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Analysis of defocus aberration characteristics on a typical passively confocal unstable resonator
Optik - International Journal for Light and Electron Optics 202 (2020) 163625
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Analysis of defocus aberration characteristics on a typical passively confocal unstable resonator
T
⁎
Xiang Zhanga, Zhengrong Denga, , Zhiqiu Hua, Xiaoxia Zhangb a Key Laboratory of Solid-state Laser and Applied Techniques, School of Photoelectric Technology, Chengdu University of Information Technology, Chengdu 610225, China b School of Optoelectronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Laser resonator Optical aberration Wavefront properties Mode reconstruction
The analytical treatment of typical defocus aberration and its impact on the mode wavefront properties of a confocal structural laser resonator is discussed. Numerical modeling of the system is deduced to predict the influence of defocus disturbation on mode intensity and phase distribution. In addition, an experimental method by combining Zernike aberration function and wavefront reconstruction to measure the dynamic wavefront aberration for such lasers has been developed. Beam mode characteristics, including the alignment state and defocus aberration perturbation have been presented. It is shown that the typical Zernike aberration such as A3, A5 and A10 caused by intracavity perturbation significantly affected the beam properties, PSF profiles and circle energy distribution in the far field. The experimental results show that defocus aberration characteristics can be used as an effective basis for dynamic compensation of typical aberration in such lasers. Furthermore, corresponding strategies are proposed to implement the active correction of typical optical aberration in such resonators.
1. Introduction Unstable resonators of such a confocal structure operating on the fundamental TEM00 mode produce, as known, can obtain high quality diffraction limited beams [1–3]. However, due to the influence of ambient vibration, laser oscillators with the structure of virtual confocal unstable cavity are easy to suffer from the tilt misalignment of cavity mirrors to bring much aberration [4–6]. In addition, thermal accumulation in lasers will lead to unconfocal condition for such resonators and produce a lot of beam distortion. Phase-tilt and defocus are the main aberrations among these factors, which have significant influence on the output power, mode wavefront, and long-distance transmission of beam. It will also seriously limit the further application of such lasers [7–9]. Relevant studies have shown that some relevant optical aberrations have obvious effect on the beam quality of lasers. With the increase of output power, the defocus aberration will also increase obviously. In addition, the defocus is still the main wavefront aberration even after being corrected by the phase compensating system in the case of a representative cavity geometric perturbation. The reason is that typical high-energy lasers, e.g. the chemical laser with positive confocal unstable cavity works usually in a high frequency vibration and fast speed air flow condition [10,11]. Thus the cavity mirror is easily perturbed and further deformed by strong heat radiation, and the main beam distortion is lower order aberration. The change of cavity length will cause defocus aberration, meanwhile, the thermal deformation of cavity mirrors will also result in significant mode aberration of laser beam. The intracavity compensation for defocus aberration has intrinsic advantages; it will not only compensate a wide dynamic range
⁎
Corresponding author. E-mail address: [email protected] (Z. Deng).
https://doi.org/10.1016/j.ijleo.2019.163625 Received 29 July 2019; Accepted 13 October 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 202 (2020) 163625
X. Zhang, et al.
Fig. 1. Schematic diagram of the typical unstable resonator with a virtual confocal form.
for optical aberration, but also can improve the mode properties and beam quality of lasers [12,13]. It has currently become one of the important research topics to seek dynamic compensation ways for defocus aberration inside the cavities. However, modeling the typical aberration properties and further experimental analysis is rather limited. 2. Theoretical model 2.1. Defocus aberration caused by variation of cavity length The model of the typical unstable cavity with a virtual confocal structure is shown in Fig. 1. The focal length of the cavity mirror M1 and M1 is f1 and f2, respectively. The reflector M1 and M2 have a common virtual focal point F outside the cavity. According to geometrical optics principles, it is a zero focal power system in the ideal case, however, the focal length will be rewritten as a ΔL change of the cavity length
f = f1 f2 / ΔL
(1)
By discussing the change of beam focus caused by the change of cavity length, the relationship between the focus position and the defocus aberration of beam can be further illustrated. A very narrowly beam can be introduced into the cavity to simulate the variation in the equivalent focal length of the out-coupling beam if the resonator length is changed by using the ABCD transmission matrix. Suppose the curvature radius of the two mirrors are R1 and R2, respectively. Cavity length of the resonator is expressed as L = (R1 + R2)/2 + ΔL, where ΔL denotes the change in the cavity length, which is negative when it is increased. The original thin beam is propagated back and forth in the cavity and the beam diameter will be continuously expanded. Finally, the stable eigenmode will be coupled out the resonator. According to the theory of ABCD transfer matrix for resonators
