Polymer optical fibers (POFs), due to their flexibility and ease of use, have very interesting characteristics for short-haul communication links, as well as for other applications in fields such as optical sensing, illumination and passive optical devices [1]. In addition, they constitute a promising research field as active devices, in combination with organic dopants (either organic dyes or organic semiconductors). The aim is to achieve efficient fiber lasers and amplifiers, and even optical-gain switches [2-6], in short fiber distances. High-gain amplifiers have been reported in distances shorter than 1 m, due to the high emission and absorption cross sections of organic dopants. The probability of light interacting with a molecule of dopant is quantified by the cross section for the relevant type of interaction, for example, absorption or emission. The cross section depends on the wavelength of the light. The total probability of interaction also depends on the concentration of dopant molecules per unit volume. For example, in the case of the dye rhodamine B (RB), the cross sections are about 10,000 times greater than that of Er in glass fibers. Optical amplification in dye-doped POFs was reported for the first time in 1993, for the case of a graded-index (GI) fiber doped with RB [7]. More recently, a preform technique was presented and used to fabricate RB-doped step-index (SI) POFs [6]. The optical feedback for the gain medium in active POFs is usually provided by the cylindrical surface of the optical fiber, which acts as a cavity.

As is well-known, POFs are more suitable than glass fibers for using organic dopants, because of their lower manufacturing temperature as compared to that of glass fibers. Besides, POFs are preferable over other solid hosts because of the higher density of light present in fiber cores, which increases the probability of stimulated emissions. The maximum gain usually takes place near the peak wavelength of the emission cross section [8]. For example, by doping SI or GI POFs with RB and pumping them optically, it is possible to achieve amplification in the visible region. The total gain depends mainly on the dopant concentration, on the fiber characteristics and on the energy of the pump pulses. High gains can be achieved in both SI and GI active POFs, although with some peculiarities that will be reviewed in this paper. The type of POF chosen will also depend on the rest of the optical link used to connect the emitter and the receiver. The main advantage of passive GI POFs is that their bandwidth is typically 2 orders of magnitude larger than that of SI POFs, but they are not so cheap [9].

Organic semiconductors usually yield lower gains than dyes, but they have other interesting properties. For example, whereas dye-doped fiber lasers and amplifiers have to be pumped optically, organic semiconductors open the possibility of electrical pumping in the future, since they are capable of charge transport [10]. In the case of POFs doped with conjugated organic semiconductors, such as the commonly-studied poly(9,9-dioctylfluorene) (PFO), ultrafast-switching capabilities have been observed [5,11]. Specifically, by impinging a second laser pulse at the correct wavelength, the stimulated emission can be annihilated, with almost instantaneous recovery. However, the gains reported so far with PFO and some other fluorine oligomers have been much smaller than those achieved with organic dyes. This is because organic dyes have higher optical cross sections, which allow higher gains, and also because they can usually be dissolved in higher concentrations in poly(methyl-methacrylate) (PMMA), which is the typical material of the core [12]. As will be shown in this paper, the length of a POF-based amplifier can be shortened by dissolving the dopant in high concentrations. By using dyes, large gains can be achieved in very sort fiber distances. For example, experimental gains reported for dye-doped POF amplifiers (POFAs) are 27 dB for a 50-cm RB-doped POF [7] or 18 dB for an 8-cm rhodamine 6G-doped POF [13]. Either longitudinal or lateral optical pumping can be employed [2].

In this paper, we analyze POFs that are pumped longitudinally with short light pulses of several energies. The dye chosen for the analysis is RB. Apart from its particularly large cross sections, it has high photochemical stability, which is important in practice to withstand multiple excitation cycles with a pulsed pump laser [14,15]. Moreover, the photostability achieved with RB-doped POFs is greater than that accomplished with most dyes [16,17]. If high-energy photons are used (as in the proximity of the ultraviolet region), photo bleaching can occur [14], which can limit the maximum energy of the pump pulses for some dyes. This is not the case of RB, which is suitable to obtain high gains in the wavelengths of low attenuation in POFs, from the final part of the green color to the beginning of the red.

