## Abstract

A computer model using a finite element technique was written to model the behavior of a chalcogenide fiber Raman laser. The model demonstrates the feasibility of a middle infrared fiber Raman laser pumped at 5.59-µm by a carbon monoxide laser and operating at a wavelength of 6.46-µm. This wavelength may be of interest in surgical applications since it corresponds to the amide II absorption. Calculations show slope efficiencies can approach 80% with moderate threshold powers.

©2003 Optical Society of America

Chalcogenide glasses are a class of glasses that posses many unique optical properties. A chalcogenide glass is formed from one or more of the chalcogen elements, S, Se, and Te, with other elements such as Ge, As, or Sb added to form a stable glass. These glasses can transmit in the near, middle, and out into the far IR depending on the particular glass composition due to their low multi-phonon absorption energies. These glasses are chemically durable, possess large glass forming regions, and can be fabricated into mechanically strong low-loss core-clad optical fiber [1].

Recent studies at the Naval Research Laboratory have looked at Raman amplification in a chalcogenide glass fiber [2]. It was shown that the Raman gain coefficient of As-Se fiber is more than 300 times greater than the Raman gain coefficient of silica at a wavelength of 1.5-µm. Although this experiment was performed in the near IR, the results can be extrapolated to the mid-IR since the Raman gain coefficient scales inversely with wavelength. Figure 1 depicts a typical loss spectrum for a large core As-Se fiber. The absorption at 4.5-µm is due to H-Se, however with a slight glass modification this absorption can be significantly reduced [3]. Elsewhere the loss of the fiber remains below 0.4-dB/m in the 3–7 µm range.

Laser sources in the mid-infrared are of great interest due to many potential applications in such areas as chemical sensing and laser surgery. For example, studies have shown that the best wavelength to ablate soft tissue with minimal collateral damage is 6.45-µm [4]. The large Raman gain coefficient of As-Se in addition to its low loss in the mid-IR makes this glass composition a promising material for a mid-IR fiber Raman laser. This paper will report on efforts to study the potential of an As-Se mid-IR fiber Raman laser through the use of a computer model.

The equations that govern the evolution of the pump and Stokes waves are as follows [5]:

In the above equations, I_{P} and I_{S} are the intensities of the pump and Stokes waves respectively, α_{P} and α_{S} are the absorption coefficients at the pump and Stokes wavelengths respectively, and gR is the Raman gain coefficient. With some slight modification these two equations can be used to model a fiber Raman laser using a finite element technique. With this technique the fiber is divided into N sections each of length ΔL. This is depicted in Fig. 2. In the figure, P_{n} is the power of the pump wave flowing into section n, S_{n} is the power of the Stokes wave flowing into section n in the same direction as the pump wave, and S’_{n} is the power in the Stokes wave flowing into section n in the opposite direction of the pump wave. If the length, ΔL, is small enough such that the change in power across any one section is small, then Eq. (1) can be rewritten in a discrete form. For the power in the forward flowing Stokes wave, the equation becomes:

In the above equation, the coefficient, A_{eff}, is the effective mode area of the pump and stokes waves in the fiber. A similar equation can be written for the Stokes wave traveling in the opposite direction.

Finally, Eq. (2) can be rewritten to model the change in power of the pump wave as follows:

This equation includes the effects of linear loss as well as pump depletion due to the power that is lost to the Stokes wave and to vibrations in the material.

The computer model stores the power of the pump wave, the forward traveling Stokes wave, and the backward traveling Stokes wave in three arrays each containing N elements. Each array element corresponds to a section in the laser as shown in Fig. 2. The Stokes waves’ arrays are initialized with a small positive number to provide a seed for the Raman amplification. This is necessary since the model presented above does not take into account spontaneous Raman scattering. In a real laser cavity, the spontaneous Raman scattering provides the seed light for the eventual lasing action. Despite the lack of consideration for spontaneous Raman scattering, the results of the model should still be valid except near threshold conditions where a large portion of the light emitted from the laser results from spontaneous Raman scattering or amplified spontaneous Raman scattering.

To perform the calculations, the computer model steps through the array elements from n=1 to n=N using Eqs. (3), (4), and (5) to calculate the powers flowing into adjacent elements. Special consideration is taken for the first and last element. For the first element, n=1, P_{1} is always equal to the pump power launched into the fiber and S_{1} is equal to S’_{0}, as calculated from Eq. (4) with n=1, multiplied by the reflectivity of the high reflector. For the last element, n=N, S’_{N} is equal to S_{N+1}, as calculated from Eq. (3) with n=N, multiplied by the reflectivity of the output coupler. The output of the laser is equal to S_{N+1} multiplied by the transmissivity of the output coupler. The program steps through the arrays from n=1 to n=N and then backwards from n=N to n=1 making the above calculations. Although it is not strictly necessary to perform the calculations in both directions it does decrease the number of iterations needed to reach convergence. If the parameters such as pump power and mirror reflectivities are chosen such that the fiber Raman laser will lase, then after many iterations the calculated values will converge yielding the output power of the laser. Conversely, if threshold conditions are not met, then the calculated values will diverge toward zero.