1 0 ⎤ ⎡ 1 (R1 + R2)/2 + ΔL ⎤ ⎡ 1 0⎤ ⎡ A B ⎤ = ⎡ 1 (R1 + R2)/2 + ΔL ⎤ ⎡ 1 1 ⎣ C D⎦ ⎣0 ⎦ ⎣− 2/ R1 1 ⎦ ⎣ 0 ⎦ ⎣− 2/ R2 1 ⎦ (4ΔL2 − R12 + 2R2 ΔL)/ R1 R2 [R12 − (R2 + 2ΔL)2]/2R1⎤ =⎡ ⎢ ⎥ 4ΔL/(R1 R2) (−R2 + 2ΔL)/ R1 ⎣ ⎦
(2)
The corresponding ΔL = 0 when the cavity length is -aligned, and Eq. (2) is simplified as
− R1/ R2 (R12 − R22)/2R1⎤ ⎡A B⎤ = ⎡ − R2 / R1 ⎥ ⎣C D⎦ ⎢ ⎦ ⎣ 0
(3)
at θ0 = 0 for a thin plane wavefront, the transfer matrix can be written as
x x ⎡ 1 ⎤ = ⎡ A B ⎤ ⎡ 0 ⎤ = ⎡− (R1/ R2) x 0 ⎤ θ θ C D ⎦ ⎣ 0 ⎦ ⎣0 ⎣ 1⎦ ⎣ ⎦
(4)
In fact, when the cavity length is aligned, the beam diameter for each round trip in the cavity will be magnified by M times, and the paraxial thin beam will propagate more times inside the cavity. Corresponding parameters to the beam can be obtained. The relationship between the beam radius ω0 and transmission times is
n = [ln(r2/ ω0)]/ln M
(5)
When, ΔL ≠ 0 , the light beam propagates for the first time in the cavity 2 x x ⎡ 1 ⎤ = ⎡ A B ⎤ ⎡ 0 ⎤ = ⎡ 4ΔL / R1 R2 + 2ΔL/ R1 + M ⎤ x 0 ⎥ 4 / R1 R2 ΔL ⎣ θ1⎦ ⎣ C D ⎦ ⎣ θ0 ⎦ ⎢ ⎣ ⎦
(6)
There is no light coupling out the resonator in the case, and the ray extension line will converge on the z axis. The position of the convergence light point is given by 2
Optik - International Journal for Light and Electron Optics 202 (2020) 163625
X. Zhang, et al.
Xf = −x1/ θ1 ≈ −R12 /(4ΔL)
(7)
where ΔL ≪ |Ri|, (i = 1, 2), we can see that Xf is independent of x0 in Eq. (7), it does mean that the paraxial beam will converge at one point on the axis after the first round trip in the resonator. The beam arrives at mirror M2 for the second round trip, the equation is written by
x x m2 + F (ΔL/ R) ⎤x ⎡ 2 ⎤ = ⎡ A B ⎤⎡ A B ⎤⎡ 0⎤ = ⎡ 2 2⎥ 0 3 2 ⎣ θ2 ⎦ ⎣ C D ⎦ ⎣ C D ⎦ ⎣ θ0 ⎦ ⎢ ⎣16ΔL /(R1 R2) − 4ΔL/ R1 − 4ΔL/ R2 ⎦
(8)
The position of convergence point is given by
Xf ≈ −R12 /(4ΔLm2 + 4ΔL)
(9)
It can be seen from Eq. (9) that Xf (output beam focal length) is inversely proportional to the parameter ΔL (which is approximately linear to the Zernike defocus aberration A3). It can be deduced that the relationship still holds even after the beam propagates several times in the cavity. Thus the parameter Xf -satisfies inversely proportional relation to Zernike defocus aberration. 2.2. Defocus caused by thermal effect The curvature radius of cavity mirrors will change due to the thermal deformation in laser irradiation, which will cause typical defocus aberration. The influence of the aberration on mode features is obvious especially in high energy lasers. We can simplify the problem of thermal effect to the variation of the equivalent vector height of mirrors. In fact, the focus change of the output beam caused by thermal distortion of cavity mirrors is equivalent to the result caused by the change of the cavity length. The results prove that the effect of thermal deformation can be fitted by the mathematical function given by the equation in the case of uniform heat radiation [10]
Δ = k (x 2 − R2)2
(10)
To ensure the beam quality of mode, the misalignment of cavity length and the vector height of cavity mirrors shall be controlled within a certain range, in which the length misalignment will affect the stability of cavity structures, and the change of vector height is related to the thermal condition inside lasers and the material of mirrors. Suppose the x coordinate is perpendicular to the optical axis. In addition, the function of -self-reproducing mode is described by Fresnel diffraction integral, e.g. for the optical cavity with aligned mirrors, the equation can be expressed by
U (x , y ) = γ
ik 4π
−ikρ
∬ U (x′, y′) e ρ
(1 − cos θ) ds′
(11)
Where, γ is the eigenvalue which shows the mode diffractive loss in the cavity, U (x , y ) and U (x ′, y′) represents the mode properties function on the two mirrors, respectively. Parameter ρ is the vector length between the two points on the two opposing mirrors, which can be further expressed in the form of relatively complex rectangular coordinates. The equation can be numerically solved by the appropriate algorithm [14,15]. The cavity mode profiles can be accurately obtained in the case of mirror thermal deformation and defocus condition. 2.3. Calculation results In the numerical calculation, the characters of initial field in the cavity can be assumed to be an arbitrary distribution according to the self-reproducing theory of laser mode. The geometric length of the cavity is 0.5 m, and the radius of curvature is 10 m for the concave reflector. The diameter of reflectors, geometric structural factors g1 and g2 are 1 cm, 1.0 and 0.95, respectively. In Fig. 2, mode characteristics with different extent of cavity reflector deformation are revealed. From the result, it illustrates that beam mode presents an ideal distribution of Gaussian function in such a resonator. However, the profiles of eigenmode will deviate obviously from the distribution of ideal case with mirror thermal distortion. For example, when the vector height of each cavity reflector has a change of 1/15 wavelength, there are some extreme points in the relative amplitude distribution, which obviously departs from the ideal case of fundamental transverse mode. It proves that the height deformation the mirrors should be controlled within the range of 1/20 wavelength as far as possible in order to avoid the eigenmode distortion of the beam. If the vector height of the mirror exceeds 1/10 wavelength, the stable self-reproducing mode can not be formed in the resonator. 3. Experimental setup 3.1. Experimental schematic and design Our approach constructs an unstable resonator with a special virtual confocal structure, collimated by a He-Ne laser, which is shown in Fig. 3. Such a structure of the folded resonator is usually used in high energy lasers . The total resonator cavity is mainly composed of two folded planar mirrors, an intracavity concave and convex reflector, and the sharp beam scraper used as an OC 3
