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OPTICS LETTERS / Vol. 39, No. 20 / October 15, 2014
Characterization and χ(3) measurements of thin films by third-harmonic microscopy Cristina Rodríguez* and Wolfgang Rudolph Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA *Corresponding author: [email protected] Received August 19, 2014; accepted September 8, 2014; posted September 22, 2014 (Doc. ID 221272); published October 15, 2014 Third-harmonic (TH) generation from a thin layer on a substrate is analyzed in reflection and transmission geometry taking into account interference effects of fundamental and TH waves in the film, the backward-generated TH, and the pump-beam profile. Conditions are derived for both geometries where the signal from the film dominates, which is important for TH microscopy. The analysis results are applied to retrieve from experiment nonlinear susceptibilities χ 3 of hafnia/silica mixture (Hf x Si1−x O2 ), alumina (Al2 O3 ), and scandia (Sc2 O3 ) thin films. © 2014 Optical Society of America OCIS codes: (180.4315) Nonlinear microscopy; (310.0310) Thin films. http://dx.doi.org/10.1364/OL.39.006042
Since the introduction of third-harmonic (TH) microscopy [1] this technique has found applications ranging from the imaging of living biological cells [2] to the characterization of laser-induced plasmas [3] and morphology of optical films [4]. TH microscopy is particularly suited to visualize interfaces between materials that differ in their linear (refractive index) and nonlinear (third-order susceptibility χ 3 ) optical properties [1]. The interpretation of images of thin samples, e.g., films or membranes, is hampered by the fact that the TH signal, often detected in transmission, is a sum of fields from the layer of interest and the surrounding material within the focal volume including the substrate. It has been suggested that by detecting the TH in reflection the contribution from the substrate could be considerably suppressed [5]. When using this detection geometry, however, special attention has to be paid to the fact that not only forward-generated TH contributes to the total signal measured, but also backward-generated TH [6] can have a significant contribution. Due to the suitability of TH microscopy for material characterization [7], several studies have dealt with the problem of determining χ 3 of materials from TH measurements. There are two common ways of doing this. The first one involves comparing the measured and simulated ratio of TH generated from two interfaces, involving a layer of a material with unknown χ 3 , e.g., [8,9]. The second one entails measuring the absolute TH conversion efficiency, and comparing it to the predicted value, assuming the TH from the materials surrounding the layer of interest can be neglected [10]. Among these studies, some considered the importance of multiple reflections of fundamental and TH waves inside the film [11,12] for the case of incident plane waves. In this Letter we analyze the relative TH contribution of a thin film and the substrate in both transmission and reflection geometry and define parameter ranges in which the signal from the film dominates. To this end we take into account (i) the beam profile of the focused input field, (ii) interference effects of both fundamental and TH waves in the film, and (iii) forward- and backward-generated TH. We apply our model to determine 0146-9592/14/206042-04$15.00/0
nonlinear optical susceptibilities χ 3 of various dielectric films from TH measurements. A schematic diagram of TH signal generation and detection is depicted in Fig. 1. A laser beam (incident electric field F 0 ) is focused onto a film (medium 1) deposited on a substrate (medium 2) and the TH signals are detected in transmission (T) and reflection (R) geometry after appropriate filtering using photomultiplier tubes. The field transmission and reflection coefficients, refractive indices, and wave vectors for the fundamental (TH) are tij , r ij , nj , and kj (τij , ρij , ηj , and κ j ), where i, j 0, 1, 2 refers to air, film, and substrate, respectively. The TH fields propagating to the left and right, E l and E r , are produced by nonlinear conversion in the film only, while the fields E R and E T also include contributions from the substrate. A general matrix approach can be used to treat nonlinear optical signal conversion of focused fundamental beams in a stack of layers of different optical properties taking into account multiple reflections of the interacting fields [13]. Here we will use these results for the special case of a single film of thickness much smaller than the Rayleigh range 2d ≪ z0 , and a substrate of thickness L ≫ z0 , and assume an incident Gaussian beam of beam waist w0 . Because of this geometry we can neglect the contribution of the field reflected at the output face of the substrate to E R . The TH field E T in the spatial frequency (ρ) domain is the sum of the field produced in the film, E r , multiplied by the spatial frequency spectrum and the Fresnel propagator, and the field produced in the front surface of the substrate [13]:
Fig. 1. TH generation in a thin film (1) on a semi-infinite substrate (2) in reflection and transmission from an incident fundamental field F 0 . © 2014 Optical Society of America
October 15, 2014 / Vol. 39, No. 20 / OPTICS LETTERS
πw20 −π2 w2 ρ2 ∕3 iaρ2 L 0 e e Er 3 ZL −iΔk−2 z iaρ2 L−z ; −b dzP 2 ρ; ze e
E l Hτ10
E T ρ τ20
(1)
−2iΔϕ1 d 3r 12 e−4ik1 d e−iΔϕ1 d sincΔϕ 1 d1 ρ01 r 12 e
−
3r 12 e−4ik1 d e−iΔϕ1 d sincΔϕ−1 dr 12 e−2iΔϕ1 d ρ12 e−4iκ1 d
− where a 2π 2 ∕κ 2 , b 9iω2 ∕2c2 κ 2 χ 3 2 , Δk2 3k2 − κ 2 is the wave-vector mismatch of copropagating fundamental and TH waves in the substrate, and P 2 ρ; z HTfF 3 r; zg is the Hankel transform of the nonlinear polarization in the substrate. Since 2d ≪ z0 , we may also neglect the ρ dependence of the phase of fundamental and harmonic fields introduced by the film [13]. The latter implies that for each spectral frequency component ρ the thin-film response is that for a normally incident plane wave. The fundamental field after the film subject to Fresnel 2 2 2 2 diffraction is Fρ; z F 0 t01 t12 e−2ik1 d πw20 e−π w0 ρ eiaρ z−d ∕ −4idk 1 . 1 r 01 r 12 e The TH amplitude after the film can be written as [13]
Er Hτ12 e−2iκ1 d
−
× fe−iΔk1 d sincΔk−1 d1 − ρ01 r 312 e−2iΔk1 d
−
−2iΔϕ1 d 3r 12 e−4ik1 d e−iΔϕ1 d sincΔϕ − ρ01 1 dr 12 e
−
3r 12 e−4ik1 d e−iΔϕ1 d sincΔϕ−1 d1 − ρ01 r 12 e−2iΔϕ1 d −
3 −2iΔk1 d e−iΔk1 d sincΔk − ρ01 g; 1 dr 12 e
(2)
where
H
−
× fe−iΔk1 d sincΔk−1 dr 312 e−2iΔk1 d ρ12 e−4iκ1 d
0
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3 −9iω2 χ 3 1 dt01 F 0 ; 2 −4idκ 1 κ 1 c 1 ρ01 ρ12 e 1 r 01 r 12 e−4idk1 3
3 −2iΔk1 d e−iΔk1 d sincΔk g: 1 d1 ρ12 r 12 e
(5)
Depending on the reflection coefficients at the film interfaces, some of the terms in Eqs. (2) and (5) can be neglected. This model can be used to determine χ 3 values of films by comparingR the measured TH power in transmission or reflection, jE R;T ρj2 ρdρ, to that obtained from the substrate alone for which the third-order susceptibility is known. The measurements were performed with the pulse train from a titanium:sapphire laser oscillator (790 nm, 50 fs, 113 MHz, linear polarization) focused onto the sample with a UV aspheric lens (NA 0.5). The samples were scanned through the focus of the incident beam and the TH signals in reflection and transmission were spectrally filtered and detected with photomultiplier tubes (cf. Fig. 1). The results presented here represent the maximum TH obtained from the longitudinal scans. Several high-quality dielectric films (highly uniform and with scattering well below 1%) on fused silica were studied including ternary oxides Hf x Si1−x O2 , with different composition x [14]. The measurement results are shown in Fig. 2(a), and the retrieved nonlinear susceptibilities in Fig. 2(b). The latter are summarized in Table 1. The known jχ 3 2 j of fused silica,
(3)
∓ and Δk∓ 1 3k1 ∓κ 1 (Δϕ1 k1 ∓κ 1 ) is the wave-vector mismatch for the forward- (−) and backward- () generated TH resulting from the mixing of co- (counter-) propagating fundamental fields in the film. A similar procedure is used to obtain the total TH field ER propagating to the left. Again, because of 2d ≪ z0 , for each spectral frequency component the nonlinear film response is that for a normally incident plane wave. The field E R ρ is the sum of the TH produced in the film and backward-generated TH in the substrate and transmitted through the film [13]:
E R ρ
πw20 −π2 w2 ρ2 ∕3 τ21 τ10 e−2iκ1 d 0 e El 3 1 ρ01 ρ12 e−4iκ2 d ZL 2 ×b dzP 2 ρ; ze−iΔk2 z eiaρ d−z ; 0
(4)
where Δk 2 3k2 κ 2 is the wave-vector mismatch between the fundamental and TH waves for the backward-generated TH in the substrate. The TH amplitude E l is given by [13]
Fig. 2. (a) Measured TH signal from Hf x Si1−x O2 films sputtered on fused silica as a function of x. For x 1 the signal in transmission was about 20 times larger than that in reflection. (b) Measured ratio of third-order susceptibility of film and fused silica substrate as a function of the bandgap of the film material. The solid line is from a model explained in the text. The Hf x Si1−x O2 films represented half-wave layers at 1064 nm.
