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Thermally-affected Cosine-Gauss and Parabolic-Gauss beams and comparisons of Helmholtz–Gauss beam families
Optics Communications 341 (2015) 160–172
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Thermally-affected Cosine-Gauss and Parabolic-Gauss beams and comparisons of Helmholtz–Gauss beam families H. Nadgaran n, R. Fallah Physics Department, Shiraz University, Shiraz 71454, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 3 June 2014 Received in revised form 23 November 2014 Accepted 2 December 2014 Available online 6 December 2014
This article firstly investigates the thermal effects on the generation of Cosine-Gauss and Parabolic-Gauss beams and compares with the impacts of thermal effects on Bessel–Gauss and Mathieu–Gauss beams reported in our previous works. For this investigation, the intensity distribution of the beams against radial distances at different distances from the beam waist under various pump powers and pump beam waists has been studied. The results show that the thermal effects have severe impact on the laser output, so that true distinguishing of thermally-affected and non-thermal intensity profiles is unlikely feasible. The comparisons show that the thermal effects has minimal impact on the generation of Bessel– Gauss beams. The role of pump beam waist on the strength of the thermal effects was also studied. & 2014 Elsevier B.V. All rights reserved.
Keywords: Thermal effects Laser special beams Helmholtz–Gauss equation Heat equation
1. Introduction Solution to Helmholtz equation can be classified to four distinct families. These solutions that are ideal non-diffracting beams are based on the proper coordinate systems adopted. They are parabolic beams in parabolic cylindrical coordinates [1,2], Mathieu beams in elliptic cylindrical coordinates [1,3], Bessel beams in circular cylindrical coordinates [1,4], and cosine beams in Cartesian coordinates. The transverse intensity distribution of these four families remain unchanged upon their propagation in free space carrying infinite energy due to their inherent infinite extent that is why they are called non-diffracting beams. In order to remove this physically unreasonable behavior of infinite extent, the modified version of these families are usually built up by apodizing them with a radial Gaussian apodization, that is, their paraxial apodized versions called Helmholtz–Gauss (HzG) beams is composed of an ideal non-diffracting beams modulated by a radial Gaussian envelope. Following our previous works, in which we studied the thermal effects on the generation of Bessel–Gauss (BG) [5] and Mathieu–Gauss (MG) beams [6], we have been motivated to study the cases of Cosine-Gauss (CG) and parabolic-Gauss (PG) beams under induced thermal loads. Cosine-Gauss beams are obtained as exact solutions to the paraxial wave equation propagating in free space and also in complex optical systems [7,8]. The name of Cosine-Gauss beams are usually brought with the more n
Corresponding author. Fax: þ 987116460839 E-mail address: [email protected] (H. Nadgaran).
http://dx.doi.org/10.1016/j.optcom.2014.12.007 0030-4018/& 2014 Elsevier B.V. All rights reserved.
general beams of cosh-Gauss, Hermite–Gauss (HG), Hermite-cosine-Gauss (HCG), Hermite-sine-Gauss (HSG) and Hermite–Gauss beams. The above mentioned four families of special type of beams are utilized in applications in which efficient extraction of energy is required [9]. Various studies in the field of special beams can be found in the literature. These investigations cover the propagation properties of HzG beam families through a paraxial optical ABCD system with a hard-edge aperture [3,10]. In particular propagation characteristics of CG beams in turbulent atmosphere [11] and analytical expression for propagation of non-paraxial CG beams in free space [12], were also studied. PG beams are obtained as exact solutions to the wave equation in parabolic cylindrical coordinates. Recently Lopez-Mariscal et al. presented experimentally the existence of parabolic-Gauss beams [13]. Thermal effects play an important and major role in high power solid-state lasers. In these lasers, particular attention has been paid to induced heat load in the laser cavity since the laser pump energy is only partly converted to laser light and most of the pump energy is lost in the form of heat [14,15]. This induced heat leading to thermo-mechanical and thermo-optical effects such as thermal lensing, photo-elastic effects, thermal stress and thermal fracture [16,17]. The most serious problems are then the distortion of the beam and degradation in the beam profile due to thermal lensing and depolarization loss due to stress induced birefringence [18,19]. Thermal lens modeling and measurement in end-pumpedsolidstate lasers have been developed well [20–22]. As almost every important parameters of laser radiation is directly influenced by induced thermal loads, the propagation characteristics of laser beams under severe thermal loads is necessary for designers and
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manufactures to have a true, clear and reliable insight to the laser output features and their conversion. Knowledge of thermal effects can also lead to proper design and construction of laser cooling systems. To account for induced thermal loads in investigation of special beam propagation characteristics, one needs to consider laser beam propagation in an inhomogeneous thermal media. Here in this work, the impact of thermal effects on the generation and behavior of Cosine-Gauss and Parabolic-Gauss beams is studied. Also to evaluate the effect of heat on the generation of special beams, we have compared the results of this work with the results of our previous works.
→ Complex field amplitude of HzG beams, H1 r1 ; k1 at the input
(
)
plane of z ¼z1 can be written as in Eq.(1) [23], where index 1 refers to the input parameters of the Gaussian beam traveling axially through an ABCD system.
(
)
(
)
(1)
equation of:
(
)
(2)
→ and r1 = (x1, y1) and ω1 show the transverse coordinate and beam waist of the Gaussian envelope, respectively. As Eq. (1) shows the → non-diffracting beam F r1 ; k1 , has been apodized by a Gaussian
(
)
beam in order to make it physically sensible, and this can be expressed as a superposition of plane waves, that is: →
F (r1; κ1) =
π
∫−π A(φ)exp(iκ1(xcos(φ) + y sin(φ))dφ
(3)
where A(φ) is an arbitrary angular spectrum of the ideal non-diffracting beam. Using Huygens diffraction integral, the transverse distribution of HzG beams at the output plane of z¼ z2 of an ABCD optical system would be given as [23]:
⎛ ikr 2 ⎞ ⎛ k k B ⎞ exp(ikς) → U (r2) = exp⎜⎜ 2 ⎟⎟exp⎜ 1 2 ⎟ F( r ; κ ) ⎝ 2ik ⎠ A + (B/q1) 2 2 ⎝ 2q2 ⎠
Aq1 + B Cq1 + D
, κ2 =
κ1q1 Aq1 + B
(5)
It is reminded that ω , q and k both at z ¼z1 and z ¼z2 obey the following relation:
ωj =
2iqj kj
, j = 1, 2 (6)
2.1. Cosine-Gauss (CG) beams One of the simplest non-diffracting beams in Cartesian coordinates is the ideal cosine field: →
)
⎛ ∂2 ⎞ → ∂2 ⎜ 2 + 2 + k12⎟F r1 ; k1 = 0 ⎝ ∂x ⎠ ∂ y
q2 =
(
)
F r ; k1 = Acos(κ1y)
→ where F r1 ; k1 is the solution of the two dimensional Helmholtz
(
optical path length measured along the optical axis from z¼ z1 to z¼ z2. The following transformation laws relates the beam input parameters to beam output parameters as [24]:
In all above relations, q complex parameter is defined as 1/q = (1/R) + (2i/kω2)[24], where R and k are the radius of wave front curvature and the wave vector of the Gaussian apodization, respectively.
