1

Enhancing second harmonic generation in gold

nanoring resonators filled with lithium niobate

Dennis Lehr*1, Jörg Reinhold1, Illia Thiele1, Holger Hartung1, Kay Dietrich1, Christoph Menzel1,

Thomas Pertsch1, Ernst-B. Kley1, Andreas Tünnermann1,2

1Institute of Applied Physics, Abbe Center of Photonics, Friedrich-Schiller-University Jena,

Max-Wien-Platz 1, 07743 Jena, Germany

²Fraunhofer Institute for Applied Optics and Precision Engineering IOF, Albert-Einstein-

Straße 7, 07745 Jena, Germany

Plasmonic nanorings provide the unique advantage of a pronounced plasmonic field

enhancement inside their core. If filled with a polarizable medium it may significantly enhance

its optical effects. Here, we demonstrate this proposition by filling gold nanorings with lithium

niobate. The generated second harmonic signal is compared to the signal originating from an

unpatterned lithium niobate surface. Measurements and simulation confirm an enhancement of

about 20. Applications requiring nanoscopic localized light sources like fluorescence

spectroscopy or quantum communication will benefit from our findings.

Keywords: Lithium niobate, nanorings, second harmonic generation, ion beam enhanced etching,

plasmonics

2

Second harmonic generation (SHG), i.e. doubling the frequency of light by interaction with a

nonlinear material1 is an optical process with great technological importance. As the required

interaction length between light and the nonlinear material is large, devices employing this

process tend to be bulky. Integration in switches and optical modulators requires miniaturization

of optical components. Designs that are more compact are achieved by optical waveguides. They

enhance the nonlinear interaction length through lateral field confinement and, hence, increase

SHG compared to bulky systems2. However, strong nonlinear signals are generated just upon

propagation over distances large compared to the wavelength.

For a range of recent applications, this lateral miniaturization is not sufficient. Second

harmonic (SH) light sources within three-dimensionally confined volumes at the sub-wavelength

scale are required3. They have been proposed as nonbleaching localized probes for fluorescence

spectroscopy4,5 or as integrated single photon sources for quantum communication6. Furthermore,

such structures provide more flexibility when designing devices as phase matching does not play

any role. To realize efficient generation of SH at sub-wavelength volumes, light confinement and

field enhancement as provided by nanoscale resonators is required. Obvious choices for such

resonators are metallic nanostructures exploiting plasmonic resonances7,8,9. Placing metallic

nanostructures on top of nonlinear substrates or embedding them into a material with high

intrinsic optical nonlinearity enables high nonlinear responses at moderate power – in the case of

second order nonlinearity, Lithium niobate (LN) is an excellent choice3,10,11,12.

In fact, any finite resonant metallic nanostructure simply placed on top of a nonlinear material

can provide an enhanced nonlinear response13. Assuming the linear refractive index of the

substrate is larger than the index of the surrounding medium, the first order low-frequency

resonance of the metallic nanostructure exhibits a field enhancement in close vicinity to the

3

particle-substrate interface. Hence, the nonlinear field generated in that region will be strong

compared to the nonlinear response of the same substrate without the resonant nanostructure3,14,15.

However, two important aspects have to be kept in mind. First, for an optimum performance the

plasmonic resonator has to be carefully designed, where it is obviously advantageous to embed

the resonator into the nonlinear material or to place the nonlinear material inside the resonator to

maximize the overlap between plasmonically enhanced field and the nonlinear material12,16.

Second, for any kind of application and potential mass fabrication we have to rely on efficient

and reproducible fabrication techniques. Self-organization processes to fabricate core-shell

particles filled with a nonlinear medium have been developed and demonstrated for this

purpose17,18,19,20.

Our approach goes beyond by relying on a unique combination of heavily standardized wafer-

level processes and electron beam lithography (EBL). As nanoparticles, we propose the use of

gold nanorings, which are very similar to core-shell particles exhibiting a strong field inside their

core21. Plasmonic nanorings for application at near infrared (NIR) frequencies can be generated

deterministically with high quality and throughput by means of character projection (CP)22 EBL

and double patterning (DP)23,24, i.e. pattern definition with an area of multiple dm² is possible

within hours. By pre-patterning the crystalline LN substrate and subsequent fabrication of the

metallic nanorings, we obtain nanorings filled with crystalline LN.

Within this letter, we describe the implemented fabrication process to efficiently realize the

proposed LN-filled nanorings utilizing the methods mentioned above. To evaluate our samples

we performed measurements of the linear as well as the nonlinear properties and compare the

generated SH signal to an unpatterned LN substrate. The experiment is complemented with

numerical finite difference time domain simulations supporting the experimental results.

