## Abstract

We introduce a rigorous theory of third-harmonic generation in optical waveguides and apply it to design a micro-fiber waveguide for efficient generation of third-harmonic radiation from infrared lasers. Phase-matching with efficient mode overlap is achieved in micro-fibers having a diameter roughly equal to half of the fundamental wavelength. Using a typical solid-state or fiber laser for pumping, high conversion efficiency is possible in only a few centimeters of a micro-fiber.

©2005 Optical Society of America

## 1. Introduction

Generation of harmonics is a convenient way of producing new wavelengths of light. In particular, third-harmonic generation (THG) from infrared sources (such as Nd:YAG, Nd:YLF, and Nd:YVO_{4} solid-state lasers) is often used as a source of ultraviolet (UV) light at ~350 nm.

At present, nonlinear crystals (for example BBO) are utilized for harmonic generation. Such crystals have strong second-order nonlinearity, which is useful for parametric processes, such as sum or difference frequency generation, and second harmonic generation. In order to produce third-harmonic radiation, a two-stage process is employed. First, the second harmonic of the fundamental frequency is generated (ω + ω = 2ω) in a nonlinear crystal. Then, the second harmonic is combined with the remaining fundamental beam in another crystal to make the third harmonic (2ω + ω = 3ω). Thus, two crystals and many optical components, such as mirrors and lenses for focusing the beam, are required in such a system making it complicated and expensive. In addition, maintaining the alignment of all components over time is critical. Laser damage could also be an issue, especially for the components exposed to intense UV light.

On the other hand, direct THG is possible using third-order nonlinearity present in all media. However, such direct conversion has not become practical due to the low third-order nonlinear susceptibility of available optical materials and the difficulty in achieving phase-matching between the fundamental wave and its third harmonic.

In this paper, we show that efficient phase-matched THG is feasible in specially-designed glass micro-fibers. Due to waveguiding properties of such micro-fibers, the fundamental and the third-harmonic waves can efficiently overlap and interact over the whole length of the waveguide. Because of the small diameters of the micro-fibers, high pump intensity can be achieved with relatively low input power, which compensates for the weak third-order nonlinearity of glass. As a result, only a few centimeters of such a micro-fiber are required for converting a large portion of the pump light into its third harmonic. Mode-locked or Q-switched fiber lasers can serve for pumping such frequency converters. This opens the possibility for an all-fiber system for generating third-harmonic UV light, which would also require much fewer optical components and eliminate any need for alignment. Therefore, such an all-fiber THG system would be a cheaper and a more robust alternative to existing nonlinear-crystal technologies.

## 2. THG in glass: fundamental equations

To analyze the process of THG in glass, we will first derive the basic equations describing the interaction of a fundamental wave and a third-harmonic wave in an arbitrary waveguide having third-order nonlinearity. Such interaction is typically considered in bulk materials by assuming both waves to be infinite plain waves. This approximation also works for waveguides with low core-cladding index contrast. In this work, however, we are considering micro-waveguides with a size on the order of the wavelength of light and a high refractive index contrast. In this case, the standard plain-wave approach is inadequate. For correct description of the mode interaction, all components of the electric and magnetic fields must be taken into account. To simplify the analysis, we are considering only quasi-continuous waves. This approximation is justified for pulses longer than ~1 ns, typically produced by Q-switched lasers.

The total electric field **E** and magnetic field **H** at any location of the waveguide can be represented as a sum of the mode fields at each frequency *ω*_{j}
(*j*=1 for the fundamental wave and *j*=3 for the third harmonic, “*c.c*.” represents a complex conjugate):

$$\mathbf{H}\left(\mathbf{r},t\right)=\sum _{j=\mathrm{1,3}}\frac{{A}_{j}\left(z\right)}{2}{Z}_{0}^{-\frac{1}{2}}{\mathbf{G}}_{j}\left({\mathbf{r}}_{\perp}\right)\mathrm{exp}\left[i\left({\beta}_{j}z-{\omega}_{j}t\right)\right]+c.c.$$

$$\mathbf{r}=\left(x,y,z\right),{\mathbf{r}}_{\perp}=\left(x,y\right)$$

$${\beta}_{j}=\frac{{\omega}_{j}}{c}{n}_{j}^{\mathit{eff}}({\omega}_{j}),\phantom{\rule{.9em}{0ex}}{Z}_{0}=\sqrt{\frac{{\mu}_{0}}{{\epsilon}_{0}}}$$

The factor ${Z}_{0}^{1/2}$ is introduced so that the transverse electric and magnetic mode field distributions **F**
_{j} and **G**
_{j} have the same units, which is convenient for numeric calculations. Note that **F**
^{j} and **G**
_{j} are vectors and, in general, they are complex. The mode normalization is chosen as follows:

where *dS* ≡ *dx dy* and *A*
_{∞} is the infinite cross-section of the waveguide. Therefore, the units of **F**
_{j} and **G**
_{j} are [m^{-1}]. With this normalization, the total field power is:

As we can see from Eq. (3), the amplitudes of the modes are directly related to their powers, ∣*A*_{j}
(*z*)∣^{2} =*P*_{j}
(*z*), which means that the units of the *A*_{j}
are [ W^{1/2}].