2. Helmholtz–Gauss (HzG) beams
⎛ →2 ⎞ r → → H1 r1 ; k1 = exp⎜− 1 2 ⎟F r1 ; k1 ⎜ ω ⎟ 1 ⎝ ⎠
161
(4)
where q2 and κ2 are the complex parameter and complex transverse wave number at the output plane, respectively. ς is the
(7)
where κ1 is the transverse wave number and A is the normalization constant. The cosine field when apodized by a Gaussian envelope can then be written of:
⎛ r2 ⎞ CG(r) = N exp⎜⎜ − 2 ⎟⎟cos(κ1y) ⎝ ω1 ⎠
(8)
where N is the normalization constant. 2.2. Bessel–Gauss (BG) beams In recent years, the existence of Bessel–Gauss beams has been studied theoretically and experimentally. By solving the wave equation in circular cylindrical coordinates, The BG beam can be written as follows[5]:
⎛ r2 ⎞ BG m(r) = Aexp⎜⎜− 2 ⎟⎟Jm (κ1r)exp(imθ) ⎝ ω1 ⎠
(9)
where A is the normalization constant and Jm (.) is the mth-order Bessel functions. 2.3. Mathieu–Gauss (MG) beams Mathieu beams constitute another family of non-diffracting
Fig. 1. (a) Heat density distribution versus r in the plane of z ¼ 0 and (b) versus Z at r¼ 0 for ωp ¼ 100 μ m for different pump powers.
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Fig. 2. (a) Heat density distribution versus r in the plane of z¼ 0 and (b) versus Z at r¼ 0 and P ¼5 W for different pump beam waists. Table 1 The focal length of thermal lens for ωp = 100 μm . Pump power
1W
3W
5W
Focal length
28 cm
9.3 cm
5.6 cm
∞
MG2en(r) = 2πNmeC2m ∑ where
Table 2 The focal length of thermal lens for P¼ 5 W. Pump beam waist
50 μm
100 μm
200 μm
300 μm
Focal length
3.9 cm
5.6 cm
22.8 cm
2.38 m
beams that are solutions of the Helmholtz equation in elliptic coordinates. The closed-form expression of the n′th-order even and odd MG beams is given as [6]:
MG2en
normalization constants and the even and odd angular Mathieu functions ce( ) and se( ), respectively. Even-order MG beams can also be written as an expansion of BG beams [6]:
Nme
Mathieu functions are described by the nth-order even and odd radial Mathieu functions Je( ) and Jo( ). N2en and N2on + 1 are the
(11)
2.4. Parabolic-Gauss (PG) beams The transverse field of the even and odd parabolic beams is written as [13]:
PG e(r , a) =
⎛ iκ 2 z ⎞ exp⎜⎜− 1 ⎟⎟Pe( 2κ1 ξ ; a)Pe( 2κ1 η; − a) π 2 ⎝ 2k ⎠ Γ1
2
⎛ r2 ⎞ exp⎜⎜− 2 ⎟⎟ ⎝ ω1 ⎠ PG o(r , a) =
(10)
⎛ r2 ⎞ A2j ( − 1)2j exp⎜⎜ − 2 ⎟⎟cos(2jθ)J2j (κ1r) ⎝ ω1 ⎠
is the normalization constant.
⎛ r2 ⎞ = N2enexp⎜⎜− 2 ⎟⎟Je2n(ξ , q)Ce2n(η, q) ⎝ ω1 ⎠
⎛ r2 ⎞ MG2on + 1 = N2on + 1exp⎜⎜− 2 ⎟⎟Jo2n + 1(ξ , q)Cs2n + 1(η, q) ⎝ ω1 ⎠
j=0
(12)
⎛ iκ 2 z ⎞ exp⎜⎜− 1 ⎟⎟Po( 2κ1 ξ ; a)Po( 2κ1 η; − a) π 2 ⎝ 2k ⎠ Γ3
2
⎛ r2 ⎞ exp⎜⎜− 2 ⎟⎟ ⎝ ω1 ⎠
(13)
Fig. 3. a) Intensity distributions of CG beams versus radial distance for z ¼2 cm from the beam waist, under pump power of 1 W. Solid plots represent the non-thermal. CG beam, dashed plots shows CG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot.
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163
Fig. 4. a) Intensity distributions of CG beams versus radial distance for z ¼ 2 cm from the beam waist, under pump power of 3 W. Solid plots represent the non-thermal. CG beam, dashed plots shows CG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot.
Fig. 5. a) Intensity distributions of CG beams versus radial distance for z ¼ 2 cm from the beam waist, under pump power of 5 W. Solid plots represent the non-thermal. CG beam, dashed plots shows CG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot.
Where Γ1 = Γ = ((1/4) + ia/2), Γ3 = Γ = ((3/4) + ia/2), the parabolic cylindrical coordinates (ξ, η) are given by x = η2 − ξ 2 /2 and y = ξη , also in the above equation Pe(.) and Po(.)
(
)
are the even and odd solutions to the parabolic cylindrical differential equation and its Taylor expansion with v ¼0 is given by[2]:
p(v , a) =
∞
∑ n = 0 cn
vn , n!
cn + 2 = acn −
n (n − 2) cn − 2 4
(14)
The first two coefficients for Pe are c0 = 1 and c1 = 0, whereas for Po they are c0 = 0 and c1 = 1.
3. Thermal model Finite thermal conductivity and non-uniformity of a pump radiation usually causes the pump-induced heat in laser gain medium leading to variations of gain cross section which in turn makes the gain medium to act as a lens. This is known as the thermal lens effect, where the optical path difference (OPD) function of an ideal
thermal lens is parabolic. This ideal parabolic OPD suffers from a spherical aberration in strongly pumped laser rods. This deviation from parabolic OPD leads to radial variation of the thermal lens’ focal length (f), which then becomes a quadratic function of r. Here in this work a pump of Gaussian profile (GP) was employed as follows:
⎛ ⎞ 2r 2 SGP (r , z) = Q GPexp⎜⎜− 2 ⎟⎟exp(−αz) ⎝ ωp ⎠
(15)
where α is the absorption coefficient of laser crystal, Q GP is constant given in the Appendix of reference [5]:
Q GP =
αηqηabsP 2 2 − π ω (1 − e αl )(1 − e−2(a2 / ωp2))
(16)
where P is the total pump power, ηq is the fraction of pump power converting to heat, ωp is the pump beam waist and ηabs is the fraction of absorbed pump beam. Ɩ and ɑ show the laser crystal length and radius, respectively.
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Fig. 6. a) Intensity distributions of CG beams versus radial distance for z ¼5 cm from the beam waist, under pump power of 1 W. Solid plots represent the non-thermal. CG beam, dashed plots shows CG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot.
Fig. 7. a) Intensity distributions of CG beams versus radial distance for z ¼5 cm from the beam waist, under pump power of 3 W. Solid plots represent the non-thermal. CG beam, dashed plots shows CG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot.
As far as the pump power is concerned, the laser crystal as a homogeneous gain medium is converted to an inhomogeneous graded index (GRIN) one by severe influence of the induced heat. This leads to refractive index variation of the gain medium. Therefore assuming the GRIN medium as an ABCD affected medium, one has to calculate the element of the ABCD matrix of the GRIN medium for its full characterization. To do this, solution of the following heat equation should be obtained:
∇2T (r , z) = −
S(r , z) K
∂n [T (r , z) − T (r = 0, z)] + (n0 − 1)εzz ∂T
Δn(r , z) =
(18)
where T1 is the ambient or coolant temperature and h is the heat transfer coefficient of the medium. Based on the temperature
(19)
where n0 is the index of refraction at the center of the crystal and єzz is the strain tensor component of the crystal [5]:
εzz = (1 + v)αT [T (r , z) − T (0, z)] (17)
where K is the thermal conductivity of the GRIN medium and S(r, z) is the heat source density in watts/volume. In order to have a proper boundary conditions for edge and faces of the crystal, the Newton's law of heat transfer was used:
→ −K ⌢ n . ∇ T (r , z)|boundary = h(T (r , z) − T∞)
distribution of the laser crystal, the crystal refractive index change, Δn, can be calculated as:
(20)
where υ is Poisson ratio and αT is the thermal expansion coefficient of the laser crystal. The temperature dependent change of refractive index can cause thermally induced phase-shifts as [25]:
Δφ = k
l
∫0 Δn(r, z) dz
For a lens-like medium, the focal length is given by [25]:
(21)
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Fig. 8. a) Intensity distributions of CG beams versus radial distance for z ¼ 5 cm from the beam waist, under pump power of 5 W. Solid plots represent the non-thermal. CG beam, dashed plots shows CG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot.
