excitation in thin metallic slabs and nanowires [11]. The coherence properties directly affect the characteristics of electromagnetic fields, such as their behavior upon propagation [12]; thus, the possibilities to flexibly control them could be valuable for many types of applications. The existing schemes for modifying the field coherence are typically based on either intricate filtering systems for increasing the coherence of incoherent light, or decreasing the correlations by randomizing the phase of the field [13–16]. The potential of surface plasmons to 0146-9592/13/235000-04$15.00/0
increase the field correlations could allow the manipulation of the coherence properties with small and simple nanophotonic components instead of complicated optical systems. Moreover, the local nature of plasmons could also allow spatially varying control of the coherence properties leading to the generation of new types of nonuniformly correlated fields [17,18]. In this Letter, we use rigorous numerical simulations to conduct a theoretical examination of coherence changes induced in periodic metallic gratings with narrow slits. Our aim is to find the optimal parameters to exploit the surface plasmon resonances in metallic gratings and increase the coherence of optical beams, which typically is not possible with single optical elements. In contrast to previous studies that concentrated on the field correlations between specific points in systems consisting of a finite number of slits or holes [7–9], we consider the changes of the coherence functions and the effective degree of coherence of an incident Gaussian Schell-model (GSM) beam transmitted through periodic linear grating structures. In the following, we will first introduce the basic concepts and the methods used in the numerical simulations. The relation of the grating parameters to the coherence properties are then studied by numerical simulations and the changes in the coherence functions are illustrated using a GSM beam as an example. Let us consider a partially coherent field with a monochromatic scalar component Ur; ω that gives the field realization U in the spatial position r at the frequency ω in the space–frequency domain. The spatial coherence properties of the field are described by the cross-spectral density W r1 ; r1 ; ω hU r1 ; ωUr2 ; ωi;
(1)
that measures the field correlations between two spatial points [12]. The spectral density of the field is given by Sr; ω W r; r; ω and the spectral degree of coherence of the field is defined as W r1 ; r2 ; ω : μr1 ; r2 ; ω p Sr1 ; ωSr2 ; ω © 2013 Optical Society of America
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Furthermore, the global coherence properties of the field can be described by the effective degree of spectral coherence [19,20] Sr1 ; ωSr2 ; ωjμr1 ; r2 ; ωj2 d3 r 1 d3 r 2 RR ; (3) 3 3 D Sr1 ; ωSr2 ; ωd r 1 d r 2
that gives the average spatial coherence weighted by the spectral density in the domain D for a single frequency. To facilitate the numerical analysis of the interaction of the partially coherent field and the grating structure, the cross-spectral density can be expressed as an incoherent sum of fully coherent modes [12] W r1 ; r2 ; ω
X λn ωϕn r1 ; ωϕn r2 ; ω;
(4)
n
where λω are the eigenvalues and ϕr; ω are the eigenvectors of the Fredholm integral equation of the second kind: Z D
W r1 ; r2 ; ωϕn r1 ; ωd3 r 1 λn ωϕn r2 ; ω:
(5)
The cross-spectral density function is non-negative definite and Hermitian, which, in turn, ensures that the eigenvalues λ are real and non-negative. Moreover, the eigenfunctions are orthonormal in the sense Z D
ϕm r; ωϕn r; ωd3 r δmn ;
(6)
where δmn denotes the Kronecker delta symbol. The eigenvalues may be interpreted as energy distribution between the distinct modes represented by the eigenfunctions. Employing the coherent mode representation, we can propagate each coherent mode individually and then sum the resulting diffracted modes to obtain the crossspectral density function of the scattered partially coherent field. In this work, we apply Fourier modal method (FMM) [21] in the determination of the diffraction amplitudes of each coherent mode. If the incident angular spectrum of the field is known, we can solve the reflected and transmitted angular spectra numerically using FMM. In the case of partially coherent fields, we simply calculate the coherent modes ϕn r; ω of the incident field and the angular spectrum of each mode. The coherent modes behind the grating can then be calculated separately through the transmitted angular spectrum of each mode solved numerically using FMM. The output cross-spectral density is obtained by summing the modes according to Eq. (4), weighted by the eigenvalues λn of the input field. Finally, the effective degree of coherence of the output beam can be calculated using Eq. (3). A more detailed description of the computational approach can be found, e.g., in [22] and [23]. The studied structure, shown in Fig. 1, consists of a binary gold grating on a fused silica substrate. In our simulations, we have used the refractive index data for gold and fused silica from [24] and [25], respectively. Since we are interested in plasmonic effects, we focus on
Fig. 1. Linear binary gold grating with period d, height h, and slit width c on a fused silica substrate.