4

To confirm our approach and to identify suitable geometrical ring parameters with respect to

their linear and their nonlinear optical properties we performed numerical simulations with the

free Finite Difference Time Domain implementation Meep25. The simulations are performed in

three steps. First, the field at the fundamental frequency (FF) is calculated applying a

monochromatic plane wave excitation. Second, the nonlinear polarization is calculated using this

fundamental field26. Eventually, the propagation of the second harmonic fields is simulated upon

excitation of the system by the nonlinear polarization. This stepwise calculation of the SH fields

is applicable only if the evolution of the fundamental fields can be decoupled from all other

harmonics. The decoupling is achieved in the undepleted-pump approximation (UPA)27

neglecting the feedback of higher harmonics on the linear polarization, i.e. the nonlinear

polarization has to be small compared with the linear polarization at the FF. The UPA is

commonly assumed for plasmonically enhanced optical nonlinearities28,29,30 due to the small

volumes of nonlinear interaction and valid within the power range considered here.

The filled gold nanorings are periodically arranged on a square lattice in the y-z-plane on top

of a crystalline LN substrate (x-cut) where tabulated data for the permittivity of gold is taken

from Ref. 31. The rings are characterized by their inner diameter (80 nm), outer diameter (120

nm), height (100 nm), and period (260 nm). The chosen period prevents the occurrence of

diffraction orders outside the substrate throughout our study. This is particularly valid for the SH

frequencies such that transmission and reflection are fully characterized by the zeroth diffraction

order for all frequencies. The nanorings are illuminated preferably with light polarized in z-

direction. Here, the electric field of the incoming light and, hence, the dipolar moment of the

localized surface plasmon polariton resonance is aligned to the largest principal diagonal element

5

of the second-order susceptibility tensor 𝜒!!!(!) = −8.34 ∙ 10!!! m

V. This will generate the largest

SH signal.

The results of the simulation are presented in Figure 1. The reflection obtained for plane wave

excitation at the FF (Figure 1b) indicates a pronounced plasmonic resonance at 315 (950

wavelength).

At normal incidence, about 50% of the incoming light is reflected. At this fundamental

resonance, the field is mainly concentrated at the nanoring-substrate interface, where a

considerable portion of the field is inside the core as well (Figure 1a). The z-component of the

electric field, which is dominating the contribution to the SH process, is increased by a factor of

4 to 8 inside the core and peaks at 15 near the lower boundary of the gold ring (Figure 1a). To

evaluate the enhancement independently from the illumination intensity, we calculated the SH

enhancement12 τ which compares the SH intensity reflected by the gold ring array 𝐼!"#$% to the

intensity reflected by the unpatterned surface of the substrate 𝐼!"# (Figure 1c).

𝜏 =𝐼!"#$%𝐼!"#

Within our simulations, we calculated the SH intensity reflected from the unpatterned substrate

analytically27. We chose the bare, unpatterned LN surface instead of the structured surface, i.e.

LN pillars without Au rings, as reference of choice as it allows us to determine the enhancement

of surface SHG by applying a resonant nano-pattern. The physical interpretation of our results

would not change by using LN pillars as reference. In fact, the SH enhancement would even

become larger, because the SH signal obtained from LN pillars is smaller than the SH signal

obtained from the unpatterned surface by roughly a factor of 4. This decrease may be explained

by the antireflective behavior of nonresonant subwavelength nanostructure.

6

Please note, that the lines connecting the individual monochromatic simulations in Figure 1 are

just guides to the eye with unspecified line shapes. Clearly, the SH enhancement τ peaks at the

fundamental plasmonic resonance and reaches values of up to 60 for normal incidence. In fact,

the resonance position estimated from the linear reflection is at slightly larger frequencies

compared to the peak in τ. Furthermore, the resonance width in the linear reflectance spectrum

(Figure 1b) is larger than the width of the SH enhancement τ (Figure 1c). However, both the shift

as well as the change in width can be attributed to the nonlinear response and the shift between

the spectral near field and far field intensities7,32. The scattering of the data points around the

guide to the eye lines close to 300 THz (1000 nm wavelength) is due to a localized higher order

resonance of the LN filled rings at 600 THz (500 nm wavelength). Note, that this dip in the

enhancement occurs for all 4 scenarios shown in Figure 1c, i.e. at normal as well as oblique

incidence. The quite sharp, resonant absorption at 600 THz leads to this decrease of the SH

signal and, hence, the SH enhancement, which can be observed solely from the simulations. This

as well as other higher order resonances are broadened and eventually invisible in the experiment

due to deviations of the fabricated rings from the idealized shape.