Fig. 9. a) Intensity distributions of PG beams versus radial distance for z ¼ 2 cm from the beam waist, under pump power of 1 W at θ¼0°. Solid plots represent the nonthermal. PG beam, dashed plots shows PG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot for a¼ 1.
fth = −
k r2 2Δφ
(22)
The exact solution of Eq.(17) using the above boundary conditions is as follows [26]:
T (r , z) =
∞
∑i = 1 ti(z)J (vir) + T∞ °
(23)
where J (vr i )′s are the zero-order Bessel functions with vi′s as the ° positive roots of the equation hJ (va) − KJ1 (va) = 0, ti(z) has been ° fully given in [5]. By expansion the Bessel function of Eq.(23) and use of paraxial ray approximation, one can end up with the crystal refractive index as a second order function of r, that is [5]:
⎛ r 2γ (z)2 ⎞ ⎟ n(r , z) = n0⎜1 − 2 ⎠ ⎝
(24)
γ (z) is an important function showing the variation of the
medium refractive index along z coordinate. In general for a lenslike medium, the ABCD transfer matrix is written as [27]:
⎡ A B ⎤ ⎡cos(γl) sin(γl)/γ ⎤ ⎥ ⎢⎣ ⎥⎦ = ⎢ C D ⎣⎢−γ sin(γl) cos(γl) ⎥⎦
(25)
This ABCD matrix represents thermal effects imposed on the gain medium and its detailed calculation has been given by Sabaeian et al. [5]. By constructing the ABCD transfer matrix, we can simulate the HzG beams suffered by induced heat.
4. Results and discussion Here in this section, we first present the effects of induced heat load on the generation of CG and PG beams. Then the study will be continued by comparison of these results with that of other four special beams. For this purpose, an end pumped cylindrical Nd: YAG laser crystal [28] with I ¼5 mm and radius of ɑ¼ 1.5 mm was
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Fig. 10. a) Intensity distributions of PG beams versus radial distance for z¼ 2 cm from the beam waist, under pump power of 3 W at θ¼ 0°. Solid plots represent the nonthermal. PG beam, dashed plots shows PG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot for a¼ 1.
Fig. 11. a) Intensity distributions of PG beams versus radial distance for z ¼2 cm from the beam waist, under pump power of 5 W at θ ¼0°. Solid plots represent the nonthermal. PG beam, dashed plots shows PG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot for a¼ 1.
assumed. The laser was longitudinally pumped by a diode laser at wavelength of λ p ¼808 nm and pump beam waist of ωp ¼100 μm. CG and PG beams were generated with beam waist of ωl ¼300 μm, laser wavelength of λl ¼1064 nm where transverse wave vector was taken as κt ¼22144 cm−1[28]. Figs. 1 and 2 show the variations of induced heat density against r and z for different pump powers and different pump beam waists. One can notice the decay rate of heat density when pump beam waist is increasing (Fig. 2a) showing the importance of pump beam waist in generating induced heat and making different thermal lens with different strength. Table 1 shows the focal length of the thermally-induced lens calculated for different pump powers. It is shown that by increasing the pump power, the thermal focal length would decrease, leading to stronger lens as expected. Table 2 shows the focal length of the thermally-induced lens calculated for different pump beam waists. It is shown that by
increasing the pump beam waist, the thermal focal length would increase, leading to weaker thermal lens. Figs. 3–8 Show the transverse intensity distributions of CG beams versus radial distance for 1, 3 and 5 W pump powers, respectively. From these figures and Table 1, the following conclusions can be drawn: Under pump power of 1 W, there can be seen no considerable intensity profile changes under induced heat. The non-thermal model of intensity profile almost coincides with the thermal ones under 1 W pump power for different z, a distance from the beam waist. Our calculations shows that the focal length of the induced thermal lens in this case is 28 cm, that is very weak thermal lens is built up and the laser crystal dose not suffer much from the induced heat load. The fact that high power end-pumped lasers suffers a spherical aberration originated from a non-uniform temperature distribution suggests that in this work also one can no longer encounters a
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167
Fig. 12. a) Intensity distributions of PG beams versus radial distance for z ¼5 cm from the beam waist, under pump power of 1 W at θ¼0°. Solid plots represent the nonthermal. PG beam, dashed plots shows PG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot for a¼ 1.
Fig. 13. a) Intensity distributions of PG beams versus radial distance for z¼ 5 cm from the beam waist, under pump power of 3 W at θ¼ 0°. Solid plots represent the nonthermal. PG beam, dashed plots shows PG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot for a¼ 1.
perfects lens with refractive index distribution of parabolic profile. The existence of spherical aberration makes the thermal lens’ focal length to depend on ωl through [29]:
⎡ ⎛ ω ⎞2⎤ fth (ωl) = f ⎢1 + x⎜⎜ l ⎟⎟ ⎥ °⎢ ⎝ ωp ⎠ ⎥⎦ ⎣
(26)
where f is the focal length of the thermal lens without spherical ° aberration, ωl is the laser mode beam waist, ωp is the pump beam waist and x is a parameter representing the strength of the spherical aberration. This equation shows that spherical aberration makes beams with different mode waist to see different thermal lens's focal length. When the laser crystal is high power pumped, spherical aberration becomes more and more pronounced resulting in notable diffraction loss for fundamental mode laser beam and degrading the beam quality by coupling fundamental mode laser to higher modes. Moreover, from efficiency point view, systems with aberrated thermal lensing are less efficient compared to
systems without aberration. Since thermal aberration is essentially a loss that is dependent on the pump power. Upon increasing the pump power, severe deviation from the non-thermal intensity profile can be realized showing that the laser media is considerably affected by the induced thermal lens. The most thermally affected CG intensity distributions can be learned from Fig.8 when compared to Figs. 6 and 7. It is seen that upon increasing the pump power, the side lobes of the central spot are gradually grown so that the two spots shown in Fig. 8b is actually the two modified lobes, that is no longer resembles the nonthermal profile( Fig. 8c). This piece of information has its very impact impotence in distinguishing true thermally affected profile from a non-thermal one. Here in this example (Fig. 8) the designer and the experimenter should be aware that the bright spots seen in Fig. 8b are not the main spot of the CG beam rather they are side spots which have grown due to thermal lens of f ¼5.6 cm. From focal length point of view, it is seen that a larger quantity of induced heat can build stronger thermal lens which in turn can
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Fig. 14. a) Intensity distributions of PG beams versus radial distance for z¼ 5 cm from the beam waist, under pump power of 5 W at θ¼ 0°. Solid plots represent the nonthermal. PG beam, dashed plots shows PG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot for a¼ 1.