If both the fundamental and the third-harmonic modes are bound (with the fields vanishing at infinity), we can apply the reciprocity theorem [1] to calculate the derivatives of the amplitudes *dA*_{j}
/*dz* due to the nonlinear interaction of the two fields over the length of the waveguide [2,3]. Assuming the absorption at both wavelengths to be negligible, we obtain:

where ⟨‥⟩ is time averaging and **P**
_{NL} is the nonlinear component of polarization. In an isotropic medium such as glass, the nonlinear polarization is related to the real electric field **E** (defined in Eq. (1)) as [4]:

In simple glass-air waveguides (such as circular fibers) we assume the third-order susceptibility *χ*
^{(3)}(**r**) to be *z*-independent and distributed over the fiber cross-section as fllows:

By carefully arranging the terms in Eq. (5) and selecting only those that match the fundamental and the third-harmonic frequencies, then substituting **P**
_{NL} into Eq. (4), and integrating over the fiber cross-section taking into account Eq. (6), we arrive at the following coupled-mode equations:

$$\frac{\partial {A}_{3}}{\partial z}=i{n}^{\left(2\right)}{k}_{1}\left[\left({6J}_{2}{\mid {A}_{1}\mid}^{2}+3{J}_{5}{\mid {A}_{3}\mid}^{2}\right){A}_{3}+{J}_{3}^{*}{A}_{1}^{3}{e}^{-\mathit{i\delta \beta z}}\right],$$

where *δβ* = *β*
_{3} - 3*β*
_{1} is the propagation constant mismatch, ${k}_{1}=\frac{{\omega}_{1}}{c}=\frac{2\pi}{{\lambda}_{1}}$ is the propagation constant of the fundamental wave in vacuum, ${n}^{\left(2\right)}=\frac{3}{16}{\chi}^{\left(3\right)}{Z}_{0}\approx 2.7\xb7{10}^{-20}\frac{{m}^{2}}{W}$ is the nonlinear refractive index coefficient, and *J*_{i}
are nonlinear overlap integrals defined as:

Here ${\hat{\mathbf{e}}}_{i}=\frac{{\mathbf{F}}_{i}}{\mid {\mathbf{F}}_{i}\mid}$ are the normalized vector fields of the modes. Note that, because **F**
_{j} and **G**
_{j} are measured in [m^{-1}], the units of *J*_{i}
are [m^{-2}].

Since the integrals *J*
_{1}, *J*
_{2}, and *J*
_{5} are real by definition, we can show from Eqs. (7) that the total energy of the field is conserved, as expected in a non-absorbing medium:

In addition, assuming that the modes are not elliptically-polarized, the mode fields can be defined in such a way that the transverse components are purely real and the *z*-components are purely imaginary [1]. This means that the overlap integral *J*
_{3} is real as well.

In the case of small amount of third harmonic generated (∣*A*
_{3}∣<<∣*A*
_{1}∣), the approximate solution of Eqs. (7) is:

where *P*
_{0} is the input fundamental wave power and *δ$\stackrel{\u0303}{\beta}$* = *δβ* + 3*k*
_{1}
*n*
^{(2)} (2*J*
_{2}-*J*
_{1}) *P*
_{0} is the propagation constant mismatch modified by the competing effects of nonlinear self-phase-modulation (SPM) and cross-phase-modulation (XPM). The average efficiency of the third harmonic generation is:

We can observe from Eq. (10) that for efficient THG in waveguides the following conditions must be satisfied:

- The phase mismatch must be small:
*δ$\stackrel{\u0303}{\beta}$*<<*k*_{1}. - The overlap integral
*J*_{3}must be sufficiently large.

For a more detailed analysis, we can simplify the form of the coupled-mode equations (Eqs. (7)) by making the following substitutions:

$${\nu}_{1}=2\frac{{J}_{2}}{{J}_{1}}-1,\phantom{\rule{.9em}{0ex}}{\nu}_{2}=\frac{{J}_{3}}{{J}_{1}},\phantom{\rule{.9em}{0ex}}{\nu}_{3}=\frac{{J}_{5}}{{J}_{1}}-2\frac{{J}_{2}}{{J}_{1}}$$

$$\Phi =\mathit{\delta \beta}\phantom{\rule{.2em}{0ex}}z+{\phi}_{3}-3{\phi}_{1}$$

$${L}_{\mathit{NL}}=\frac{1}{\left({n}^{\left(2\right)}{k}_{1}{J}_{1}{P}_{0}\right)},\phantom{\rule{.9em}{0ex}}\xi =\frac{z}{{L}_{\mathit{NL}}}$$