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OPTICS LETTERS / Vol. 39, No. 20 / October 15, 2014
Table 1. Measured Third-Order Susceptibilities of Dielectric Films Normalized to That of the Fused Silica 3 a Substrate, jR12 j jχ 3 1 ∕χ 2 j for λ 790 nm Material Al2 O3 Sc3 O2 HfO2 Hf 0.66 Si0.34 O2 Hf 0.3 Si0.7 O2 Hf 0.02 Si0.98 O2 SiO2
2d (nm) 100 206 267 293 326 354 bulk
n 1.76 1.97 2.09 1.86 1.64 1.52 1.45
η 1.83 2.22 2.30 2.00 1.72 1.53 1.50
Eg (eV) 6.2 5.7 5.6 5.9 6.3 7.5 8.3
jR12 j 8.2 17.8 20.9 9.5 3.2 0.7 1
a Film thicknesses, refractive indices for the fundamental and TH wavelengths, and the material bandgap Eg are also shown.
2.0 0.2 × 10−22 m2 ∕V2 [15], can be used to obtain numerical χ 3 values for the films. Because of the short interaction length and moderately short pulses we can neglect dispersive effects and apply the model introduced above. As the HfO2 content in the samples decreases the contribution from the substrate to the transmitted TH becomes more important, until, for the sample with the lowest HfO2 content (1.6%), the detected signal matches to within a few percent the TH generated from the substrate (fused silica) alone [cf. Fig. 2(a)]. Roughly speaking, the contribution from the substrate to the total TH detected in transmission can be neglected if 3 3 0 0 χ 3 1 d ∕χ 2 z0 ≫ 1, where the products χ 1 d and 3 χ 2 z0 are an approximate measure of the magnitude of the signal generated in the film and substrate, respectively. Here, d0 is the smaller of the film thickness and the coherence length for forward-generated TH in the film. This is the desirable parameter range for TH microscopy of, for example, thin cell samples on substrates. The contribution of the substrate to the measured TH is considerably reduced when detecting in reflection and can safely be neglected for the ternary oxide films if x ≥ 0.2. In more general terms, the signal from the film dominates if χ 3 z1 ∕χ 3 2 z2 ≫ 1, where the products 3 31 χ 1 z1 and χ 2 z2 are an approximate measure of the magnitude of the signal generated in the film and substrate, respectively. Here z1 (z2 ) is the bigger (smaller) of d0 ρ12 (Rayleigh range) and the coherence length for backward-generated TH in the film (substrate). The coherence length for forward- (−) and backward- () generated TH in the film is d∓ coh λ∕6∕n2 ∓η2 . This condition assumes that forward-generated TH in the substrate and reflected at the substrate’s backsurface does not contribute to the measured signal. To ensure this in the experiment, we placed our samples (film on 1-mm substrates) on fused silica blocks, using indexmatching oil opaque at 266 nm. Any forward-generated TH in the substrate that could potentially get reflected at the substrate’s backsurface is absorbed by the oil. To test this technique, we measured in reflection the TH from bare fused silica substrates of varying thickness and found the signal to be thickness independent. This is as expected when assuming that only backwardgenerated TH is detected. The data in Fig. 2(b) are plotted as a function of the bandgap E g of the film. The solid line represents results
from a simple dispersion model for nonlinear susceptibilities [16], χ 3 ∝
E2g
−
E23
1 ; − 2iℏE3 γE2g − E21 − 2iℏE1 γ3
(6)
where En nhν and γ is a phenomenological damping coefficient. Good agreement of model and experiment was obtained for 2iℏEn γ ≪ E2g − E2n , meaning 1∕γ ≳ 1 fs. The latter is reasonable considering the off-resonant character of the interaction with the wide-gap materials (3hν < E g ). As a test and to illustrate the effect of various model components, using the values of jχ 3 1 j for the Hf x Si1−x O2 films obtained from the transmission data, we calculated the TH signal in reflection as a function of x using Eq. (4), with and without taking into account the backwardgenerated TH in both film and substrate [see Fig. 3(a)]. There is good agreement of model and experimental data only if backward-generated TH is considered. The experimentally obtained ratio of signals detected in reflection between the sample with x 10 and a bare substrate was 1.6, which shows the non-negligible contribution of the backward-generated TH from the substrate. Consequently, when detecting in reflection, the backward TH is not always negligible. In thin films, where the reflected forward and backward components interfere with each other, varying the film thickness on the order of the coherence length d coh can increase the contribution of the
Fig. 3. (a) Comparison of measured and calculated TH generation in reflection from Hf x Si1−x O2 films on fused silica substrates shown as a function of x with and without the backward-generated TH (BTH) taken into account. (b) Measured TH signal in reflection from HfO2 films on fused silica as a function of film thickness and comparison to the model with and without film interference (F–I) effects. The calculated TH is normalized to minimize the mean-square deviation with respect to the experimental data.
