## Abstract

Harmonic generation by tightly-focused Gaussian beams is finding important applications, primarily in nonlinear microscopy. It is often naively assumed that the nonlinear signal is generated predominantly in the focal region. However, the intensity of Gaussian-excited electromagnetic harmonic waves is sensitive to the excitation geometry and to the phase matching condition, and may depend on quite an extended region of the material away from the focal plane. Here we solve analytically the amplitude integral for second harmonic and third harmonic waves and study the generated harmonic intensities vs. focal-plane position within the material. We find that maximum intensity for positive wave-vector mismatch values, for both second harmonic and third harmonic waves, is achieved when the fundamental Gaussian is focused few Rayleigh lengths beyond the front surface. Harmonic-generation theory predicts strong intensity oscillations with thickness if the material is very thin. We reproduced these intensity oscillations in glass slabs pumped at 1550nm. From the oscillations of the 517nm third-harmonic waves with slab thickness we estimate the wave-vector mismatch in a Soda-lime glass as Δ*k _{H}*= -0.249

*μm*

^{-1}.

© 2015 Optical Society of America

## 1. Introduction

In harmonic-generation microscopies [1-3], and in particular second harmonic (SH) [4-6] and third harmonic (TH) [7-9] microscopies, the harmonic wave is excited by a tightly focused Gaussian laser beam, and the image information is carried by the intensity variations as this beam is scanned through the sample. Harmonic generation by focused beams is also common in short-pulse measurements [10], and of course, in various wavelength conversion schemes. Several researchers extended the focused beam theory to include vector field excitation, either in a general context [11,12], or in relations to microscopy [13].

Although it is often assumed that the signal is generated near the focal region ([11] talks about inverse of the wave-vector mismatch as a measure for the near-focus interaction length), it is well known that harmonic wave generation is the result of coherent interaction over an extended volume, and depends on material properties (susceptibility and chromatic dispersion) and on excitation geometry. The integral predicting the amplitude of the harmonic wave, which is found in textbooks (e.g. [14], Eq. (2).10.11b), was studied already in the late 1960’s in a classic paper by Boyd and Kleinman [15] and later by Ward and New, who studied TH generation in various gases by a focused laser beam [16]. Ward and New introduced analytic solutions of the amplitude integrals for Bulk excitation configuration [16].

In this paper we study intensity oscillations of SH and TH as predicted by the amplitude integral in a Gaussian excitation geometry. We derived new analytic expressions for the amplitude integral for each of the two harmonic waves, and present harmonics intensity maps vs. integration limits using these expressions. The maps show how harmonics intensities depend on the integration limits in a finite medium. Our calculated maps show that generally, for both SH and TH waves and a typical negative wave-vector mismatch, maximum harmonic intensity is achieved at Bulk excitation (Gaussian beam focused midway between surfaces) if the window is thin (one Rayleigh length) and by Surface or near-surface excitation if the window is thick (few Rayleigh lengths). For these negative wave-vector materials at some *anti-resonant* thicknesses, high harmonics are practically not generated, regardless of focal plane position within the material. The theoretical expressions also show that the harmonic intensity is affected quite strongly by regions a few Rayleigh-lengths away from the focal plane, and contribute to strong oscillations in the intensity. Note that the intensity oscillations discussed here differ from the sub-wavelength oscillations of back-propagated TH waves predicted by Olivier and Beaurepaire [13].

We have compared our results with experiments, where we have reproduced TH intensity oscillations quite close to those predicted by the amplitude integral for relatively thin windows (window thickness on the order of one Rayleigh length). The period of these oscillations offers a good estimate to the wave-vector mismatch characteristic to the material used. The estimate can be refined by a more careful fit of the theoretical curve to the measured curve.

We start by evaluating the amplitude integrals for SH waves in section 2 and for TH waves in section 3 where we analyze two geometrical situations, whether the pump beam waist is at the surface of a slab of nonlinear material or at its center. We present experimental results in section 4 and then summarize.

## 2. Amplitude integral for Gaussian-excited second harmonic waves

The amplitude of a Gaussian-excited second harmonic wave at an axial plane ${z}_{2}$ is proportional to an amplitude integral ${J}_{2}(\Delta {k}_{H},{z}_{R},{z}_{1},{z}_{2})$ which is the result of the coherent buildup of the harmonic wave from the input facet ${z}_{1}$ up to ${z}_{2}$ [14,16]. The amplitude integral is given as ([14], Eq. (2).10.11b with small notation changes):

*b*in [14,16]), that relates to the beam waist radius ${w}_{0}$ by:

Two well defined and interesting cases of excitation geometry are described by a “Bulk” amplitude integral and by a “Surface” amplitude integral. In [12], these two geometries are termed “centered” and “offset”, respectively. For these two cases, the full thickness of the material, assumed to be shaped as a flat window, is $2\cdot {\zeta}_{2}\cdot {z}_{R}$. In the case of a Bulk amplitude integral, the Gaussian waist is located midway between the two window’s surfaces, and in the case of a Surface amplitude integral, the Gaussian waist is located at one of the window’s surfaces:

For infinite limits (${\zeta}_{2}=\infty $) the Bulk amplitude integral (Eq. (5)) evaluates as [12,16]:

Mathematically, we see two “steps” in ${J}_{2bulk}({z}_{R},u,\infty )$ at $\Delta k=0$ (or $u=0$). A $\pi \cdot {z}_{R}$ step at ${0}_{-}\to 0$, and another $\pi \cdot {z}_{R}$step at $0\to {0}_{+}$. For positive wave-vector mismatch ($\Delta k$or the dimensionless $u$ - associated with **anomalous** dispersion), the Bulk SH amplitude integral exponentially decays with increasing mismatch (cf. Figure 2).

