Thermal, stress, and thermo-optic effects in high ... - IEEE Xplore

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 37, NO. 2, FEBRUARY 2001

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Thermal, Stress, and Thermo-Optic Effects in High Average Power Double-Clad Silica Fiber Lasers David C. Brown, Member, IEEE, and Hanna J. Hoffman

Abstract—Yb : Glass fiber lasers have matured to the point where the average power scaling of such devices to the kilowatt level and beyond can be realistically pursued. In this paper, we present a comprehensive study of thermal, stress, and average power scaling in double-clad silica fiber lasers. We show that careful management of thermal effects in fiber lasers will determine the efficiency and success of scaling-up efforts. Index Terms—Birefringence in fiber lasers, diode-pumped fiber lasers, fiber lasers, scaling of fiber lasers, thermal effects in fiber lasers, thermal heating in fiber lasers, Yb-doped fiber lasers.

I. INTRODUCTION

F

IBER lasers have been the subject of considerable attention recently [1]–[3] and ytterbium-doped fibers have now reached continuous wave (CW) average power levels in excess of 100 W [4]. Such lasers are beginning to find a number of commercial and military applications, for example, as sources for thermal printing, welding, marking, medical applications, and remote sensing. Among the performance benefits of Yb : Glass fiber lasers are high optical-optical and wallplug efficiencies, single-transverse-mode output, low noise output, and wide tunability [5]. There is now a great deal of interest in further scaling the output of Yb : Glass fiber output average power levels another order of magnitude to the kilowatt level [6]. In this regime, bulk Nd- and Yb-doped crystalline lasers encounter difficult thermal management issues, and fiber lasers would, therefore, offer substantial advantages because of more favorable surface to volume ratios. However, while it is often assumed [5] that thermal effects will play little or no role in the average power scaling of fiber lasers, such assumptions have not been experimentally tested or theoretically validated in the kilowatt power domain. Here, we present recent results from theoretical modeling of Yb : Glass fiber lasers scaled to the kilowatt level and beyond, describing what role thermal and stress effects might play at these high powers. Our analysis indicates that simple assumptions of straightforward scaling cannot be justified because standard fiber designs will lead to elevated core temperatures as average power is increased to high levels, adversely affecting the fiber laser performance. We focus, in this paper, on the case where the deposition of heat power into the fiber is uniform; that is, we assume no variation, where is the propagation direction in the fiber. The

Manuscript received May 19, 2000; revised September 26, 2000. The authors are with Advanced Laser Systems, Brackney, PA, 18812 USA, and Palo Alto, CA 94303 USA. Publisher Item Identifier S 0018-9197(01)00884-3.

obtained results can, however, be easily extended to the case of nonuniform heat deposition or pump light absorption as has been treated previously for bulk lasers [7]. We also concentrate on double-clad fibers in which a core doped with Yb ions is surrounded by a lower index cladding which is, in turn, surrounded by a second cladding of lower index. To model thermal and stress effects, we assume that heat is deposited only in the doped core; the core and two cladding regions are assumed to have the same thermal and mechanical properties, since they are often composed of similar glasses with properties that vary only slightly with respect to one another. Thus, parameters such as thermal conductivity, Poisson’s ratio, Young’s modulus, and thermal expansion and change in index with temperature are all assumed to be identical in the three regions, and constant with temperature. Some current glass fibers have a core region surrounded by polymer materials, which typically have an order of magnitude lower thermal conductivity than glass as well as substantially different mechanical properties. Such structures are best analyzed using finite-element methods. The use of polymer cladding materials will, however, only exacerbate the thermal effects discussed in this paper; in particular, core temperatures will further increase. We also realize that most thermal and mechanical parameters do vary with temperature, which can lead to important modifications of the linear solutions presented here, as has been found for the case of YAG [8]–[10]. Nevertheless, though the scope of this work is limited to the linear approximation, results are expected to be useful for determining, to first-order, important trends in thermal, stress, and thermo-optical effects for silica fiber lasers at high pump powers. We begin, in Section II, by solving the heat diffusion equation for double clad fibers, leading to explicit expressions for the thermal distributions in the core and cladding regions, and for the average temperature. As we will see, there are two distinct regions of the fiber that need to be addressed: the core region and the cladding regions. In what follows, we refer to the cladding region as that entire region outside the core, without identifying separate cladding layers. This is justified when the thermal and mechanical properties of the various cladding regions can be assumed to be identical, as was stated above. We also ignore the outer protective cladding region which is applied to most optical fibers, because the outer cladding will only have a secondary effect on the laser performance. Furthermore, in this paper, we consider only the case where the core and cladding regions are concentric. This assumption can also be readily modified in a more advanced treatment of scaling effects and is not expected to affect the substance of our conclusions. In Section III, we derive explicit expressions for the radial, tangential, and stresses