considering TM polarization where the y-component of the magnetic field specifies completely the properties of the electromagnetic field. The cross-spectral density W that describes completely the coherence properties of the field is then actually the y; y-component of the magnetic cross-spectral density tensor [12]. Furthermore, we use the typical assumption that the incident field is a GSM beam, so its cross-spectral density is of the form 2 x x2 x − x 2 W x1 ;x2 ; ω W 0 exp − 1 2 2 exp − 1 2 2 ; (7) w0 2σ 0 where W 0 is constant, w0 is the beam half-width of the beam, and σ 0 is the coherence length [12]. Now we can proceed to study the effect of the grating parameters on the transmission and coherence changes of a partially coherent beam. We consider a GSM beam with w0 10 μm, σ 0 3 μm, at the wavelength λ 633 nm. The effective degree of coherence of the input beam μ¯ i ≈ 0.54. Figure 2(a) shows the transmission intensity of the beam energy and (b) the numerically calculated effective degree of coherence of the beam transmitted through a grating as a function of the period d and the height h. The slit width is kept as constant c 130 nm and the output beam is analyzed at the distance of 3 μm behind the grating. The effective degree of coherence is clearly increased with certain parameter combinations. The highest value μ¯ o ≈ 0.73 is obtained (a)
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OPTICS LETTERS / Vol. 38, No. 23 / December 1, 2013
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where ε1 and ε2 are the permittivities of the materials and k0 is the wave vector in free space. Typically, transmission minima occur when the period of the grating equals the wavelength of the surface plasmons corresponding to a homogeneous interface as λsp Rfksp g [4]. This is seen in Fig. 2 for the period d λsp 390 nm, while the transmission and coherence maxima are found at the edge of this plasmonic bandgap.
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Figure 3 further illustrates the spectral dependence of the transmission and coherence changes for different grating periods compared to the plasmon wavelength. Other grating parameters used in the simulation are h 81 nm and c 130 nm. It is seen that at all wavelengths the maxima of both the transmission and the effective degree of coherence are closely located beside the plasmon curve calculated for a homogeneous interface. On the other hand, it should also be remembered that the GSM beam has a finite angular spectrum and the field components incident from different angles experience slightly different plasmon-related transmission resonances. Thus, the angular filtering performed by the grating may also play a role in the coherence changes in the same way as in some previous works considering coherence increases caused by different grating structures [5,22]. While the effective degree of coherence of the GSM beam chosen as an example is clearly increased with certain grating parameters, we still need to consider the changes in the coherence functions to get a more complete picture of the effect. Figure 4 compares the coherence functions of the original incident GSM beam with those of the beam transmitted through a grating with parameters d 424 nm, h 81 nm, and c 130 nm at wavelength λ 633 nm. The coherence functions are calculated at the distance of 3 μm behind the grating. It is seen that the cross-spectral density and the degree of coherence are widened behind the grating, and the widening increases toward the edges of the beam. Following the simple model in [9], it may be assumed that, with suitable grating parameters, part of the incident field is coupled to surface plasmons that spread over several adjacent grating slits; thus, the coherence of
x [nm]
when the period of the grating d 424 nm and the height h 81 nm. Also, the transmission of the beam has local maxima in the same parameter areas. In addition, the transmission increases when the grating period is reduced, because the slit width is kept constant and the relative amount of metal is decreasing. However, this effect does not increase the coherence. Furthermore, the maximum areas are periodically repeated when the height of the grating is increased by constant amounts. This phenomenon is related to the vertical (or cavity) resonances occurring in the slits [4]. On the other hand, some parameter combinations also show a slight decrease in the effective degree of coherence. However, in this example, the effect is relatively weak, especially in the areas with significant transmission. In the earlier studies on the coherence changes in finite slit and hole arrays [8,9], the enhancement of coherence has been explained by the coupling of the incident field to the plasmons propagating on the surface of the structure, which increases the correlations between separate points. Assuming a homogeneous interface between the gold and silica, we can calculate the wave vector of the propagating surface plasmons from the dispersion relation [1]
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0.1 0.2 0.3 0.4 0.5 z [mm]
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Fig. 4. (a) Cross-spectral density, (c) the degree of coherence, and (e) the spreading upon propagation of the incident GSM beam, and the corresponding properties, (b), (d), and (f), respectively, of the beam transmitted through the grating.
