Fiber-based saturable-absorber action based on a focusing Kerr effect

Fiber-based saturable-absorber action based on a focusing Kerr effect

Optics Communications 367 (2016) 292–298 Contents lists available at ScienceDirect Optics Communications journal homepage: www.elsevier.com/locate/o...

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Optics Communications 367 (2016) 292–298

Contents lists available at ScienceDirect

Optics Communications journal homepage: www.elsevier.com/locate/optcom

Invited Paper

Fiber-based saturable-absorber action based on a focusing Kerr effect Long Wang, Joseph W. Haus n Electro-optics Program, University of Dayton, 300 College Park, Dayton, OH 45469, USA

art ic l e i nf o

a b s t r a c t

Article history: Received 18 September 2015 Received in revised form 21 January 2016 Accepted 22 January 2016 Available online 5 February 2016

We report numerical simulations on a fiber compatible, self-focusing, saturable absorber device. Two fiber ends are separated by a bulk, nonlinear medium. An optical beam transmitted from one tapered fiber end, propagates through the nonlinear medium and couples back into the other tapered fiber end. The fiber mode distributions at the ends of the tapered fibers are calculated using a Finite Difference Method (FDM). We apply the beam propagation method to simulate the diffraction and nonlinearity in the nonlinear medium. As a function of initial beam power and the fiber mode design, the coupling efficiency plots are calculated and compared for different nonlinear mediums. Our simulations identify the optimum contrast between low and high input powers. & 2016 Elsevier B.V. All rights reserved.

Keywords: Self-focusing Kerr effect Saturable absorber Mode-locked fiber lasers Fiber modes Critical power Diffraction and nonlinearity

1. Introduction Laser beam self-focusing as a result of an intensity-dependent increase in the refractive index has been studied for many decades; usually in the context of catastrophic focusing and filamentation of intense beams. The phenomenon is described by third-order nonlinear processes, which will be simply denoted as χ (3) processes after the nonlinear coefficient with its tensorial indices suppressed. Efficient mixing of multiple frequency beams using χ (3) processes require the phase-matching which introduces additional conditions. However, the self-induced nonlinear effect (often referred to as Kerr effect) occurs naturally since the phasematching condition is automatically satisfied [1–3]. The application of self-focusing as a passive device for pulse formation was first reported for Ti:sapphire lasers in 1991 by Spence’s group [4]. It was called lasers Kerr-lens mode locking and achieved pulses as short as 60 fs pulses. Since that time there has been rapid progress in generation of ultra-short pulses based on the same technique with various cavity designs. The reason ultrashort pulses can be generated with Kerr effect is due to the ultrafast response time of the electronic effect, which in many materials is usually estimated to be 1–2 fs. Mode locking action using a self-focusing χ (3) nonlinearity in conjunction with a hard aperture is schematically shown in Fig. 1; when inserted in a laser cavity its action leads to Kerr lens mode locking with a hard aperture. At low intensities the optical beam is severely clipped by n

Corresponding author. E-mail address: [email protected] (J.W. Haus).

http://dx.doi.org/10.1016/j.optcom.2016.01.063 0030-4018/& 2016 Elsevier B.V. All rights reserved.

the hard aperture. However, an intense optical beam going through a Kerr medium maintains its temporal profile, but it is to be spatially contracted in the transverse direction depending on the local intensity. The degree of the beam’s contraction depends on the instantaneous intensity of the pulse, which means higher intensity experiences higher degree of beam contraction at the aperture. When the pulse passes through a hard aperture, it experiences lower loss at high intensities due to the nonlinear focusing effect of the beam and higher loss at low intensities where the aperture clips the more of the beam, which has broadened due to diffractive spreading. Thus the pulse after the aperture becomes shorter in time as the low intensity wings of the pulse are filtered out of the cavity. When the pulse shortening due to the aperture is balanced by spectral broadening due to nonlinear effects, such as self-phase modulation, and by cavity dispersion effects the laser cavity may exhibit stable operation. The combination of the Kerr medium plus the aperture has an effect that is called saturableabsorber action; in other words the losses are lower at higher intensity.

2. Saturable absorber action design In this paper we study a new approach to achieving a saturable-absorber action by harnessing the Kerr effect on the transverse beam shape, i.e. self-focusing effect in a compact, optical fiber environment. The saturable-absorber’s design illustrated in Fig. 2 shows two tapered fiber ends separated by a space with a nonlinear Kerr medium between them. In Fig. 2, both the transmitting and receiving fibers are tapered to modify the mode sizes

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Fig. 1. Illustration of Kerr lens intensity-sensitive pulse control with a hard aperture.