0018–9197/01$10.00 © 2001 IEEE

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in the two regions; these expressions are subsequently used in Section IV to calculate the radially varying indices of refraction in the core and cladding regions, including changes in index of , as well as stress. In Section V, we derefraction due to rive an analytic expressions for the average power output scaling of a fiber laser as a function of pump input, and based on the stress-fracture limit, and calculate a critical index limit at which the index variation from the center to the edge of the core region is equal to that of typical step-index fibers. This allows one to characterize the core center temperature variation as a function of critical fiber parameters. We conclude that it is possible, using present technology, that the core center temperature can reach a substantial fraction of the melting temperature for fused silica fibers if measures are not taken to minimize it. We then examined, in Section VI, the ramifications of a radially varying temperature distribution on the achievable inversion density and gain in the quasi-three-level fiber material Yb : SiO , and find that managing the core center temperature is critical to maintaining good laser efficiency as power is scaled up. Implications of the modeling results are discussed in Section VII, with respect to, and along with, recommendations for future work in this area. II. FIBER TEMPERATURE DISTRIBUTION We begin by finding the radial thermal distribution for the fiber geometry shown in Fig. 1. The radial coordinate is and is the tangential angle. The quantities and are the core and cladding radii, respectively. Knowing the temperature distribution in a fiber is necessary in order to determine the radially , as well as to calcuvarying index of refraction due to late stresses that contribute to the change in index of refraction through the stress-optic effect. We write the steady-state heat equation for an isotropic medium as (Region I)

(1)

Fig. 1. Geometry of modeled fiber showing active (doped) inner core of radius a and cladding region with radius b. Radial coordinate is r and ' is the azimuthal angle.

where is the convective or film coefficient, and the coolant temperature. For most fiber lasers, the coefficient would be used to describe cooling at the fiber surface in contact with unforced air. Straightforward solution of (1) and (2), subject to the aforementioned boundary conditions, results in the following expressions for the temperature in Regions I and II: (4) (5) Equation (4) shows that in the pumped core region, the temperature varies quadratically with , while (5) reveals that the temperature falls off logarithmically with . The center tempercan be related to the coolant temperature by use of ature

for the core Region I, and

(6) (Region II)

(2)

for the cladding region II. In (1), denotes the thermal conthe heat density. Note that in writing (1) and ductivity and (2), we have ignored any azimuthal ( ) variations in the temperature, and thus effectively assume cylindrical symmetry. We is constant, which also assumed that the heat power density is a good approximation in a long weakly doped fiber. In (1) and are the temperatures in regions I and and (2), II, respectively. The temperatures and their derivatives must be ; that is, continuous across the boundary and . In addimust tion, the temperature in the center of the rod for . Yet another boundary condition is satisfy , where we assume Newton’s law of cooling; thus, that for must satisfy

The terms in the bracket represent, from left to right, the temperature drop in the core region, cladding region, and the temperature drop due to the thermal resistance of the convective boundary region. Equations (4) and (5) can then be represented as in terms of the coolant temperature

(7)

(8) We can calculate the average temperature

as

(9) (3)

BROWN AND HOFFMAN: HIGH AVERAGE POWER DOUBLE-CLAD SILICA FIBER LASERS

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IV. INDEX OF REFRACTION DISTRIBUTIONS

Integrating (9) and using (7) and (8) leads to (10) III. STRESS DISTRIBUTIONS Because optical fibers are high aspect ratio devices, where the length is much greater than a typical fiber outside diameter , we can invoke the plane-strain approximation everywhere. It can then be [11] in which the strain shown that the radial, tangential, and stresses , , and can be found from (11)

(12) and in the case that the fiber end faces are free of traction

Having derived expressions for the radial temperature distributions, (7) and (8), and for the stress distributions, (14)–(19), we now turn our attention to the calculation of the radially varying radial and tangential index of refraction distributions. For comparison to modern finite-element codes, we prefer to cast the calculation of the induced change in the index of refraction in terms of material stresses, rather than strains. We begin by writing the changes in the and indices of refraction as (21) where linear index of refraction; change in index due to the change in index with ); temperature ( stress-induced index changes for the and components of the electric field. As before, the Roman numerals refer to Regions I and II. Here, using the following equation: we will calculate

(13) , and are Young’s modulus, thermal expansion coefficient, and Poisson’s ratio, respectively. Solution of (11)–(13), using the expressions previously derived for the temperatures in Regions I and II, (7) and (8), results in the following final expressions for the stresses in Regions I and II: (14)

(22) For Regions I and II, (22) becomes (23) and (24)

(15) (16)

The calculation of the stress-induced index changes are treated in the Appendix. Using (23) and (24), as well as (A9)–(A12), we write the solutions of (21) for the total index change in Regions I and II

(17) (18)

(25)

(19) The quantity by

in (14)–(19) is the materials constant, given

(26)

(20) It can be shown that (14)–(19) satisfy the boundary conditions , , , and the continuity conditions , , and . Equations (14)–(19) will be used in Section IV below to calculate the radially varying index of refraction due to the stressoptic effect. Note that the equations as derived above for the fiber stresses could also have been obtained by use of the deviation of the temperature distribution from the average and the Airy stress potential [12].

