Extraction characteristics of a one dimensional Talbot cavity with stochastic propagation phase

P. Peterson; A. Gavrielides; M. P. Sharma

doi:10.1364/OE.8.000670

1 Introduction

One method of accomplishing strong global coupling of an array of fiber lasers is through diffractive coupling employing Talbot imaging. This type of coupling has been studied for semiconductor laser arrays, CO₂ laser arrays, but not yet extensively for solid state fiber laser arrays. In the following we study steady state diffractive coupling of fibers when each has a stochastic linear propagation phase. This model incorporates the entire range of gains, losses, and arbitrary fill factors, all in a statistical environment. Additionally, we assess the change on the far-field performance for internal and external phase corrections as a function of Talbot cavity length.

We restrict our literature review to within approximately the last ten years. The earliest work was both experimental and theoretical and concerned supermode control[1]. Later this same theme occurred in semiconductor arrays and concentrated on threshold, and mode discrimination as a function of the fill factor in the half-Talbot plane[2, 3]. Subsequent to this, another study emphasized lateral mode control and studied the change in the lateral mode threshold as a function of external cavity length[4] using microlenses[5]. In this general area[6] concurrent work included phase adjustments using liquid crystal arrays[7, 8]. Then, after a short pause there was a lot of general work in 1D and 2D Talbot arrays[9, 10, 11, 12, 13, 14]. Talbot cavities are still of interest and are now being applied, for example, to multicore fiber lasers[15].

Our simulations differ from previous work since we integrate the forward and reverse coupled nonlinear steady state complex field differential equations for N emitters in the Rigrod approximation. Each emitter has a random propagation phase originating from different cavity lengths, and dispersion effects. Integration of these coupled differential equations yields the emitter amplitudes, phases, as well as the far-field patterns all for a specific rms phase. Coupling between emitters is through a boundary condition utilizing a Talbot cavity reflectivity. The reflection coefficient is obtained from the overlap between an emitter and the sum of all the Fresnel propagated fields, as has been done before.[1] The small-signal region was previously studied[1] through the solution of a threshold eigenvalue equation. Here, we reconsider threshold except now in an stochastic environment. We derive an expression which shows how the cavity losses increase with the rms phase. Additionally, we analytically show the decrease in the emitter amplitude and the corresponding decrease in the far-field intensity with increasing rms phase.

The two cavity lengths of particular interest are the half and quarter-Talbot planes. The first length is the out-of-phase solution, and the second is the in-phase solution; in the Appendix we discuss the Talbot cavity reflection coefficient in these planes. The former has a far-field pattern with a null on-axis and the latter has a maximum on-axis. Even though this indicates that the quarter-Talbot plane gives the best far-field performance we show that with external phasing the half-Talbot plane gives a slight increase in the extracted far-field performance. We also show that phasing all the emitters internal to the cavity can drive the resonator below threshold. Further more, when stochastic effects are included we show that an array with a large rms phase is more likely to lock in the half-Talbot plane.

2 Theory

We consider N gain elements coupled through a Talbot cavity terminated by a monolithic mirror of field reflectivity r_t . The j^th gain element has outcoupling field reflectivity r_j and gain g_j . The period of the array is d and the fiber emitting aperture diameter is a_j Further we allow each laser to run on a wavelength λ_j . With these assumptions the the j^th gain element supports a forward $E_{j}^{+}$ field and a reverse field $E_{j}^{-}$ , both described by

\frac{d E_{j}^{\pm}}{d z} = \pm \frac{g j}{1 + ∣ E_{j}^{\pm} ∣^{2} + ∣ E_{j}^{\mp} ∣ 2} E_{j}^{\pm} \pm i k_{j} E_{j}^{\pm}

where z is the coordinate along the fiber axis. Equation (1) can be brought into an integrable form by separating the amplitude and phase according to

E_{j}^{\pm} (z) = A_{j}^{\pm} (z) \exp (ϕ_{j}^{\pm} (z)) .