Fig. 2. Four elements in the free space region (fast SA): Kerr-type nonlinear medium, End 1, End 2 and the separation between them.

medium thickness can be increased and the laser power is spread over a larger area decreasing the nonlinearity required to cancel diffractive spreading. Note that the transmission and coupling efficiency are interchangeable in this paper. To understand the pulse transmission through the Kerr medium region, let us first assume that the beam transmitting through the medium is a continuous wave. In this paper we study the SA device characteristics; for simplicity we assume fiber 1 and fiber 2 have fiber modes with the same radii at the ends. For this situation the coupling or transmission from fiber 1 to fiber 2 can be maximized if the diffraction in the nonlinear medium can be exactly balanced by the nonlinearity. For this case the beam power is called critical power in Watts given by the following estimated expression [5,6]:

Pcr = 0. 148

Fig. 3. Illustration of the saturable absorber in a fiber laser cavity whose action can lead to mode-locking. Coupler 1 transmits wavelength nearby 972 nm and 1550 nm. Coupler 2 splits the input into two parts: one through output and one back into the fiber cavity. In the fiber cavity, the black fiber is the Er:Yb doped gain fiber and the blue fibers are SMFs.

and are placed face to face with a certain separation in between. The gap is filled with a nonlinear medium (for example CS2 or chalcogenide glass As2 S3). The transmitting fiber is denoted as fiber 1, the receiving fiber is denoted as fiber 2 and their separation is denoted as D . This design can occur in many fiber lasers designs, with one shown in Fig. 3, and the fiber mode radius is adjusted by tapering the fiber ends in Fig. 2. The Kerr effect medium is located in the free space region in dashed rectangle. In this paper we use the term “free space” to refer to beam propagation without lateral boundaries. The black line in the figure stands for the Er:Yb doped fiber and the blue lines are normal single mode fibers. An isolator is used to ensure that the pulse is propagating clockwise. In Fig. 2, the beam coming out from fiber 1 is reshaped in the nonlinear medium due to the net balance of diffraction and Kerr effect; the beam coupling into fiber 2 is affected by the beam size. Comparing Fig. 2 to Fig. 1, the similarity between the end of fiber 2 and the previously discussed hard aperture is apparent, and the analogy is also closely connected to the pulse shortening process in the fiber. Inserting the saturable absorber (SA) device in a fiber laser cavity it can lead to mode-locking action; a potential fiber laser cavity design is illustrated in Fig. 3. The aperture in our device has a soft shape determined by the fiber mode. The transmission amplitude between fiber 1 and fiber 2 depends on several parameters: the initial and final fiber mode radii, the separation between the fibers, the nonlinear material and the pulse power carried in the wave. The main reason for using a tapered fiber is to enlarge the mode radius, which reduces the diffractive effect in the nonlinear medium by increasing the Rayleigh range. Therefore the

λ2 , nn2

(1)

where n is the refractive index, λ is the vacuum wavelength in meters (1.55 microns is used throughout the paper) and n2 is the Kerr nonlinearity in units of m2/W. In this paper, all the powers used are taken to be smaller than the critical power, since beyond the critical power the beam becomes unstable and collapsing to a filament can occur [7]. The critical power for beam trapping and stability in all dimensions was discussed, for instance, in Ref. [8]. For a two dimensional beam the trapped beam shape is called the Townes profile [9] and it is unstable and can collapse to a filament. Our system operates well below the critical power; as the beam power increases the nonlinearity reduces the beam spread. For instance, in a fiber laser cavity the peak power coupling loss for our device must be sufficient to overcome all other cavity losses, so that laser mode-locking may occur. In other words, a beam carrying higher power is going through the fiber-based device with higher transmission rate than the lower intensities. This behavior is analogous to the action of a fast SA [2,10–12]. The beam from fiber 1 is assumed to be the mode inside multiplied by a constant and is calculated using the Finite Difference Method (FDM) [13]. The transmission rate through the free space region as a function of beam power is calculated using the beam propagation method (BPM) with diffraction [1] and the terminologies of free space and SA will refer to the same thing in this paper. As shown in Fig. 2, the SA configuration consists of four parameters: the fiber 1 end denoted as ‘end 1’, fiber 2 end as ‘end 2’, the separation in between the two ends and the nonlinear medium. We place fiber 1 and fiber 2 face to face without any angular tilt and lateral shift. To adjust the SA characteristics, one can change any one of the four parameters and the corresponding transmission rate would be different. Also, the exact transmission rate should take into consideration the beam shape transitioning through the taper region and the reflection on the interface of different materials. While the critical power is independent of the mode diameter, the Rayleigh range depends on it and so by using different mode diameters the thickness of the SA medium will be affected. So a wider mode has a lower peak power and the longer Rayleigh range means the nonlinear focusing can be weaker to