The paper is organized as follows. The model describing amplification in active SI and GI fibers is explained and discussed in Section 2. The first issue investigated in Section 3 is the spectral shift of the emission with traveled distance in active POFs, which can be employed for making tunable fiber lasers and amplifiers [18]. As will be shown, when the emission and absorption cross sections overlap significantly, the absorption of the fiber can have a strong influence on the output spectrum, which can present a shift either towards longer or towards shorter wavelengths (i.e., “red shift” or “blue shift” respectively) [8]. This paper clarifies to what extent it is possible to tune the spectral emission of active POFs by changing the fiber length. We also study the influence of the pump power on the output spectrum, with and without signal. In the latter case the light emission process is called Amplified Spontaneous Emission (ASE) [19]. The analysis of the ASE provides information about the wavelengths at which high power is achieved in a POF laser. One of the main novelties of our work is that the spectral study has been made computationally for the first time, as far as we know. After the spectral study, we analyze the influence of the distribution of the dopant concentration and the effect of the numerical aperture in doped GI POFs. Specifically, we pay special attention to the radial distributions of the light power density and of the dopant concentration in the fiber core. The emission features of SI POFs and GI POFs are compared, which allows us to reach conclusions about the influence of the type of fiber. Section 3.2 shows results without input signal (i.e., POF lasers) and Section 3.3 results with signal (i.e., POFAs). All these computational analyses can be very useful at the early design stages of POF lasers and amplifiers.

We will start with a brief description of the most important optical transitions that occur in the dopants embedded in the fiber core. As shown in Figure 1, organic dyes have two main electronic energy states, S₀ and S₁, with many energy vibrational levels within each one of them [20]. Due to the fact that transitions between vibrational levels in each electronic energy state are very rapid and nonradiative, photon emissions are well described as taking place from the ground level of state S₁ to one of the vibrational levels of state S₀, while photon absorptions are well described as taking place from the ground energy level of state S₀ to one of the vibrational levels of state S₁. In consequence, the optical properties in the doped fiber are usually analyzed by means of time-dependent rate equations between two main energy levels (from E₂ to E₁ in Figure 1 for emitted photons) [12].

It is important to note that in our analyses, the independent variables are not only the time t and the position z along the fiber symmetry axis [12], but also the wavelength λ. This allows us to carry out computational simulations of spectral phenomena, such as emission spectral shifts and changes in emission spectral width. To introduce this dependence, we make use of the λ-dependent absorption and emission cross sections σ^a(λ) and σ^e(λ) of the dopant, and we divide the range of wavelengths of interest into discrete subintervals centered at wavelengths λ_k.

We consider that the doped POF is pumped longitudinally at z = 0, with the z axis oriented in the direction of the fiber symmetry axis (see Figure 2). The pump wavelength is λ_p and the input signal is centered at the wavelength λ_s.

We assume that the pump laser illuminates the whole cross section of the doped fiber, which, in turn, emits light at random directions. A fraction β of such emissions can propagate towards the output fiber end in guided directions and, therefore, contribute to stimulated emissions along the fiber. β can be calculated from the solid angle corresponding to the maximum inclination with respect to the z axis of guided light rays. For an SI fiber whose core refractive index is n₁ and whose cladding refractive index is n₂, we can estimate the value of β, which is constant, as the quotient between the solid angle in guided directions and in all directions. In the case of GI fibers, we can first calculate the value of β at the fiber axis (β₀) in the same way, as follows: (1) β 0 = ∫ 0 θ c 2 π sin θ d θ ∫ 0 π 2 π sin θ d θ = 1 − cos θ c 2 = n 1 − n 2 2 n 1 ≈ N A 0 2 4 n 1 2 but in this case β is not constant, since the core refractive index decreases from n₁ at the fiber axis to n₂ at the core-cladding interface, so β depends on the radial distance r and an average value of β should be calculated. In Equation (1), θ_c is the complementary critical angle, and NA₀ is the numerical aperture of the fiber at its symmetry axis.