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the transmitted field increases as they are coupled out. The plasmons excited at the low intensity edges of the beam and that propagate toward the beam center, are weaker and, therefore, contribute less to the increase of coherence than the plasmons traveling from the beam center toward the edges. Nevertheless, the coherence functions are not strongly distorted from the original GSM shapes and the intensity distribution of the transmitted beam remains nearly Gaussian. Figure 4 illustrates the spreading of the beams propagating in air. As expected, the beam transmitted through the grating spreads significantly less upon propagation due to its higher degree of coherence. From a practical point of view, gratings with suitable parameters for increased coherence can be fabricated, e.g., with standard electron beam lithography. While we have demonstrated results for only one GSM beam with specific parameters, our further numerical simulations imply similar behaviors for a wide range of beams with different widths and degrees of coherence. The main drawback for applications is that the transmission of the beam through the metallic grating is relatively low despite the transmission resonances of the structure. Moreover the ideal parameters for coherence and transmission maxima do not exactly coincide, as in the previous examples. Thus, depending on the objectives, one should choose parameter combinations that give reasonable values to both the transmission of the beam and the increase of its coherence. It should also be remembered that linear grating structures work only with TM-polarized light. Three-dimensional periodic structures, such as arrays of holes or nanoantennas, could possibly produce similar effects for both polarization components simultaneously. In conclusion, we have studied the coherence changes in a GSM beam transmitted through a binary linear gold grating with narrow slits. It has been shown that, with specific combinations of the grating parameters, the effective degree of coherence is significantly increased. The changes can also be seen in the coherence functions and the propagation characteristics of the beam. Our results imply that metallic subwavelength gratings with unique resonance and plasmonic properties offer a novel way for manipulating the coherence properties of partially coherent fields.
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This research has been supported by the strategic funding of the University of Eastern Finland. References 1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003). 2. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, Nat. Mater. 9, 193 (2010). 3. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, Nature 391, 667 (1997). 4. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, Rev. Mod. Phys. 82, 729 (2010). 5. J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen, Nature 416, 61 (2002). 6. N. Kuzmin, G. W. ‘t Hooft, E. R. Eliel, G. Gbur, H. F. Schouten, and T. D. Visser, Opt. Lett. 32, 445 (2007). 7. C. H. Gan, G. Gbur, and T. D. Visser, Phys. Rev. Lett. 98, 043908 (2007). 8. C. H. Gan and G. Gbur, Plasmonics 3, 111 (2008). 9. C. H. Gan, Y. Gu, T. D. Visser, and G. Gbur, Plasmonics 7, 313 (2012). 10. J. Lindberg, T. Setälä, M. Kaivola, and A. T. Friberg, J. Opt. Soc. Am. A 23, 1349 (2006). 11. H. J. Hyvärinen, J. Turunen, and P. Vahimaa, Appl. Phys. B, 101, 273 (2010). 12. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995). 13. P. De Santis, F. Gori, G. Guattari, and C. Palma, J. Opt. Soc. Am. A 3, 1258 (1986). 14. D. Mendlovic, G. Shabtay, and A. W. Lohmann, Opt. Lett. 24, 361 (1999). 15. T. Shirai and E. Wolf, J. Opt. Soc. Am. A 21, 1907 (2004). 16. A. S. Ostrovsky, G. Martínez-Niconoff, V. Arrizón, P. MartínezVara, M. A. Olvera-Santamaría, and C. Rickenstorff-Parrao, Opt. Express 17, 5257 (2009). 17. H. Lajunen and T. Saastamoinen, Opt. Lett. 36, 4104 (2011). 18. Z. Tong and O. Korotkova, J. Opt. Soc. Am. A 29, 2154 (2012). 19. M. J. Bastiaans, J. Opt. Soc. Am. A 1, 711 (1984). 20. P. Vahimaa and J. Tervo, J. Opt. A 6, S41 (2004). 21. J. Turunen, in Micro-Optics, Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, 1997). 22. P. Vahimaa and J. Turunen, J. Opt. Soc. Am. A 14, 54 (1997). 23. J. Tervo, H. J. Hyvärinen, T. Saastamoinen, I. Vartiainen, and J. Turunen, Proc. SPIE 8171, 81710F (2011). 24. D. R. Lide, CRC Handbook of Chemistry and Physics, 64th ed. (CRC Press, 1984), p. E-370. 25. http://www.crystran.co.uk/optical‑materials/silica‑glass‑sio2.