Optik - International Journal for Light and Electron Optics 202 (2020) 163625
X. Zhang, et al.
Fig. 2. Beam mode profiles in the typical passive cavity. (a) normalized amplitude profiles of transverse mode as a function of mirror thermal deformation; (b) phase distribution in different cases.
Fig. 3. Schematic diagrams of analyzing the influence of intracavity defocus aberration on beam characteristics.
mirror to output beam. The OC mirror is inclined to an angle of 45 degrees from the cavity optical axis. Such structure is a typical standing wave cavity. He-Ne probing laser is adopted to align the resonator. In addition, it is available to a reference light source in the H–S wavefront reconstruction. The unstable cavity is constructed by the convex and concave mirror with a virtual confocal way. Curvature radius of the cavity mirror is 21 m and 7.5 m respectively, thus the resonator length is (R2 − R1)/2 = 6.75m .
4
Optik - International Journal for Light and Electron Optics 202 (2020) 163625
X. Zhang, et al.
Table 1 Expressions of typical Zernike polynomials and the aberration name. No.
Name
Expressions
0 1 2 3
Piston Tilt X Tilt Y Defocus
1 2ρ cos θ 2ρ sin θ
4
Astigmatism X
6 ρ2 cos 2θ
5
Astigmatism Y
6 ρ2 sin 2θ
6
Coma X
8 (3ρ3 − 2ρ2 )cos θ
7
Coma Y
8 (3ρ3 − 2ρ2 )sin θ
8
Trefoil X
8 ρ3 cos 3θ
9
Trefoil Y
8 ρ3 sin 3θ
10
Spherical
5 (6ρ4 − 6ρ2 + 1)
3 (2ρ2 − 1)
3.2. Wavefront described by Zernike polynomials Zernike polynomials can be used to characterize wavefront aberration of laser beam [16–18]. It is proved that Zernike polynomial is orthogonal in the circular domain, so that such polynomials is very convenient to fit the aberrated mode wavefront. In addition, each order Zernike aberration be expressed independently from the original distorted wavefront, so that the correction for the aberrated beam is facilitated by using appropriate adaptive optical device. The wavefront distortion can be described by a series of sum forms of Zernike polynomials, which is written as
φ (x , y ) =
∞
∑k =1 ak Zk (x , y)
(12)
Where, ak and k denotes the aberration order and Zernike coefficient of each mode, respectively. Zk represents the aberration function sequence of Zernike polynomial. In the experiment, Zernike coefficients can be obtained by a CCD sampling of beam spot lattice with
Fig. 4. Distribution of mode wavefront characteristics. (a) original image of intensity spot lattice with 24 × 24 arrays detected by HSe method; (b) profiles of beam aberration achieved by model wavefront reconstruction. In the case, PV = 0.969λ and RMS = 0.224λ, and the result is the average of 50 frames sampling images; (c) 2D mode properties with 30 mm defocus of the cavity length; (d) Coupling out mode distribution near the OC mirror. 5
Optik - International Journal for Light and Electron Optics 202 (2020) 163625
X. Zhang, et al.
Fig. 5. Result of different aberration weight. Zernike polynomial orders vs. aberration coefficients achieved by wavefront fitting.
24 × 24 arrays and further directslope algorithms. The expressions of typical Zernike polynomials for the first eleven order function are shown in Table 1. 3.3. Experimental results and discussion In the experiment layout, we employ a probing He-Ne laser into the folded confocal unstable resonator to collimate it . Such a laser is used as a beacon light, and a passive cavity structure is adopted. So the influence of many other intracavity disturbance can be
Fig. 6. Comparison of beam mode PSF profiles. (a) the cavity is adjusted to the ideal design condition including cavity length and alignment; (b) the case of a 30 mm defocus change in the cavity length. 6
Optik - International Journal for Light and Electron Optics 202 (2020) 163625
X. Zhang, et al.
Fig. 7. Characteristics of the circle energy as a function of the diffraction limit multiple in the far-field. (a) the case of 10 mm misalignment of the cavity length; (b) the case when the defocus variation of cavity length is 30 mm.