Fig. 15. Intensity distributions of special beams versus radial distance for z ¼ 2 cm from the beam waist. Solid plots represent the non-thermal. beam, dashed plots shows thermally-affected. beam under GP. From the top: for pump powers of 1, 3, 5 W, respectively.
potentially allow the grown up of the side lobes making the designer and experimenter confused. One more aspects of Fig. 8a is that the CG intensity distributions were plotted for z¼ 5 cm from the beam waist and as the calculated focal length is about 5.6 cm, one can say that the CG profile shown in this figure is just around the focal point of the thermal lens making much deviations of the corresponding non-thermal intensity profile. Figs. 9–14 Show the transverse intensity distributions of PG beams versus radial distance for 1, 3 and 5 W pump powers, respectively. The following points can be noticed:
compared to Figs. 13 and 12. It is seen that upon increasing the pump power, the central spots are gradually grown so that the side spots shown in Figs. 12b and 13b is actually disappeared, that is no longer resembles the non-thermal one (Fig. 14c). 3. From these figures, one can see the eventual intensity enhancement of the main peak together with the peak shift toward lower r when the pump power is increased.
5. Comparisons 1. Under pump power of 1 W, there can be seen no considerable change in intensity profile and laser spots under induced heat. 2. Upon increasing the pump power, the most thermally affected PG intensity distributions can be seen in Fig. 14 when
We now compare the results of thermal effects on the generation of HzG beams. For this purpose, we have plotted the intensity distribution of zeroth order BG and MG beams versus
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169
Fig. 16. Thermally-affected. laser spots under GP profile for z ¼2 cm from the beam waist. From the top: for pump powers of 1, 3, 5 W.
Fig. 17. Intensity distributions of special beams versus radial distance for z ¼ 5 cm from the beam waist. Solid plots represent the non-thermal. beam, dashed plots shows thermally-affected. beam under GP. From the top: for pump powers of 1, 3, 5 W, respectively.
radial distance at z¼2.5 cm from the beam waist for the different pump powers and different pump beam waists. Fig. 15 shows CG, zeroth-order BG, MG and PG beam profiles for z ¼2 cm from the beam waist under pump powers of 1, 3 and 5 W. From Fig. 13 one can realize that upon increasing the pump power, the laser intensity profiles gradually shift toward r ¼0 showing
closer peaks. Moreover, the thermally-affected profiles show higher intensities versus r. whereas the above mentioned shifts are almost the same for CG, BG, MG and PG beams. Fig. 16 shows the corresponding laser spots for the same four families of special beams. The figure shows no considerable change in laser spots for z¼ 2 cm detection from the beam waist.
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Fig. 18. Thermally-affected. laser spots under GP profile for z¼ 5 cm from the beam waist. From the top: for pump powers of 1, 3, 5 W, respectively.
Fig. 19. Intensity distributions of special beams versus radial distance for z¼ 5 cm from the beam waist for p¼ 5 W. The rows indicate the pump beam waists of 100, 200, 300 μm, respectively.
Fig. 17 shows the same profiles as in Fig. 15 for z¼ 5 cm detection from the beam waist. The figure clearly shows that upon increasing the pump power, considerable alterations in peak shifts for all beams are recognized, whereas the peak amplitudes for CG, BG, MG and PG beams of 5 W do not resemble the non-thermal profile and the 1, 3 W profiles. Fig. 18 shows the corresponding laser spots for z¼5 cm. Here
remarkable spot changes are seen for all special beams by increasing the pump power. For the case of CG beams one can see two-part spots for 5 W case which is no longer similar to its 1 and 3 W cases. For the case of BG beams systematic increase of the spot radius is clearly notable. The situation is also understood for MG beams where the middle spot is gradually disappeared upon increasing the pump power. PG beams shows even stronger effect
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171
Fig. 20. Thermally-affected. laser spots under GP profile for z ¼ 5 cm from the beam waist for P¼ 5 W. The rows indicate the pump beam waists of 100, 200, 300 μm, respectively.
due to complete disappearance of the side lobes. The rows in Fig. 19 show the transverse intensity distributions of special beams versus radial distance at z ¼5 cm from the beam waist for pump beam waists of 100, 200, 300 μm, respectively. The figure clearly shows that as the pump beam waist increases the effect of induced heat diminishes. This point holds true for all special beam profiles, as the third row clearly indicates. As a result, for large pump beam waist, the beams behave as though no heat effect, even for large delivered powers is present. Fig. 20 shows the corresponding laser spots for z¼ 5 cm. One can clearly learn that the strength of the thermal effects is gradually reduced, so that for large pump beam waist, the spot shapes and sizes can be regarded as similar to the non-thermal case. In summary by increasing the pump power: 1. CG beams: a. gradual increase of peak amplitude at z ¼2 cm. b. gradual disappearance of the main peak at z ¼5 cm. c. no considerable change for beam spots at z ¼2 cm. d. considerable change of laser spots for z ¼5 cm. 2. BG beams: a. gradual increase of the peak amplitude at z¼ 2 cm. b. gradual appearance of a middle peak ( reported experimentally[17]). c. no consider change for laser spots at z¼ 2 cm. d. much pronounced change for laser spots at z ¼5 cm. 3. MG beams: a. gradual increase of the peak amplitudes at z¼ 2 cm. b. gradual disappearance of the main peak in expense of rise in side peaks. c. no change in laser spots at z¼2 cm. d. much notable change in laser spots at z ¼5 cm. 4. PG beams: a. gradual increase of peak amplitude at z ¼2 cm. b. gradual appearance of the main peak at z¼ 5 cm.
c. no considerable change for beam spots at z¼ 2 cm. d. considerable change of laser spots for z¼ 5 cm.
6. Conclusion Based on our previous works on thermal effects on the generation of special types of beams. We have studied thermal effects on the generation of Cosine-Gauss and Parabolic-Gauss beams and compared the results with that of our previous works. The comparison shows that thermal effects has greater impact on the generation of CG, BG, MG and PG beams at pump power of 5 W and ωp ¼100 μm at z¼5 cm from the beam waist. Moreover, the most influential factor such as pump beam waist in strengthening the thermal effects were also studied. It is shown that large pump beam waist can eventually makes the laser system similar to nonthermal one despite high pump power received. Therefore, a designer and experimenter needs to account for thermal effects, otherwise he/she might be confused to identify true and reliable laser intensity profiles from their non-thermal ones.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
J.C. Gutierrez-Vega, M.A. Bandres, J. Opt. Soc. Am. A 22 (2005) 289. M.A. Bandres, J.C. Gutierrez-Vega, S. Chavez-Cerda, Opt. Lett. 29 (2004) 44. J.C. Gutierrez-Vega, M.A. Bandres, J. Opt. Soc. Am. A 24 (2007) 215. J. Durnin, J. Opt. Soc. Am. A 4 (1987) 651. M. Sabaian, H. Nadgaran, Opt. Commun. 281 (2008) 672. H. Nadgaran, M. Servatkhah, M. Sabaeian, Opt. Commun. 283 (2010) 417. Guoquan Zhou, Rong Qu, Lu Sun, J. Modern Opt. 55 (15) (2008) 2485. L.W. Caperson, A.A. Tover, J. Opt. Soc. Am. A 15 (1998) 954–961. Nanrun Zhou, Guihua Zeng, Opt. Commun. 232 (2004) 49–59. J.C. Gutierrez-Vega, M.D. Iturbe-Castillo, S. Chavez-Cerda, Opt. Lett. 25 (2000) 1493. [11] Halil Tanyer Eyyuboglu, JOSA A 22 (8) (2005) 1527. [12] Kang Xiao-Ping, L.ü Bai-Da, Chin. Phys. Lett. 23 (2006) 2430.