(27)

(28) We can also define the birefringence

, given by (29)

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which may be calculated by using (25)–(28), yielding

takes the approximate value of W/cm for a 50-m long silica fiber, and is almost always violated in fiber lasers. If we ignore stress effects, which are small compared to the term in (36) and (37), then both focal lengths are equal , where is the and given by is the total heat power in the fiber. If we fiber length and then calculate the heat power or heat density required so that the ), obtained focal length is just equal to the fiber length ( we find for a 4.6- m core radius and 50-m length fiber typical of those in use today, that only 1.19 W of heat is required and is just W/cm . that the corresponding value of If a Yb : SiO fiber is pumped with 180 W at 915 nm and lases at 1120 nm [4], the quantum defect or approximate heat fraction is 0.183. Thus, for complete absorption in the same fiber, the is about amount of heat power is 32.9 W and the value of 9912 W/cm . These values are far in excess of those required for the fiber focal length to be just equal to the fiber length, and we thus conclude that, for any practical power levels, the focal lengths calculated using (36) and (37) are, in fact, much shorter than typical fiber lengths and that focal lengths have no meaning when applied to fiber lasers. This conclusion is not surprising, given the guided-wave nature of clad fibers.

(30) and (31) Equations (30) and (31) show that the fiber birefringence depends only on the thermally induced stresses. Finally, by using (25)–(28) and substituting (14)–(19) for the radial, tangential, and stress components, we arrive at the final equations describing the radially varying indices of refraction in a fiber

(32)

V. HIGH AVERAGE POWER SCALING CONSIDERATIONS (33)

(34)

Having derived the fundamental relationships necessary to describe thermal and stress effects in fiber lasers, we now turn our attention to the calculation of various performance limits we have found that have relevance to the scaling up of fiber laser output power. We begin, in analogy with bulk lasers, by . This is followed by calculating the stress fracture limit calculating the critical index limit, which has no analogy in bulk . Of great interest in the present work is to calculate lasers the fiber melting limit, or that performance point at which the center fiber core temperature becomes just equal to the melting . point of the glass matrix A. Stress Fracture Limit

(35) and , apart from a conWe observe that both stant term, vary quadratically with the fiber radius, in agreement with previously published work on bulk rod laser amplifiers [9], [12]. For region II, however, it is noted that and vary in a complicated way that involves both a logarithmic function and an inverse square. Using the quadratic dependence on of the indices in the fiber core, it can be shown [9] that in analogy to bulk associated solid-state lasers, we can define focal lengths with the radial and azimuthal polarizations, given by (36) and (37) These equations are only valid under the conditions and . It is not difficult to show that

It is customary [12], [13] with bulk solid-state lasers to consider thermally-induced material fracture that normally is initiated at the material surface due to the presence of cracks, scratches, and voids. Bulk crystalline materials are stronger in compression than in tension, making the surfaces particularly susceptible to fracture due to the presence of tangential and tensile forces. Since most glassy materials are also weaker under tensile forces, and we expect the surface of fibers to be weaker than the bulk, also due to cracks, scratches, and voids, we calculate the tensile-limited stress for fibers. From (17) or is given by (19), the surface stress (38) , where is the mean surface tensile If we now set strength [13], we can derive the maximum heat power density that can be tolerated as (39)


To make the equation more transparent, we now use , where is known as the rupture modulus, and write

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and the corresponding maximum extractable optical power can be found from

(40)

(45)

is the maximum tolerable heat power per unit where length. Substitution of (39) in (40) yields

The maximum heat power/length is seen to depend only on , while the maximum extractable power/length is seen to depend on and .