ruled out in the process of a practical laser output, so the typical defocus aberration can be finely analyzed. The actual oscillation wavelength of the resonator is 1315 nm. However, in the case of low gain, the mode properties of active cavity can be better reflected by the case of passive cavity. The measured intensity lattice of the beam spot can be processed to obtain phase profiles in the full aperture by Zernike polynomials method. The reconstructed wavefront results are illuminated in Fig. 4. Plane wavefront distribution of eigenmode can be obtained near the OC mirror according to laser mode principle in such a confocal cavity. In the arrangement, we adopt a standard collimation light source to calibrate Hartmann-Shack sensor before wavefront analysis. In addition, He-Ne laser is used as a reference probing light to align the resonator. From the calculation, the phase profile shown in Fig. 4(b) still contains obvious distortions even if a relatively small 30 mm change of the resonator length. The value of PV (peak-valley) for the average of 100 frame images arrived to 0.969 wavelength compared to the standard planewave. The RMS (root mean square) of the mode wavefront is 0.224λ correspondingly in the defocus condition, where wavelength is 633 nm, as is shown in Fig. 4(c). In addition, a real-time analysis in wavefront properties is obtained by programming optimization, and the speed of wavefront reconstruction is six frames per second. Fig. 5 compares the confocal unstable cavity model in defocus state with experimental results, where the aberration coefficient is plotted against Zernike polynomial orders of the beam incident on the BS. The experiment proves that the A3 coefficient (defocus aberration) and A10 (spherical aberration) are the significant components in the out-coupling beam distortion, which arrive to -0.175 and 0.084 respectively. In addition, the two coefficients A4 and A5, where represent X and Y directional astigmatism are 0.042 and -0.063 respectively in the defocus state. Fig. 6 presents the variation of the PSF distribution as a function of the cavity length change. Fig. 6(a) shows PSF is close to the ideal diffraction distribution in Fraunhofer far-field. In the case, the resonator is completely adjusted to alignment without defocus perturbation. The results are the average value of sampled images for 50 frames. However, the distribution of PSF deviates obviously from the ideal diffractive profile in the far field in Fig. 6(b), showing obvious ripple shape at a 30 mm change of the cavity length. It illustrates typical intracavity defocus aberration is included in the output beam. Corresponding beam quality parameter, namely, the diffraction factor is calculated. The smaller is the factor, the result is better. Fig. 7 shows the variation of circle energy as a function of the diffraction limit of the output beam. It appears in the latter case that the defocus aberration has a significant effect on the beam quality, where the red curve (marked with the cross sign) represents the case of the diffraction of an ideal plane wave without aberration in the far-field. It illustrates that the diffraction limit factor has reduced by 1.6 times from 1.61 to 2.57 at the position of 0.84 circle energy, when the defocus degree of cavity length changes from 10 mm to 30 mm in the case of the same coaxial state. Hence there are situations in which the measurement of real-time and accurate 7
Optik - International Journal for Light and Electron Optics 202 (2020) 163625
X. Zhang, et al.
beam aberration can provide many benefits for further precise wavefront correction. In fact, the change of cavity length and the thermal effect of gain medium will dynamically affect beam mode characteristics of such lasers. 4. Conclusions In summary, we present an analytical treatment of typical defocus aberration and its impact on the mode wavefront properties in an unstable resonator with typical confocal structure. Numerical calculation modeling of the system is found to predict the influence of defocus disturbation on mode intensity and phase distribution. In addition, an experimental method by combining Zernike aberration function and wavefront reconstruction algorithms to measure the dynamic wavefront aberration for such lasers has been developed. Beam characteristics, including the alignment state and defocus aberration perturbation have been presented. It is shown that the typical Zernike aberration caused by intracavity defocus significantly affected the beam properties, PSF profiles and circle energy distribution. The experiment proves that such structural resonator has a higher misalignment sensitivity caused by typical aberration perturbation, e.g. the change of equivalent cavity length. An accurate real-time measurement of wavefront aberration will provide effective information for valuable predicting the working efficiency of lasers. The result can be used as a useful reference for researching the optimum design of such resonators and further improving beam quality by dynamic compensation of typical aberration in laser unstable cavities. Acknowledgments This work was supported by the Postdoctoral Research Foundation of China Grants 2014M562294; and partial supported by the National Natural Science Fund of China Grants 61275039. The authors would like to thank Prof. L.K. Su for helpful discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
