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Phase Resolved Observation of the Gouy Phase Shift of Surface Plasmon Polaritons Tobias Birr, Tim Fischer, Andrey B. Evlyukhin, Urs Zywietz, Boris N. Chichkov, and Carsten Reinhardt ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.6b00999 • Publication Date (Web): 20 Mar 2017 Downloaded from http://pubs.acs.org on March 26, 2017
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Phase Resolved Observation of the Gouy Phase Shift of Surface Plasmon Polaritons Tobias Birr,
†
Tim Fischer,
†
Chichkov,
†Laser
Andrey B. Evlyukhin,
†
†
Urs Zywietz,
and Carsten Reinhardt
†
Boris N.
∗, †, ‡
Zentrum Hannover e.V., Hollerithallee 8, D-30419 Hannover, Germany
‡Hochschule
Bremen, Neustadtswall 30, D-28199 Bremen
E-mail:
[email protected] Abstract
vided a simple derivation of the Gouy phase shift for multimode beams, which for fundamental mode in two dimensional case φGouy,2D can be expressed by 3
The space-resolved Gouy phase shift for focused surface plasmon polaritons (SPPs) in the optical regime is experimentally demonstrated for the rst time. SPPs are excited by CW laser radiation focused onto a dielectric ridge placed on a cover glass surface coated by a silver lm. The Gouy phase measurements are performed using interference between a reference plane wave and leakage radiation. The measured Gouy shift has a value of π/2 corresponding to the twodimensional nature of surface plasmon polaritons.
φGouy,2D =
z 1 arctan , 2 zR
(1)
where z is the propagation direction and zR = n π w02 / λ is the Rayleigh length for the Gaussian beam with the wavelength λ, focus width w0 , and the medium refractive index n. The Gouy shift has already been measured and reported for surface plasmons and phonon polaritons in the THz range. 48 The direct phase resolved observation in the optical range is still missing. This paper provides the rst direct observation of the Gouy shift by comparison of the phase resolved focused and plane wave SPP leakage radiation images, as can be seen in Fig. 1. Since SPPs propagate at the interface of a metal and dielectric medium and represent a pure two dimensional system the total measured phase shift of π/2 is observed in this work. The images were recorded with the leakage radiation microscopy technique which enables phase and time sensitive imaging of the SPPs similar to that introduced by Gorodetski et al. 9 Simple theoretical modelling provides straightforward explanation of the observed results and can be applied for all kinds of beam propagation phenomena. Because the Gouy shift occurs in all kinds of focused waves, it plays a fundamental role in
Keywords Gouy phase shift, Surface plasmon-polaritons, Leakage radiation microscopy In the late 19th century, Louis Georges Gouy has theoretically predicted and experimentally veri ed the existence of a longitudinal phase shift for light waves passing through the focal point with respect to the phase front of collimated light beams. 1,2 This phase shift became widely known as the Gouy phase shift. The origin can be explained with the uncertainty relation in the focus region, which is ∆k∆x = 1/2 for the two dimensional case. 3 Since the convergent waves have a nite spatial extent, when passing a caustic, the wave vector has to consist of a superposition of transversal and longitudinal components. Feng and Winful pro-
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Exp
to the phase front of the exciting beam at the ridge surface. 21 When the laser focus is far away from the excitation ridge, the beam curvature can be neglected and the paraxial part of the SPP beam can be treated as a plane wave. When the laser beam focus is slightly behind the excitation ridge, the excited SPP beam will be converging and focussed. 22,23 The LRM setup, shown in Fig. 2, consists of a 40 x microscope objective with 0.65 NA for focusing the laser beam on the sample, an 40 x oil immersion microscope objective with 1.3 NA for image generation, and a CMOS-Camera. An additional 4-f setup is implemented to provide a possibility for blocking out the excitation laser beam on the recorded images. For enabling phase resolved measurements, about 10 % of the excitation beam intensity is coupled out to a bypass line as a probe beam (PB), before entering the LRM setup. The bypass line oers the same optical path length as the LRM setup. At the output both beams are combined in front of the CMOS. For variation of the optical path length in the bypass line, two mirrors are mounted on a linear micrometer stage. A further mirror behind the linear stage in the bypass line is used for the adjustment of the incident angle of the probe beam onto the CMOS, to align it to the incident angle of the leakage radiation (LR). For maintaining a homogeneous illumination of the whole CMOS, the PB is expanded with an Galilei telescope. In this con guration, the PB and the LR will interfere directly on the CMOS. Thus, the recorded intensity resembles a time integrated superposition of the leakage radiation and the probe beam. This can be expressed by
