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Description and characterization of plasmonic Gaussian beams
View the table of contents for this issue, or go to the journal homepage for more
2017 J. Opt. 19 085001
(http://iopscience.iop.org/2040-8986/19/8/085001)
Home Search Collections Journals About Contact us My IOPscience
Received 24 March 2017, revised 12 May 2017Accepted for publication 5 June 2017Published 14 July 2017
AbstractIn this work, we present for the first time a detailed description and experimental characterizationof plasmonic Gaussian beams (PGBs), as well as the analytical expression to describe their fieldand intensity distributions. The propagation parameters of the PGBs, such as the divergenceangle, Rayleigh range, beam width function, and the beam waist are determined experimentallyin accordance to the proposed model. The radius of curvature of the wavefront and the Gouyphase shift of PGBs can also be predicted using this method.
Keywords: Gaussian beams, plasmonics, diffraction
(Some figures may appear in colour only in the online journal)
1. Introduction
Plasmonics is a branch of nano-optics which deals with theexcitation and manipulation of surface plasmon polaritons(SPPs) [1]. Single metallic ridges can couple free-propagatinglight of a broad range of wavelengths (in the plasmonic range)into SPPs by illuminating with a focused Gaussian beam. Thisexcitation technique was first introduced and characterized in2003 as an efficient and local alternative to couple light intoSPPs, which generates an SPP beam propagating perpend-icular to the ridge structure [2]. Since then, the method hasbeen widely used as a standard SPP excitation mechanism[3–7], but less attention has been directed to the descriptionand characteristics of the beam.
Generation and characterization of different types ofplasmonic beams have gained the attention of differentresearch groups in the last decade. Self-accelerating, andquasi diffraction-free beams, such as the Airy [8, 9] and theBessel beam [10], have been studied analytically and char-acterized experimentally. In this direction, we aim for thestudy and characterization of the propagation parameters ofplasmonic beams excited as mentioned above. It was alreadyobserved that if SPPs are excited with a transverse-electro-magnetic (TEM00) Gaussian beam, the divergence of the
generated SPP must equal the divergence of the excitationlight [11]. This fact served as a motivation for the presentwork and lead us to study the properties and parameterstransferred from the Gaussian beam to the SPP. Ourhypothesis is that the generated SPP beam inherits not onlythe divergence of the source, but all of the well-knownGaussian properties, such as the Rayleigh range, beam widthfunction, intensity distribution, radius of curvature of thewavefront, and the Gouy phase shift [12]. Henceforth, SPPbeams generated in the aforementioned configuration will bereferred to as plasmonic Gaussian beams (PGBs) alongthe text.
In this work, we find the expressions that describe thefield and intensity distribution of PGBs as solutions of theparaxial wave equation. Next, we characterize experimentallythe PGBs with leakage-radiation microscopy (LRM), usingmicroscope objectives for excitation with three differentnumerical apertures (NA = 0.10, 0.25, and 0.40).