To investigate the contribution of LN inside the rings to the SH signal, we performed identical

simulations but replaced the material inside the rings with an artificial material having identical

linear but vanishing nonlinear properties (𝜒(!) = 0). In this case, the SH enhancement τ breaks

down from almost 60 to below 15. This confirms our presumption that filling the rings with LN

is crucial for getting a significant enhancement.

Note, that the results are shown for normal incidence as well as for oblique incidence in TE-

configuration (electric field vector parallel to the sample surface). The latter are required for

comparison with the experiment, since the experimental measurements are performed at oblique

7

incidence to spatially separate light reflected from the front- and backside of the substrate. The

resonance position is independent on the angle of incidence and, hence, the resonance is clearly a

localized surface plasmon polariton resonance.

From our point of view, an efficient and reproducible fabrication technique is a necessity for a

real world application. We therefore developed a fabrication process aiming towards this

objective.

Main steps within our process comprise the creation of crystalline LN pillars by CP-EBL and

ion beam enhanced etching (IBEE). CP-EBL ensures high-throughput and pattern accuracy.

IBEE allows selective etching of LN while preserving the crystallinity of LN33. Afterwards DP is

utilized which allows metallization of the pillars to form the nanorings.

As first step within the process chain, the LN substrate is consecutively coated with 250 nm

silicon dioxide (SiO2), 20 nm chromium (Cr) and 110 nm chemically amplified negative tone

resist (TOK OEBR-CAN034). Resist pillars with a period of 260 nm and a diameter of 120 nm

are patterned by EBL (Figure 2a). This is a serial process and requires rasterizing the electron

beam across the sample to define the desired pattern. If the patterned area is increased, it easily

becomes the most time consuming and limiting process step within a lithographic process chain.

We therefore utilize a Vistec SB350 OS EBL system capable of CP, i.e. imaging of a fixed mask

into the electron beam resist. This increases the raster size to the size of the mask (typically 2.5 x

2.5 µm²) and reduces the overall writing time by up to three orders of magnitude compared to

classical EBL22. In our case, a writing time in the order of dm² in hours is achieved.

Next, the resist pattern is transferred into the Cr layer and the underlying SiO2 layer by

inductively coupled plasma reactive ion etching (ICP-RIE) using the etchants Chlorine and

8

Fluoroform, respectively (Figure 2b, 3a). The Cr layer primarily serves as hard etch mask for the

SiO2 layer.

The resulting SiO2 pattern is transferred into the LN substrate by IBEE. The sample was

therefore irradiated with Argon ions with energies of 120 keV and 60 keV and a fluence of

1015 cm-2 and 1.2x1015 cm-2, respectively. The pillar shaped SiO2 mask is protecting the

underlying LN crystal from irradiation damage. In the unmasked region, this irradiation leads to

a damage of the crystalline structure from the surface down to a depth of 100 nm. After

irradiation, the mask is wet-chemically removed. By wet etching of the damaged LN-Volume in

50% aqueous KOH at a temperature of 65 °C LN pillars are formed34 (Figure 2c, 3b).

To yield gold nanorings, the LN pillars are coated with 40 nm gold by means of physical vapor

deposition. To coat the pillars homogenously, the sample is tilted by 45° and continuously

rotated during evaporation (Figure 2d). Finally, the gold is removed from all horizontal surfaces

by ion beam etching (Figure 2e, 3c). With respect to the comparability between our theoretical

study and our sample, the dimensions of the fabricated sample were chosen to match the

dimensions of the theoretical model. We demonstrated the process on 100 mm fused silica wafer

with a patterned area of one cm². For applications requiring larger areas, our described method

and the used tools easily allow the fabrication of multiple dm² sized substrates.

To characterize the fabricated samples and to verify the proposed enhancement scheme

experimentally we performed optical measurements of the linear reflection spectra and the SH

enhancement τ (Figure 4). For the SH enhancement we measured the SH signal from the

nanopatterned surface and compared it to the measured SH signal of the unpatterned substrate.