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overcome diffraction. The gain in the laser cavity is set below the threshold for continuous wave lasing. A pulse circulating inside the laser cavity can be sustained since it lowers the losses in the SA. The nonlinear medium is assumed to react instantaneously to the pulse. We slice the pulse into many temporal segments, one segment of the pulse is viewed as a continuous wave and the analysis of transmission through the SA for each slice is similar to the case of a continuous wave; the dispersion added to the pulse in this short section is negligible. Due to chromatic dispersion the pulse width is broadened when it is propagating around the laser cavity; when the pulse passes through the SA it is treated as a series of continuous waves; each slice of the pulse is independently passing through the SA. Similar to the continuous wave analysis, different slices of the pulse containing varying powers will pass through the SA with different transmission amplitudes; the center part of the pulse with higher power experiences higher transmission rate than that in the wings of the pulse. Thus, the pulse width is shortened by the SA action. Once the shortening process through the SA is balanced by the broadening effects from the rest of the cavity, the laser may approach a steady state solution. We consider two nonlinear materials in the following two sections.

3. Numerical study of the saturable absorber with carbon disulfide In this section, the nonlinear medium is chosen to be carbon disulfide ( CS2). From Table 4.1.2 in [6], the refractive and nonlinear indices for CS2 are n0=1.63 and n2 = 3.2 × 10−14 cm2/W , respectively. The nonlinearity value is reported for pulses longer than the 2 ps relaxation time of the molecular reorientation contribution. For shorter pulses (110 fs) the nonlinearity is smaller and reported values are about one order of magnitude smaller [14]. The nonlinear coefficient is related to the nonlinear coefficient χ (3) discussed earlier. The nonlinear refractive index is defined as a function of the intensity I as

(2)

n = n0 +n2 I,

The CS2 is a liquid at room temperature with a strong nonlinearity compared to many other materials. The tapered fiber ends for the SA are chosen as a pair. For example, the fiber-end pair 10–10 means that both the emitting and accepting fiber ends are tapered to be of 10 μm as the outer cladding radius. Note that both fibers are tapered from the same normal, single-mode fiber, which is assumed to have an outer radius of 62.5 μm and a core radius of 4.1 μm . All the fiber pairs used for simulation are presented in Table 1 denoted by fiber radii. Non-equal choices for the fiber ends are also possible and can be introduced to further explore the design space for optimization of the device’s operation. Consider the 20–20 fiber pair where both fiber 1 and fiber 2 have a 20 μm outer radius. The coupling efficiency, η (P , D), as a function of continuous beam power and separation between fiber 1 and fiber 2 is plotted in Fig. 4(a). For simulations beyond the critical power the beam collapses (for this fiber the beam peak Table 1 Fiber end pairs used for simulation with nonlinear material CS2 . Number