The amplitude integral depends explicitly on four variables – the Rayleigh length (${z}_{R}$), the wave-vector mismatch (given here by the dimensionless parameter *u* for a fixed${z}_{R}$), and the two integration limits (${z}_{1},{z}_{2}$) (Eq. (1)). In the following we keep the Rayleigh length fixed at${z}_{R}=50\mu m$. We are left with three variables. Bulk amplitudes and Surface amplitudes, shown by Fig. 2, are special cases of only two variables as the lower integration limit ${z}_{1}$ is either zero or is symmetrical to the upper integration limit with respect to the focal plane. Similar to [19], we define a normalized window thickness (${L}_{wN}$) as

The curves of Fig. 3 display absolute value squared of Bulk amplitudes and Surface amplitudes. This time the wave-vector mismatch (*u*) is fixed and the variable is the normalized window thickness (given in these cases by${\zeta}_{2}$, cf. Equation (5) and Eq. (7)).

The curves in Fig. 3(a) clearly show strong intensity oscillations for “thin” windows (windows’ thickness on the order of one Rayleigh length). Observe the curves in “c” showing higher intensities generated in the Bulk configuration (vs. the Surface configuration) for a small positive wave-vector mismatch ($u=+1$).

The last illustrative figure for SH generated intensities in a Gaussian excitation configuration is Fig. 4. The figure presents three SH intensity maps with the integration limits as the horizontal and vertical axes. Note that the maps are symmetric with respect to the diagonals and one redundant half is preserved in the shown maps only for convenience and aesthetic reasons. The maps clearly show the geometrical location of maximum intensity. By adjusting the Rayleigh length to a given window thickness (or vice-versa) and by adjustingthe position of the focal plane (relative to, say, the front surface of the window), these maximum intensities may be reached. Looking at the top-left quarter of Fig. 4(a), for example, sitting at the first peak of Bulk excitation (i.e. along the diagonal red arrow) and moving the focal plane away from the window’s center (moving along the other diagonal, in the direction of the dashed line, keeping the total window thickness constant), the map indicates quick reduction in generated SH power. Such power reduction was indeed observed by Bjorkholm during his 1965 SH experiments in ADP crystals [20].

## 3. Amplitude integral for Gaussian-excited third harmonic waves

The third harmonic amplitude integral ${J}_{3}(\Delta {k}_{H},{z}_{R},{z}_{1},{z}_{2})$ ([14], Eq. (2).10.11b with small notation changes) is given as:

As it turns out, divided into purely real and purely imaginary integrals, the amplitude integral of the third harmonic wave of Eq. (9) also has an analytic solution in terms of the $\u2102i(u,\zeta )$ and $\mathbb{S}i(u,\zeta )$ functions.

Bulk TH amplitude and Surface TH amplitude are defined as in Eq. (5) with ${J}_{3}$ replacing${J}_{2}$. For infinite integration interval (${\zeta}_{2}=\infty $), the TH Bulk integral evaluates as [12,16]:

As for the Surface integral, given an infinite medium, we find for perfect wave-vector match ($u=0$) (cf. (Fig. 5)):

For positive wave-vector mismatch $\Delta {k}_{H}$ (associated with **anomalous** dispersion), the Bulk TH amplitude integral peaks at $u=1$ giving a max value of $(2\cdot \pi /e)\cdot {z}_{R}\simeq 2.3\cdot {z}_{R}$ (Fig. 5). Compare this max of $2.3\cdot {z}_{R}$ for the TH amplitude integral with the $2\cdot \pi \cdot {z}_{R}$ for the max SH amplitude integral (at $u={0}_{+}$).

Graphical illustrations for TH intensities are given by Figs. 5-7. An important difference between SH intensities and TH intensities is the location of maximum intensity with respect to the wave-vector mismatch. While for SH waves and thick windows, maximum intensity occurs at perfect wave-vector match ($u=0$, Fig. 2(a)), TH wave intensity peaks at small positive wave-vector mismatch ($u=1$,[14] - Fig. 2.10.2, and Fig. 5(a)).

