Performance of a simplified slit spatial filter for large laser systems Han Xiong, Xiao Yuan,* Xiang Zhang, and Kuaisheng Zou Institute of Modern Optical Technologies, Key Lab of Advanced Optical Manufacturing Technologies, Soochow University, Suzhou, Jiangsu, 215006, China * [email protected]
Abstract: A new-type slit spatial filter system with three lenses was proposed, in which the focal spot was turned into focal line by adding cylindrical lenses to increase focal area and then lower the focal intensity. Its performances on image relay, aperture matching and spatial filtering are comprehended by detailed theoretical calculations and numerical simulation. According to transmission spatial filter in national ignition facility, we present a replaceable slit spatial filter, which can reduce the overall length of laser system, improve the beam quality and suppress or even avoid the pinhole (slit) closure in the spatial filter. ©2014 Optical Society of America OCIS codes: (070.6110) Spatial filtering; (110.4850) Optical transfer functions; (120.4570) Optical design of instruments.
References and links 1.
N. N. Akhmediev, V. I. Korneev, and R. F. Nabiev, “Modulation instability of the ground state of the nonlinear wave equation: optical machine gun,” Opt. Lett. 17(6), 393–395 (1992). 2. J. E. Murray, D. Milam, C. D. Boley, K. G. Estabrook, and J. A. Caird, “Spatial filter pinhole development for the National Ignition Facility,” Appl. Opt. 39(9), 1405–1420 (2000). 3. J. E. Murray, D. Milam, C. D. Boley, K. G. Estabrook, and F. Bonneau, “Spatial filter issues,” Proc. SPIE 3492, 496–503 (1999). 4. J. E. Murray, K. G. Estabrook, D. Milam, W. D. Sell, B. M. Van Wonterghem, M. D. Feit, and A. M. Rubenchik, “Spatial filter issues,” Proc. SPIE 3047, 207–212 (1996). 5. S. Sato, and H. Ashida, “Vacuum-cored hollow waveguide for transmission of high-energy, nanosecond Nd:YAG laser pulses and its application to biological tissue ablation,” Opt. Lett. 25(1), 49–51 (2000). 6. W. Yu, G. Wang, S. Wang, and Y. Li, “Diagnosis and compensation of wavefront distortion of laser beams,” Chin. J. Lasers 10(4), 220–224 (1983). 7. A. C. Erlandson, “Spatial filter for high average power lasers,” US Patent: 8,320,056, 1–15 (2012). 8. A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. A 18(8), 1897–1904 (2001). 9. J. Chen, “Propagation and transformation of flat-topped multi-Gaussian beams in a general nonsymmetrical apertured double-lens system,” J. Opt. Soc. Am. A 24(1), 84–92 (2007). 10. Y. Gao, B. Zhu, D. Liu, and Z. Lin, “Propagation of flat-topped multi-Gaussian beams through a double-lens system with apertures,” Opt. Express 17(15), 12753–12766 (2009). 11. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chaps. 3 and 5.
1. Introduction Since the existence of spatial imperfections, such as dust, scratches and impurities in the optical component and optical paths, intense laser beam will inevitably encounter modulation of spatial frequencies. These modulations will grow in the laser systems and lead to fast nonlinear growth in the beam as well as small-scale self-focusing [1], degrading the laser beam quality and resulting in damages in the laser medium and optical components, which is considered one of the main limitations in designing and building high-power laser systems, especially in inertial confinement fusion lasers that have higher intensities. Traditional spatial filter (SF) consists of two convex lenses and a pinhole placed at their common focal plane. The pinhole is used to improve output beam quality by clearing off the middle and high spatial frequencies. However, the intensity in the focal spot in large laser system is so powerful that efficient cutoff radius of pinhole is always irradiated by intense laser and then results in pinhole closure [2]. Besides, high vacuum environment in SF system [3] is required
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22211
to avoid air breakdown caused by high intensity [4,5]. Therefore, it is of great significance for large laser system development if the intensity of focal spot in SF could be significantly dropped down and keeps the efficient filtering effects. Fortunately, Slit SF is able to meet these requirements, which can be obtained by adding cylindrical lenses in SF. But the study on slit SF is still rare so far. In 1983, W. Yu, G. Wang, S. Wang and Y. Li [6] in Shanghai Institute of Optics and Fine Mechanics used double slits to filter holographic image of astigmatic beam. But the relevant characteristics of the slits about spatial filtering were not mentioned, and high power laser was not involved either. In 2012, A. C. Erlandson at the Lawrence Livermore National Laboratory applied for a patent [7] about slit SF, which used four cylindrical lenses to replace the original spherical lenses in the pinhole SF. This method turns the original focal spot into a focal line to enlarge the focal area, which is beneficial in solving of pinhole closure since the lowering of focal intensity. However, this slit SF will raise the cost and make it more difficult to align and maintain due to the complex angular adjustment raised by cylindrical lenses and slit apertures. It will be helpful for the practical application of slit SF with fewer lenses in slit SF or using spherical lenses instead of cylindrical lenses as far as possible. In this article, we propose a new-type slit SF with the combination of two onedimensional cylindrical lenses and a spherical lens. The characteristics of this new-type filter on image relaying, aperture matching and spatial filtering are comprehended by both theoretical calculations and numerical simulation. Besides, we use this slit SF model to simulate national ignition facility (NIF) system, and better performances were shown compared to the original transmission spatial filter (TSF) in NIF system. 