Inserting this into eq. (1) gives the phase solution

ϕ_{j}^{\pm} (L_{j}) - ϕ_{j}^{\pm} (0) = \pm k_{j} L_{j}

where the individual cavity length is L_j . Later we incorporate the different λ_j ’s, and L_j ’s into a random variable. An important consequence of eq. (2) is a constant C_j that satisfies

C_{j} = A_{j}^{+} (z) A_{j}^{-} (z) = A_{j}^{+} (L_{j}) A_{j}^{-} (L_{j}) = A_{j}^{+} (0) A_{j}^{-} (0) .

This allows elimination of $A_{j}^{\mp}$ in the amplitude differential equation. Thus, our working amplitude differential equation becomes

\frac{d A_{j}^{\pm}}{d z} = \pm \frac{g j}{1 + A_{j}^{\pm^{2}} + \frac{C_{j}}{A_{j}^{\pm^{2}}}} A_{j}^{\pm}

which can be integrated to give

\ln \frac{A_{j}^{+} (L_{j})}{A_{j}^{+} (0)} + \frac{1}{2} [A_{j}^{+^{2}} (L_{j}) - A_{j}^{+^{2}} (0)] - \frac{C_{j}}{2} [\frac{1}{A_{j}^{+^{2}} (L_{j})} - \frac{1}{A_{j}^{+^{2}} (0)}] = g_{j} L_{j} .

A similar equation can be written for $A_{j}^{-}$ by using eq. (4).

To complete the formulation of the differential equations we turn to a discussion of the two-point boundary conditions. At the outcoupling end, z=0, we have that

$E_{j}^{+}$ (0)=r_j $E_{j}^{-}$ (0). At the Talbot end of the laser array, z=L, continuity of the electric fields requires that the reverse field is composed as $E_{i}^{-}$ (L_i )=∑R _i,j $E_{j}^{+}$ (L_j ). R _i,j is the complex reflection matrix developed by evaluating the overlap integral between the Fresnel propagated electric field of the j^th emitter integrated over the aperture of the of the i^th laser. This has been calculated before[1, 2], but we analyze its behavior in the Appendix and identify the optimum fill factor range for various Talbot planes.

We now cast the feedback condition into a form which explicitly displays the phase. Using eqs. (3,4)the feedback equation can be rewritten as

\frac{1}{r_{i}} \frac{A_{i}^{+^{2}} (0)}{A_{i}^{+} (L_{i})} \exp (i ϕ_{i}^{-} (L_{i})) = Σ R_{i, j} A_{j}^{+} (L_{j}) \exp (i ϕ_{j}^{+} (L_{j})) \exp (i k_{j} z + ϕ_{0}) .

We close this section by considering the effects of the random phase on the small signal gain region as well as on the saturated gain region. Specifically, we quantify the increase in threshold gain, and the decrease in the extracted laser amplitudes due to uncertainty in the individual laser phases. These features are confirmed with our numerical simulations. It is at this point we drop all reference to the different lengths L_j and incorporate these effects in a random phase ϕ_j associated with the j ^th emitter. In the small-signal region the field grows, according to eq. (6), exponentially as $A_{j}^{+}$ (L)= $A_{j}^{+}$ (0)exp(g_jL). Inserting this into eq. (7) gives

E_{i}^{+} (L) = r \exp (2 g L) Σ R_{i, j} E_{j}^{+} (L) \exp [i (ϕ_{j} - ϕ_{i})]

where we display the random phases ϕ_j on the right-hand-side, and we have absorbed the Talbot cavity propagation phase factor into R _i,j. If the random phases are eliminated this equation is identical to the eigenvalue equation studied earlier[1] with the eigenvalue λ̄ _i related to the threshold gain ḡ _i by ḡ _i =ln(1/r)+ln(1/|λ̄ _i |). In reality the phases ϕ_j are random due to different gain lengths and different dispersion. In averaging eq. (8) over the random phases we assume that the average of the product equals the product of the averages. Thus, the average takes the form

< E_{i}^{+} (L) > = r \exp (2 g L) Σ R_{i, j} < E_{j}^{+} (L)) > < \exp [i (ϕ_{j} - ϕ_{i})] > .