It is important to address the limitations of the computer model presented above. The model is time independent and thus is only valid for CW pumping in steady state conditions. Secondly, the model does not take into account cascaded stimulated Raman scattering. That is, some of the Stokes shifted light in the signal beam will be Stokes shifted again into a second order Stokes beam. Under certain circumstances this can be a significant source of unaccounted for loss due to the high intracavity power at the signal wavelength. To minimize this loss it is important to eliminate any feedback at the second order wavelength and to reduce the intracavity power in the signal beam. Using the highest output coupling percentage possible can reduce the intracavity power in the signal beam while antireflection coatings or angle cleaved end faces can be used to reduce feedback at the second order wavelength.

As mentioned previously, the wavelength of 6.45-µm is of interest for laser surgery applications. Since the Raman shift of As-Se is 240-cm^{-1}, a Raman laser operating at this wavelength would need to be pumped with a 5.59-µm laser. Commercially available carbon monoxide (CO) lasers can output tens of watts at this wavelength and could be used to pump such a Raman laser. The computer model described above was used to estimate what type of performance one might expect from such a laser.

In the design of the optical fiber, two important properties are the core size and the numerical aperture that together determine the mode field size. Since the Raman gain depends on the intensities of the pump and Stokes waves, a small mode field diameter is important. Additionally, it is important for the fiber to be single mode at the relevant wavelengths to maximize the mode overlap between the pump and Stokes waves. The single mode cutoff wavelength is given by:

where the value, a, is the core radius and NA is the numerical aperture. The cutoff wavelength is chosen to be below both the pump and Stokes wavelengths. For this model, a cutoff wavelength of 5-µm was chosen. With the cutoff wavelength set, either the core diameter or the numerical aperture can be chosen and Eq. (6) will determine the other value. In this case, the NA was set to 0.60 and which requires the core radius to be 3.2-µm to maintain a single mode cutoff wavelength of 5-µm. The mode field radius can then be determined from the following equation:

where the value, V, is the normalized frequency or V-number, V=(2*π*a/λ)NA. In this case, for a NA of 0.60 and a core radius of 3.2-µm the mode field radius is 4.0-µm.

Along with the mode field radius, the computer model requires a value for the pump and Stokes wavelengths, the Raman gain coefficient, the fiber length, the fiber loss, the reflectivity of the high reflector, and the reflectivity of the output coupler. The Raman gain coefficient was calculated based on the measurement in Ref. [2]. The value used was 6.2×10^{-12}-m/W based on the gain coefficients scaling with wavelength. The fiber length was kept at 1-m and the loss from Fig. 1 is 0.3-dB/m at these wavelengths. For simplicity the high reflector was assumed to reflect 100% and the output coupling was varied.

Figure 3 shows the calculated Raman laser output verses pump laser input for various output coupling percentages. For output coupling of 10%, 20%, 30%, and 40% the Raman laser has slope efficiencies of 37%, 53%, 62%, and 67% respectively. The thresholds are 1.0-W, 1.5-W, 2.1-W, and 2.7-W respectively. The high thresholds are due in part to the large cavity losses. Although, 0.3-dB/m is a typical loss for As-Se fiber drawn at NRL, in sort lengths we have observed losses as low as 0.1-dB/m and the theoretical lower limit is orders of magnitude below that value [6]. Figure 4 shows the calculated Raman laser output if the fiber losses were reduced to 0.1-dB/m. For the same output coupling percentages the slope efficiencies become 60%, 71%, 76%, and 79%, respectively. The thresholds become 0.6-W, 1.1-W, 1.6-W, and 2.3-W, respectively.

As can be seen from the calculations above, relatively high slope efficiencies can be obtained in a chalcogenide Raman laser. Reducing the loss of the fiber is one way to increase the efficiencies and reduce the threshold powers. Other methods can be employed to further reduce the threshold powers for the laser. For example, increasing the NA of the fiber will further reduce the mode field radius yielding a higher gain in the fiber. The index of refraction of As-Se is 2.8 allowing for relatively small change in composition between the core and clad to affect large changes in the NA of the fiber. Core-clad fibers with numerical apertures approaching 1.0 are certainly possible. If the large NA is not desirable for beam divergence considerations then making tapered fiber is also possible. In this case a low NA fiber is tapered down providing an adiabatic transition from a large core low NA fiber to a small core large NA fiber. This technique achieves high intensities in the fiber while maintaining a low NA output.

In summary, the model described in this letter has shown that chalcogenide glass fiber Raman lasers are viable mid-IR laser sources. For a system pumped by 5.59-µm laser slope efficiencies approaching the quantum defect limit of 87% are possible. For example, from Fig. 4 pumping at 10-W results in a laser output just under 8-W. For lower power applications where a low threshold is required other techniques can be applied to reduce threshold powers by increasing intracavity intensities.

## References and links

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**5. **G. P. Agrawal, *Nonlinear Fiber Optics* (Academic, 1995), Chap. 8.

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