OPTICS LETTERS / Vol. 38, No. 23 / December 1, 2013
Increase of spatial coherence by subwavelength metallic gratings Toni Saastamoinen* and Hanna Lajunen Institute of Photonics, University of Eastern Finland, FI-80101 Joensuu, Finland *Corresponding author: [email protected] Received July 5, 2013; revised October 16, 2013; accepted October 22, 2013; posted October 23, 2013 (Doc. ID 193469); published November 21, 2013 We study the coherence changes in partially coherent beams transmitted through binary metallic gratings. The interaction of Gaussian Schell-model beams with grating structures supporting surface plasmons is rigorously modeled using the Fourier modal method and the coherent mode representation of partially coherent fields. Our numerical results show that, by choosing suitable parameters for the grating, the degree of coherence of the beam can be significantly increased. The studied approach offers new possibilities to alter the coherence properties of fields using nanophotonic components. © 2013 Optical Society of America OCIS codes: (030.1640) Coherence; (050.6624) Subwavelength structures; (240.6680) Surface plasmons. http://dx.doi.org/10.1364/OL.38.005000
The escalating ability to fabricate elaborate metallic nanostructures has opened up new ways to manipulate light based on plasmonic effects, and has led to a wide range of novel applications, such as nanoscopic field confinement, waveguiding, and detection [1,2]. One of the key discoveries in the field of plasmonics is the extraordinary transmission of light through metallic film with subwavelength holes [3], which has inspired numerous studies on the transmission resonances of electromagnetic fields incident on periodic slit and hole arrays [4]. On the whole, these works have shown that it is possible to tailor the spectral transmission properties of the metallic subwavelength structures by varying their geometrical parameters. While most studies on metallic subwavelength structures concentrate on the changes of the spectral intensity in transmission, there are fewer studies considering the effects of similar structures on the coherence properties of electromagnetic fields. However, it has been observed that surface plasmon polaritons generated in periodic microstructures can also enhance the coherence of light emitted by thermal sources [5]. Surface plasmons have also been experimentally shown to increase coherence between two independent optical sources [6]. Furthermore, the effect of plasmons on field correlations between specific points has been theoretically studied for systems consisting of two and three slits [7,8], and finite hole arrays [9] in metallic films. Previous theoretical studies have also considered also the effect of plasmon resonances in metallic nanocylinders on the coherence and polarization characteristics of the nearfield [10] and the influence of the state of coherence on plasmon excitation in thin metallic slabs and nanowires [11]. The coherence properties directly affect the characteristics of electromagnetic fields, such as their behavior upon propagation [12]; thus, the possibilities to flexibly control them could be valuable for many types of applications. The existing schemes for modifying the field coherence are typically based on either intricate filtering systems for increasing the coherence of incoherent light, or decreasing the correlations by randomizing the phase of the field [13–16]. The potential of surface plasmons to 0146-9592/13/235000-04$15.00/0
increase the field correlations could allow the manipulation of the coherence properties with small and simple nanophotonic components instead of complicated optical systems. Moreover, the local nature of plasmons could also allow spatially varying control of the coherence properties leading to the generation of new types of nonuniformly correlated fields [17,18]. In this Letter, we use rigorous numerical simulations to conduct a theoretical examination of coherence changes induced in periodic metallic gratings with narrow slits. Our aim is to find the optimal parameters to exploit the surface plasmon resonances in metallic gratings and increase the coherence of optical beams, which typically is not possible with single optical elements. In contrast to previous studies that concentrated on the field correlations between specific points in systems consisting of a finite number of slits or holes [7–9], we consider the changes of the coherence functions and the effective degree of coherence of an incident Gaussian Schell-model (GSM) beam transmitted through periodic linear grating structures. In the following, we will first introduce the basic concepts and the methods used in the numerical simulations. The relation of the grating parameters to the coherence properties are then studied by numerical simulations and the changes in the coherence functions are illustrated using a GSM beam as an example. Let us consider a partially coherent field with a monochromatic scalar component Ur; ω that gives the field realization U in the spatial position r at the frequency ω in the space–frequency domain. The spatial coherence properties of the field are described by the cross-spectral density W r1 ; r1 ; ω hU r1 ; ωUr2 ; ωi;