October 15, 2014 / Vol. 39, No. 20 / OPTICS LETTERS
backward component to the total reflected TH from a few percent to over 100% When detecting in transmission the backward component can always be neglected. In bulk materials, the backward-generated TH detected in reflection geometry is always much smaller than the forward-generated TH detected in transmission. Figure 3(b) shows the measured TH in reflection for HfO2 films of different thickness and model results using the χ 3 from Fig. 2(b). Obviously, the correct account of interference is necessary to reproduce the measurements. If this is done, the same χ 3 values (normalized to the value for the fused silica substrate) are obtained from transmission and reflection data. We also measured the TH signal in transmission and reflection produced by stacks of HfO2 and SiO2 films and found good agreement with the model [13], considering the uncertainties (≈5%) in the thickness of the individual layers. In summary, we analyzed TH generation from thin films in transmission and reflection. We applied a theoretical model that takes into account interference effects of fundamental and TH waves in the film, the backwardgenerated TH, and the pump-beam profile (focusing) to determine third-order nonlinear susceptibilities, χ 3 , of several oxide films from measurements. The bandgap dependence of χ 3 of these dielectric films agrees with predictions from a simple nonlinear oscillator model. We derived relations involving key parameters such as the nonlinear susceptibilities of film and substrate, the film thickness, the coherence length for forward- and backward-generated TH, the Rayleigh range of the incident beam in the substrate, and the reflection coefficient at the film-substrate interface, to determine when the TH from the film dominates that from the substrate. TH in reflection can considerably reduce the signal from the substrate compared to TH in transmission, which is particularly important for TH microscopy of thin structures (e.g., cell membranes) and the determination of nonlinear susceptibilities of thin films.
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We are grateful to Prof. C. Menoni (Colorado State University) and Prof. D. Ristau (Laser Zentrum Hannover eV) and their groups for providing high-quality thin-film samples. This work was supported by JTO/ONR (N00014-071-1068) and JTO/ARO (W911NF 11-1-007). References 1. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, Appl. Phys. Lett. 70, 922 (1997). 2. J. Squier, M. Muller, G. Brakenhoff, and K. R. Wilson, Opt. Express 3, 315 (1998). 3. C. Rodriguez, Z. Sun, Z. Wang, and W. Rudolph, Opt. Express 19, 16115 (2011). 4. R. A. Weber, C. Rodríguez, D. N. Nguyen, L. A. Emmert, D. Patel, C. Menoni, and W. Rudolph, Opt. Eng. 51, 121807 (2012). 5. G. Berkovic, Chem. Phys. Lett. 241, 355 (1995). 6. C. Chang, H. Chen, M. Chen, W. Liu, W. Hsieh, C. Hsu, C. Chen, F. Chang, C. Yu, and C. Sun, Opt. Express 18, 7397 (2010). 7. J. M. Schins, T. Schrama, J. Squier, G. J. Brakenhoff, and M. Müller, J. Opt. Soc. Am. B 19, 1627 (2002). 8. R. Barille, L. Canioni, L. Sarger, and G. Rivoire, Phys. Rev. E 66, 067602 (2002). 9. V. Shcheslavskiy, G. Petrov, and V. V. Yakovlev, Appl. Phys. Lett. 82, 3982 (2003). 10. G. I. Petrov, V. Shcheslavskiy, V. V. Yakovlev, I. Ozerov, E. Chelnokov, and W. Marine, Appl. Phys. Lett. 83, 3993 (2003). 11. D. Neher, A. Wolf, C. Bubeck, and G. Wegner, Chem. Phys. Lett. 163, 116 (1989). 12. F. Krausz, E. Wintner, and G. Leising, Phys. Rev. B 39, 3701 (1989). 13. C. Rodríguez and W. Rudolph, “Modeling third-harmonic generation from layered materials using nonlinear optical matrices,” Opt. Express (to be published). 14. L. O. Jensen, M. Mende, H. Blaschke, D. Ristau, D. Nguyen, L. Emmert, and W. Rudolph, Proc. SPIE 7842, 784207 (2010). 15. U. Gubler and C. Bosshard, Phys. Rev. B 61, 10702 (2000). 16. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).