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Figure 1: Experimental demonstration of the spatially resolved two dimensional Gouy phase shift for SPPs. Leakage radiation images for focused and plane wave SPPs are shown in the upper and lower parts, respectively. The phase resolution was achieved by interference with a probe laser beam. Regions of positive and negative interference are shown by red and blue colors. modern physics. In optical systems, this effect is important for the correct dimensioning of laser cavities, 10 nonlinear eects, 11 attosecond pulse generation, 12 and optical coherence tomography. 13 In SPP systems it should be taken into account for observations of localized nonlinear eects and development of integrated circuits, bio-sensors, and data storage devices. 14
Experimental details For experimental observations of the Gouy phase shift, the SPP system, consisting of a 55 nm thin silver lm thermally evaporated on a standard microscopic cover glass (Menzel), is investigated by leakage radiation microscopy (LRM). 15,16 To excite SPPs, a TM laser beam is focussed on a polymer ridge with a width and height of about 300 nm underneath the silver lm. The polymer ridge structures are produced from Ormosil 17 by means of microscopeprojection-photolithography. 1820 As excitation laser source a titanium-sapphire oscillator (Kapteyn Murnane Labs) is used in CW mode at 780 nm center wavelength. When the excitation laser beam has Gaussian shape, the excited SPP beam will resemble a two dimensional projection of this shape corresponding
ICMOS (x, z, t) ∝ Z
τfov
|ELR (x, z, t) + EPB (t)|2 dt,
(2)
0
where z is the SPP propagation direction and x is perpedicular to z . Because the exposure time of the CMOS is much longer than the SPP propagation time for the eld of view (fov) τSPP = lLfov /cSPP , the integral gives a time average signal of the interference pattern. Since
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Figure 2: Scheme of the used setup for phase resolved measurements: about 10 % of the excitation beam intensity is coupled out to a bypass line as a probe beam (PB), before entering the Leakage radiation microscope setup and is recombined with the leakage radiation (LR) in front of the CMOS camera. For maintaining a homogeneous illumination of the whole CMOS, the PB is expanded with an Galilei telescope. An additional 4-f setup is implemented to provide the possibility for blocking out the excitation laser beam on the recorded images. In numerical calculations, the expressions for the respective electric elds are rst superposed as shown in Eq. 2. The square of the absolute values is than directly computed into threedimensional (x, z, t) matrices, which are nally integrated over the time axis. This method allows to use the superior vector computation of numPy (Python), Matlab or Octave. 25,26 As spatial steps of the mentioned matrices, the resolution of the experimental taken LR images are used. To calculate a su cient time discretization, the Courant factor, normally applied for FDTD simulations, 27 is taken into account.
the probe beam is illuminating the CMOS homogeneously, it can be considered as a time dependent plane wave (3)
EPB (t) ∝ exp (−i ω t),
with the the angular frequency ω . The leakage radiation images the SPP electrical eld and can therefore be represented as two dimensional distribution of a Gaussian beam. The electrical eld of the Gaussian SPP beam at the metal surface can be presented by 24
ELR,foc (x, z, t) ∝
exp −
x2
Results and discussion
+ ikSPP z − iωt
iz w02 (1 + ) z rR iz 1+ zR
,
The SPP resembles a two dimensional projection of the incident laser beam. The SPP beam can be focused or collimated to a plane wave by altering the distance of the laser beam focus with respect to the polymer ridge at the sample. 22,23 The LR images corresponding to both cases are compared in Fig. 1. Here, a perfect overlap of the wave fronts for the plane and focused SPP waves is clearly visible in front of the focus position. After the focus, one can observe the phase mismatch between the wave fronts
(4)
where w0 is the beam waist in the focus and zR is the Rayleigh length. The SPP wave vector kSPP is assumed to be complex, to include propagation losses into the model. The plane SPP wave can be expressed by
ELR,pl (x, z, t) ∝ exp (i kSPP z − i ω t),