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[13] C. López-Mariscal, M.A. Bandres, J.C. Gutierrez-Vega, Opt. Eng. 45 (6) (2006) 68001. [14] W. Koechner, Solid-state Laser Engineering, fifth ed., Springer-Verlag, Berlin, Heidelberg, 1999. [15] R. Weber, B. Neuenschwander, H.P. Weber, Opt. Mater. 11 (1999) 245. [16] S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, P. Georges, IEEE J. Quantum Electron. 40 (2004) 1217–1234. [17] D.C. Brown, IEEE J. Quantum Electron 34 (1998) 2393–2402. [18] M. Schmid, Th Graf, H.P. Weber, J. Opt. Am. B 17 (2000) 1398. [19] H. Nadgaran, M. Sabaian, Pramana – J. Phys. 67 (2006) 119. [20] M.E. Innocenzi, H.T. Yura, C.L. Fincher, R.A. Fields, Appl. Phys. Lett. 56 (1990) 1831–1833. [21] A.K. Cousins, IEEE J. Quantum Electron. 28 (1992) 1057–1069. [22] B. Ozygus, J. Erhard, Appl. Phys. Lett. 67 (1995) 1361–1362.
[23] M. Guizar-Sicairos, J.C. Gutierrez-Vega, Opt. Lett. 31 (2006) 2912. [24] R.I. Hernandez-Aranda, J.C. Gutierrez-Vega, M. Guizar-Sicairos, M.A. Bandres, Opt. Express 14 (2006) 8974. [25] M.P. MacDonald, Th Graf, J.E. Balmer, H.P. Weber, Opt. Commun. 178 (2000) 383–393. [26] B.A. Usievich, V.A. Sychugov, F. Pigeon, A. Tishchenko, IEEE J. Quantum Electron. 37 (2001) 1210. [27] A.E. Siegman, Lasers, University Science Books, Mill Valley, CA, 1986. [28] A. Hakola, S.C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, J. Turuen, Opt. Commun. 238 (2004) 335. [29] Zhigang Zhao, Sunqiang Pan, Zhen Xiang, Yantao Dong, Jianhong Ge, Chong Liu, Jun Chen, Opt. Express 20 (10) (2012) 10605.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Thermally-affected Cosine-Gauss and Parabolic-Gauss beams and comparisons of Helmholtz–Gauss beam families H. Nadgaran n, R. Fallah Physics Department, Shiraz University, Shiraz 71454, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 3 June 2014 Received in revised form 23 November 2014 Accepted 2 December 2014 Available online 6 December 2014
This article firstly investigates the thermal effects on the generation of Cosine-Gauss and Parabolic-Gauss beams and compares with the impacts of thermal effects on Bessel–Gauss and Mathieu–Gauss beams reported in our previous works. For this investigation, the intensity distribution of the beams against radial distances at different distances from the beam waist under various pump powers and pump beam waists has been studied. The results show that the thermal effects have severe impact on the laser output, so that true distinguishing of thermally-affected and non-thermal intensity profiles is unlikely feasible. The comparisons show that the thermal effects has minimal impact on the generation of Bessel– Gauss beams. The role of pump beam waist on the strength of the thermal effects was also studied. & 2014 Elsevier B.V. All rights reserved.
Keywords: Thermal effects Laser special beams Helmholtz–Gauss equation Heat equation
1. Introduction Solution to Helmholtz equation can be classified to four distinct families. These solutions that are ideal non-diffracting beams are based on the proper coordinate systems adopted. They are parabolic beams in parabolic cylindrical coordinates [1,2], Mathieu beams in elliptic cylindrical coordinates [1,3], Bessel beams in circular cylindrical coordinates [1,4], and cosine beams in Cartesian coordinates. The transverse intensity distribution of these four families remain unchanged upon their propagation in free space carrying infinite energy due to their inherent infinite extent that is why they are called non-diffracting beams. In order to remove this physically unreasonable behavior of infinite extent, the modified version of these families are usually built up by apodizing them with a radial Gaussian apodization, that is, their paraxial apodized versions called Helmholtz–Gauss (HzG) beams is composed of an ideal non-diffracting beams modulated by a radial Gaussian envelope. Following our previous works, in which we studied the thermal effects on the generation of Bessel–Gauss (BG) [5] and Mathieu–Gauss (MG) beams [6], we have been motivated to study the cases of Cosine-Gauss (CG) and parabolic-Gauss (PG) beams under induced thermal loads. Cosine-Gauss beams are obtained as exact solutions to the paraxial wave equation propagating in free space and also in complex optical systems [7,8]. The name of Cosine-Gauss beams are usually brought with the more n
Corresponding author. Fax: þ 987116460839 E-mail address: [email protected] (H. Nadgaran).
http://dx.doi.org/10.1016/j.optcom.2014.12.007 0030-4018/& 2014 Elsevier B.V. All rights reserved.
general beams of cosh-Gauss, Hermite–Gauss (HG), Hermite-cosine-Gauss (HCG), Hermite-sine-Gauss (HSG) and Hermite–Gauss beams. The above mentioned four families of special type of beams are utilized in applications in which efficient extraction of energy is required [9]. Various studies in the field of special beams can be found in the literature. These investigations cover the propagation properties of HzG beam families through a paraxial optical ABCD system with a hard-edge aperture [3,10]. In particular propagation characteristics of CG beams in turbulent atmosphere [11] and analytical expression for propagation of non-paraxial CG beams in free space [12], were also studied. PG beams are obtained as exact solutions to the wave equation in parabolic cylindrical coordinates. Recently Lopez-Mariscal et al. presented experimentally the existence of parabolic-Gauss beams [13]. Thermal effects play an important and major role in high power solid-state lasers. In these lasers, particular attention has been paid to induced heat load in the laser cavity since the laser pump energy is only partly converted to laser light and most of the pump energy is lost in the form of heat [14,15]. This induced heat leading to thermo-mechanical and thermo-optical effects such as thermal lensing, photo-elastic effects, thermal stress and thermal fracture [16,17]. The most serious problems are then the distortion of the beam and degradation in the beam profile due to thermal lensing and depolarization loss due to stress induced birefringence [18,19]. Thermal lens modeling and measurement in end-pumpedsolidstate lasers have been developed well [20–22]. As almost every important parameters of laser radiation is directly influenced by induced thermal loads, the propagation characteristics of laser beams under severe thermal loads is necessary for designers and
H. Nadgaran, R. Fallah / Optics Communications 341 (2015) 160–172
manufactures to have a true, clear and reliable insight to the laser output features and their conversion. Knowledge of thermal effects can also lead to proper design and construction of laser cooling systems. To account for induced thermal loads in investigation of special beam propagation characteristics, one needs to consider laser beam propagation in an inhomogeneous thermal media. Here in this work, the impact of thermal effects on the generation and behavior of Cosine-Gauss and Parabolic-Gauss beams is studied. Also to evaluate the effect of heat on the generation of special beams, we have compared the results of this work with the results of our previous works.
→ Complex field amplitude of HzG beams, H1 r1 ; k1 at the input
(
)
plane of z ¼z1 can be written as in Eq.(1) [23], where index 1 refers to the input parameters of the Gaussian beam traveling axially through an ABCD system.