The last step is to complete the average of the two exponents. The diagonal term, (i=j), is straight forward. However, the off-diagonal average<exp[i(ϕ_j -ϕ_i )]>requires more work. First, since i≠j and the emitters are independent this average becomes the product <exp(iϕ_j )><exp(iϕ_i )]>. The last step is to complete the ensemble average of just one of the exponentials. This process has been developed for the turblenece transfer function[16], and for atomic decay processes[17]. Without going into these details we just quote the result that <exp(iϕ_j ) >=exp(-σ ²/2) when the phase satisfies Gaussian statistics, has zero mean, with a mean square phase of σ ². Thus eq. (9) becomes

< E_{i}^{+} (L) > = r \exp (2 g L) [R_{i, i} < E_{i} (L) > + \exp (- σ^{2}) Σ_{j \neq i}^{N} R_{i, j} < E_{j}^{+} (L) >] .

Eq. (10) clearly shows that the effect of the random phases is to increase the cavity losses by decreasing the coupling between gain elements. Thus, the threshold gain increases compared to the non-random case which is obtained by setting σ ² to zero, see ref. (1). In eq. (10) the amplitude of R _i,j contains different losses for different supermodes and the phase condition determines the supermode frequency.

Next we show that above threshold the fluctuating phases cause a decrease in the gain amplitudes <A_j >. To this end we turn to the saturated gain region where eq. (6) has the approximate solution

\frac{A_{j}^{+^{2}} (0)}{A_{j}^{+} (L)} \approx A_{j}^{+} (L) \frac{1 - 2 r g L + r A_{j}^{+^{2}} (L)}{1 + r A_{j}^{+^{2}} (L)} .

Inserting this into eq. (7) gives

E_{i}^{+} (L) \approx \frac{1 + r ∣ E_{i}^{+} (L) ∣^{2}}{1 - 2 r g L + r ∣ E_{i}^{+} (L) ∣^{2}} r Σ R_{i, j} E_{j}^{+} (L) \exp [i (ϕ_{j} - ϕ_{i})],

which can be solved through iteration for different phase realizations. When this equation is averaged the right-hand-side is diminished by exp(-σ²) and as a consequence the amplitudes $A_{j}^{+}$ are reduced from the nonfluctuating phase solution, as mentioned.

Finally, as a measure of locking we define the coherence function C as

C = ∣ Σ E_{j} (0) ∣^{2} ⁄ Σ ∣ E_{j} (0) ∣^{2}

which is related to the far-field on-axis intensity I(0)by I(0)=C∑|A_m |². For completely coherent emitters this function has a value N equal to the number of emitters. If, however, the phases are only random the average of the coherence function takes a simple form, particularly if the amplitude spectrum is also flat. With these restrictions and the previous averaging procedure eq. (13) averages to

< C > = \frac{1}{N} < \sum_{i, j}^{N} \exp [i (ϕ_{j} - ϕ_{i})] > = \frac{1}{N} [N + N (N - 1) \exp (- σ^{2})],

which approaches N exp(-σ²) for N large. We will numerically verify this behavior in the next section.

3 Numerical Simulations

In this section we solve eq. (1) subject to the two-point boundary conditions at z=0, $E_{j}^{+}$ (0)=r_j $E_{j}^{-}$ (0), and at z=L, $E_{i}^{-}$ (L)=∑R _i,j $E_{j}^{+}$ (L) through an iteration scheme. Specifically, at z=0, $E_{j}^{+}$ (0)is guessed and then the constant C_j is formed through C_j = $A_{j}^{+}$ (0)²/r. Integration of eq. (1) gives $E_{j}^{+}$ (L). In this process the phases just follow eq. (3). Next these values determine the initial condition on the reverse wave through $E_{i}^{-}$ (L)=∑R _i,j $E_{j}^{+}$ (L) and a new constant C_i formed using C_i = $A_{i}^{-}$ (L) $A_{i}^{+}$ (L). Integrating back to z=0 yields $E_{i}^{+}$ (0) and C_i (0). The entire process is repeated until convergence is achieved. This technique is far superior to a shooting method, especially for a large number of equations.