(41)

Equation (41) shows that the maximum heat power/length depends on the fiber rupture modulus and the square of the radius ratio, but is independent of the convective coefficient . Since is always greater than or equal to , the maximum power/length , and is , which is identical occurs when to that previously obtained for bulk rod amplifiers [12], [13]. For current fiber designs, is typically larger than a by at least a factor of ten, so to first order, the denominator in equation (41) can be ignored, and the maximum tolerable heat/length is inde(of the pendent of or . We may also use the heat fraction absorbed pump power) [14] to find an expression for the maxat the fracture imum extractable power per unit length limit, given by (42)

C. Melting Limit Numerical calculations, to be discussed later in Section VII, have shown that under certain scaling scenarios, heating of the core region is a major issue to be dealt with. In particular, when scaling up the pump power in conventional fiber lasers, we find that the maximum on-axis core temperature can become a significant fraction of, or in some cases even exceed, the melting temperature of fused silica fibers. To calculate this , and calculate the maximum limit, we use (7) evaluated at , such that the maximum temperature heat power/length is just equal to the melting temperature . The result is (46)

with the corresponding maximum extractable power per unit being length (47)

has been tabulated for a variety of The rupture modulus optical materials, including fused silica [15]. B. Critical Index Limit beIn typical step-index fiber lasers, the index mismatch tween the core and surrounding cladding layer is . This magnitude of index variation is also typical of the index difference between the center and the edge of a graded index fiber. It is clear from (32) and (33) that the index of refraction in a pumped fiber laser varies quadratically with the radius in the core region. This type of index variation is, of course, fairly common in graded-index fiber lasers. It, therefore, seems legitimate to ask what heat power density or equivalently, heat per unit length, is needed to induce an index variation between the core center and the edge of a pumped fiber laser just equal to the index mismatch of conventional fiber lasers. We refer to this limit as the critical index limit. If we ignore stress-induced changes in the index of refraction, which are typically small , using (32) or compared to index changes due to in the (33), one can show that the induced critical index core region is given by (43) The maximum heat power/length

is then given by (44)

In Section VII, we will show that of the heat power/length limits considered in this section, (42), (45), and (47), the one with the lowest limit is (47). VI. THERMO-OPTICAL CONSIDERATIONS Numerical calculation of the maximum on-axis temperature and the average temperature in common fiber laser designs shows that the expected quadratic temperature profile can have a deleterious effect upon laser performance, particularly in quasi-three-level materials like Yb : SiO , where the ground state can accumulate significant population. As has been shown in bulk crystalline Yb : YAG lasers, the loss from the ground-state population must be overcome to reach lasing threshold [16], which degrades the laser efficiency. In the present case of a fiber, the situation is more complicated in that pumping a fiber results in a CW radially dependent temperature profile, described by (8). As a consequence of the temperature being highest in the center of the fiber core region, and minimum at the edge, we observe that there is a radially dependent temperature-induced loss that is strongest in the fiber center and decreasing toward the edge of the pumped core region. To describe this mathematically, we calculate here the radial gain including the thermal population of the upper and lower laser levels. Because the spectroscopy of Yb ions in glass is only partially known [17], we will in this treatment use effective rather than spectroscopic gain cross-sections, level and lower level calculating the upper

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Boltzmann populations for the manifolds as a whole [18]. The rate equation describing the upper laser level manifold can be shown to be equal to inversion density

the entire fiber core region, the condition has to be fulfilled. It can also be seen that for some outer regions of the fiber core, there may be net positive gain while in the central core region the gain may be negative. and , we use To calculate the quantities the following relationships [18]:

(48) where excitation density; upper level effective lifetime; upper level equilibrium population density at temperature . Similarly, we can write (49) for the lower level manifold population by using the conserva, where is the tion of ions equation total ion density. The excitation density can be calculated from , where is the inversion density the equation the laser photon energy. By subtracting (48) and (49), and and adding a stimulated-emission term, we arrive at the inversion equation (50) , Here, stimulated emission cross-section, and If we assume steady-state pumping, take , we find and solve for

, is the effective is the laser intensity. ,

(51)

Equation (51) shows that the inversion density is dependent . This equation is similar to upon the radial temperature those derived previously [16], [19] for the quasi-three-level in Nd : YAG at 946 nm. There, the system , where is the occupation factor value of manifold, is calculated of the terminal laser level in the for the ambient temperature and held constant regardless of whether or not pumping takes place or its magnitude. That is, pump-induced changes in temperature are not taken into is the thermal population of the entire account. Here, Yb : SiO terminal level, and its value is dependent upon , which varies with and whose the local temperature magnitude depends upon the pump level. To find the gain, we multiply (51) by the effective gain crossin the standard form section , and writing the gain , we find the radial small-signal gain coefis given by ficient (52) is the standard saturation intensity. Equation and (60) is the main result of this section, and shows that the radial gain in a fiber is dependent upon the temperature through the . For there to be unity or net positive loss term must be fulfilled. It gain, the condition is then readily seen that to have unity or net positive gain across