We utilized a tunable Spectra-Physics Mai Tai Ti:Sapphire Laser source. In the appropriate

wavelength range, the laser delivers pulses with a duration of about 300 fs. In the whole

9

wavelength range, the average power was set to 10 mW. The beam was focused on the surface

using a lens with a focal length of 30 cm. The resulting spot size (full width half maximum) was

about 0.1 mm². This corresponds to a maximum intensity of 4 GW/cm² and an average fluence

of 10 µJ/cm². The electric field vector was parallel to the z-axis of the LN crystal – the same

configuration as in the simulation.

We tilted the sample by 45° to measure the signal originating from the nanostructured surface

(denoted as “FF1 + SH1” in Figure 4) and separate it from the bulk signal. At this angle, the bulk

signal obtained upon propagation through the substrate and possible multiple back reflections at

the substrate-air interfaces is bypassing the nanoring array (inset Figure 4). This ensures that no

additional SH signals are superimposing the SH signal originating from the nanostructured

surface.

The FF signals for the linear reflection spectrum and the SH signals for the SH enhancement

were measured with the same setup – in particular with the same detector, the same laser source

and the same input intensity. Except for the optical filters, which are placed between the sample

and the detector. For detecting FF signals, we used neutral density filters just to reduce the

detection signal. For detecting the SH signals, chromatic filters are used to block all wavelengths

except the corresponding SH wavelength. As a proof we checked the input power dependence of

the signals for each wavelength: The FF signals were proportional to the input power and the SH

signals increased quadratically by increasing the input power.

The input beam passed an optical chopper. The choppered signal was detected using a

Hamamatsu Photosensor Module H5784-20 and the Stanford Research Lock-In Amplifier

SR850. The spectra (Figure 4) were measured for discrete wavelengths by scanning the center

10

wavelength of the laser pulse with a 5 nm step size, which is less than the spectral bandwidth of

the laser.

The measured linear spectrum indicates a plasmonic resonance at a frequency of 365 THz

(820 nm wavelength). At this resonance, 55% of the incoming light is reflected. This value is

close to the simulated spectrum. The resonance position is slightly shifted by about 40 THz to

higher frequencies. This minor shift may be attributed to geometrical and dispersive deviations to

the model used in our theoretical study. Most notably, the real shape of the fabricated sample

does not perfectly match the ideal shape of the model. For example, the footing, which is clearly

visible in figures 3b, is not considered in our simulations.

The SH enhancement in the spectral range of the local plasmon resonance, which was found in

the simulation, is experimentally confirmed. At 350 THz (860 nm wavelength) the SH

enhancement reaches a maximum of τmax = 18, which is close to the expected value of 23

determined by our simulations. These deviations may also be attributed to geometrical properties

of the fabricated sample - in particular, sharp edges that are assumed in simulation but rounded

in the sample produce stronger local field enhancement. Thus, a higher SH enhancement is

expected from the simulation. Furthermore, the small shift of 15 THz (40 nm wavelength

difference) between the maxima in the linear reflection and in the SH enhancement factor, which

was observed in the simulation, appears in the experiment as well.

From our experimental and theoretical results we can conclude, that the crystallinity of LN

inside the rings is in fact preserved during the process. Otherwise, τmax is expected to be less than

five because the nonlinearity is decreased considerably upon transition from crystalline to the

noncrystalline regime. We also verified that the introduced and performed process is adequate

for encapsulation of LN with gold nanorings to enhance SHG plasmonically.

11

In conclusion, we proposed gold nanorings filled with crystalline LN as a nano-optical device

to enhance SHG at NIR frequencies. Like self-organized core-shell particles, these nanorings

provide the unique advantage of a pronounced field enhancement inside their core, which

significantly increases SHG. We developed a fabrication process to provide an efficient and

reproducible fabrication technique - a necessity for many applications - which allows the

deterministic fabrication of LN-filled nanorings. This is achieved by utilizing methods like

character projection EBL, ion beam enhanced etching, and double patterning. Through our

theoretical study, we confirmed our presumption that filling the nanorings with LN is crucial for

getting a significant enhancement. Measurements of the linear reflection spectra as well as of the

SH signal were performed. They are in good agreement with the numerical simulations, hence,

verifying our numerical results and the adequateness of the fabrication process. The

measurements of the SH enhancement confirmed the theoretically predicted τmax of about 20 for

tilted illumination. At normal incidence, one can expect τmax to reach values of 50 to 60.

Moreover, the implemented fabrication process is well suited to perform an effective fabrication

with an efficiency of dm² in hours and allows tuning of the geometrical parameters over a wide

range23,24. Even fabrication of nanorings with spatially varying geometrical properties side-by-

side is generally possible. We are confident that our results will directly contribute to the

development of efficient nanoscopic localized light sources employing enhanced SHG in sub-

wavelength scale volumes. Beyond that, it is possible to fill the rings with other polarizable

materials than LN, which we assume allows enhancing light-matter interactions at the nanoscale

for a broad range of applications.