OPTICS LETTERS / Vol. 39, No. 20 / October 15, 2014
Characterization and χ(3) measurements of thin films by third-harmonic microscopy Cristina Rodríguez* and Wolfgang Rudolph Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA *Corresponding author: [email protected] Received August 19, 2014; accepted September 8, 2014; posted September 22, 2014 (Doc. ID 221272); published October 15, 2014 Third-harmonic (TH) generation from a thin layer on a substrate is analyzed in reflection and transmission geometry taking into account interference effects of fundamental and TH waves in the film, the backward-generated TH, and the pump-beam profile. Conditions are derived for both geometries where the signal from the film dominates, which is important for TH microscopy. The analysis results are applied to retrieve from experiment nonlinear susceptibilities χ 3 of hafnia/silica mixture (Hf x Si1−x O2 ), alumina (Al2 O3 ), and scandia (Sc2 O3 ) thin films. © 2014 Optical Society of America OCIS codes: (180.4315) Nonlinear microscopy; (310.0310) Thin films. http://dx.doi.org/10.1364/OL.39.006042
Since the introduction of third-harmonic (TH) microscopy [1] this technique has found applications ranging from the imaging of living biological cells [2] to the characterization of laser-induced plasmas [3] and morphology of optical films [4]. TH microscopy is particularly suited to visualize interfaces between materials that differ in their linear (refractive index) and nonlinear (third-order susceptibility χ 3 ) optical properties [1]. The interpretation of images of thin samples, e.g., films or membranes, is hampered by the fact that the TH signal, often detected in transmission, is a sum of fields from the layer of interest and the surrounding material within the focal volume including the substrate. It has been suggested that by detecting the TH in reflection the contribution from the substrate could be considerably suppressed [5]. When using this detection geometry, however, special attention has to be paid to the fact that not only forward-generated TH contributes to the total signal measured, but also backward-generated TH [6] can have a significant contribution. Due to the suitability of TH microscopy for material characterization [7], several studies have dealt with the problem of determining χ 3 of materials from TH measurements. There are two common ways of doing this. The first one involves comparing the measured and simulated ratio of TH generated from two interfaces, involving a layer of a material with unknown χ 3 , e.g., [8,9]. The second one entails measuring the absolute TH conversion efficiency, and comparing it to the predicted value, assuming the TH from the materials surrounding the layer of interest can be neglected [10]. Among these studies, some considered the importance of multiple reflections of fundamental and TH waves inside the film [11,12] for the case of incident plane waves. In this Letter we analyze the relative TH contribution of a thin film and the substrate in both transmission and reflection geometry and define parameter ranges in which the signal from the film dominates. To this end we take into account (i) the beam profile of the focused input field, (ii) interference effects of both fundamental and TH waves in the film, and (iii) forward- and backward-generated TH. We apply our model to determine 0146-9592/14/206042-04$15.00/0
nonlinear optical susceptibilities χ 3 of various dielectric films from TH measurements. A schematic diagram of TH signal generation and detection is depicted in Fig. 1. A laser beam (incident electric field F 0 ) is focused onto a film (medium 1) deposited on a substrate (medium 2) and the TH signals are detected in transmission (T) and reflection (R) geometry after appropriate filtering using photomultiplier tubes. The field transmission and reflection coefficients, refractive indices, and wave vectors for the fundamental (TH) are tij , r ij , nj , and kj (τij , ρij , ηj , and κ j ), where i, j 0, 1, 2 refers to air, film, and substrate, respectively. The TH fields propagating to the left and right, E l and E r , are produced by nonlinear conversion in the film only, while the fields E R and E T also include contributions from the substrate. A general matrix approach can be used to treat nonlinear optical signal conversion of focused fundamental beams in a stack of layers of different optical properties taking into account multiple reflections of the interacting fields [13]. Here we will use these results for the special case of a single film of thickness much smaller than the Rayleigh range 2d ≪ z0 , and a substrate of thickness L ≫ z0 , and assume an incident Gaussian beam of beam waist w0 . Because of this geometry we can neglect the contribution of the field reflected at the output face of the substrate to E R . The TH field E T in the spatial frequency (ρ) domain is the sum of the field produced in the film, E r , multiplied by the spatial frequency spectrum and the Fresnel propagator, and the field produced in the front surface of the substrate [13]:
Fig. 1. TH generation in a thin film (1) on a semi-infinite substrate (2) in reflection and transmission from an incident fundamental field F 0 . © 2014 Optical Society of America
October 15, 2014 / Vol. 39, No. 20 / OPTICS LETTERS
πw20 −π2 w2 ρ2 ∕3 iaρ2 L 0 e e Er 3 ZL −iΔk−2 z iaρ2 L−z ; −b dzP 2 ρ; ze e
E l Hτ10
E T ρ τ20
(1)
−2iΔϕ1 d 3r 12 e−4ik1 d e−iΔϕ1 d sincΔϕ 1 d1 ρ01 r 12 e
−
3r 12 e−4ik1 d e−iΔϕ1 d sincΔϕ−1 dr 12 e−2iΔϕ1 d ρ12 e−4iκ1 d
− where a 2π 2 ∕κ 2 , b 9iω2 ∕2c2 κ 2 χ 3 2 , Δk2 3k2 − κ 2 is the wave-vector mismatch of copropagating fundamental and TH waves in the substrate, and P 2 ρ; z HTfF 3 r; zg is the Hankel transform of the nonlinear polarization in the substrate. Since 2d ≪ z0 , we may also neglect the ρ dependence of the phase of fundamental and harmonic fields introduced by the film [13]. The latter implies that for each spectral frequency component ρ the thin-film response is that for a normally incident plane wave. The fundamental field after the film subject to Fresnel 2 2 2 2 diffraction is Fρ; z F 0 t01 t12 e−2ik1 d πw20 e−π w0 ρ eiaρ z−d ∕ −4idk 1 . 1 r 01 r 12 e The TH amplitude after the film can be written as [13]
Er Hτ12 e−2iκ1 d
−
× fe−iΔk1 d sincΔk−1 d1 − ρ01 r 312 e−2iΔk1 d
−
−2iΔϕ1 d 3r 12 e−4ik1 d e−iΔϕ1 d sincΔϕ − ρ01 1 dr 12 e
−
3r 12 e−4ik1 d e−iΔϕ1 d sincΔϕ−1 d1 − ρ01 r 12 e−2iΔϕ1 d −
3 −2iΔk1 d e−iΔk1 d sincΔk − ρ01 g; 1 dr 12 e
(2)
where
H
−
× fe−iΔk1 d sincΔk−1 dr 312 e−2iΔk1 d ρ12 e−4iκ1 d
0
6043
3 −9iω2 χ 3 1 dt01 F 0 ; 2 −4idκ 1 κ 1 c 1 ρ01 ρ12 e 1 r 01 r 12 e−4idk1 3
3 −2iΔk1 d e−iΔk1 d sincΔk g: 1 d1 ρ12 r 12 e
(5)
Depending on the reflection coefficients at the film interfaces, some of the terms in Eqs. (2) and (5) can be neglected. This model can be used to determine χ 3 values of films by comparingR the measured TH power in transmission or reflection, jE R;T ρj2 ρdρ, to that obtained from the substrate alone for which the third-order susceptibility is known. The measurements were performed with the pulse train from a titanium:sapphire laser oscillator (790 nm, 50 fs, 113 MHz, linear polarization) focused onto the sample with a UV aspheric lens (NA 0.5). The samples were scanned through the focus of the incident beam and the TH signals in reflection and transmission were spectrally filtered and detected with photomultiplier tubes (cf. Fig. 1). The results presented here represent the maximum TH obtained from the longitudinal scans. Several high-quality dielectric films (highly uniform and with scattering well below 1%) on fused silica were studied including ternary oxides Hf x Si1−x O2 , with different composition x [14]. The measurement results are shown in Fig. 2(a), and the retrieved nonlinear susceptibilities in Fig. 2(b). The latter are summarized in Table 1. The known jχ 3 2 j of fused silica,
(3)
∓ and Δk∓ 1 3k1 ∓κ 1 (Δϕ1 k1 ∓κ 1 ) is the wave-vector mismatch for the forward- (−) and backward- () generated TH resulting from the mixing of co- (counter-) propagating fundamental fields in the film. A similar procedure is used to obtain the total TH field ER propagating to the left. Again, because of 2d ≪ z0 , for each spectral frequency component the nonlinear film response is that for a normally incident plane wave. The field E R ρ is the sum of the TH produced in the film and backward-generated TH in the substrate and transmitted through the film [13]:
E R ρ
πw20 −π2 w2 ρ2 ∕3 τ21 τ10 e−2iκ1 d 0 e El 3 1 ρ01 ρ12 e−4iκ2 d ZL 2 ×b dzP 2 ρ; ze−iΔk2 z eiaρ d−z ; 0
(4)
where Δk 2 3k2 κ 2 is the wave-vector mismatch between the fundamental and TH waves for the backward-generated TH in the substrate. The TH amplitude E l is given by [13]
Fig. 2. (a) Measured TH signal from Hf x Si1−x O2 films sputtered on fused silica as a function of x. For x 1 the signal in transmission was about 20 times larger than that in reflection. (b) Measured ratio of third-order susceptibility of film and fused silica substrate as a function of the bandgap of the film material. The solid line is from a model explained in the text. The Hf x Si1−x O2 films represented half-wave layers at 1064 nm.