Additionally, if the dopant concentration and the light density distributions vary with r, which only happens in the case of GI fibers, we have to take the overlapping between both distributions into account. More detailed explanations about this can be found in Appendix. In a typical GI fiber, light is more concentrated towards the fiber symmetry axis, where the dopant concentration is also higher than average. In consequence, stimulated emissions are favored and this effect will be taken into account by means of an overlapping factor γ that will be greater than 1 and which will be explained later. In SI fibers, γ = 1, because the dopant concentration is constant (or also because the light power density is uniform in each fiber cross section).

The complete physical system is described by a set of three partial differential equations with initial and boundary conditions. The unknown functions to be determined are P(t,z,λ) (light power as a function of time, position and wavelength) and N₂ (electronic population in the excited level E₂). We also take into account the trivial relationship N₁ = N−N₂ between N₂ and the population N₁ in level E1, where N is the dopant concentration. The following differential equation governs the evolution of the pump power P_p: (2) ∂ P p ∂ z = − σ a ( λ p ) N 1 P p γ − 1 v z ∂ P p ∂ t

The first right-hand term in Equation (2) represents the absorption of the pump. The product σ^a(λ_p) N₁ is the absorption coefficient α_p of the material at λ_p. There is an initial increase of N₂ due to the pump that reduces N₁ and, therefore, it also reduces the absorption coefficient α=N₁ σ^a. The last term in Equation (2) represents the propagation of the pump.

The variation of N₂ with time, for each wavelength subinterval centered at λ_k, is governed by: (3) ∂ N 2 ∂ t = − N 2 τ − ( σ e ( λ k ) h ( c / λ k ) A core ) N 2 P γ + ( σ a ( λ p ) N 1 h ( c / λ p ) A core ) P p γ + ( σ a ( λ k ) h ( c / λ k ) A core ) N 1 P γ

The first right-hand term in Equation (3) represents the spontaneous decay, where τ is the spontaneous lifetime of the dopant in PMMA. The second term accounts for the stimulated decay, where σ^e(λ_k) is the stimulated emission cross section at λ_k. Both terms are associated, respectively, with spontaneous and stimulated emissions of photons. The last two terms are associated, respectively, with the absorption of photons of P_p (excitation by the pump) and of P (reabsorption). Reabsorption (the last term) is usually much smaller than excitation by the pump, because P is usually orders of magnitude smaller than P_p and because σ^a(λ_p) is larger than σ^a(λ_k). A_core is the cross section of the fiber core and h is Planck's constant.

The equation to calculate the evolution of P, for each subinterval centered at λ_k, is: (4) ∂ P ∂ z = σ e ( λ k ) N 2 P γ − σ a ( λ k ) N 1 P γ − 1 v z ∂ P ∂ t + N 2 τ ( h c λ k ) σ s p e ( λ k ) β A core

The first right-hand term in Equation (4) accounts for the emissions stimulated by incoming photons. The second term represents the attenuation due to material absorption. The third term represents the propagation of the power inside the fiber with speed v_z. The last term accounts for the spontaneous emissions, where σ_sp^e(λ_k) is the spontaneous emission cross section at λ_k, in the sense of determining the probability distribution of the wavelengths of spontaneously emitted photons. More specifically, a_sp^e(λ_k) is the probability for a spontaneous emission to take place at the wavelength subinterval around λ_k. Above threshold, the last term in Equation (4), corresponding to spontaneous emissions, tends to be much smaller than the term corresponding to stimulated emissions. However, the former is necessary to simulate the appearance of the first photons in a fiber laser. Note that we have neglected the possible occurrence of absorption of the pump power by excited molecules. The reason is that this phenomenon, which could be important for other applications [21], can be neglected for our type of analysis to a first approach [12]. Besides, in our model we do not include possible nonlinear-absorption effects that could occur at very high power densities, such as the absorption of more than one photon simultaneously or via a multistep process, or the decrease in absorption due to optical saturation or optical bleaching. Such effects would require changing the equations described above. Finally, our model does not take into account modal dispersion in the active POF. However, this approach is reasonable, since the fiber lengths we will work with are very short (no longer than 1 m and even much shorter in some cases).