O. Svelto, Principles of Lasers, 5th ed., World Book Publishing Press, Beijing, 2014. W. Koechner, Solid-State Laser Engineering, 4th ed., Springer-Verlag, Berlin, 1996. D.Y. Shen, D.Y. Fan, Mid-infrared Lasers and Its Application, National Defense Industry Press, Beijing, 2015. C. Pargmann, T. Hall, F. Duschek, K.M. Grünewald, J. Handke, COIL emission of a modified negative branch confocal unstable resonator, Appl. Opt. 46 (2007) 7751–7756. N. Hodgson, H. Weber, Influence of spherical aberration of the active medium on the performance of Nd:YAG lasers, IEEE J. Quantum Electron. 29 (1993) 2497–2507. D. Wen, D. Li, J. Zhao, Generalized sensitivity factors for optical-axis perturbation in nonplanar ring resonators, Opt. Express 19 (2011) 19752–19757. Y.J. Cheng, C.G. Fanning, A.E. Siegman, Transverse-mode astigmatism in a diode-pumped unstable resonator Nd:YVO4 laser, Appl. Opt. 47 (2008) 1130–1134. B.L. Pan, J. Yang, Y.J. Wang, M.L. Li, Thermal effects in high-power double diode-end-pumped Cs vapor lasers, IEEE J. Quantum Electron. 48 (2012) 485–489. X. Zhang, B. Xu, W. Yang, Influence on outcoupled mode by introducing intracavity mirror tilt perturbation, Chin. J. Lasers 33 (2006) 303–310. Q. Zhuang, F.T. Sang, D.Z. Zhou, Shortwave Chemical Laser, National Defence Industry Press, Beijing, 1997. D.L. Yu, F.T. Sang, Y.Q. Jin, Y.Z. Sun, Output beam drift and deformation in high-power chemical oxygen iodine laser, Opt. Laser Technol. 35 (2003) 245–249. K.E. Oughstun, Intracavity adaptive optic compensation of phase aberrations. II: passive cavity study for a small Neq resonator, J. Opt. Soc. Am. 71 (1981) 1180–1192. R. Bauer, A. Paterson, C. Clark, D. Uttamchandani, W. Lubeigt, Output characteristics of Q-switched solid-state lasers using intracavity MEMS micromirrors, IEEE J. Sel. Top. Quantum Electron. 21 (2015) 908–916. D.B. Rensch, A.N. Chester, Iterative diffraction calculations of transverse mode distributions in confocal unstable laser resonators, Appl. Opt. 12 (1973) 997–1010. A.E. Siegman, H.Y. Miller, Unstable optical resonator loss calculations using the prony method, Appl. Opt. 9 (1970) 2729–2735. X. Zhang, C.H. Wang, H. Xian, J. Liu, Wavefront measurement of laser beam traversing a supersonic flow by using H–S wavefront sensor, High Power Laser Part. Beams 17 (2005) 1003–1007. W.H. Southwell, Wave-front estimation from wave-front slope measurements, J. Opt. Soc. Am. 70 (1980) 998–1006. A.V. Goncharov, M. Nowakowski, M.T. Sheehan, C. Dainty, Reconstruction of the optical system of the human eye with reverse ray-tracing, Opt. Express 16 (2008) 1692–1703.
8
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Analysis of defocus aberration characteristics on a typical passively confocal unstable resonator
T
⁎
Xiang Zhanga, Zhengrong Denga, , Zhiqiu Hua, Xiaoxia Zhangb a Key Laboratory of Solid-state Laser and Applied Techniques, School of Photoelectric Technology, Chengdu University of Information Technology, Chengdu 610225, China b School of Optoelectronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Laser resonator Optical aberration Wavefront properties Mode reconstruction
The analytical treatment of typical defocus aberration and its impact on the mode wavefront properties of a confocal structural laser resonator is discussed. Numerical modeling of the system is deduced to predict the influence of defocus disturbation on mode intensity and phase distribution. In addition, an experimental method by combining Zernike aberration function and wavefront reconstruction to measure the dynamic wavefront aberration for such lasers has been developed. Beam mode characteristics, including the alignment state and defocus aberration perturbation have been presented. It is shown that the typical Zernike aberration such as A3, A5 and A10 caused by intracavity perturbation significantly affected the beam properties, PSF profiles and circle energy distribution in the far field. The experimental results show that defocus aberration characteristics can be used as an effective basis for dynamic compensation of typical aberration in such lasers. Furthermore, corresponding strategies are proposed to implement the active correction of typical optical aberration in such resonators.
1. Introduction Unstable resonators of such a confocal structure operating on the fundamental TEM00 mode produce, as known, can obtain high quality diffraction limited beams [1–3]. However, due to the influence of ambient vibration, laser oscillators with the structure of virtual confocal unstable cavity are easy to suffer from the tilt misalignment of cavity mirrors to bring much aberration [4–6]. In addition, thermal accumulation in lasers will lead to unconfocal condition for such resonators and produce a lot of beam distortion. Phase-tilt and defocus are the main aberrations among these factors, which have significant influence on the output power, mode wavefront, and long-distance transmission of beam. It will also seriously limit the further application of such lasers [7–9]. Relevant studies have shown that some relevant optical aberrations have obvious effect on the beam quality of lasers. With the increase of output power, the defocus aberration will also increase obviously. In addition, the defocus is still the main wavefront aberration even after being corrected by the phase compensating system in the case of a representative cavity geometric perturbation. The reason is that typical high-energy lasers, e.g. the chemical laser with positive confocal unstable cavity works usually in a high frequency vibration and fast speed air flow condition [10,11]. Thus the cavity mirror is easily perturbed and further deformed by strong heat radiation, and the main beam distortion is lower order aberration. The change of cavity length will cause defocus aberration, meanwhile, the thermal deformation of cavity mirrors will also result in significant mode aberration of laser beam. The intracavity compensation for defocus aberration has intrinsic advantages; it will not only compensate a wide dynamic range
⁎
Corresponding author. E-mail address: [email protected] (Z. Deng).
https://doi.org/10.1016/j.ijleo.2019.163625 Received 29 July 2019; Accepted 13 October 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 202 (2020) 163625
X. Zhang, et al.
Fig. 1. Schematic diagram of the typical unstable resonator with a virtual confocal form.