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OPTICS LETTERS / Vol. 39, No. 20 / October 15, 2014
Table 1. Measured Third-Order Susceptibilities of Dielectric Films Normalized to That of the Fused Silica 3 a Substrate, jR12 j jχ 3 1 ∕χ 2 j for λ 790 nm Material Al2 O3 Sc3 O2 HfO2 Hf 0.66 Si0.34 O2 Hf 0.3 Si0.7 O2 Hf 0.02 Si0.98 O2 SiO2
2d (nm) 100 206 267 293 326 354 bulk
n 1.76 1.97 2.09 1.86 1.64 1.52 1.45
η 1.83 2.22 2.30 2.00 1.72 1.53 1.50
Eg (eV) 6.2 5.7 5.6 5.9 6.3 7.5 8.3
jR12 j 8.2 17.8 20.9 9.5 3.2 0.7 1
a Film thicknesses, refractive indices for the fundamental and TH wavelengths, and the material bandgap Eg are also shown.
2.0 0.2 × 10−22 m2 ∕V2 [15], can be used to obtain numerical χ 3 values for the films. Because of the short interaction length and moderately short pulses we can neglect dispersive effects and apply the model introduced above. As the HfO2 content in the samples decreases the contribution from the substrate to the transmitted TH becomes more important, until, for the sample with the lowest HfO2 content (1.6%), the detected signal matches to within a few percent the TH generated from the substrate (fused silica) alone [cf. Fig. 2(a)]. Roughly speaking, the contribution from the substrate to the total TH detected in transmission can be neglected if 3 3 0 0 χ 3 1 d ∕χ 2 z0 ≫ 1, where the products χ 1 d and 3 χ 2 z0 are an approximate measure of the magnitude of the signal generated in the film and substrate, respectively. Here, d0 is the smaller of the film thickness and the coherence length for forward-generated TH in the film. This is the desirable parameter range for TH microscopy of, for example, thin cell samples on substrates. The contribution of the substrate to the measured TH is considerably reduced when detecting in reflection and can safely be neglected for the ternary oxide films if x ≥ 0.2. In more general terms, the signal from the film dominates if χ 3 z1 ∕χ 3 2 z2 ≫ 1, where the products 3 31 χ 1 z1 and χ 2 z2 are an approximate measure of the magnitude of the signal generated in the film and substrate, respectively. Here z1 (z2 ) is the bigger (smaller) of d0 ρ12 (Rayleigh range) and the coherence length for backward-generated TH in the film (substrate). The coherence length for forward- (−) and backward- () generated TH in the film is d∓ coh λ∕6∕n2 ∓η2 . This condition assumes that forward-generated TH in the substrate and reflected at the substrate’s backsurface does not contribute to the measured signal. To ensure this in the experiment, we placed our samples (film on 1-mm substrates) on fused silica blocks, using indexmatching oil opaque at 266 nm. Any forward-generated TH in the substrate that could potentially get reflected at the substrate’s backsurface is absorbed by the oil. To test this technique, we measured in reflection the TH from bare fused silica substrates of varying thickness and found the signal to be thickness independent. This is as expected when assuming that only backwardgenerated TH is detected. The data in Fig. 2(b) are plotted as a function of the bandgap E g of the film. The solid line represents results
from a simple dispersion model for nonlinear susceptibilities [16], χ 3 ∝
E2g
−
E23
1 ; − 2iℏE3 γE2g − E21 − 2iℏE1 γ3
(6)
where En nhν and γ is a phenomenological damping coefficient. Good agreement of model and experiment was obtained for 2iℏEn γ ≪ E2g − E2n , meaning 1∕γ ≳ 1 fs. The latter is reasonable considering the off-resonant character of the interaction with the wide-gap materials (3hν < E g ). As a test and to illustrate the effect of various model components, using the values of jχ 3 1 j for the Hf x Si1−x O2 films obtained from the transmission data, we calculated the TH signal in reflection as a function of x using Eq. (4), with and without taking into account the backwardgenerated TH in both film and substrate [see Fig. 3(a)]. There is good agreement of model and experimental data only if backward-generated TH is considered. The experimentally obtained ratio of signals detected in reflection between the sample with x 10 and a bare substrate was 1.6, which shows the non-negligible contribution of the backward-generated TH from the substrate. Consequently, when detecting in reflection, the backward TH is not always negligible. In thin films, where the reflected forward and backward components interfere with each other, varying the film thickness on the order of the coherence length d coh can increase the contribution of the
Fig. 3. (a) Comparison of measured and calculated TH generation in reflection from Hf x Si1−x O2 films on fused silica substrates shown as a function of x with and without the backward-generated TH (BTH) taken into account. (b) Measured TH signal in reflection from HfO2 films on fused silica as a function of film thickness and comparison to the model with and without film interference (F–I) effects. The calculated TH is normalized to minimize the mean-square deviation with respect to the experimental data.