Fiber pair radii

1 2 3 4 5

62.5–62.5 40–40 30–30 20–20 10–10

intensity at the critical power is 1.17 GW/cm2 in CS2) and would normally form a filament or damage the material and thus this regime is avoided. For each separation, the transmission increases with increased beam power up to a beam power 67.08kW , which is the critical power calculated using Eq. (1) [6]. For many applications, such as mode-locking, the absolute coupling efficiency is important, but also the transmission difference between high and low beam powers is important, i.e. δη = η (P , D)−η (0.287, D); this quantity is displayed in Fig. 4(b). Note that the lowest power of the pulse used for simulation is 0.287W . For each separation, the transmission difference is obtained by subtracting the coupling efficiency of the lowest power ( 0.287W in our simulation) from the absolute transmission. When the transmission difference is higher, then the SA action is easier to achieve. For example, with a separation of 2.003mm , the continuous beam with power of 64.02kW can propagate through the SA with transmission rate of 83.01% and the transmission rate for continuous beam with power of 0.287W is 15.48% as shown on Fig. 4(a). The difference between the two continuous beams is 67.53% as shown in Fig. 4(b). Similarly, at the 2.003mm separation, the transmission can be calculated for beams with other powers and transmission difference can be calculated by subtracting the transmission of lowest power (0.287W ) from that of the specified beam. Similar calculations can be carried out for all separations. As discussed above the transmission curves so far are calculated using continuous beams can be used for pulses for a fast SA. For instance, assuming that a pulse oscillating in the laser cavity has peak instantaneous power of 64.02kW and then transmission rate difference between the center slice and the low power slice ( 0.287W ) is the same as the one between the two continuous beams (67.53%). Even for the same pulse, different separations produce different transmission curves. Thus the separation between fiber 1 and fiber 2 can be adjusted to achieve optimal SA action. For example, the pulse with peak instantaneous power of 64.02kW minus the low power transmission (data tips in Fig. 4(a)), we find that fiber ends separated by 2.003mm has with the transmission difference, δη , of 67.53% (data tip in Fig. 4(b)). This value is very close to the largest transmission difference, which occurs for the same pulse when the fiber ends are separated by 1.865mm ; the corresponding transmission difference is 67.79% (not shown). For each application design, the stable pulse that can be supported is different and thus the peak instantaneous power of the pulse is also different. From the analysis above, for pulses with different peak instantaneous powers, optimum separations between the two fiber ends are different. By setting both fiber 1 and fiber 2 radii to be 20 μm as the outer radius, and the nonlinear medium to be CS2, the best separation for different pulses can be extracted from Fig. 4(b). The values are plotted in Fig. 5(a). As shown, as the peak instantaneous power of a pulse increases, the optimum separation between the fibers ends increases accordingly. The coupling efficiency difference between the peak power and the low power of a pulse at optimum separation is plotted in Fig. 5(b), where the transmission difference between low and high intensity increases with increased pulse peak power. Although Fig. 5 illustrates the operational characteristics only for the fiber-end pair 20–20., we extract similar information from simulations for other fiber-end pairs and plot them on the same plot in Fig. 6. In Fig. 6(b) the optimum transmission rate difference as a function of the peak power is almost the same no matter which fiber pair is used. On the other hand the optimum separations between fiber ends are different for different fiber pairs, as shown in Fig. 6(a). The fiber pair that allows largest separation is the pair of 30–30 and 20–20. While the transmission curve is insensitive to the fiber pairs, experimental devices allowing larger separation between fiber ends are more tolerant to fabrication errors.

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Fig. 4. Simulation for fiber pair 20–20 with carbon disulfide ( CS2 ): (a) the coupling efficiency η as a function of separation D and beam power; (b) the coupling efficiency difference between high power and low power beams. On (a), η for two beams with power of 0.287 W and 64.02 kW are shown at D = 2.003mm and the difference, δη = η (P )−η (0) , between them is shown on (b).

Fig. 5. Results for the nonlinear medium CS2. (a) Optimum separation as a function of pulse peak power; (b) Optimum coupling efficiency difference, δη , between the peak power and the low power region as a function the pulse peak power.

Fig. 6. Optimum SA action information for different fiber pairs with the nonlinear material CS2. (a) The optimum separation as a function of different pulses; (b) Optimum coupling efficiency difference, δη , between the peak power and the low power region as a function the pulse peak power.

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Fig. 7. Results for the nonlinear medium As2 S3. (a) Optimum separation as a function of pulse peak power; (b) Optimum coupling efficiency difference, δη , between the peak power and the low power region as a function the pulse peak power.

Fig. 8. Coupling difference, δη , for fiber pair 20–20 with nonlinear medium of (a) CS2; (b) As2 S3.

4. Numerical study of the saturable absorber with As2 S3

5. Optimal saturable absorber design

Our results from the previous section reveal that the optimum coupling efficiency is almost irrelevant to the radii of the fiber-end pairs and optimum separation are larger for fiber pairs of 20–20 and 30–30. By replacing CS2 by chalcogenide glass As2 S3, we find that optimum transmission rate and the separation follow a similar pattern. Thus in this section we only simulate the fiber pair of 20–20 to illustrate the essential result for As2 S3. From [6] we know the refractive index and nonlinear index for As2 S3 are

The analysis in previous sections show that for each pulse there is an optimum SA action separation where the pulse center and wings transmission rate difference is maximized. However, the steady-state pulse shape and energy remains to be determined by inserting the device into a specific laser cavity. By adjusting the fiber ends separation towards the optimum point found by our simulations, the steady pulse would adjust accordingly which in turn would result in another different optimum point for the pulse in the cavity. Thus in practice, it is an evolving process in which design of SA is adjusted according to the cavity design. We find that for both CS2 and As2 S3, the optimum separation is close to 3.0mm when the pulse peak power is close to the critical power. However the device could efficiently operate at peak powers far below the critical power and still maintain a sufficient low power to high power transmission contrast; this would certainly be the case, for instance, in mode-locked laser cavity applications. When the question arises: which nonlinear medium should be used for the device? The answer depends on the estimate of the pulse peak power. For example, if we choose CS2 as the nonlinear medium then a steady-state pulse with 6.88kW peak instantaneous power would have high to low power transmission difference of 0.8225% at 3 mm compared to 3.315% at 0.8853 mm,