(
)
(
)
(1)
equation of:
(
)
(2)
→ and r1 = (x1, y1) and ω1 show the transverse coordinate and beam waist of the Gaussian envelope, respectively. As Eq. (1) shows the → non-diffracting beam F r1 ; k1 , has been apodized by a Gaussian
(
)
beam in order to make it physically sensible, and this can be expressed as a superposition of plane waves, that is: →
F (r1; κ1) =
π
∫−π A(φ)exp(iκ1(xcos(φ) + y sin(φ))dφ
(3)
where A(φ) is an arbitrary angular spectrum of the ideal non-diffracting beam. Using Huygens diffraction integral, the transverse distribution of HzG beams at the output plane of z¼ z2 of an ABCD optical system would be given as [23]:
⎛ ikr 2 ⎞ ⎛ k k B ⎞ exp(ikς) → U (r2) = exp⎜⎜ 2 ⎟⎟exp⎜ 1 2 ⎟ F( r ; κ ) ⎝ 2ik ⎠ A + (B/q1) 2 2 ⎝ 2q2 ⎠
Aq1 + B Cq1 + D
, κ2 =
κ1q1 Aq1 + B
(5)
It is reminded that ω , q and k both at z ¼z1 and z ¼z2 obey the following relation:
ωj =
2iqj kj
, j = 1, 2 (6)
2.1. Cosine-Gauss (CG) beams One of the simplest non-diffracting beams in Cartesian coordinates is the ideal cosine field: →
)
⎛ ∂2 ⎞ → ∂2 ⎜ 2 + 2 + k12⎟F r1 ; k1 = 0 ⎝ ∂x ⎠ ∂ y
q2 =
(
)
F r ; k1 = Acos(κ1y)
→ where F r1 ; k1 is the solution of the two dimensional Helmholtz
(
optical path length measured along the optical axis from z¼ z1 to z¼ z2. The following transformation laws relates the beam input parameters to beam output parameters as [24]:
In all above relations, q complex parameter is defined as 1/q = (1/R) + (2i/kω2)[24], where R and k are the radius of wave front curvature and the wave vector of the Gaussian apodization, respectively.
2. Helmholtz–Gauss (HzG) beams
⎛ →2 ⎞ r → → H1 r1 ; k1 = exp⎜− 1 2 ⎟F r1 ; k1 ⎜ ω ⎟ 1 ⎝ ⎠
161
(4)
where q2 and κ2 are the complex parameter and complex transverse wave number at the output plane, respectively. ς is the
(7)
where κ1 is the transverse wave number and A is the normalization constant. The cosine field when apodized by a Gaussian envelope can then be written of:
⎛ r2 ⎞ CG(r) = N exp⎜⎜ − 2 ⎟⎟cos(κ1y) ⎝ ω1 ⎠
(8)
where N is the normalization constant. 2.2. Bessel–Gauss (BG) beams In recent years, the existence of Bessel–Gauss beams has been studied theoretically and experimentally. By solving the wave equation in circular cylindrical coordinates, The BG beam can be written as follows[5]:
⎛ r2 ⎞ BG m(r) = Aexp⎜⎜− 2 ⎟⎟Jm (κ1r)exp(imθ) ⎝ ω1 ⎠
(9)
where A is the normalization constant and Jm (.) is the mth-order Bessel functions. 2.3. Mathieu–Gauss (MG) beams Mathieu beams constitute another family of non-diffracting
Fig. 1. (a) Heat density distribution versus r in the plane of z ¼ 0 and (b) versus Z at r¼ 0 for ωp ¼ 100 μ m for different pump powers.
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Fig. 2. (a) Heat density distribution versus r in the plane of z¼ 0 and (b) versus Z at r¼ 0 and P ¼5 W for different pump beam waists. Table 1 The focal length of thermal lens for ωp = 100 μm . Pump power
1W
3W
5W
Focal length
28 cm
9.3 cm
5.6 cm
∞
MG2en(r) = 2πNmeC2m ∑ where
Table 2 The focal length of thermal lens for P¼ 5 W. Pump beam waist
50 μm
100 μm
200 μm
300 μm
Focal length
3.9 cm
5.6 cm
22.8 cm
2.38 m
beams that are solutions of the Helmholtz equation in elliptic coordinates. The closed-form expression of the n′th-order even and odd MG beams is given as [6]:
MG2en
normalization constants and the even and odd angular Mathieu functions ce( ) and se( ), respectively. Even-order MG beams can also be written as an expansion of BG beams [6]:
Nme
Mathieu functions are described by the nth-order even and odd radial Mathieu functions Je( ) and Jo( ). N2en and N2on + 1 are the
(11)
2.4. Parabolic-Gauss (PG) beams The transverse field of the even and odd parabolic beams is written as [13]:
PG e(r , a) =
⎛ iκ 2 z ⎞ exp⎜⎜− 1 ⎟⎟Pe( 2κ1 ξ ; a)Pe( 2κ1 η; − a) π 2 ⎝ 2k ⎠ Γ1
2
⎛ r2 ⎞ exp⎜⎜− 2 ⎟⎟ ⎝ ω1 ⎠ PG o(r , a) =
(10)
⎛ r2 ⎞ A2j ( − 1)2j exp⎜⎜ − 2 ⎟⎟cos(2jθ)J2j (κ1r) ⎝ ω1 ⎠
is the normalization constant.
⎛ r2 ⎞ = N2enexp⎜⎜− 2 ⎟⎟Je2n(ξ , q)Ce2n(η, q) ⎝ ω1 ⎠
⎛ r2 ⎞ MG2on + 1 = N2on + 1exp⎜⎜− 2 ⎟⎟Jo2n + 1(ξ , q)Cs2n + 1(η, q) ⎝ ω1 ⎠
j=0
(12)
⎛ iκ 2 z ⎞ exp⎜⎜− 1 ⎟⎟Po( 2κ1 ξ ; a)Po( 2κ1 η; − a) π 2 ⎝ 2k ⎠ Γ3
2
⎛ r2 ⎞ exp⎜⎜− 2 ⎟⎟ ⎝ ω1 ⎠
(13)
Fig. 3. a) Intensity distributions of CG beams versus radial distance for z ¼2 cm from the beam waist, under pump power of 1 W. Solid plots represent the non-thermal. CG beam, dashed plots shows CG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot.
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163
Fig. 4. a) Intensity distributions of CG beams versus radial distance for z ¼ 2 cm from the beam waist, under pump power of 3 W. Solid plots represent the non-thermal. CG beam, dashed plots shows CG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot.
Fig. 5. a) Intensity distributions of CG beams versus radial distance for z ¼ 2 cm from the beam waist, under pump power of 5 W. Solid plots represent the non-thermal. CG beam, dashed plots shows CG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot.
Where Γ1 = Γ = ((1/4) + ia/2), Γ3 = Γ = ((3/4) + ia/2), the parabolic cylindrical coordinates (ξ, η) are given by x = η2 − ξ 2 /2 and y = ξη , also in the above equation Pe(.) and Po(.)
(
)
are the even and odd solutions to the parabolic cylindrical differential equation and its Taylor expansion with v ¼0 is given by[2]:
p(v , a) =
∞
∑ n = 0 cn
vn , n!
cn + 2 = acn −
n (n − 2) cn − 2 4
(14)
The first two coefficients for Pe are c0 = 1 and c1 = 0, whereas for Po they are c0 = 0 and c1 = 1.
3. Thermal model Finite thermal conductivity and non-uniformity of a pump radiation usually causes the pump-induced heat in laser gain medium leading to variations of gain cross section which in turn makes the gain medium to act as a lens. This is known as the thermal lens effect, where the optical path difference (OPD) function of an ideal
thermal lens is parabolic. This ideal parabolic OPD suffers from a spherical aberration in strongly pumped laser rods. This deviation from parabolic OPD leads to radial variation of the thermal lens’ focal length (f), which then becomes a quadratic function of r. Here in this work a pump of Gaussian profile (GP) was employed as follows:
⎛ ⎞ 2r 2 SGP (r , z) = Q GPexp⎜⎜− 2 ⎟⎟exp(−αz) ⎝ ωp ⎠
(15)
where α is the absorption coefficient of laser crystal, Q GP is constant given in the Appendix of reference [5]:
Q GP =
αηqηabsP 2 2 − π ω (1 − e αl )(1 − e−2(a2 / ωp2))
(16)
where P is the total pump power, ηq is the fraction of pump power converting to heat, ωp is the pump beam waist and ηabs is the fraction of absorbed pump beam. Ɩ and ɑ show the laser crystal length and radius, respectively.