Each laser can have a separate gain g_j , reflectivity r_j , and length L_j . The latter is included as a random propagation phase added to the total phase after each forward and reverse propagation. This allows an assessment of the loaded cavity performance in the presence of random phase. For a specific ensemble of phases {ϕ_j }, convergence is achieved when the variables do not change between the N and the N-1 iteration. Thus, we introduce the probability of locking P as the number of converged cases divided by the total number of attempts for a set of M ensembles all of which have the same mean and rms phase. This is a measure of how easily an array with a given rms phase will lase. In the following, the number of emitters, N, is 6, the number of ensembles M=30, and each representation is iterated at least 60 times. The Monte Carlo technique is embodied in creating the M=30 ensembles all with differentt phase distributions but each with the same average and rms phase, then applying these to the differential equation.

Fig. 1. First (black), second (red), and third (green) nearest neighbor coupling amplitudes as a function of fill factor.

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In the Appendix we derive the coupling matrix R _i,j and illustrate some of its phase properties. Here, we consider the amplitudes. Fig. (1) shows the amplitude of the first, |R _3,4|² (black curve); second, |R _2,4|² (red curve); and third, |R _1,4|² (green curve), nearest neighbor coupling in the half-Talbot plane as a function of the fill factor. This figure shows that when the fill factor, defined by f=a/d where a is the emitting aperture width and d the period, is greater than .16 coupling is dominated by just the nearest neighbors. However, for f=.08 the coupling between first nearest neighbors and second nearest neighbors are comparable, .18 compared to .12, with non-negligible contribution from the third nearest neighbors at .08. For f<.08 the coupling becomes even more uniform. Thus, the filling factor should not be greater than .08 in the half-Talbot plane for strong coupling. This amplitude behavior is manifested in the Talbot cavity operation. Specifically, in the region where the coupling is dominated by just the nearest neighbors f>.16 the cavity experiences very little loss and the cavity supermodes become less distinct. Thus, as f increases the loss decreases and so does the threshold gain[1]. In contrast, for f < .08, more of the energy couples to the end emitters and exits the cavity so that the loss increases and consequently the threshold gain[1]. Another consequence of this amplitude behavior is that if the coupling is just between nearest neighbors f>.16 the probability of locking is smaller than the strong coupling case for a given rms phase. In passing we mention that the same functions for the quarter-Talbot plane shows that the form factor should be less than about .04.

As a baseline simulation we chose a relative high gain g_jL_j =gL=4, outcoupling r_j =r=.8, and a Talbot mirror reflectivity, r_t , of unity. The integration of dz is from zero to one. The remaining parameters are the array fill factor f=ω ₀/d, and the position of the Talbot mirror. For the latter we concentrate on the half-Talbot plane z=z_t /2=d ²/λ. This is the out-of-phase plane where the return image is displaced by d/2 and there is a null on-axis in the far-field. We choose a period of 150µm and an aperture of diameter 10/√(2)µm. This gives a fill factor of 0.47 and a Talbot distance of z_t =3.cm for λ=1.55µm.