12

Figure 1. Simulated linear and nonlinear optical properties of periodically arranged gold

nanorings with LN inside their core for z-polarized plane wave excitation (E0). a) Z-component

of the electric field at resonance frequency normalized to electric field of the illuminating wave.

The cross-sections intersect the center point of the unit cell. Incident frequency: 315 THz, launch

13

angle: 0°; b) Reflection spectrum for normal and oblique incidence; c) SH enhancement factor τ

for normal and oblique incidence, as well as with and without nonlinearity inside the core.

Figure 2. Sketches of intermediate fabrication steps for the fabrication of LN-filled nanorings; a)

Resist pillars after EBL; b) Cr / SiO2 pillars after ICP-RIE as mask for IBEE; c) Pillars of

crystalline LN; d) LN pillars coated with gold; e) LN pillars encapsulated with a gold ring after

removing gold from all horizontal surfaces by IBE.

Figure 3. a) SEM Images showing the mask for IBEE (Cr / SiO2 pillars); b) LN pillars after

IBEE; c) Final pattern: LN pillars encapsulated with a gold ring.

14

Figure 4. Measured SH enhancement factor and linear reflection spectrum of the fabricated

sample. The inset depicts the chosen measurement geometry. The illuminating beam is

impinging at an angle of 45°. This way, the surface reflection containing the SH signal of the

nanostructure is spatially isolated from the bulk signal which is contained in higher order

backside reflections bypassing the nanoring array.

AUTHOR INFORMATION

Corresponding Author

*Email: dennis.lehr@uni-jena.de

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval

to the final version of the manuscript. The authors contributed equally.

Funding Sources

15

The authors gratefully acknowledge support by the German Federal Ministry of Education and

Research (PhoNa 03IF2101A, OpMiSen 16SV5577) and by German Research Foundation (PE

1524/5-2). C. Menzel gratefully acknowledges financial support by the Carl Zeiss Stiftung.

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENT

We wish to thank D. Voigt, N. Sergeev, W. Gräf, M. Banasch, H. Schmidt, H.-J. Fuchs, W.

Rockstroh, S. Ratzsch and T. Käsebier for their assistance during sample fabrication.

ABBREVIATIONS

LN, Lithium niobate; SHG, second harmonic generation; NIR, near infrared; CP, character

projection; EBL, electron beam lithography; DP, double patterning; IBEE, Ion Beam Enhanced

Etching; τ, SH enhancement factor; SiO2, Silicon Dioxide; Cr, Chromium; ICP-RIE, inductively

coupled plasma reactive ion etching; FF, fundamental frequency; SH, second harmonic.

REFERENCES

[1] Franken, P.A.; Hill, A.E. Peters, C.W.; Weinreich, G. Phys. Rev. Lett. 1961, 7, 118-119.

[2] Yamamoto, K. Second Harmonic Generation (Waveguide). In Encyclopedic Handbook of

Integrated Optics; Iga, K., Kokubun, Y., Eds.; CRC Press, 2010; pp 359-371.

[3] Dutto, F.; Heiss, M.; Lovera, A.; López-Sánchez, O.; Fontcuberta i Morral, A.; Radenovic,

A. Nano Lett. 2013, 13, 6048-6054.

[4] Genevet, P.; Tetienne, J.-P.; Gatzogiannis, E.; Blanchard, R.; Kats, M. A.; Scully, M. O.;

Capasso, F. Nano Lett. 2010 10, 4880-4883.

16

[5] Pantazis, P.; Maloney, J.; Wu, D.; Fraser, S. E. PNAS 2010, 107, 14535-14540.

[6] Castelletto, S. A.; Scholten, R. E. Eur. Phys. J.-Appl. Phys 2008, 41, 181-194.

[7] Metzger, B.; Hentschel, M.; Lippitz, M.; Giessen, H. Opt. Lett. 2012, 37, 4741-4743.

[8] Reinhold, J.; Shcherbakov, M.R.; Chipouline; A.; Panov, V.I.; Helgert; C.; Paul, T.;

Rockstuhl, C.; Lederer, F.; Kley, E.-B.; Tünnermann, A.; Fedyanin, A.A.; Pertsch, T. Phys. Rev.