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Fig. 6. a) Intensity distributions of CG beams versus radial distance for z ¼5 cm from the beam waist, under pump power of 1 W. Solid plots represent the non-thermal. CG beam, dashed plots shows CG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot.
Fig. 7. a) Intensity distributions of CG beams versus radial distance for z ¼5 cm from the beam waist, under pump power of 3 W. Solid plots represent the non-thermal. CG beam, dashed plots shows CG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot.
As far as the pump power is concerned, the laser crystal as a homogeneous gain medium is converted to an inhomogeneous graded index (GRIN) one by severe influence of the induced heat. This leads to refractive index variation of the gain medium. Therefore assuming the GRIN medium as an ABCD affected medium, one has to calculate the element of the ABCD matrix of the GRIN medium for its full characterization. To do this, solution of the following heat equation should be obtained:
∇2T (r , z) = −
S(r , z) K
∂n [T (r , z) − T (r = 0, z)] + (n0 − 1)εzz ∂T
Δn(r , z) =
(18)
where T1 is the ambient or coolant temperature and h is the heat transfer coefficient of the medium. Based on the temperature
(19)
where n0 is the index of refraction at the center of the crystal and єzz is the strain tensor component of the crystal [5]:
εzz = (1 + v)αT [T (r , z) − T (0, z)] (17)
where K is the thermal conductivity of the GRIN medium and S(r, z) is the heat source density in watts/volume. In order to have a proper boundary conditions for edge and faces of the crystal, the Newton's law of heat transfer was used:
→ −K ⌢ n . ∇ T (r , z)|boundary = h(T (r , z) − T∞)
distribution of the laser crystal, the crystal refractive index change, Δn, can be calculated as:
(20)
where υ is Poisson ratio and αT is the thermal expansion coefficient of the laser crystal. The temperature dependent change of refractive index can cause thermally induced phase-shifts as [25]:
Δφ = k
l
∫0 Δn(r, z) dz
For a lens-like medium, the focal length is given by [25]:
(21)
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165
Fig. 8. a) Intensity distributions of CG beams versus radial distance for z ¼ 5 cm from the beam waist, under pump power of 5 W. Solid plots represent the non-thermal. CG beam, dashed plots shows CG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot.
Fig. 9. a) Intensity distributions of PG beams versus radial distance for z ¼ 2 cm from the beam waist, under pump power of 1 W at θ¼0°. Solid plots represent the nonthermal. PG beam, dashed plots shows PG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot for a¼ 1.
fth = −
k r2 2Δφ
(22)
The exact solution of Eq.(17) using the above boundary conditions is as follows [26]:
T (r , z) =
∞
∑i = 1 ti(z)J (vir) + T∞ °
(23)
where J (vr i )′s are the zero-order Bessel functions with vi′s as the ° positive roots of the equation hJ (va) − KJ1 (va) = 0, ti(z) has been ° fully given in [5]. By expansion the Bessel function of Eq.(23) and use of paraxial ray approximation, one can end up with the crystal refractive index as a second order function of r, that is [5]:
⎛ r 2γ (z)2 ⎞ ⎟ n(r , z) = n0⎜1 − 2 ⎠ ⎝
(24)
γ (z) is an important function showing the variation of the
medium refractive index along z coordinate. In general for a lenslike medium, the ABCD transfer matrix is written as [27]:
⎡ A B ⎤ ⎡cos(γl) sin(γl)/γ ⎤ ⎥ ⎢⎣ ⎥⎦ = ⎢ C D ⎣⎢−γ sin(γl) cos(γl) ⎥⎦
(25)
This ABCD matrix represents thermal effects imposed on the gain medium and its detailed calculation has been given by Sabaeian et al. [5]. By constructing the ABCD transfer matrix, we can simulate the HzG beams suffered by induced heat.
4. Results and discussion Here in this section, we first present the effects of induced heat load on the generation of CG and PG beams. Then the study will be continued by comparison of these results with that of other four special beams. For this purpose, an end pumped cylindrical Nd: YAG laser crystal [28] with I ¼5 mm and radius of ɑ¼ 1.5 mm was
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Fig. 10. a) Intensity distributions of PG beams versus radial distance for z¼ 2 cm from the beam waist, under pump power of 3 W at θ¼ 0°. Solid plots represent the nonthermal. PG beam, dashed plots shows PG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot for a¼ 1.
Fig. 11. a) Intensity distributions of PG beams versus radial distance for z ¼2 cm from the beam waist, under pump power of 5 W at θ ¼0°. Solid plots represent the nonthermal. PG beam, dashed plots shows PG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot for a¼ 1.
assumed. The laser was longitudinally pumped by a diode laser at wavelength of λ p ¼808 nm and pump beam waist of ωp ¼100 μm. CG and PG beams were generated with beam waist of ωl ¼300 μm, laser wavelength of λl ¼1064 nm where transverse wave vector was taken as κt ¼22144 cm−1[28]. Figs. 1 and 2 show the variations of induced heat density against r and z for different pump powers and different pump beam waists. One can notice the decay rate of heat density when pump beam waist is increasing (Fig. 2a) showing the importance of pump beam waist in generating induced heat and making different thermal lens with different strength. Table 1 shows the focal length of the thermally-induced lens calculated for different pump powers. It is shown that by increasing the pump power, the thermal focal length would decrease, leading to stronger lens as expected. Table 2 shows the focal length of the thermally-induced lens calculated for different pump beam waists. It is shown that by
increasing the pump beam waist, the thermal focal length would increase, leading to weaker thermal lens. Figs. 3–8 Show the transverse intensity distributions of CG beams versus radial distance for 1, 3 and 5 W pump powers, respectively. From these figures and Table 1, the following conclusions can be drawn: Under pump power of 1 W, there can be seen no considerable intensity profile changes under induced heat. The non-thermal model of intensity profile almost coincides with the thermal ones under 1 W pump power for different z, a distance from the beam waist. Our calculations shows that the focal length of the induced thermal lens in this case is 28 cm, that is very weak thermal lens is built up and the laser crystal dose not suffer much from the induced heat load. The fact that high power end-pumped lasers suffers a spherical aberration originated from a non-uniform temperature distribution suggests that in this work also one can no longer encounters a
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167
Fig. 12. a) Intensity distributions of PG beams versus radial distance for z ¼5 cm from the beam waist, under pump power of 1 W at θ¼0°. Solid plots represent the nonthermal. PG beam, dashed plots shows PG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot for a¼ 1.
Fig. 13. a) Intensity distributions of PG beams versus radial distance for z¼ 5 cm from the beam waist, under pump power of 3 W at θ¼ 0°. Solid plots represent the nonthermal. PG beam, dashed plots shows PG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot for a¼ 1.