We begin by considering two forms of phase corrections and compare these results when the only phase present is the Talbot phase. One phase compensation scheme is internal cavity phasing. Numerically this is acheived by forcing the individual phases ϕ_j (L) of eq.(3) equal to zero at each step of the integration. This case is instructive since it can drastically alter the performance of the resonator by driving it below threshold or by decreasing the near-field amplitudes, as we will show. Experimentally internal phasing is difficult to implement since changing an individual phase alters all other phases. The other type of phase correction is external to the outcoupling mirror. This is simulated by allowing the integrator to converge without any constraints and then afterwards setting the phase to zero. These two cases are compared with the uncorrected Talbot resonator. Figures (2a,2b) show the on-axis intensity as a function of z/z_t . In order to show the above mentioned threshold behavior fig (2a)i s for a gain gL=8 while fig. (2b)is for half that value. Both figures show that the uncorrected Talbot cavity displays several in-phase solutions at z=z_t /4, .75z_t , z_t , and at several other intermediate cavity lengths. Also, these figures show that .4< z/z_t < .63 the on-axis intensity is near zero. This is a manifestation of the out-of-phase solution and this region narrows as the number of emitters increases or as the fill factor decreases. For external phasing, both figures show that the maximum on-axis intensity occurs for z/z_t =.5 and that the on-axis intensity is slightly greater than in the quarter-Talbot plane. The major difference between these two figures is the behavior of the internally phased resonator (green curve). Fig. (2a) shows that the on-axis intensity is more erratic and is always less than or equal to the externally corrected resonator (red curve). In fact, in fig (2b), for gL=4, the internally corrected resonator has dropped below threshold except for very small cavity lengths, see the truncated green line. In principle this behavior could be predicted by solving our threshold condition with the constraint that the phases ϕ_j =0.

All of this behavior is reflected in the coherence functions. Specifically, the coherence function for the externally corrected phase is C=N=6; that of the internally corrected is erratic bouncing between 4 and 6, while the uncorrected C looks like that in fig. (2a) Next, we turn to the transverse characteristics. In all cases the far-field is given by the diffractive envelope exp(-π ² x ²/ $ω_{0}^{2}$ )times the interference term $\sum_{m = 0}^{N}$ A_m exp(iθ_m )×exp(iπmx/d). The amplitude A_m is a solution of eq. (5) and θ_m is a random phase. Thus, the divergence and separation between main lobes for the coherent case goes as the inverse of the period, while the incoherent divergence goes as the inverse of the laser apertures, see the averaging process used in eq. (10).

Finally, we include a random propagation phase to the uncorrected phase case for the half-Talbot plane. Fig. (3a) shows the near-field amplitude for emitter 3 as a function of the rms phase; fig. (3b)shows the probability of locking again as a function of the rms phase; and fig. (3c)shows the far-field coherence function defined by eq. (13). Both the amplitude and the locking probability decrease as the rms phase increases. However, the coherence function remain near zero and increases as the rms phase increases; this indicates the dominance of the anti-phase solution which certainly does not satisfy the exponential behavior indicated by eq. (14).

Fig. 2. On-axis intensity as a function of cavity length normalized to the Talbot distance. The black curve is without correction, the red is with external phasing, the green is with internal phasing. (a) is for a gain of 8 and (b) is for a gain of 4.

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Fig. 3. (a) the amplitude of emitter number 3, (b) the locking probability, and (c) the coherence function all as function of the rms phase in the half-Talbot plane.

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Fig. 4. (a) the amplitude of emitter number 3, (b) the locking probability, and (c) the coherence function all as function of the rms phase in the quarter-Talbot plane.

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The results for the quarter Talbot plane do not fair so well in a stochastic environment. Fig. (4a)shows the mode amplitudes are close to the half-Talbot amplitudes but do show a greater spread. Fig. (4b)shows that the major difference is in the locking probability. Specifically, the quarter-Talbot configuration is much more sensitive to rms phase variations as compared to fig. (3b). That is, in order to have successful lasing the fiber OPDs must not have large variations. Fig. (4c)shows the that the coherence function distribution is bounded by eq. (14) the solid line. It is in the in-phase distribution which most closely satisfies the conditions imposed on eq. (14).

In closing, we list other results of our simulations. Changing the fill factor has a marked effect on the locking probability. Specifically, increasing the fill factor forces nearest neighbor operation which means a decreasing locking as the rms phase increases. Next, as the number of emitters increases the corresponding tolerable rms phase decreases to maintain a constant locking probability.