2. Simulation and the characteristics of slit SF The slit SF system includes two cylindrical lenses, a spherical lens and two slit apertures placed in front of and after the common focal planes, as shown in Fig. 1. In the slit SF, the original focal spot is transformed into two independent and orthogonal focal lines, as shown in Fig. 2. Focal intensity is significantly reduced with the larger focal areas, and the spatial frequencies aside the focal lines are filtered with the slits placed in the focal planes. Therefore, the pinhole closure in pinhole SF can be effectively eased.
Fig. 1. Schematic diagram of slit SF with the combination of two cylindrical lenses and a spherical lens.
Since the introduction of cylindrical lenses, the positions of the front (back) focal planes in x- and y-directions will get apart from each other, as shown in Fig. 3, where Y-FFP means the front focal plane in y-direction, and X-BFP means the back focal plane in x-direction. In order to make understand the image relaying function of slit SF and find out the front (back) focal plane of the filtering system, theoretical analysis will be adopted based on RayleighSommerfeld diffraction integral and lens transformation.
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22212
Fig. 2. Focal lines at the focal planes. (a)3-dimensional view and (b) vertical view of spatial spectrum in plane of slit-I. (c) 3-dimensional view and (d) vertical view of spatial spectrum in plane of slit-II.
Fig. 3. The marked positions of focal planes.
A super-Gaussian square beam expressed in the form of a similar multi-Gaussian function [8–10] is incident on the silt SF system. The incident plane is chosen before cylindrical lens-I by a distance of z1, and the output plane is chosen behind cylindrical lens-II by a distance of z2. Both the distances z1 and z2 are undefined constants. The incident beam used here is a multi-Gaussian function: R
U i ( x, y ) =
( x − mW ) 2 R ( y − nW ) 2 exp − W2 W2 n =− R
exp −
m =− R
R
R
exp(−m ) exp(−n 2
m =− R
2
,
(1)
)
n =− R
where R is the order of multi-Gaussian function, W is waist radius of each Gaussian component in multi-Gaussian function. According to Rayleigh-Sommerfeld diffraction integral and lens transformation theory [11], the output laser beam can be obtained as: U o ( x, y ) =
exp(ikS ) R
R
exp(−m ) exp(−n
m =− R
2
2
)
−ikW 2 N (2 z1 − ikW 2 ) + 2 z2 N
(2)
n =− R
ik ( x + NmW ) ik ( y + NnW ) × exp 2 exp 2 , 2 2 m =− R N (2 z1 − ikW ) + 2 z2 n =− R N (2 z1 − ikW ) + 2 z2 R
2
R
2
where S is the whole optical path of the system, and k is wave number. At the condition of N2·z1 + z2 = 0, Eq. (2) can be converted into:
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22213
( x N + mW ) 2 R ( y N + nW ) 2 − exp exp − W2 W2 m =− R n =− R , U o ( x, y ) = R R 2 2 N exp(−m ) exp(− n ) R
m =− R
(3)
n =− R
When magnification N is a unity, the form of Eq. (3) is similar to Eq. (1). Since the range of x and y are symmetrical with respect to the coordinate origin, it takes no effect on Eq. (3). If magnification N is not a unity, the amplitude item in Eq. (3) will decrease by N times (intensity will decrease by N 2 times) in the situation of beam expansion of N times, and the distribution of laser beam will increase by N times in both x- and y-directions (beam area will increase by N2 times), which coincides with the principle of optical transmission that the product of intensity and beam area is constant. So it meets the requirements of image relaying and aperture matching. The condition N2·z1 + z2 = 0 can be further converted into z1 = z2 = 0, which implies the front and back focal planes of the system are just in the positions of cylindrical lens-I (marked with S-FFP in Fig. 3) and cylindrical lens-II (marked with S-BFP in Fig. 3), respectively. For conventional pinhole SF, the output beam can be similar to the input beam at the condition of:
z1 N 2 − z2 = ( N − 1) Nf ,
(4)
z1 N 2 − N 2 f = z2 − Nf ,
(5)
or in another form of:
where f is the focal length of the first lens in pinhole SF. It can be seen from Eq. (4) that z1 must be equal to z2 when N is a unity. Set the both sides of Eq. (5) to zero, and then the solutions can be obtained: z1 = f ,
(6)
z2 = Nf ,
(7)
Equations (6) and (7) show the positions of the front and back focal plane of pinhole SF, which is consistent with the image relay characteristics of the conventional pinhole SF.