(53) (54) is the occupation factor of the entire manifold, where that of the manifold. is the total occupation and . The occupation factors are calfactor, given by culated from (55) is the energy of the th level in the manifold and where is Boltzmann’s constant. Using the energy level values of [19] for Yb : SiO , we can now calculate the radially dependent gain/loss. It should be noted that in [19] , not all of the energy levels of Yb are present and some must be assumed to be degenerate. We discuss this in more detail in Section VII. VII. DISCUSSION Based upon the theoretical results shown in Sections II–VI, we now present numerical calculations and discuss the ramifications of the obtained results. For simplicity, we concentrated in this work on Yb : SiO fiber designs like those used in the recent demonstration of a CW 110-W Yb fiber laser [4]. We, thus, assume a fiber consisting of a 9.2- m core region with 1.5 wt-% Yb doping. The core region is surrounded by a rectangular inner m m, which cladding region with dimensions of is in turn surrounded by a polymeric outer cladding region with outside diameter of about 630 m. As mentioned previously in this article, to simplify the analysis, we assume that the core and cladding regions are circular and concentric, and that they have the same thermal and mechanical properties. The geometry we have modeled is shown in Fig. 1; the monolithic fiber has a doped inner core region of radius a (equal to 4.6 m), while the outer radius is and takes on the value 315 m. The fiber length is assumed to be 50 m. We assume in the following that the pump wavelength is 915 nm, and the lasing wavelength 1120 nm [4]. takes the value 0.183 due to the Therefore, the heat fraction quantum defect alone. We have, thus, ignored any nonradiative effects that may be present and which could increase . A. Temperature Effects In Fig. 2, we show the temperature as a function of the fiber radial coordinate for 180 W of pump power (assumed totally absorbed) and a convective heat transfer coefficient of W/cm -K which is typical of fibers passively cooled in air. was assumed to be The ambient air or coolant temperature 298 K and the thermal conductivity was taken as W/cm-K [20]. Equations (7) and (8) were used to generate this figure. It can be seen that the temperature in the core region is


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Fig. 2. Temperature in fiber as a function of radial coordinate for 180 W of 915-nm pump power and convective coefficient of 1 10 W/cm -K.

Fig. 4. Fiber center temperature as a function of pump power for a core region radius of 4.6 m, cladding region radii of 200, 315, and 500 m, and convective coefficient of 1 10 W/cm -K.

Fig. 3. Temperature in fiber as a function of radial coordinate for 1800 W of 915-nm pump power and convective coefficient of 1 10 W/cm -K.

We have also calculated the change in the center temperature is varied, and as a funcas the radius of the cladding region tion of the pump power. Fig. 4 shows the fiber center temperaas a function of the pump power from 0 to 1800 W for ture an assumed constant core radius of 4.6 m and for three values of the outer fiber radius, 200, 315, and 500 m. For this calculation, the convective coefficient was kept constant at W/cm -K. As Fig. 4 shows, the largest center temperature rise is associated with the smallest fiber outside radius, 200 m, while the minimum temperature is obtained for 500 m. This is a consequence of the temperature drop associated with the convective layer, as may be seen from calculating the fiber center temperain the parentheses domture from (6), where the term inates. It is not difficult to see from Fig. 4 that if small radius fibers are used, the center temperature can become very large, indeed. has been found to be very sensitive to the The value of value of . In Fig. 5, the center temperature is again plotted as a function of pump power for the same fiber outer radii as in Fig. 4, but with the coefficient value increased to W/cm -K. Center temperatures are now seen to be substantially lower than those of Fig. 4. This is, again, a consequence in (6). These calculations indicate of the same term the importance of implementing forced, rather than convective, cooling as fiber lasers are scaled up to higher average powers. In Figs. 2 and 3, a common feature is that the thermal gradients from the fiber center to the outside edge are rather small, and even smaller from the center to the edge of the core region. As we will see in what follows, this leads to small thermally induced stresses values and, consequently, to small variations in the index of refraction and birefringence. The center temperature, or the average temperature of the core region, however, plays an important role in determining operating laser efficiency, as discussed later. Minimizing the core average temperature rather than the thermal gradient is, therefore, a major technical challenge in scaling fiber lasers to higher average output power.

2

2

quadratic in , whereas outside the core region, it falls off logarithmically. Fig. 3 also shows temperature as a function of radial coordinate, but with the pump power increased by a factor of ten to 1800 W. This is approximately the pump power needed to demonstrate a 1-kW fiber laser. Of interest here is the result that the center temperature in the core rises from about 331 to 634 K, which is a significant fraction of the melting temperature for SiO , 1982K. The maximum temperature shown in Fig. 3 is conservative, since the outer cladding regions of fibers tend to be polymeric materials with a thermal conductivity lower than that of glass by a factor of ten. In addition, most fibers are surrounded by plastic outer sheaths that will further increase the thermal resistance and thus the center temperature. Both of these factors can significantly increase the fiber center temperature well beyond the levels shown in Fig. 3.

2

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Fig. 5. Fiber center temperature as a function of pump power for a core region radius of 4.6 m, cladding region radii of 200, 315, and 500 m, and convective coefficient of 1 10 W/cm -K.