4 Conclusion

We simulated the extraction of a lateral array of coupled emitters in an external Talbot cavity in the Rigrod approximation. In our model all the emitters are coupled through linear equations imposed by the Talbot cavity boundary condition. These differential equations are solved with and without a linear random propagation phase. For just deterministic phases we evaluated the far-field performance with inter-cavity phasing, and external phasing. These results were then compared with the uncorrected phase case. From this, we showed that the best far-field performance is for external phasing in the half-Talbot plane. However, performance in the quarter-Talbot is about the same for uncorrected phasing or external phasing; since this plane is the in-phase solution. Additionally, we showed that internal phasing can be detrimental since it can reduce the near-field or even drive the resonator below threshold.

In the random phase cases we showed that the half-Talbot plane gives the best far-field performance and that in the quarter-Talbot plane the random phases reduce the resonator extraction.

Appendix

In the following we develop the propagation of a one-dimensional array of Gaussian laser sources. Then we derive a simple expression for the complex reflection matrix. The finite laser array is modeled as a one-dimensional line of N Gaussians of period d. Each Gaussian has a radius ω ₀, where ω ₀ is the e ^-1 of the electric field. The initial electric field distribution is then

E_{0} (x) = \sum_{m = 0}^{N} E_{m} \exp [- \frac{{(x - m d)}^{2}}{{ω_{0}}^{2}}] .

If the laser aperture is a then a=2ω ₀/√2.

At a propagation distance z the Fresnel propagated electric field is

E_{P} (x, z) = \frac{i \exp (i k z)}{⋋ z} \iint E_{0} (x, y) \exp (\frac{i k}{2 z} [{(x - x^{'})}^{2} + {(y - y^{'})}^{2}]) d x^{'} d y^{'}

Integration gives

E_{P} (x, z) = \exp (i k z) \sqrt{\frac{1}{1 + i z ⁄ z_{0}}} \sum_{m = 0}^{N} E_{m} \exp [i \frac{k}{2 (z - i z_{0})} {(x - m d)}^{2}]

where z ₀= ${π ω}_{0}^{2}$ /λ.

Next we consider the reflection coefficient R _i,j . This is given by the overlap integral between the electric field propagated through a distance z and the initial electric field distribution at z=0. Thus,

R_{i, j} = \frac{{\int G}_{i} {(x, y, 0)}^{*} G_{j} (x, y, z) d x d y}{\int G_{i} {(x, y, 0)}^{*} G_{i} (x, y, 0) d x d y}

where the integrals extend over all space. Inserting the above two equations and integrating gives

R_{i, j} = \exp (i k z) \sqrt{\frac{1}{1 + i z ⁄ 2 z_{0}}} \exp [i \frac{k}{2 (z - i 2 z_{0})} {(x_{i} - x_{j})}^{2})]

where x_m =md. This equation can be cast in a form which clearly shows the uniqueness of the Talbot distance z_t =2d ²/λ. In the usual configuration z, z_t >> z ₀, in our case z_t /z ₀≈10³, and eq. (A5) becomes

R_{i, j} \approx \exp (i k z) \sqrt{\frac{1}{1 + i z / 2 z_{0}}} \exp [i \frac{π}{2} {(i - j)}^{2} \frac{z_{t}}{z}] .

The two most common Talbot planes are at z=z_t /2, z_t /4 which are determined by the phase of R _i,j . In the half-Talbot plane the phase factor of the coupling coefficient is an alternating ±1, which is the out-of-phase solution and in the quarter-Talbot plane the phase is always +1 which is the in-phase solution. The former displays a half-period shift in the half-Talbot plane which results in a far-field null on axis.

References and links

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Extraction characteristics of a one dimensional Talbot cavity with stochastic propagation phase

Abstract

1 Introduction

2 Theory

3 Numerical Simulations

4 Conclusion

Appendix

References and links

Cited By

Figures (4)

Equations (21)

Optics Express