Here *u*
_{1} and *u*
_{3} are real and positive numbers representing (when squared) the fraction of the total energy in each wave. Eqs. (7) can now be represented in terms of *u*
_{1}(*ξ*), *u*
_{3}(*ξ*), and Φ(*ξ*):

We can show that the solution for the third harmonic is:

The best possible scenario for THG is when ⟨sinΦ⟩_{z} = 1, which implies that

throughout the whole length of the waveguide. In this case, the power of the third harmonic is:

We define a characteristic length of the interaction *L*_{THG}
as the length needed to obtain 50% conversion:

To understand the requirements for satisfying the resonance condition (16), we need to revisit Eqs. (7). It is clear from the second equation that, when no third harmonic is present at the input, the phases of both waves at *z* = 0 always satisfy the following relationship:

Therefore,

Using relationship (20) and Eq. (13.4), we conclude that the resonance (16) occurs when two conditions are met:

Eq. (22) provides the condition, which ensures that the effect of the SPM and XPM on phase matching is independent of the relative fraction of power in each wave. Eq. (22) sets the required detuning from the exact propagation-constant resonance in order to compensate for the SPM and XPM. It is equivalent to *δ$\stackrel{\u0303}{\beta}$* =0 in Eq. (11). When Eq. (21) is not strictly satisfied (*ν*
_{1} ≈ *ν*
_{3}), the optimum detuning is:

Note that since the overlap integrals depend on the polarization states of the modes, the SPM/XPM correction in Eqs. (21–23) is polarization-dependent as well. We will study such polarization effects in a separate work.

## 3. Phase-matching

We will now limit our discussion to a simple case of micro-fibers – uniform silica rods suspended in the air or vacuum, with a diameter comparable to the wavelength. Other types of waveguides can be analyzed in a similar fashion by using the formulas developed in Section 2.

For efficient harmonic generation (Eq. (11)), the fundamental and the third-harmonic waves must be phase-matched: *δ$\stackrel{\u0303}{\beta}$* = 0. Assuming that the SPM/XPM corrections to the propagation constants (Eq. (23)) are small, this condition is equivalent to: *n*_{eff}
(*ω*) = *n*_{eff}
(3*ω*).
Ideally, we would like to use the fundamental modes of both waves, simply because their symmetrical and nearly Gaussian field distributions are easily compatible with other laser sources. However, it is difficult to achieve phase-matching of these two fundamental modes for two reasons:

- The refractive index of the glass increases with decreasing wavelength, so the fundamental wave always sees a lower index than the third-harmonic wave.
- The third-harmonic mode is more strongly confined within the waveguide, which increases its effective index even further.

One possible solution is to use higher modes of the third-harmonic wave. Any mode of the third-harmonic wave, other than its fundamental mode HE_{11}(3*ω*), can be phase-matched with the fundamental mode of the fundamental wave HE_{11}(*ω*) by choosing an appropriate fiber diameter. This is true because any high mode of the third-harmonic wave will experience a cutoff when the fiber diameter becomes sufficiently small. At this point, the effective index of such a mode will become equal to that of the air surrounding the fiber (~1), and so it will be lower than the index of the mode of the fundamental wave, which does not have a cutoff. On the other hand, as the diameter of the fiber is increased, the indices of both modes will approach the refractive indices of the glass at the corresponding wavelength, so the effective index of the third-harmonic mode will eventually become higher than that of the mode of the fundamental wave. Since the effective indices change continuously with the fiber diameter, a point of intersection therefore must occur in between these two extremes.

Even though many of the modes of the third-harmonic wave can be phase-matched to the fundamental wave, most of them are not usable for efficient third-harmonic generation due to their poor overlap integrals *J*
_{3}. This is especially true for high-order modes, which typically have a large number of the field oscillations in their transverse profiles, thus producing a very small overlap with the nearly-Gaussian mode of the fundamental wave. This type of high-order-mode phase-matching for THG was explored in photonic crystal fibers [5,6]. Naturally, the observed conversion efficiency was well below 1%. In contrast to that approach, here we propose a waveguide configuration, in which low-order modes having a large overlap can be phase-matched, which results in efficient third-harmonic generation.

Figure 1 shows the effective indices of the fundamental mode of the fundamental wave (λ_{1}=1.064 μm) and the modes of the third-harmonic wave (λ_{3}=0.355 μm) for a uniform circular silica fiber. Losses for both waves are neglected. Only HE_{1n} and EH_{1n} modes of the third harmonic are considered due to their angular symmetry.