Fig. 6. Stress in fiber as a function of radial coordinate for 180 W of 915-nm pump power and convective coefficient of 1 10 W/cm -K.

2

2

B. Stress Effects In Fig. 6, we plotted the three principal stress components , , and as a function of the radial coordinate for a Yb : SiO fiber with 180-W pumping, a 315- m outer raW/cm -K. Equations (14)–(19) dius, and an value of and were used for these calculations. The values derived for , as well as the values for , found in Section IV were used. For the thermal expansion coefficient , we use the value /K [15], while for , we use the calculated value W-cm/kg. In the fiber core region, all stresses of are compressive, while in the cladding region, the stress is partly compressive and tensile. We see that the required boundary conditions reviewed in Section III are all satisfied. At the fiber outer m), the radial stress is zero while the tanboundary ( gential and longitudinal stresses are equal and tensile. From Fig. 6, it can be noted that the calculated stress values are very small, and will thus have a negligible effect upon the index of refraction. The reason for the small stress values is that the thermal gradient between the center and the edge of the core region or the cladding region is very small. From Fig. 2, we see that the temperature difference between the center and outside diameter of the fiber is less than 1 K. The thermal gradient across the core region is, therefore, less than 0.125 K. Increasing pumping level by a factor of ten does not change these results significantly. The resulting gradients are still small as are the corresponding stress values. As a consequence, we expect that changes in the index of refraction due to the stress-optic effect will be negligible and thermally induced birefringence to be absent in fiber lasers, even at output power levels well beyond 1 kW. C. Index of Refraction Using (40)–(43), we plotted the radial and the tangential indices of refraction as a function of the radial coordinate in Fig. 7. The pump power was again 180 W, the fiber outside raW/cm -K. We dius 315 m, and the coefficient

Fig. 7. Change in index of refraction as a function of radial coordinate for 180 W of 915-nm pump power and convective coefficient of 1 10 W/cm -K.

2

used the measured values for the index of refrac[20]. Because the radial and tangention, and and tial stresses are so small, the curves for are essentially identical, and the radially varying index is due only. Note that the magnitude of the change is to , and that across the core region the variasmall, up to . These tion from the center to the edge is only about changes in the index of refraction are small, in fact about three orders of magnitude smaller than the typical center-edge index variation found in graded index fibers or the index changes of 0.002 to 0.003 typical of step-index fibers. Using the index variations shown in Fig. 7 at the center and edge of the core region, it can be shown that, for a 50-m fiber, the core center-edge phase difference amounts to about 22.30 waves of distortion at 1120 nm. Evaluating the effect of this phase difference on beam propagation in a single-mode fiber is, however, outside the scope of the work reported here.


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D. Calculation of Limits We use (42) to calculate the maximum extractable power per unit length at the fracture limit, using the rupture modulus value W/cm [15] for a 50- m diameter flaw size. For the same 4.6- m core radius and 315- m outer radius fiber used in the previous sample calculations, we find kW/cm. For a 50-m long fiber at the stress fracture limit, the fiber extractable power is found to be 7.56 MW. This rather large value indicates that thermally-induced fiber fracture is improbable even at the -kW output power levels feasible in the foreseeable future. This is a consequence of the small stress levels found in fiber lasers. It should be pointed out, however, that if large defects are found on the fiber outside surface, fracture limits may be much lower, requiring modification of the above conclusion. The maximum extractable power at the critical index limit , we find can be calculated from (45). Using kW/cm, and the total extractable power to be 14.3 MW for a 50-m long fiber. Again, the calculation indicates that index variations from the center to the edge in the core region approximately equal to the core/cladding index mismatch found in present day step-index fibers will not be approached, even at the kilowatt power level. Finally, using (47), we calculate the maximum power per unit for the center fiber temperature to become length equal to the melting point of SiO , 1983 K, with assumed W/cm -K. We find W/cm, so to be for a 50-m long fiber, the maximum output will be about 2400 W. This limit is thus the one that is reached first as power is scaled up. The limit can, however, be much lower if the heat fraction is larger than that assumed in this work.The use of polymeric cladding regions and/or an external protective cladding layer will increase thermal resistance and elevate the fiber center temperature still further. We also point out, however, that as we have mentioned previously, increasing the value of reduces the fiber center temperature, which can significantly raise the melting limit. E. Inversion and Gain The spectroscopy of Yb in SiO is not well-known. For the purposes of numerical calculations we used the Yb : SiO enground-state manifold ergy level assignments of [17]. The (0 cm ), (550 cm ), (550 cm ), and levels are (960 cm ). These energy levels correspond to the levels , , and of [17], and we assume that level is degenerate and conand . For the upper level sists of two discrete levels, (10 260 cm ), (10 260 cm ), manifold, the levels are (10 900 cm ), which correspond to the levels and and of [17] and we assume that and are degenerate. By use of (51)–(53), assuming that the excitation density is uniform and can be calculated as , and using (7) to calculate the temperature distribution in the core shown in Fig. 8 for a region, we generated the plot of ions/cm . The variation across the Yb doping density of core region is quadratic in , but it is very small, typically a few ions/cm . For Fig. 8, about 7% of the available ion times density appears as terminal level thermal population. From (51),

Fig. 8. Lower level thermal population as a function of radial coordinate for 10 1800 W of 915-nm pump power and a convective coefficient of 1 W/cm -K. Doping density is 4 10 ions/cm .