2. Analytical description
Gaussian beams are solutions to the paraxial wave equation[12], and the two-dimensional (2D) expression of the field
Journal of Optics
J. Opt. 19 (2017) 085001 (5pp) https://doi.org/10.1088/2040-8986/aa7724
propagating in the z-direction can be expressed as
y y
j
=-
´ - + -
⎛⎝⎜
⎞⎠⎟
⎡⎣⎢
⎛⎝⎜
⎞⎠⎟
⎤⎦⎥
( )( ) ( )
( )( ) ( )
x zw
w z
x
w z
k z kx
R zz
, exp
exp i2
, 1
G 00
2
2
0 0
2
where w(z), commonly known as the beam width or radius, isthe radial distance from the propagation axis of the beam atwhich the field amplitude has decayed 1/e times (or 1/e2 ofthe intensity), and ψ0 is the field amplitude in the origin. Thebeam waist w0 = w(0) corresponds to the minimal value of w(z), and x is the transverse coordinate. Here, k0 = 2π/λ0corresponds to the free-space wavevector, and R(z) and j(z)are the radius of curvature of the wavefront and the 2D Gouyphase shift, respectively. The beam width is a function of thepropagation coordinate, and can be expressed as
lp
= +⎛⎝⎜
⎞⎠⎟( ) ( )w z w
z M
w1 . 20
2
02
2
In practice, the beams generated in laser systems mayvary on how well they approximate to a Gaussian TEM00
mode (figure 1). In this context, the factor M2 is introduced inequation (2) to account for the beam quality. The M2 factorcorresponds to the ratio of the beam parameter products(BPP) (i.e. the product of the beam waist and the far-fieldbeam divergence) of the real case divided by the ideal caseM2 = w0,realθreal/w0,idealθideal. The ideal case correspondstoM2 = 1, and w0 = w0,ideal is diffraction limited. For the realcase (M2 > 1), the beam waist is wider and determined by theBPP. Gaussian beams are also characterized by the Rayleighrange zR = πw0
2/λM2, defined as the distance along z wherethe beam width is √2 larger than w0, the divergence angle
θ ≈ λM2/πw0, that corresponds to the radial angle at whichthe beam spreads asymptotically, the 2D Gouy phase shift j(z) = ½ atan(z/zR), and the radius of curvature of the wave-front
pl
= +⎡⎣⎢⎢
⎛⎝⎜
⎞⎠⎟
⎤⎦⎥⎥( ) ( )R z z
w
z M1 . 30
2
2
2
In the case of a 2D Gaussian beam, the intensity distributioncan be expressed as
=-⎛
⎝⎜⎞⎠⎟
⎛⎝⎜
⎞⎠⎟( )
( ) ( )( )I x z I
w
w z
x
w z, exp
2, 40
02
2
where the second factor describes the intensity decay alongthe propagation coordinate caused by the divergence of thebeam (diffraction), and the exponential term corresponds tothe Gaussian transverse distribution.
So far, we have outlined the most representative para-meters and expressions that describe the propagation of real(non-ideal) Gaussian beams, and special attention wasdirected to the 2D case (cylindrical wave). Now, we focus onthe expressions the SPP beam. The field of a plane SPP wave,invariant in the x-coordinate, propagating in the z-directionalong a metal–insulator interface can be defined as
y y b= +( ) [ ( )] ( )y z z k y, exp i i , 5ySPP 0
where the y-coordinate corresponds to the directionperpendicular to the interface, β = β′ + iβ″ is the complexpropagation constant, and ky is the out-of-plane evanescentwavevector. The intensity distribution of the plane SPP wavereads simply as
=-
-⎛⎝⎜
⎞⎠⎟( ) ( ) ( )I y z I
z
Lk y, exp exp 2 , 6y0
SP
where LSP = 1/(2β″) is known as the propagation length ofthe SPP, and is defined as the distance along the propagationwhere the intensity has decreased by a factor of 1/e.Combining equations (1) and (5), we can define the field ofa PGB as
y y
b b j
=-
´ + - +
⎛⎝⎜
⎞⎠⎟
⎡⎣⎢
⎛⎝⎜
⎞⎠⎟
⎤⎦⎥
( )( ) ( )
( )( )
( )
x y zw
w z
x
w z
z k yx
R zz
, , exp
exp i i2
,
7y
PGB 00
2
2
2
which also satisfies the paraxial wave equation, since theevanescent term exp(−kyy) is independent of the transverseand propagation coordinates (x, z). Using this result, theintensity distribution is determined by
=-
-
´-
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
( )( )
( )
( )( )
I x y z Iw
w z
z
Lk y
x
w z
, , exp exp 2
exp2
. 8
y00
SP
2
2
Figure 1. Beam width of a Gaussian beam as a function of thepropagation coordinate z. The beam waist w0, Rayleigh range zR,divergence angle θ and the radius of curvature R(z) are depicted inthe graph. This example is obtained using w0 = 3.5 μm and λ0 =740 nm. The axes are scaled for clarity.