(1)
that measures the field correlations between two spatial points [12]. The spectral density of the field is given by Sr; ω W r; r; ω and the spectral degree of coherence of the field is defined as W r1 ; r2 ; ω : μr1 ; r2 ; ω p Sr1 ; ωSr2 ; ω © 2013 Optical Society of America
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Furthermore, the global coherence properties of the field can be described by the effective degree of spectral coherence [19,20] Sr1 ; ωSr2 ; ωjμr1 ; r2 ; ωj2 d3 r 1 d3 r 2 RR ; (3) 3 3 D Sr1 ; ωSr2 ; ωd r 1 d r 2
that gives the average spatial coherence weighted by the spectral density in the domain D for a single frequency. To facilitate the numerical analysis of the interaction of the partially coherent field and the grating structure, the cross-spectral density can be expressed as an incoherent sum of fully coherent modes [12] W r1 ; r2 ; ω
X λn ωϕn r1 ; ωϕn r2 ; ω;
(4)
n
where λω are the eigenvalues and ϕr; ω are the eigenvectors of the Fredholm integral equation of the second kind: Z D
W r1 ; r2 ; ωϕn r1 ; ωd3 r 1 λn ωϕn r2 ; ω:
(5)
The cross-spectral density function is non-negative definite and Hermitian, which, in turn, ensures that the eigenvalues λ are real and non-negative. Moreover, the eigenfunctions are orthonormal in the sense Z D
ϕm r; ωϕn r; ωd3 r δmn ;
(6)
where δmn denotes the Kronecker delta symbol. The eigenvalues may be interpreted as energy distribution between the distinct modes represented by the eigenfunctions. Employing the coherent mode representation, we can propagate each coherent mode individually and then sum the resulting diffracted modes to obtain the crossspectral density function of the scattered partially coherent field. In this work, we apply Fourier modal method (FMM) [21] in the determination of the diffraction amplitudes of each coherent mode. If the incident angular spectrum of the field is known, we can solve the reflected and transmitted angular spectra numerically using FMM. In the case of partially coherent fields, we simply calculate the coherent modes ϕn r; ω of the incident field and the angular spectrum of each mode. The coherent modes behind the grating can then be calculated separately through the transmitted angular spectrum of each mode solved numerically using FMM. The output cross-spectral density is obtained by summing the modes according to Eq. (4), weighted by the eigenvalues λn of the input field. Finally, the effective degree of coherence of the output beam can be calculated using Eq. (3). A more detailed description of the computational approach can be found, e.g., in [22] and [23]. The studied structure, shown in Fig. 1, consists of a binary gold grating on a fused silica substrate. In our simulations, we have used the refractive index data for gold and fused silica from [24] and [25], respectively. Since we are interested in plasmonic effects, we focus on
Fig. 1. Linear binary gold grating with period d, height h, and slit width c on a fused silica substrate.
considering TM polarization where the y-component of the magnetic field specifies completely the properties of the electromagnetic field. The cross-spectral density W that describes completely the coherence properties of the field is then actually the y; y-component of the magnetic cross-spectral density tensor [12]. Furthermore, we use the typical assumption that the incident field is a GSM beam, so its cross-spectral density is of the form 2 x x2 x − x 2 W x1 ;x2 ; ω W 0 exp − 1 2 2 exp − 1 2 2 ; (7) w0 2σ 0 where W 0 is constant, w0 is the beam half-width of the beam, and σ 0 is the coherence length [12]. Now we can proceed to study the effect of the grating parameters on the transmission and coherence changes of a partially coherent beam. We consider a GSM beam with w0 10 μm, σ 0 3 μm, at the wavelength λ 633 nm. The effective degree of coherence of the input beam μ¯ i ≈ 0.54. Figure 2(a) shows the transmission intensity of the beam energy and (b) the numerically calculated effective degree of coherence of the beam transmitted through a grating as a function of the period d and the height h. The slit width is kept as constant c 130 nm and the output beam is analyzed at the distance of 3 μm behind the grating. The effective degree of coherence is clearly increased with certain parameter combinations. The highest value μ¯ o ≈ 0.73 is obtained (a)
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where ε1 and ε2 are the permittivities of the materials and k0 is the wave vector in free space. Typically, transmission minima occur when the period of the grating equals the wavelength of the surface plasmons corresponding to a homogeneous interface as λsp Rfksp g [4]. This is seen in Fig. 2 for the period d λsp 390 nm, while the transmission and coherence maxima are found at the edge of this plasmonic bandgap.