for optical aberration, but also can improve the mode properties and beam quality of lasers [12,13]. It has currently become one of the important research topics to seek dynamic compensation ways for defocus aberration inside the cavities. However, modeling the typical aberration properties and further experimental analysis is rather limited. 2. Theoretical model 2.1. Defocus aberration caused by variation of cavity length The model of the typical unstable cavity with a virtual confocal structure is shown in Fig. 1. The focal length of the cavity mirror M1 and M1 is f1 and f2, respectively. The reflector M1 and M2 have a common virtual focal point F outside the cavity. According to geometrical optics principles, it is a zero focal power system in the ideal case, however, the focal length will be rewritten as a ΔL change of the cavity length
f = f1 f2 / ΔL
(1)
By discussing the change of beam focus caused by the change of cavity length, the relationship between the focus position and the defocus aberration of beam can be further illustrated. A very narrowly beam can be introduced into the cavity to simulate the variation in the equivalent focal length of the out-coupling beam if the resonator length is changed by using the ABCD transmission matrix. Suppose the curvature radius of the two mirrors are R1 and R2, respectively. Cavity length of the resonator is expressed as L = (R1 + R2)/2 + ΔL, where ΔL denotes the change in the cavity length, which is negative when it is increased. The original thin beam is propagated back and forth in the cavity and the beam diameter will be continuously expanded. Finally, the stable eigenmode will be coupled out the resonator. According to the theory of ABCD transfer matrix for resonators
1 0 ⎤ ⎡ 1 (R1 + R2)/2 + ΔL ⎤ ⎡ 1 0⎤ ⎡ A B ⎤ = ⎡ 1 (R1 + R2)/2 + ΔL ⎤ ⎡ 1 1 ⎣ C D⎦ ⎣0 ⎦ ⎣− 2/ R1 1 ⎦ ⎣ 0 ⎦ ⎣− 2/ R2 1 ⎦ (4ΔL2 − R12 + 2R2 ΔL)/ R1 R2 [R12 − (R2 + 2ΔL)2]/2R1⎤ =⎡ ⎢ ⎥ 4ΔL/(R1 R2) (−R2 + 2ΔL)/ R1 ⎣ ⎦
(2)
The corresponding ΔL = 0 when the cavity length is -aligned, and Eq. (2) is simplified as
− R1/ R2 (R12 − R22)/2R1⎤ ⎡A B⎤ = ⎡ − R2 / R1 ⎥ ⎣C D⎦ ⎢ ⎦ ⎣ 0
(3)
at θ0 = 0 for a thin plane wavefront, the transfer matrix can be written as
x x ⎡ 1 ⎤ = ⎡ A B ⎤ ⎡ 0 ⎤ = ⎡− (R1/ R2) x 0 ⎤ θ θ C D ⎦ ⎣ 0 ⎦ ⎣0 ⎣ 1⎦ ⎣ ⎦
(4)
In fact, when the cavity length is aligned, the beam diameter for each round trip in the cavity will be magnified by M times, and the paraxial thin beam will propagate more times inside the cavity. Corresponding parameters to the beam can be obtained. The relationship between the beam radius ω0 and transmission times is
n = [ln(r2/ ω0)]/ln M
(5)
When, ΔL ≠ 0 , the light beam propagates for the first time in the cavity 2 x x ⎡ 1 ⎤ = ⎡ A B ⎤ ⎡ 0 ⎤ = ⎡ 4ΔL / R1 R2 + 2ΔL/ R1 + M ⎤ x 0 ⎥ 4 / R1 R2 ΔL ⎣ θ1⎦ ⎣ C D ⎦ ⎣ θ0 ⎦ ⎢ ⎣ ⎦
(6)
There is no light coupling out the resonator in the case, and the ray extension line will converge on the z axis. The position of the convergence light point is given by 2
Optik - International Journal for Light and Electron Optics 202 (2020) 163625
X. Zhang, et al.
Xf = −x1/ θ1 ≈ −R12 /(4ΔL)
(7)
where ΔL ≪ |Ri|, (i = 1, 2), we can see that Xf is independent of x0 in Eq. (7), it does mean that the paraxial beam will converge at one point on the axis after the first round trip in the resonator. The beam arrives at mirror M2 for the second round trip, the equation is written by
x x m2 + F (ΔL/ R) ⎤x ⎡ 2 ⎤ = ⎡ A B ⎤⎡ A B ⎤⎡ 0⎤ = ⎡ 2 2⎥ 0 3 2 ⎣ θ2 ⎦ ⎣ C D ⎦ ⎣ C D ⎦ ⎣ θ0 ⎦ ⎢ ⎣16ΔL /(R1 R2) − 4ΔL/ R1 − 4ΔL/ R2 ⎦
(8)
The position of convergence point is given by
Xf ≈ −R12 /(4ΔLm2 + 4ΔL)
(9)
It can be seen from Eq. (9) that Xf (output beam focal length) is inversely proportional to the parameter ΔL (which is approximately linear to the Zernike defocus aberration A3). It can be deduced that the relationship still holds even after the beam propagates several times in the cavity. Thus the parameter Xf -satisfies inversely proportional relation to Zernike defocus aberration. 2.2. Defocus caused by thermal effect The curvature radius of cavity mirrors will change due to the thermal deformation in laser irradiation, which will cause typical defocus aberration. The influence of the aberration on mode features is obvious especially in high energy lasers. We can simplify the problem of thermal effect to the variation of the equivalent vector height of mirrors. In fact, the focus change of the output beam caused by thermal distortion of cavity mirrors is equivalent to the result caused by the change of the cavity length. The results prove that the effect of thermal deformation can be fitted by the mathematical function given by the equation in the case of uniform heat radiation [10]
Δ = k (x 2 − R2)2
(10)
To ensure the beam quality of mode, the misalignment of cavity length and the vector height of cavity mirrors shall be controlled within a certain range, in which the length misalignment will affect the stability of cavity structures, and the change of vector height is related to the thermal condition inside lasers and the material of mirrors. Suppose the x coordinate is perpendicular to the optical axis. In addition, the function of -self-reproducing mode is described by Fresnel diffraction integral, e.g. for the optical cavity with aligned mirrors, the equation can be expressed by
U (x , y ) = γ
ik 4π