2

2

Fig. 9. Lower laser level occupation factor as a function of pump power at 915 nm, for h equal to 1 10 (0) and 1 10 W/cm -K ( ).

2

2

we see that a significant thermal population in the terminal laser level requires one to pump harder to achieve a desired smallsignal gain/cm. We have also calculated the lower level occupation factor as a function of the in the center of the core region and pump power and for two values of W/cm -K, as shown in Fig. 9. For a pump power of 180 W and W/cm -K, the occupation factor is found to be about 0.013. Increasing the pump power to 1800 W, however, increases the occupation factor to about 0.067 for the same coefficient value. If, however, is increased by two orders of magnitude (using, for example, some method of forced cooling), the occupation factor remains close to zero over the entire pump power range. We conclude that the use of forced cooling, or some other method of more aggressive heat removal, will provide a significant benefit in reducing the terminal level thermal population. This will maximize the small-signal gain, and therefore increase the laser efficiency.

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VIII. SUMMARY In this paper, we have presented a comprehensive first-order theoretical treatment of thermal, stress, thermo-optic, and scaling effects in Yb : SiO fiber lasers. To make the problem tractable and to provide analytical solutions as a starting point in the analysis of such effects in fiber lasers, we used the simplified fiber geometry shown in Fig. 1, which reduced the double-clad structure to just two distinct regions. The results presented here can easily be extended to more complicated structures with adjacent regions exhibiting different thermal, mechanical, and optical properties by use of any of a number of commercially available finite-element codes. It is also straightforward to include thermal and mechanical properties that vary with temperature [8]–[10]. The results of our analysis are notable in that we find no inherent limitations to fiber scaling other than the need to manage fiber core temperature. While other considerations, such as optically induced damage and various nonlinear effects will undoubtedly play an important role in determining the scalability of fiber designs, we have shown that, if more advanced fiber structures are used allowing superior heat removal properties than is currently done, and/or forced cooling is applied coupled with modified fiber designs, there are no obvious obstacles to scaling fiber output power well beyond the kilowatt range. To summarize, based on the analysis presented here of thermal, stress, and thermo-optic effects in Yb-doped fiber lasers as power is scaled to the kilowatt range, we determined that it is critical to minimize core heating to maintain efficiency and avoid damage. However, such adverse effects occur at much higher powers in fiber lasers than those typically found for bulk lasers at comparable pump power, validating the superior power handling and thermal management capabilities of fiber lasers. Larger fiber cladding diameters are expected to be necessary for better heat dissipation at elevated power levels as are holey fiber structures that may allow some forced air cooling. On the other hand, the use of existing cladding layers and outer protective claddings that use low thermal conductivity materials, such as polymers, should be avoided, if possible, at high powers. APPENDIX CALCULATION OF STRESS-INDUCED RADIAL VARIATION OF THE INDEX OF REFRACTION From [8], we can write (A1)

are the photoelastic coefficients [21] and the the radially dependent stress components in Regions I and II. For computational purposes, we shall represent the indices in the compact form

Here,

(A2)

by using the suffix notation of Nye [21]. In this paper we will consider only the isotropic material fused silica (SiO ), for which the thermal, mechanical, and elasto-optic coefficients are well-known. For isotropic materials, the matrix representation is particularly simple, and given by [21] of

(A3)

, and, thus, there are only two inwhere dependent coefficients. For many optical materials, including fused silica, the components are not tabulated, but the related elasto-optic coefficients are [21]. The matrix representation for these components is

(A4)

is given by and there are The coefficient only two independent coefficients. The photoelastic and elastooptic coefficients and are related by (A5) where is the elastic compliance matrix, which for fused silica takes the same form as (A3) or (A4). There are also only two inand , and is calculated from dependent coefficients, . It can be shown that the compliance ma, , and trix coefficients are given by [21], where is the rigidity modulus given by . The values of and are well known for fused silica, as are the independent elasto-optic coefficients and [20]. Performing the matrix multiplication (A5), we arrive at the following expressions for the photoelastic coefficients (A6) (A7) (A8) and are known in the optics and laser literThe quantities ature as the parallel and perpendicular stress-optic coefficients, is known simply as the stress-optic coeffiwhile and are , cient. The numerical values of [20]. and have the values and kg/cm2 and [20]. By using , , and , and the relation, and to be: ships (A6)–(A8), one finds the values of , cm /kg, cm /kg, and cm /kg. These constants can also be ex(nm/cm) pressed in another commonly used form:


cm /kg, (nm/cm) cm /kg, and (nm/cm) cm /kg. , we can now evaluate Having determined the constants cor(A2) in our cylindrical coordinate system, where respond to the and components. The results are