perfects lens with refractive index distribution of parabolic profile. The existence of spherical aberration makes the thermal lens’ focal length to depend on ωl through [29]:
⎡ ⎛ ω ⎞2⎤ fth (ωl) = f ⎢1 + x⎜⎜ l ⎟⎟ ⎥ °⎢ ⎝ ωp ⎠ ⎥⎦ ⎣
(26)
where f is the focal length of the thermal lens without spherical ° aberration, ωl is the laser mode beam waist, ωp is the pump beam waist and x is a parameter representing the strength of the spherical aberration. This equation shows that spherical aberration makes beams with different mode waist to see different thermal lens's focal length. When the laser crystal is high power pumped, spherical aberration becomes more and more pronounced resulting in notable diffraction loss for fundamental mode laser beam and degrading the beam quality by coupling fundamental mode laser to higher modes. Moreover, from efficiency point view, systems with aberrated thermal lensing are less efficient compared to
systems without aberration. Since thermal aberration is essentially a loss that is dependent on the pump power. Upon increasing the pump power, severe deviation from the non-thermal intensity profile can be realized showing that the laser media is considerably affected by the induced thermal lens. The most thermally affected CG intensity distributions can be learned from Fig.8 when compared to Figs. 6 and 7. It is seen that upon increasing the pump power, the side lobes of the central spot are gradually grown so that the two spots shown in Fig. 8b is actually the two modified lobes, that is no longer resembles the nonthermal profile( Fig. 8c). This piece of information has its very impact impotence in distinguishing true thermally affected profile from a non-thermal one. Here in this example (Fig. 8) the designer and the experimenter should be aware that the bright spots seen in Fig. 8b are not the main spot of the CG beam rather they are side spots which have grown due to thermal lens of f ¼5.6 cm. From focal length point of view, it is seen that a larger quantity of induced heat can build stronger thermal lens which in turn can
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Fig. 14. a) Intensity distributions of PG beams versus radial distance for z¼ 5 cm from the beam waist, under pump power of 5 W at θ¼ 0°. Solid plots represent the nonthermal. PG beam, dashed plots shows PG beam under GP. b) thermally-affected. laser spot. c) non-thermal. (thermally-unaffected.) laser spot for a¼ 1.
Fig. 15. Intensity distributions of special beams versus radial distance for z ¼ 2 cm from the beam waist. Solid plots represent the non-thermal. beam, dashed plots shows thermally-affected. beam under GP. From the top: for pump powers of 1, 3, 5 W, respectively.
potentially allow the grown up of the side lobes making the designer and experimenter confused. One more aspects of Fig. 8a is that the CG intensity distributions were plotted for z¼ 5 cm from the beam waist and as the calculated focal length is about 5.6 cm, one can say that the CG profile shown in this figure is just around the focal point of the thermal lens making much deviations of the corresponding non-thermal intensity profile. Figs. 9–14 Show the transverse intensity distributions of PG beams versus radial distance for 1, 3 and 5 W pump powers, respectively. The following points can be noticed:
compared to Figs. 13 and 12. It is seen that upon increasing the pump power, the central spots are gradually grown so that the side spots shown in Figs. 12b and 13b is actually disappeared, that is no longer resembles the non-thermal one (Fig. 14c). 3. From these figures, one can see the eventual intensity enhancement of the main peak together with the peak shift toward lower r when the pump power is increased.
5. Comparisons 1. Under pump power of 1 W, there can be seen no considerable change in intensity profile and laser spots under induced heat. 2. Upon increasing the pump power, the most thermally affected PG intensity distributions can be seen in Fig. 14 when
We now compare the results of thermal effects on the generation of HzG beams. For this purpose, we have plotted the intensity distribution of zeroth order BG and MG beams versus
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Fig. 16. Thermally-affected. laser spots under GP profile for z ¼2 cm from the beam waist. From the top: for pump powers of 1, 3, 5 W.
Fig. 17. Intensity distributions of special beams versus radial distance for z ¼ 5 cm from the beam waist. Solid plots represent the non-thermal. beam, dashed plots shows thermally-affected. beam under GP. From the top: for pump powers of 1, 3, 5 W, respectively.
radial distance at z¼2.5 cm from the beam waist for the different pump powers and different pump beam waists. Fig. 15 shows CG, zeroth-order BG, MG and PG beam profiles for z ¼2 cm from the beam waist under pump powers of 1, 3 and 5 W. From Fig. 13 one can realize that upon increasing the pump power, the laser intensity profiles gradually shift toward r ¼0 showing
closer peaks. Moreover, the thermally-affected profiles show higher intensities versus r. whereas the above mentioned shifts are almost the same for CG, BG, MG and PG beams. Fig. 16 shows the corresponding laser spots for the same four families of special beams. The figure shows no considerable change in laser spots for z¼ 2 cm detection from the beam waist.
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Fig. 18. Thermally-affected. laser spots under GP profile for z¼ 5 cm from the beam waist. From the top: for pump powers of 1, 3, 5 W, respectively.
Fig. 19. Intensity distributions of special beams versus radial distance for z¼ 5 cm from the beam waist for p¼ 5 W. The rows indicate the pump beam waists of 100, 200, 300 μm, respectively.
Fig. 17 shows the same profiles as in Fig. 15 for z¼ 5 cm detection from the beam waist. The figure clearly shows that upon increasing the pump power, considerable alterations in peak shifts for all beams are recognized, whereas the peak amplitudes for CG, BG, MG and PG beams of 5 W do not resemble the non-thermal profile and the 1, 3 W profiles. Fig. 18 shows the corresponding laser spots for z¼5 cm. Here
remarkable spot changes are seen for all special beams by increasing the pump power. For the case of CG beams one can see two-part spots for 5 W case which is no longer similar to its 1 and 3 W cases. For the case of BG beams systematic increase of the spot radius is clearly notable. The situation is also understood for MG beams where the middle spot is gradually disappeared upon increasing the pump power. PG beams shows even stronger effect
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Fig. 20. Thermally-affected. laser spots under GP profile for z ¼ 5 cm from the beam waist for P¼ 5 W. The rows indicate the pump beam waists of 100, 200, 300 μm, respectively.
due to complete disappearance of the side lobes. The rows in Fig. 19 show the transverse intensity distributions of special beams versus radial distance at z ¼5 cm from the beam waist for pump beam waists of 100, 200, 300 μm, respectively. The figure clearly shows that as the pump beam waist increases the effect of induced heat diminishes. This point holds true for all special beam profiles, as the third row clearly indicates. As a result, for large pump beam waist, the beams behave as though no heat effect, even for large delivered powers is present. Fig. 20 shows the corresponding laser spots for z¼ 5 cm. One can clearly learn that the strength of the thermal effects is gradually reduced, so that for large pump beam waist, the spot shapes and sizes can be regarded as similar to the non-thermal case. In summary by increasing the pump power: 1. CG beams: a. gradual increase of peak amplitude at z ¼2 cm. b. gradual disappearance of the main peak at z ¼5 cm. c. no considerable change for beam spots at z ¼2 cm. d. considerable change of laser spots for z ¼5 cm. 2. BG beams: a. gradual increase of the peak amplitude at z¼ 2 cm. b. gradual appearance of a middle peak ( reported experimentally[17]). c. no consider change for laser spots at z¼ 2 cm. d. much pronounced change for laser spots at z ¼5 cm. 3. MG beams: a. gradual increase of the peak amplitudes at z¼ 2 cm. b. gradual disappearance of the main peak in expense of rise in side peaks. c. no change in laser spots at z¼2 cm. d. much notable change in laser spots at z ¼5 cm. 4. PG beams: a. gradual increase of peak amplitude at z ¼2 cm. b. gradual appearance of the main peak at z¼ 5 cm.
c. no considerable change for beam spots at z¼ 2 cm. d. considerable change of laser spots for z¼ 5 cm.
6. Conclusion Based on our previous works on thermal effects on the generation of special types of beams. We have studied thermal effects on the generation of Cosine-Gauss and Parabolic-Gauss beams and compared the results with that of our previous works. The comparison shows that thermal effects has greater impact on the generation of CG, BG, MG and PG beams at pump power of 5 W and ωp ¼100 μm at z¼5 cm from the beam waist. Moreover, the most influential factor such as pump beam waist in strengthening the thermal effects were also studied. It is shown that large pump beam waist can eventually makes the laser system similar to nonthermal one despite high pump power received. Therefore, a designer and experimenter needs to account for thermal effects, otherwise he/she might be confused to identify true and reliable laser intensity profiles from their non-thermal ones.
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