−ikρ
∬ U (x′, y′) e ρ
(1 − cos θ) ds′
(11)
Where, γ is the eigenvalue which shows the mode diffractive loss in the cavity, U (x , y ) and U (x ′, y′) represents the mode properties function on the two mirrors, respectively. Parameter ρ is the vector length between the two points on the two opposing mirrors, which can be further expressed in the form of relatively complex rectangular coordinates. The equation can be numerically solved by the appropriate algorithm [14,15]. The cavity mode profiles can be accurately obtained in the case of mirror thermal deformation and defocus condition. 2.3. Calculation results In the numerical calculation, the characters of initial field in the cavity can be assumed to be an arbitrary distribution according to the self-reproducing theory of laser mode. The geometric length of the cavity is 0.5 m, and the radius of curvature is 10 m for the concave reflector. The diameter of reflectors, geometric structural factors g1 and g2 are 1 cm, 1.0 and 0.95, respectively. In Fig. 2, mode characteristics with different extent of cavity reflector deformation are revealed. From the result, it illustrates that beam mode presents an ideal distribution of Gaussian function in such a resonator. However, the profiles of eigenmode will deviate obviously from the distribution of ideal case with mirror thermal distortion. For example, when the vector height of each cavity reflector has a change of 1/15 wavelength, there are some extreme points in the relative amplitude distribution, which obviously departs from the ideal case of fundamental transverse mode. It proves that the height deformation the mirrors should be controlled within the range of 1/20 wavelength as far as possible in order to avoid the eigenmode distortion of the beam. If the vector height of the mirror exceeds 1/10 wavelength, the stable self-reproducing mode can not be formed in the resonator. 3. Experimental setup 3.1. Experimental schematic and design Our approach constructs an unstable resonator with a special virtual confocal structure, collimated by a He-Ne laser, which is shown in Fig. 3. Such a structure of the folded resonator is usually used in high energy lasers . The total resonator cavity is mainly composed of two folded planar mirrors, an intracavity concave and convex reflector, and the sharp beam scraper used as an OC 3
Optik - International Journal for Light and Electron Optics 202 (2020) 163625
X. Zhang, et al.
Fig. 2. Beam mode profiles in the typical passive cavity. (a) normalized amplitude profiles of transverse mode as a function of mirror thermal deformation; (b) phase distribution in different cases.
Fig. 3. Schematic diagrams of analyzing the influence of intracavity defocus aberration on beam characteristics.
mirror to output beam. The OC mirror is inclined to an angle of 45 degrees from the cavity optical axis. Such structure is a typical standing wave cavity. He-Ne probing laser is adopted to align the resonator. In addition, it is available to a reference light source in the H–S wavefront reconstruction. The unstable cavity is constructed by the convex and concave mirror with a virtual confocal way. Curvature radius of the cavity mirror is 21 m and 7.5 m respectively, thus the resonator length is (R2 − R1)/2 = 6.75m .
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Table 1 Expressions of typical Zernike polynomials and the aberration name. No.
Name
Expressions
0 1 2 3
Piston Tilt X Tilt Y Defocus
1 2ρ cos θ 2ρ sin θ
4
Astigmatism X
6 ρ2 cos 2θ
5
Astigmatism Y
6 ρ2 sin 2θ
6
Coma X
8 (3ρ3 − 2ρ2 )cos θ
7
Coma Y
8 (3ρ3 − 2ρ2 )sin θ
8
Trefoil X
8 ρ3 cos 3θ
9
Trefoil Y
8 ρ3 sin 3θ
10
Spherical
5 (6ρ4 − 6ρ2 + 1)
3 (2ρ2 − 1)
3.2. Wavefront described by Zernike polynomials Zernike polynomials can be used to characterize wavefront aberration of laser beam [16–18]. It is proved that Zernike polynomial is orthogonal in the circular domain, so that such polynomials is very convenient to fit the aberrated mode wavefront. In addition, each order Zernike aberration be expressed independently from the original distorted wavefront, so that the correction for the aberrated beam is facilitated by using appropriate adaptive optical device. The wavefront distortion can be described by a series of sum forms of Zernike polynomials, which is written as
φ (x , y ) =
∞
∑k =1 ak Zk (x , y)
(12)
Where, ak and k denotes the aberration order and Zernike coefficient of each mode, respectively. Zk represents the aberration function sequence of Zernike polynomial. In the experiment, Zernike coefficients can be obtained by a CCD sampling of beam spot lattice with
Fig. 4. Distribution of mode wavefront characteristics. (a) original image of intensity spot lattice with 24 × 24 arrays detected by HSe method; (b) profiles of beam aberration achieved by model wavefront reconstruction. In the case, PV = 0.969λ and RMS = 0.224λ, and the result is the average of 50 frames sampling images; (c) 2D mode properties with 30 mm defocus of the cavity length; (d) Coupling out mode distribution near the OC mirror. 5
Optik - International Journal for Light and Electron Optics 202 (2020) 163625
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Fig. 5. Result of different aberration weight. Zernike polynomial orders vs. aberration coefficients achieved by wavefront fitting.