[9] [10] [11] [12]

(A9) [13]

(A10) [14]

(A11)

[15]

[16]

(A12) [17]

REFERENCES [1] D. L. DiGiovanni and M. H. Muendel, “High power fiber lasers and amplifiers,” Opt. Photon. News, pp. 26–30, Jan. 1999. [2] B. Rossi, “Commercial fiber lasers take on industrial markets,” Laser Focus World, pp. 143–149, May 1997. [3] D. Richardson, H. Offerhaus, J. Nilsson, and A. Grudinin, “New fibers portend a bright future for high-power lasers,” Laser Focus World, pp. 92–94, June 1999. [4] V. Dominic, S. MacCormack, R. Waarts, S. Sanders, S. Bicknese, R. Dohle, E. Wolak, P. S. Yeh, and E. Zucker, “110 W fiber laser,” in Proc. Conf. Lasers and Electro-Optics (CLEO) ’99, Baltimore, MD, 1999, Postdeadline Paper CPD11-1. [5] M. H. Muendel, “High-power fiber laser studies at the polaroid corporation,” in Proc. SPIE Conf. High-Power Lasers San Jose, CA, 1999, vol. 3264, pp. 21–29. [6] “U. S. Department of Defense Small Business Innovation Research Program Solicitation 00.1,” U.S. Dept. of Defense SBIR Program Office, Washington, DC, 1999. [7] M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett., vol. 56, pp. 1831–1833, 1990. [8] D. C. Brown, “Ultrahigh-average-power diode-pumped Nd : YAG and Yb : YAG lasers,” IEEE J. Quantum Electron., vol. 33, pp. 861–873, 1997.

[18]

[19] [20] [21]

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, “Nonlinear thermal distortion in YAG rod amplifiers,” IEEE J. Quantum Electron., vol. 34, pp. 2383–2382, 1998. , “Nonlinear thermal and stress effects and scaling behavior of YAG slab amplifiers,” IEEE J. Quantum Electron., vol. 34, pp. 2393–2402, 1998. B. A. Boley and J. H. Weiner, Theory of Thermal Stresses, reprint ed. Melbourne, FL: Kreiger, 1985. J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, “The slab geometry laser—Part I: Theory,” IEEE J. Quantum Electron., vol. QE-20, pp. 289–301, 1984. D. C. Brown and K. K. Lee, “Methods for scaling high average power laser performance,” in High Power and Solid State Lasers, 1986, vol. 622, pp. 30–41. T. Y. Fan, “Heat generation in Nd : YAG and Yb : YAG,” IEEE J. Quantum Electron., vol. 29, pp. 1457–1459, 1993. W. F. Krupke, M. D. Shinn, J. E. Marion, J. A. Caird, and S. E. Stokowski, “Spectroscopic, optical, and thermomechanical properties of neodymium- and chromium-doped gadolinium scandium gallium garnet,” J. Opt. Soc. Amer. B, vol. 3, pp. 102–113, 1986. T. Y. Fan and R. L. Byer, “Modeling and CW operation of a quasi-threelevel 946 nm Nd : YAG laser,” IEEE J. Quantum Electron., vol. QE-23, pp. 605–612, 1987. J. Y. Allain, M. Monerie, H. Poignant, and T. Georges, “High-efficiency ytterbium-doped fluoride fiber laser,” J. Non-Crystall. Solids, vol. 161, pp. 270–273, 1993. A. C. Erlandson, G. F. Albrecht, and S. E. Stokowski, “Model predicting the temperature dependence of the gain coefficient and the extractable stored energy density in Nd : Phosphate glass lasers,” J. Opt. Soc. Am. B, vol. 9, pp. 214–222, 1992. W. P. Risk, “Modeling of longitudinally pumped solid-state lasers exhibiting reabsorption losses,” J. Opt. Soc. Am. B, vol. 7, pp. 1412–1423, 1988. Handbook of Optics (sponsored by the Optical Society of America), vol. II, Devices, Measurements, and Properties, M. Bass, Ed., McGraw-Hill, New York, 1995. J. F. Nye, Physical Properties of Crystals. Oxford, U.K.: Clarendon, 1993.

David C. Brown, photograph and biography not available at the time of publication.

Hanna J. Hoffman, photograph and biography not available at the time of publication.