2
J. Opt. 19 (2017) 085001 C E Garcia-Ortiz et al
3. Experimental characterization
The experiment consists of generating PGBs by focusing thelaser beam with a microscope objective onto a gold ridgesitting on top of a gold thin-film. The gold ridges (200 nmwide and 70 nm thick) were patterned on top of a 70 nm thickgold film using e-beam lithography (figure 2(a)). The laserwas operated at a wavelength of 740 nm, linearly polarized(perpendicular to the ridges), and with a near TEM00 funda-mental Gaussian beam (measured beam quality factor
= M 1.01 0.03L2 ). The beam quality factor of the laser
was estimated by measuring the beam width at differentpositions near the focus of a lens, and a CCD camera torecord the beam width size.
The scattered light generates two symmetrical PGBs pro-pagating in the perpendicular direction to the ridge. Both beamshave almost identical properties, and only one of them is usedfor this study. To generate PGBs with different properties, threedifferent focusing objectives were used in the experiment, sincethe associated divergence angle is determined by the numerical
aperture of the microscope objective. A beam expander situatedbefore the focusing objective is used to cover the complete areaof the objective rear aperture, so that the divergence of the beamfits the acceptance angle θa= asin(NA) of the objective. Table 1synthesizes the parameters of the focused beam after eachobjective, including the minimum radius of the beam waist inthe focal plane, if the whole rear aperture is illuminated, givenby l qp=w M .Lm
20
The intensity distribution of the generated PGBs areobtained from the LRM images (figures 2(c)–(e)). The exper-imental setup and a detailed description of the principle ofoperation of LRM can be found in the [11, 13]. For each of thebeams, the beam width w was measured as a function of theposition in the propagation direction, using a Gaussian fit alongthe transverse direction x. The obtained data points describe thedivergence of the beam, and are used to find w0 and the M2
factor through numerical fitting using equation (2) (figure 2(f)).It must be noted that the value used for λ is λSP =2π/β′ = 719 nm, i.e. the wavelength of an SPP in a gold–airinterface excited with a free-space wavelength of 740 nm. In
Figure 2. (a) Schematic of the setup used to excite the PGBs. The double arrow indicates the polarization of the incident light. (b) Scanningelectron microscopy image of the gold ridge. (c)–(e) LRM images (image plane) of the intensity distribution of PGBs, propagating from leftto right, focused with the (c) 4×, (d) 10×, and (e) 20× objectives. (f) Measured beam width of the PGBs as a function of the propagationcoordinate z. The solid lines correspond to the numerical fitting of the semi-analytical expression for the beam width.
Table 1. Parameters of the focusing objectives and of the corresponding generated PGBs.
most cases, it is not possible to determine the precise position ofthe PGB waist (focal point) by naked eye observation of theLRM images. In this sense, the fitted curve also allows to findthe position of the focal point, if z is replaced by (z + δz) andleave δz as a new fitting parameter (figure 3). Using the obtainedvalues of w0 and M2, is now possible to estimate the values ofthe Rayleigh range, divergence and the radius of curvature of thewavefront along the propagation direction. Table 1 shows theseresults, which can be compared to the parameters of the focusingobjectives. As expected, the divergence angle of the PGBapproximately matches the acceptance angle of the objective, butwith an angle reduction of 14%–21%. The decrease of diver-gence is compensated with a higher PGB beam waist, with anincrease of 37%–47%. This tradeoff can be better understood ifwe introduce the relation M2 = w0θSP/wmθ, which we candefine as the PGB beam quality factor. Since the laser has anM2 ∼ 1, we can consider the BPP = wmθ as close to ideal. Inthis regard, we can estimate the value of the PGB M2 factor bysolving the relation, if w0 and θSP are known. The results aresimilar to the obtained with numerical fitting, giving values of1.09, 1.21, and 1.22, for the 4×, 10×, and 20×, respectively. Itis important to note that in this case, the excitation beam andgenerated PGBs have different wavelengths, thus this approach,although very similar, is not the exact same as for embeddedGaussian beams. The radius of curvature of the wavefront ofeach PGB can be predicted and estimated from equation (3)using the parameters obtained experimentally (figure 4(a)). It is
of special interest to see that the PGB generated with the 4×objective, has a quasi-plane-wave behavior below 5μm, makingit suitable for experiments were a plane wavefront is preferred.The objectives with higher magnification reach much lowervalues of R, close to the excitation point, and changes rapidly inshort distances. The Gouy phase shift was also estimated andshows that the phase shift can be appreciated at this scale(figure 4(b)). It is observed that a phase shift of almost 3π/8occurs within the first 5 μm, in the case of the 20× objective.For the PGB generated with the 4× objective, the phase shiftchanges more slowly and barely surpasses a shift of π/4 after50μm. This considerations can be very important whendesigning specific experiments where a precise knowledge of thephase and wavefront shape is required at every point, or innumerical simulations of plasmonic beams. For example, whenusing PGBs in interferometric plasmonic devices, abrupt chan-ges in the phase can have different outputs or responses.