The aforementioned equations serve for both SI and GI fibers. However, there are two main differences in the values of β and γ in both cases. On the one hand, the overlapping factor γ, which is used in any term in which either P or P_p appears multiplied by N₁ or N₂, is only different from 1 in GI fibers. On the other hand, an average value of β should be calculated only in GI fibers, so as to take into account that the numerical aperture NA varies with r. In the case of an SI fiber, the numerical aperture is constant along r, so the average value of β in the fiber cross section is simply β₀ as in Equation (1). In the case of a GI fiber, the values of β and γ can be calculated in the way shown in Appendix. The results are summarized in Table 1. Depending on the application, GI POFs or SI POFs can be employed, with some differences in gain above threshold of one over another: GI POFs have higher overlapping factors and lower average values of NA. Specifically, in GI POFs there is a decrease in the numerical aperture when r increases, which is a disadvantage over SI POFs regarding the total emitted power, because it implies smaller guided angles of rays and, therefore, fewer guided rays that yield amplification. On the other hand, the dye concentration in GI POFs is maximum at r = 0, where the light density in a GI fiber is also maximum, so γ > 1. This is an advantage of GI over SI fibers. We will show that the greater γ is, the higher the gain of the GI POF tends to be, which is an advantage over SI POFs, for which γ = 1.

The governing differential equations are solved for the following initial conditions: P and N₂ are 0 everywhere at t = 0. For the spectral analyses, the powers plotted in the results will represent the time-integrated ones, i.e., P ( z , λ k ) = ∫ 0 ∞ P ( t , z , λ k ) d t. Note that the output power calculated computationally in this uni-dimensional model stands for the total power in a 3-dimensional real fiber with modes.

As for the boundary conditions at z = 0, a Gaussian pulse of pump power is launched, and, in the case of amplifiers, a Gaussian pulse of signal power is added to it, their maxima taking place at the same instant.

The numerical approach used to solve the governing equations relies on finite differences. Specifically, we have developed ad-hoc finite-difference algorithms in which all three variables z, t and λ, have been discretized. The discretization of the spatial and temporal variables is uniform, i.e., with constant finite step sizes δt and δz. For example, for a distance of 7 cm and a pulse width of 5 ns, δz is on the order of 600 ×10⁻⁹ m, δt is on the order of 0.2 × 10⁻⁹ s, and roughly 8,000 spatial nodes with at least 200 temporal steps are used. For smaller pulse widths, δt is reduced. The judicious choice of these parameters is critical in order to achieve numerical convergence within the desired precision. The number of discrete wavelength subintervals is not so critical for convergence, and it is has been on the order of 50 in most of our simulations.

As for the implementation of the numerical scheme described, a few programming details follow. A matrix N2(i,j) is allocated to contain the discrete values of N₂, where index i determines t by t_i =(i − 1)δt and index j determines z by z_j = (j − 1) δz. Similarly, a 3D matrix P(i,j,k) is allocated in order to contain the light power distribution, where indices i,j are as before, and index k determines λ_k in a similar way [8]. Finally, each term of the governing equations is estimated by means of finite differences for each discrete wavelength. The finite-difference formulae are centered whenever possible, so as to achieve a higher order of precision. Both matrices P(i,j,k) and N2(i,j) are filled column by column, with columns representing values of z and rows representing values of t. A full column is calculated before going on to the next one.

Firstly, by using our computational model, let us analyze the influence of NA₀ (or β) on the lasing properties, such as slope efficiency and lasing threshold, in active SI and GI POFs. Some researchers define the threshold as the pump energy necessary for the full width at half maximum of the emitted intensity to drop to half the value observed when the fiber is pumped with low energy [5]. Others point out that the output energy above threshold depends linearly on the pump, define the slope efficiency as the slope of the fitting straight line, and the lasing threshold as its intersection with the horizontal axis. We have adopted the latter definitions in this section.