October 15, 2014 / Vol. 39, No. 20 / OPTICS LETTERS
backward component to the total reflected TH from a few percent to over 100% When detecting in transmission the backward component can always be neglected. In bulk materials, the backward-generated TH detected in reflection geometry is always much smaller than the forward-generated TH detected in transmission. Figure 3(b) shows the measured TH in reflection for HfO2 films of different thickness and model results using the χ 3 from Fig. 2(b). Obviously, the correct account of interference is necessary to reproduce the measurements. If this is done, the same χ 3 values (normalized to the value for the fused silica substrate) are obtained from transmission and reflection data. We also measured the TH signal in transmission and reflection produced by stacks of HfO2 and SiO2 films and found good agreement with the model [13], considering the uncertainties (≈5%) in the thickness of the individual layers. In summary, we analyzed TH generation from thin films in transmission and reflection. We applied a theoretical model that takes into account interference effects of fundamental and TH waves in the film, the backwardgenerated TH, and the pump-beam profile (focusing) to determine third-order nonlinear susceptibilities, χ 3 , of several oxide films from measurements. The bandgap dependence of χ 3 of these dielectric films agrees with predictions from a simple nonlinear oscillator model. We derived relations involving key parameters such as the nonlinear susceptibilities of film and substrate, the film thickness, the coherence length for forward- and backward-generated TH, the Rayleigh range of the incident beam in the substrate, and the reflection coefficient at the film-substrate interface, to determine when the TH from the film dominates that from the substrate. TH in reflection can considerably reduce the signal from the substrate compared to TH in transmission, which is particularly important for TH microscopy of thin structures (e.g., cell membranes) and the determination of nonlinear susceptibilities of thin films.
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We are grateful to Prof. C. Menoni (Colorado State University) and Prof. D. Ristau (Laser Zentrum Hannover eV) and their groups for providing high-quality thin-film samples. This work was supported by JTO/ONR (N00014-071-1068) and JTO/ARO (W911NF 11-1-007). References 1. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, Appl. Phys. Lett. 70, 922 (1997). 2. J. Squier, M. Muller, G. Brakenhoff, and K. R. Wilson, Opt. Express 3, 315 (1998). 3. C. Rodriguez, Z. Sun, Z. Wang, and W. Rudolph, Opt. Express 19, 16115 (2011). 4. R. A. Weber, C. Rodríguez, D. N. Nguyen, L. A. Emmert, D. Patel, C. Menoni, and W. Rudolph, Opt. Eng. 51, 121807 (2012). 5. G. Berkovic, Chem. Phys. Lett. 241, 355 (1995). 6. C. Chang, H. Chen, M. Chen, W. Liu, W. Hsieh, C. Hsu, C. Chen, F. Chang, C. Yu, and C. Sun, Opt. Express 18, 7397 (2010). 7. J. M. Schins, T. Schrama, J. Squier, G. J. Brakenhoff, and M. Müller, J. Opt. Soc. Am. B 19, 1627 (2002). 8. R. Barille, L. Canioni, L. Sarger, and G. Rivoire, Phys. Rev. E 66, 067602 (2002). 9. V. Shcheslavskiy, G. Petrov, and V. V. Yakovlev, Appl. Phys. Lett. 82, 3982 (2003). 10. G. I. Petrov, V. Shcheslavskiy, V. V. Yakovlev, I. Ozerov, E. Chelnokov, and W. Marine, Appl. Phys. Lett. 83, 3993 (2003). 11. D. Neher, A. Wolf, C. Bubeck, and G. Wegner, Chem. Phys. Lett. 163, 116 (1989). 12. F. Krausz, E. Wintner, and G. Leising, Phys. Rev. B 39, 3701 (1989). 13. C. Rodríguez and W. Rudolph, “Modeling third-harmonic generation from layered materials using nonlinear optical matrices,” Opt. Express (to be published). 14. L. O. Jensen, M. Mende, H. Blaschke, D. Ristau, D. Nguyen, L. Emmert, and W. Rudolph, Proc. SPIE 7842, 784207 (2010). 15. U. Gubler and C. Bosshard, Phys. Rev. B 61, 10702 (2000). 16. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).