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Figure 3 further illustrates the spectral dependence of the transmission and coherence changes for different grating periods compared to the plasmon wavelength. Other grating parameters used in the simulation are h 81 nm and c 130 nm. It is seen that at all wavelengths the maxima of both the transmission and the effective degree of coherence are closely located beside the plasmon curve calculated for a homogeneous interface. On the other hand, it should also be remembered that the GSM beam has a finite angular spectrum and the field components incident from different angles experience slightly different plasmon-related transmission resonances. Thus, the angular filtering performed by the grating may also play a role in the coherence changes in the same way as in some previous works considering coherence increases caused by different grating structures [5,22]. While the effective degree of coherence of the GSM beam chosen as an example is clearly increased with certain grating parameters, we still need to consider the changes in the coherence functions to get a more complete picture of the effect. Figure 4 compares the coherence functions of the original incident GSM beam with those of the beam transmitted through a grating with parameters d 424 nm, h 81 nm, and c 130 nm at wavelength λ 633 nm. The coherence functions are calculated at the distance of 3 μm behind the grating. It is seen that the cross-spectral density and the degree of coherence are widened behind the grating, and the widening increases toward the edges of the beam. Following the simple model in [9], it may be assumed that, with suitable grating parameters, part of the incident field is coupled to surface plasmons that spread over several adjacent grating slits; thus, the coherence of
x [nm]
when the period of the grating d 424 nm and the height h 81 nm. Also, the transmission of the beam has local maxima in the same parameter areas. In addition, the transmission increases when the grating period is reduced, because the slit width is kept constant and the relative amount of metal is decreasing. However, this effect does not increase the coherence. Furthermore, the maximum areas are periodically repeated when the height of the grating is increased by constant amounts. This phenomenon is related to the vertical (or cavity) resonances occurring in the slits [4]. On the other hand, some parameter combinations also show a slight decrease in the effective degree of coherence. However, in this example, the effect is relatively weak, especially in the areas with significant transmission. In the earlier studies on the coherence changes in finite slit and hole arrays [8,9], the enhancement of coherence has been explained by the coupling of the incident field to the plasmons propagating on the surface of the structure, which increases the correlations between separate points. Assuming a homogeneous interface between the gold and silica, we can calculate the wave vector of the propagating surface plasmons from the dispersion relation [1]
x [nm]
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0.6
0
4
25
2
25
50 0
0
50 0
0
0.4 0.2
0.4 350
400 d [nm]
450
500
Fig. 3. (a) Transmission and (b) effective degree of coherence of a GSM beam as a function of grating period d and the wavelength of the beam λ. The wavelength of surface plasmon at homogeneous gold–silica interface is plotted with a white line.
0.1 0.2 0.3 0.4 0.5 z [mm]
0.1 0.2 0.3 0.4 0.5 z [mm]
0
Fig. 4. (a) Cross-spectral density, (c) the degree of coherence, and (e) the spreading upon propagation of the incident GSM beam, and the corresponding properties, (b), (d), and (f), respectively, of the beam transmitted through the grating.
December 1, 2013 / Vol. 38, No. 23 / OPTICS LETTERS
the transmitted field increases as they are coupled out. The plasmons excited at the low intensity edges of the beam and that propagate toward the beam center, are weaker and, therefore, contribute less to the increase of coherence than the plasmons traveling from the beam center toward the edges. Nevertheless, the coherence functions are not strongly distorted from the original GSM shapes and the intensity distribution of the transmitted beam remains nearly Gaussian. Figure 4 illustrates the spreading of the beams propagating in air. As expected, the beam transmitted through the grating spreads significantly less upon propagation due to its higher degree of coherence. From a practical point of view, gratings with suitable parameters for increased coherence can be fabricated, e.g., with standard electron beam lithography. While we have demonstrated results for only one GSM beam with specific parameters, our further numerical simulations imply similar behaviors for a wide range of beams with different widths and degrees of coherence. The main drawback for applications is that the transmission of the beam through the metallic grating is relatively low despite the transmission resonances of the structure. Moreover the ideal parameters for coherence and transmission maxima do not exactly coincide, as in the previous examples. Thus, depending on the objectives, one should choose parameter combinations that give reasonable values to both the transmission of the beam and the increase of its coherence. It should also be remembered that linear grating structures work only with TM-polarized light. Three-dimensional periodic structures, such as arrays of holes or nanoantennas, could possibly produce similar effects for both polarization components simultaneously. In conclusion, we have studied the coherence changes in a GSM beam transmitted through a binary linear gold grating with narrow slits. It has been shown that, with specific combinations of the grating parameters, the effective degree of coherence is significantly increased. The changes can also be seen in the coherence functions and the propagation characteristics of the beam. Our results imply that metallic subwavelength gratings with unique resonance and plasmonic properties offer a novel way for manipulating the coherence properties of partially coherent fields.
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