As expected, we can find a fiber diameter for phase-matching the fundamental mode of the fundamental wave HE_{11}(*ω*) with any mode of the third-harmonic wave except its lowest mode HE_{11}(3*ω*). The lower the mode of the third-harmonic wave, the lower the effective index is at the phase-matching point. For each phase-matching point, the overlap integral *J*
_{3} is calculated using Eq. (8.3).

It turns out that the most efficient overlap between the modes is achieved for the following combination: HE_{11}(*ω*) → HE_{12}(3*ω*). Phase-matching occurs when the fiber diameter is ~0.52 μm. The corresponding overlap integrals (given by Eqs. (8.1–8.4)) are:

Note that these integral values result in ν_{1}/ν_{3} ≈ 1.6, which means that the condition for resonant THG given by Eq. (21) is not perfectly satisfied. Therefore, the optimum detuning from the exact phase-matching is approximately given by Eq. (23).

By using Eq. (18), we can estimate the required interaction length for 50% efficiency of THG in the ideal case of optimum detuning. With power *P*
_{1} ≈ 1 kW (typical pump power used for generating supercontinuum in similar micro-fibers [7]), *n*
_{2} = 2.5×10^{-20} m^{2}/W, and λ_{1} = 1.06 μm, we get *L*_{THG}
≈ 9600 μm. This means that efficient THG can in principle be achieved in only 1-cm-long piece of micro-fiber, thus making this process useful for practical devices. The short interaction length is also crucial in overcoming other competing nonlinear effects (such as stimulated Raman and Brillouin scattering), which increase exponentially with length.

Figure 2 shows the calculated efficiency of THG over the length of the 0.52-μm micro-fiber, for detuning *δβ*
^{*}
*L*_{NL}
= -5.08, which is close to the optimum value (-4.98) suggested by Eq. (23). While ~50% conversion is observed in 1 cm of the fiber, more than 90% efficiency is possible in a 5-cm long micro-fiber. The THG efficiency curve is close to the ideal one (Eq. (17)) due to the relatively small difference between ν_{1} and ν_{3}. However, such a high conversion could be difficult to achieve in practice due to the inherent non-uniformities of the micro-fiber along its length, which may adversely affect the phase-matching.

For pump wavelengths other than 1.064 μm, the optimum diameter of a micro-fiber for efficient THG will change: the longer the wavelength, the larger a fiber you need. As a rule of thumb, the required micro-fiber diameter is roughly half of the fundamental wavelength.

## 4. Practical considerations

Micro-fibers with diameters as small as 0.4 μm can be readily fabricated by drawing a standard fiber using a torch to melt the glass [7]. The original fiber core is reduced to only a few nanometers in the micro-fiber making its size negligible compared to the wavelength. Therefore, such a micro-fiber can be regarded as a uniform, coreless glass rod, which is the design discussed above.

Provided that the transition between the standard-size fiber and a micro-fiber is adiabatic [1,2], light can be coupled in and out of the micro-fiber with negligible loss [7]. The situation is somewhat more complicated for the proposed nonlinear mode converter because the input and output waves have different frequencies and propagate in different modes. In order to convert the output third-harmonic HE_{12}(3*ω*) mode into a HE_{11}(3*ω*) mode with a more symmetrical beam profile, we can use a long-period grating [8]. Such a grating can be fabricated either directly within the micro-fiber or in the regular-size section of the fiber, immediately after the micro-fiber. Another long-period grating can be used to filter out the remaining pump light.

For making a practical third-harmonic generator, the nonlinear micro-fiber has to be combined with a compact, low-cost pump laser and with a mode converter for transforming the HE_{12}(3*ω*) mode of the third harmonic into a more conventional beam with a nearly Gaussian profile.

Inexpensive, high-peak-power solid-state pulsed lasers are now available, for example mode-locked Microchip lasers [7,9]. Such lasers achieve a few kilowatts of peak power and are commonly used for generating supercontinuum in micro-fibers. With a proper design of a micro-fiber, the same arrangement can be employed for efficient THG. The slight drawback of using such lasers is the need in free-space coupling into the fiber. This problem could be eliminated by using a mode-locked fiber laser permanently spliced to the micro-fiber.

## 5. Conclusion

In conclusion, we have shown that efficient third-harmonic generation is feasible in specially-designed glass micro-fibers. As a rule of thumb, the required diameter for a cylindrical, uniform micro-fiber is roughly half of the fundamental wavelength. We have derived the general equations for THG, which can be used for analyzing other waveguide configurations. The possibility of efficient frequency tripling in glass opens up a way for making a new generation of inexpensive, compact, all-fiber nonlinear devices, therefore eliminating the need for nonlinear crystals and complicated bulk optics presently used for generating harmonics. We are currently working on the experimental verification of the third-harmonic generation in micro-fibers.

## References and links

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