## 4. Measured intensity oscillations of third harmonic waves

Third harmonic generation was studied in Soda-lime glass by focusing 1550nm short pulses of about 1picosecond in duration and around one µJ each at a repetition rate of 500kHz. The glass target was shaped as a wedged window, about 80 µm at its thinnest end, to allow easytesting of sample thickness effects (Fig. 8(a)). The linearly polarized laser beam was focused by a X10 objective and generated a beam with an estimated Rayleigh range (from measured input beam diameter and objective’s focal length) to be about 100µm. Generated green-color TH pulses followed the pump polarization and were conveniently visible at a wavelength of 517nm. TH intensity was measured using a standard power meter (Ophir LaserStar) after collimating the output beam and filtering it by a low-pass filter and a prism that completely eliminated the pump beam.

In Fig. 8(b) we show the power dependence of the TH that, as expected, follows a cubic power-law (dashed curve adjusted only at a single data point). The highest conversion efficiency (${P}_{TH}/{P}_{pump}$) was measured to be ~2.10^{-6}.

Our main experimental result is displayed by the curves of Fig. 9(a). To generate these curves we manually translated the wedge twice in the $x$-direction and recorded each time the detected TH power vs. the $x$ position. One time with the exciting beam focused midway between the two wedge surfaces (Bulk), and a second time with the exciting beam focused at the front surface of the wedge (Surface). The measured curves of Fig. 9(a) are juxtaposed with the calculated curves of Fig. 9(b) (Eq. (9)). While generally showing good measured-calculated agreement, two differences are noted. First – low relative intensity of the measured Surface curve as compared with the Bulk wave. We have no explanation to this difference. Second – the Bulk curve does not go down all the way to zero at the minima. We attribute this deviation from theory to the asymmetry of the focused (aberrated) fundamental beam (axial asymmetry with respect to the focal plane – cf. Figure 10).

For the Soda-lime wedge, given the measured period of intensity oscillations (first few periods), we estimate the wave-vector mismatch to be$\Delta {k}_{H}=-0.249\mu {m}^{-1}$. This estimate is close to a calculated value of $-0.243\mu {m}^{-1}$ based on tabulated data [21], particularly in view of the fact that Soda-lime refractive index varies greatly with its chemical composition [22].

Regarding the relations between the period of intensity oscillations and the wave-vector mismatch, our simulations show that with very thin windows (about one Rayleigh length), the relations are nearly (but not exactly)$\left|\Delta {k}_{H}\right|=2\cdot \pi /period$. In fact, we find for thin windows and glass-typical negative wave-vector mismatch:

Figure 10 displays two images of the generated TH beam cross-section. Figure 10(a) shows the cross-section of beams that are generated with the fundamental Gaussian beam focused nearly everywhere within the glass wedge (with large intensity variations). Figure 10(b) shows the (rather low-intensity) beam cross-section generated with the fundamental Gaussian beam focused mid-way between the wedge surfaces, at the Bulk minima. The ring-shaped image is an indication for an incomplete destructive interference. We attribute the incomplete interference (predicted to be complete by the theory - Fig. 9(b)) to the axial asymmetry of the (aberrated) fundamental Gaussian beam.

## 5. Summary

Geometrical parameters affecting intensities of Gaussian-excited harmonic waves are included in the theory by the amplitude integral (Eq. (1) and Eq. (8) for SH and TH, respectively). We derived analytic solutions to the SH and TH amplitude integrals. With these solutions in place we generated curves and maps enabling a closer look at the influence of geometry parameters on generated harmonics intensities. Such closer look is particularly important for harmonics generation in thin windows. We looked specifically at intensity oscillations in thin normal-dispersion medium characterized by a negative wave-vector mismatch. We showed that even with such negative wave-vector mismatch, high intensity harmonic waves can be generated by thin windows. In fact, looking at the integration limits maps (Fig. 4 and Fig. 7), one can select the excitation geometry to maximize the generated harmonic intensity. Essentially, the maps show that for both SH and TH, focusing the exciting Gaussian somewhere beyond the front surface will maximize the generated intensity and show less intensity-sensitivity to window thickness.

On the experimental front, we faithfully reproduced TH intensity oscillations, as predicted by the amplitude integral. From the period of the first few oscillations (with increasing window thickness) the characteristics wave-vector mismatch can be accurately estimated. For Soda-lime material we estimated a wave-vector mismatch of$\Delta {k}_{H}=-0.249\mu {m}^{-1}$. From tabulated data for the refractive index of Soda-lime at $1551nm$ and at $517nm$ we calculated$\Delta {k}_{H}=-0.243\mu {m}^{-1}$. Our estimation from the experimental data is close to the number calculated based on the tabulated data, and seems to be the correct one in view of the “cleanliness” of the experiment and in view of the uncertainty in chemical composition of Soda-lime glasses (and hence uncertainty in refractive index).

As intensities of Gaussian-excited SH and TH waves are of interest in a number of applications, a detailed account for the influence of the geometrical parameters on the generated intensities could assist in optimizing the harmonic-excitation configuration, for example, in short pulse measurement applications. In microscopy, on the other hand, it is important to properly evaluate the effective depth which is sampled, which could be significantly larger than one confocal parameter that is often naively assumed.

## Acknowledgments

This work was supported by Icore (Israeli centers of research excellence of the ISF), the Crown Photonics Center and the European ICT project FAMOS.

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