In order for the results to correspond to a realistic situation, we have used the same parameters as those reported in experiments [12,27], which have been summarized in Table 2. The pump energy can be easily calculated from these data by taking into account that, for a pulse that is Gaussian in time, the total energy of the pulse (E) can be related to its peak power (P_max) through the equation E = P_max (2π)^1/2FWHM / 2.35.

Figure 7 illustrates the influence of the fiber numerical aperture and of the pump energy on the output energy at 581 nm, for the parameters mentioned above and a core refractive index n₁ = 1.492. Qualitatively, the evolution for SI and GI POFs (Figure 7(a,b) respectively) is the same. When the pump energy is low, the output is mainly due to spontaneous emissions. As a consequence, the output energy grows with a small slope. Above threshold, population inversion occurs and spontaneously emitted photons are amplified in a single pass through the fiber. At very high pump energies, incipient saturation effects are observed, which reduce the slope of the curves (see Figure 7(b)). Above threshold, the output energy reaches higher values when NA₀ is higher, as expected. For higher values of NA₀, there are more photons that can propagate along the fiber and interact with excited molecules, generating a higher number of stimulated emissions. In consequence there is more output energy. Considering the same NA₀ for both types of fibers, it can be observed that the output energy is about four times higher in GI than in SI fibers. This means that, in the GI POF laser considered, the effect of the greater overlapping factor γ on the output energy is dominant over that of the higher average NA (or β) of the corresponding SI POF.

As can be seen in Figure 8, the pump energy needed to achieve the threshold is reduced when NA₀ is higher, and the evolution is practically linear in both types of fibers. A similar behavior (i.e., a strong dependence of threshold with β) has been observed, for example, in random lasers [28], but the analysis in fibers is reported by us for the first time, as far as we know.

In Figure 9 we can see that the slope efficiency increases with NA₀, as expected. The maximum values reached in the SI POF (solid line) are not as high as in the GI POF (dashed), showing again a dominant effect of the higher overlapping factor over the lower average value of NA in this case.

In this work we have computationally analyzed the features of light amplification in SI and GI POFs doped with organic dyes. For this purpose, we have employed ad-hoc numerical algorithms in which the independent variables are the time, the distance traveled and the wavelength (which allows spectral analyses). The effect of the non-uniform numerical aperture in GI fibers has been calculated by introducing the average value of the fraction of spontaneously generated photons that contribute to stimulated emissions. The model also takes into account the effect of the overlap between the non-uniform dye-density and power-density radial distributions in GI POFs. From the knowledge of both distributions, we have shown how to calculate an overlapping factor γ that is introduced in the terms of the governing equations in which light powers appear multiplied by dopant concentrations.

The main results that we have obtained can be summarized as follows: (i) Firstly, we have shown the influence of the overlap between the absorption and emission cross sections of the dopant on the emission spectral shifts and widths in SI POFs. In this respect, we have seen that dyes with a strong overlap, such as rhodamine B, are suitable for tuning the spectrum of laser emission by changing the length of the doped fiber conveniently; (ii) We have also shown that an increase in the numerical aperture has a beneficial effect on the threshold power (it decreases) and on the slope efficiency (it increases) in lasers based on both SI and GI POFs doped with rhodamine B; (iii) Regarding POF amplifiers, we have shown that there is an enhancement in the signal gain when γ increases, and we have compared the results with those of SI POFs, for which γ = 1. Depending on the application, GI POF amplifiers or SI POF amplifiers can be employed, with some advantage in signal gain in GI POF amplifiers over SI ones. The optimum signal wavelength and pump power as functions of the fiber length, the fiber numerical aperture and the radial distribution of the dopant have also been analyzed, for both SI and GI POFs. In both GI and SI POFAs, the signal gain does not significantly change with variations in the numerical aperture.

These analyses can be useful to designing amplifiers and lasers based on polymer optical fibers.