B. 2012, 86, 115401.

[9] Niesler, B.P.F.; Feth, N.; Linden, S.; Wegener, M. Opt. Lett. 2011, 36, 1533-1535.

[10] Sohler, W.; Hu, H.; Ricken, R.; Quiring, V.; Vannahme, C.; Herrmann, H.; Büchter, D.;

Reza, S.; Grundkötter, W.; Orlov, S.; Suche, H.; Nouroozi, R.; Min, Y. Opt. Photonics News

2008, 19, 24-31.

[11] Arizmendi, L. Phys. status solidi 2004, 201, 253–283.

[12] Barakat, E.; Bernal, M.-P.; Baida, F. I. J. Opt. Soc. Am. B 2013, 30, 1975–1980.

[13] Bar-Lev, D.; Scheuer, J. Opt. Express 2013, 21, 29165-29178.

[14] Niesler, F. B. P.; Feth, N.; Linden, S.; Niegemann, J.; Gieseler, J.; Busch, K.; Wegener,

M. Opt. Lett. 2009, 34, 1997-1999.

[15] Abb, M.; Albella, P.; Aizpurua, J.; Muskens, O. L. Nano. Lett. 2011, 11, 2457-2463.

[16] Barakat, E.; Bernal, M.-P.; Baida, F. I. Opt. Express 2010, 18, 6530-6536.

[17] Pu, Y.; Grange, R.; Hsieh, C.-L.; Psaltis, D. Phys. Rev. Lett. 2010, 104, 207402.

[18] Zhang, Y.; Grady, K. G.; Ayala-Orozco, C.; Halas, N. J. Nano. Lett. 2011, 11, 5519-5523.

17

[19] Richter, J.; Steinbrück, A.; Pertsch, T.; Grange, R. Plasmonics 2013, 8, 115-120.

[20] Yust, B. G.; Razavi, N.; Pedraza, F.; Elliott, Z.; Tsin, A. T.; Sardar, D. K. Opt. Express

2012, 20, 26511-26519.

[21] Aizpurua, J.; Hanarp, P.; Sutherland, D. S.; Käll, M.; Bryant, G. W.; Garcia de Abajo, F.

J. Phys. Rev. Lett. 2003, 90, 057401.

[22] Kley, E.-B.; Zeitner, U.; Banasch, M.; Schnabel, B. Proc. SPIE 2012, 8352, 83520M.

[23] Lehr, D.; Alaee, R.; Filter, R.; Dietrich, K.; Siefke, T.; Rockstuhl, C.; Lederer, F.; Kley,

E.-B.; Tünnermann. A. Appl. Phys. Lett. 2014, 105, 143110.

[24] Lehr, D.; Dietrich, K.; Helgert, C.; Käsebier, T.; Fuchs, H.-J.; Tünnermann, A.; Kley, E.-

B. Opt. Lett. 2012, 37, 157-159.

[25] Oskooi, Ardavan F.; Roundy, David; Ibanescu, Mihai; Bermel, Peter; Joannopoulos, J.D.;

Johnson, Steven G. Comput. Phys. Commun. 2010, 181, 687-702.

[26] Butcher, P.N.; Cotter, D. The elements of Nonlinear Optics; Cambridge University Press,

1991.

[27] Boyd, R. W. Nonlinear Optics; Elsevier/Academic Press: Amsterdam, 2008.

[28] Nakagawa, W.; Tyan, R.-C.; Fainman, Y. J. Opt. Soc. Am. A 2002, 12, 1919-1928.

[29] Paul, T.; Rockstuhl, C.; Lederer, F. Opt. Soc. Am. B 2010, 27, 1118-1130.

[30] Bai, B.; Turunen, J. J. Opt. Soc. Am. B 2007, 24, 1105-1112.

[31] Johnson, P. B.; Christy, R. W., Phys. Rev. B 1972, 6, 4370-4379.

18

[32] Menzel, C.; Hebestreit, E; Mühlig, S.; Rockstuhl, C.; Burger, S.; Lederer, F.; Pertsch, T.

Opt. Expr. 2014, 22, 9971-9982.

[33] Schrempel F.; Gischkat, T.; Hartung, H.; Kley, E.-B.; Wesch, W. Nucl. Instrum. Methods

Phys. Res., Sect. B 2006, 250, 164–168.

[34] Geiss, R.; Brandt, J.; Hartung, H.; Tünnermann, A.; Pertsch, T.; Kley, E.-B.; Schrempel,

F. J. Vac. Sci. Technol. B 2015, 33, 010601.