2.4 and 2.0 × 10−13 cm2/W , and the critical power is 7.3kW [6]; the peak intensity of the beam for the 20 micron tapered fiber radius is 122 MW/cm2, which well below the laser damage threshold. The optimum coupling and separation for As2 S3 are shown in Fig. 7. Compared to CS2, chalcogenide glass has much lower critical power. That is why for the same pulse, the optimum coupling for As2 S3 is much higher than that for CS2, which is obtained by comparing Fig. 7(b) to Fig. 5(b). While SA with As2 S3 has a higher transmission for the same pulse parameters, the nonlinear material can be made thicker than CS2 by comparing Fig. 7(a) to Fig. 5 (a). The larger distances is partially due to the larger refractive index in As2 S3 that reduces diffractive spreading of the beam.

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Fig. 9. Absolute transmission curve for a SA for D = 3mm with the nonlinear medium of (a) CS2 ; (b) As2 S3.

with data points shown in Fig. 8(a); the latter is the optimum separation since it has a higher transmission contrast. If the pulse has peak power as high as 65.55kW (not shown in the figure), the separation distance of 3.0mm has a much larger transmission rate difference, i.e. 67.94%, than the 0.8853mm separation case. The 3.0 mm separation is not significantly lower than 73.49% at the optimal separation 2.13mm showing that the design can be quite tolerant to separation errors. Similar analysis can be carried out for As2 S3. For a pulse with peak power of 1.735kW (not shown in the figure), the coupling difference at 3.0mm is about 4% while the optimum separation occurs at 1.39mm with 7.7%. If the pulse has a peak power of 6.881kW (data tips in Fig. 8(b)), the coupling difference at 3.0mm is 52.11%, slightly lower than the optimum point of 55.59% at 2.314mm . Depending on the laser pulse characteristics in the cavity, different nonlinear mediums should be adopted. If one estimates the peak instantaneous power reaches close to 67kW , one should use CS2 as the nonlinear medium to obtain the largest low to high intensity transmission differences. On the other hand, if the peak instantaneous power of the pulse is expected to be lower than 7.3kW , chalcogenide glass As2 S3 would be the choice for the nonlinear medium. The transmission for both CS2 and As2 S3 (20–20 fiber pair with separation of 3.0mm ) are plotted in Fig. 9. The lower intensity curves show smaller transmission changes, but in applications, such as mode-locked fiber lasers, the device will not depend on large transmission changes for all cavity designs.

with higher transmission rate than that of lower power beams. Even though the transmission rates are calculated using continuous beams, it can be applied to the pulse circulating in the laser cavity for pulses much longer than the material relaxation time. To apply our calculations to pulses in a laser cavity we slice the pulse into temporal segments and treat each slice as a continuous beam. Thus the transmission for one continuous beam can be used for a slice of the pulse with same power. As a result, center part of the pulse goes through the free space region with higher transmission rate than that in wings and the pulse is shortened after transmission. We compared transmission curves for different fiber designs and found that the transmission is not sensitive to the mode size of fibers. The size of the mode is variable in our studies to increase the Rayleigh range; a larger mode size weakens the diffractive broadening of the beam in the SA, which increases the separation between the two fiber ends. As an illustration of our results we assumed both fibers are tapered to have a radius of 20 μm and the separation between them was chosen to be 3.0mm ; however, the results essentially apply to other fiber radius designs, as well. The transmission curves as a function of the instantaneous beam power through the SA are calculated for two nonlinear mediums ( CS2 and As2 S3) and the results can be applied to laser cavity simulations. A future publication will study the stability of pulses in a fiber laser cavity that uses our fiber-based Kerr lens SA device. In practice the final SA design will be adjusted along with the laser cavity parameters to converge to the steady-state pulse characteristics.

6. Conclusion We propose a SA action device that is the fiber analog of a Kerr lens SA by placing a nonlinear Kerr medium between two fiber ends. The Kerr lens device has been successfully inserted as a mode-locker in solid-state lasers and this device could be similarly used as a fiber-based mode-locker. The SA devices in a ring cavity are much more compact compared to, say, nonlinear optical loop mirrors. It does not modify the polarization state and could be used together with polarization rotation mode locking. The material in the gap between the fibers reshapes the optical beam depending on the instantaneous pulse intensity. Once the fiber sizes, separation and nonlinear medium are fixed, the transmission rate between fiber ends in SA depends only on the beam power; beams with higher powers go through that region

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