(5)
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Figure 3: Numerical calculations for SPP waves were tted to the observed experimental interference pattern for (a) plane wave and (b) focused SPP beams, where regions of positive and negative interference are shown by red and blue colors. (c) The Gouy phase shift was determined by comparing the distances between the wave front maxima for the plane and focused SPP waves on the propagation axis. The conversion of these distances to the phase shifts in radians are shown as red dots. The theoretically expected arc tangent function (with zR = 1.5 µm) is added as dashed black curves for the two dimensional ( π/2 phase shift) and the three dimensional ( π phase shift) cases. corresponding to the Gouy phase shift. Numerical calculations for plane SPP waves were tted to the observed experimental interference pattern. As a tting parameter, the complex SPP wave vector was used with an initial value of 763.3 nm, given by the semi in nite approach 28 and the Lorentz-Drude model 29 for the dielectric function of silver. The result of this t is shown in Fig.3(a) and demonstrates a perfect coincidence between the measured and tted SPP plane waves. This procedure gave for the SPP wavelength the value of 752 nm. The dierence to the calculated value is about 1.5 %, proving a very good accuracy of the applied scaling factor for the microscope images. The determined SPP wave vector was further used for interpretation of experiments with focused SPP Gaussian beams. In this case to t numerical data to the experimental results, the beam waist w0 as the only tting parameter
was used. It was determined by the t to be w0 = 589.7 nm. This is in very good correlation with the expected diraction limited spot size of d = λ/2NA = 600 nm. Comparison of the experimental and numerical results is shown in Fig. 3(b), where again a perfect agreement is visible. The Gouy phase shift was determined by comparing the distances between the wave front maxima for the plane and focused SPP waves on the propagation axis. These distances were converted to the phase shifts in radians shown in Fig. 3(c) as red dots. The theoretically expected arc tangent function, see Eq. 1, with the Rayleigh length zR =1.5 µm is shown as a dashed black curves for the two dimensional (π/2 phase shift) and the three dimensional (π phase shift) cases, respectively. One can observe a perfect agreement between the measured and computed results for the two dimensional case, emphasizing the two dimen-
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sional nature of the shown phenomenon.
(5) McGowan, R. W.; Cheville, R. A.; Grischkowsky, D. Direct Observation of the Gouy Phase Shift in THz Impulse Ranging Direct Observation of the Gouy Phase Shift in THz Impulse Ranging. Appl. Phys. Lett. 2000, 76, 670672.
Conclusion Using the leakage radiation microscopy, the space-resolved Gouy phase shift for focused surface plasmon polaritons (SPPs) has been experimentally investigated for the rst time. These measurements have been performed using interference between a reference plane wave and leakage radiation. A perfect agreement between the measured and theoretically predicted arc tangent function has been demonstrated. The measured total Gouy shift of π/2 corresponds to the two-dimensional nature of surface plasmon polaritons. In the development of SPP systems, the Gouy shift has to be taken into account in design of integrated circuits, bio-sensors, data storage and nonlinear devices.
(6) Feurer, T.; Stoyanov, N. S.; Ward, D. W.; Nelson, K. A. Direct Visualization of the Gouy Phase by Focusing Phonon Polaritons. Phys. Rev. Lett. 2002, 88, 257402. (7) Zhu, W.; Agrawal, A.; Nahata, A. Measurements of the Gouy Phase Shift for Surface Plasmons. America (NY). 2007, 1, 56245624. (8) Wang, X.; Sun, W.; Cui, Y.; Ye, J.; Feng, S.; Zhang, Y. Complete Presentation of the Gouy Phase Shift with the THz Digital Holography. Opt. Express 2013, 21, 2387 2393.
ACKNOWLEDGEMENT
The authors acknowledge nancial support of this work from the Deutsche Forschungsgemeinschaft (DFG, SPP1391: "Ultrafast Nanooptics" and SFB/TRR123: "Planar Optronic Systems") and support of the Laboratorium of Nano- and Quantenengineering (LNQE). The authors further acknowledge support from Hannover School of Nanotechnology (HSN).
(9) Gorodetski, Y.; Chervy, T.; Wang, S.; Hutchison, J. A.; Drezet, A.; Genet, C.; Ebbesen, T. W. Tracking Surface Plasmon Pulses Using Ultrafast Leakage Imaging. Optica 2016, 3, 4853. (10) Feng, S.; Winful, H. G. Spatiotemporal Structure of Isodiracting Ultrashort Electromagnetic Pulses. Phys. Rev. E 2000, 61, 862873.
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