The divergence of a PGB can hinder the estimation of thepropagation length. To recover the correct value, our descrip-tion can be used to determine LSP accurately. Equation (8)describes the intensity distribution of the PGB, as well as thepropagation losses and divergence. To prove the validity of themethod, we take intensity profiles along the propagation coor-dinate z, at the axis point (x= 0). Mapping the intensity throughthe axis avoids the problem of using a more complex coordinatesystem (hyperbolic coordinate), and also the last term inequation (8) becomes unity. The intensity profiles are processedwith numerical fitting, using the parameters obtained before.LSP is left as the only fitting parameter. The results show that,despite the strong decay in the intensity due to divergence, it isstill possible to recover the information which corresponds tothe propagation losses (figure 4(c)). The obtained values wereLSP(4×)= 32± 2 μm, LSP(10×)= 32± 3 μm, LSP(20×)= 28± 3 μm. Evidently, the value of LSP must be the same for all thePGBs, since it is only dependent on the medium where itpropagates and the excitation wavelength. The results are ingood agreement with our hypothesis, with minor deviations dueto experimental noise.
4. Conclusions
In conclusion, we have presented a detailed description andcharacterization of PGBs, generated by exciting SPPs usingmicroscope objectives with different NA. We have presented ananalytical expression which describes the field and intensitydistributions of PGBs, introducing the M2 parameter to char-acterize the quality of the beam. All the parameters that governthe propagation of the PGB, such as the divergence angle,Rayleigh range, beam width and waist were characterizedexperimentally, as well as an M2 factor for each of the beams.Moreover, this approach allowed for an accurate estimation ofthe propagation losses, even for diverging plasmonic beams. Theradius of curvature of the wavefront and the Gouy phase shiftwere estimated from the experimental values and the proposedmodel. A direct measurement of these last two parameters can be
Figure 3. (a) LRM image of the intensity distribution of the PGBgenerated with the 4× objective and (b) the measured beam widthstarting from the position of the ridge. The dashed line indicated theposition of the beam waist (focal point), which is not perceptible in(a). The scale in (a) is the same as in (b).
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J. Opt. 19 (2017) 085001 C E Garcia-Ortiz et al
obtained via near-field imaging with phase detection. Suchexperiments could, in principle, be performed with a modifiedscanning near-field optical microscope with phase detection,such as the one used in [14, 15].
Acknowledgments
We acknowledge the financial support from CONACYTBasic Scientific Research Grant No. 250719 and No. 252621.We thank the help of Dr Rodolfo Cortes (GNO Mexico) forinsightful discussions.
Note added in proof. When completing our work, we foundthat the Gouy phase shift of PGBs has been measured andreported using LRM with phase detection [16].
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Figure 4. (a) Radius of curvature of the wavefront and (b) Gouy phase shift as a function of the propagation coordinate for the three PGBsexcited with the different focusing objectives. The inset in (b) shows a zoom of the central area. (c) Intensity profiles along the axis x = 0, inthe propagation direction showing the intensity decay due to propagation losses and divergence.
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