24 × 24 arrays and further directslope algorithms. The expressions of typical Zernike polynomials for the first eleven order function are shown in Table 1. 3.3. Experimental results and discussion In the experiment layout, we employ a probing He-Ne laser into the folded confocal unstable resonator to collimate it . Such a laser is used as a beacon light, and a passive cavity structure is adopted. So the influence of many other intracavity disturbance can be
Fig. 6. Comparison of beam mode PSF profiles. (a) the cavity is adjusted to the ideal design condition including cavity length and alignment; (b) the case of a 30 mm defocus change in the cavity length. 6
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Fig. 7. Characteristics of the circle energy as a function of the diffraction limit multiple in the far-field. (a) the case of 10 mm misalignment of the cavity length; (b) the case when the defocus variation of cavity length is 30 mm.
ruled out in the process of a practical laser output, so the typical defocus aberration can be finely analyzed. The actual oscillation wavelength of the resonator is 1315 nm. However, in the case of low gain, the mode properties of active cavity can be better reflected by the case of passive cavity. The measured intensity lattice of the beam spot can be processed to obtain phase profiles in the full aperture by Zernike polynomials method. The reconstructed wavefront results are illuminated in Fig. 4. Plane wavefront distribution of eigenmode can be obtained near the OC mirror according to laser mode principle in such a confocal cavity. In the arrangement, we adopt a standard collimation light source to calibrate Hartmann-Shack sensor before wavefront analysis. In addition, He-Ne laser is used as a reference probing light to align the resonator. From the calculation, the phase profile shown in Fig. 4(b) still contains obvious distortions even if a relatively small 30 mm change of the resonator length. The value of PV (peak-valley) for the average of 100 frame images arrived to 0.969 wavelength compared to the standard planewave. The RMS (root mean square) of the mode wavefront is 0.224λ correspondingly in the defocus condition, where wavelength is 633 nm, as is shown in Fig. 4(c). In addition, a real-time analysis in wavefront properties is obtained by programming optimization, and the speed of wavefront reconstruction is six frames per second. Fig. 5 compares the confocal unstable cavity model in defocus state with experimental results, where the aberration coefficient is plotted against Zernike polynomial orders of the beam incident on the BS. The experiment proves that the A3 coefficient (defocus aberration) and A10 (spherical aberration) are the significant components in the out-coupling beam distortion, which arrive to -0.175 and 0.084 respectively. In addition, the two coefficients A4 and A5, where represent X and Y directional astigmatism are 0.042 and -0.063 respectively in the defocus state. Fig. 6 presents the variation of the PSF distribution as a function of the cavity length change. Fig. 6(a) shows PSF is close to the ideal diffraction distribution in Fraunhofer far-field. In the case, the resonator is completely adjusted to alignment without defocus perturbation. The results are the average value of sampled images for 50 frames. However, the distribution of PSF deviates obviously from the ideal diffractive profile in the far field in Fig. 6(b), showing obvious ripple shape at a 30 mm change of the cavity length. It illustrates typical intracavity defocus aberration is included in the output beam. Corresponding beam quality parameter, namely, the diffraction factor is calculated. The smaller is the factor, the result is better. Fig. 7 shows the variation of circle energy as a function of the diffraction limit of the output beam. It appears in the latter case that the defocus aberration has a significant effect on the beam quality, where the red curve (marked with the cross sign) represents the case of the diffraction of an ideal plane wave without aberration in the far-field. It illustrates that the diffraction limit factor has reduced by 1.6 times from 1.61 to 2.57 at the position of 0.84 circle energy, when the defocus degree of cavity length changes from 10 mm to 30 mm in the case of the same coaxial state. Hence there are situations in which the measurement of real-time and accurate 7
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beam aberration can provide many benefits for further precise wavefront correction. In fact, the change of cavity length and the thermal effect of gain medium will dynamically affect beam mode characteristics of such lasers. 4. Conclusions In summary, we present an analytical treatment of typical defocus aberration and its impact on the mode wavefront properties in an unstable resonator with typical confocal structure. Numerical calculation modeling of the system is found to predict the influence of defocus disturbation on mode intensity and phase distribution. In addition, an experimental method by combining Zernike aberration function and wavefront reconstruction algorithms to measure the dynamic wavefront aberration for such lasers has been developed. Beam characteristics, including the alignment state and defocus aberration perturbation have been presented. It is shown that the typical Zernike aberration caused by intracavity defocus significantly affected the beam properties, PSF profiles and circle energy distribution. The experiment proves that such structural resonator has a higher misalignment sensitivity caused by typical aberration perturbation, e.g. the change of equivalent cavity length. An accurate real-time measurement of wavefront aberration will provide effective information for valuable predicting the working efficiency of lasers. The result can be used as a useful reference for researching the optimum design of such resonators and further improving beam quality by dynamic compensation of typical aberration in laser unstable cavities. Acknowledgments This work was supported by the Postdoctoral Research Foundation of China Grants 2014M562294; and partial supported by the National Natural Science Fund of China Grants 61275039. The authors would like to thank Prof. L.K. Su for helpful discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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