Fig. 4. Simulation of image relaying. In situation of 1 time aperture matching, (a), (b) and (c) are the incident beam, the relayed beam with pinhole SF and the relayed beam with slit SF, respectively. In situation of 4 times aperture matching, (d), (e) and (f) are the incident beam, the relayed beam with pinhole SF and the relayed beam with slit SF, respectively.
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22214
We choose a 20th-order super-Gaussian square beam with the aperture of 20mm × 20mm to demonstrate the image relay process. As shown in Fig. 4(a), the incident beam is modulated by a crisscross with the modulation depth of 10% and the width of 1mm. The simulation shows that the cross images relayed by pinhole SF and slit SF are both complete and distortionless, as shown in Figs. 4(b) and 4(c), respectively. Near-field modulation and near-field contrast are used herein for evaluating the image relay effects of pinhole SF and slit SF. The near-field modulation is defined as the ratio of peak intensity to average intensity, and the near-field contrast is the ratio of root-mean-square value of intensity distribution to the average, as shown: C=
1 I avg
n
(I t =1
i
− I avg ) 2 / n .
(8)
where n is the number of sampling points, Ii is the intensity of point i, and Iavg is the averaged intensity over the beam. The difference between the two output beams in near-field modulation and near-field contrast is only 0.38% and 0.15%, respectively, which means that the image relaying function of the slit SF is equivalent to that of pinhole SF. In situation of 4 times aperture matching, it can be seen that the relayed cross images are still complete and distortionless, as shown in Figs. 4(e) and 4(f), but the difference between the two output beams in near-field modulation and near-field contrast becomes to 1.64% and 0, respectively. So the image relaying function can be still satisfied even in situation of multi-times aperture matching.
Fig. 5. Simulation of spatial filtering. (a) and (b) are the incident beam and its spatial spectrum, respectively. The spatial spectrum is 3-dimensional and shown in side view, and the insertion in (b) is the spatial spectrum shown in vertical view. (c) and (d) are the filtered beam with pinhole SF and its spatial spectrum, and (e) and (f) are the filtered beam with slit SF and its spatial spectrum, respectively.
To evaluate the filtering effects, the 20th-order super-Gaussian beam with the aperture of 20mm × 20mm is used to analyze the filtering characteristics of slit SF and pinhole SF. The
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22215
near-field distribution of the incident beam is shown in Fig. 5(a), and the corresponding spatial spectrum in side view and vertical view is shown in Fig. 5(b). The Focal length of lenses used in both SF systems are 1m, both the width of slit and diameter of pinhole are 0.73mm, and the corresponding cutoff frequency is of 0.35mm−1 or 8 times diffraction limitation (DL). The near-field distribution of the beams filtered with pinhole and slit SFs are shown in Figs. 5(c) and 5(e), respectively. Figures 5(d) and 5(f) are the spatial spectra of the beams filtered with pinhole and slit SFs, respectively. The results show that the frequencies beyond 0.35mm−1 of the beam output from slit SF are cleaned off, which is in accordance with that of the pinhole SF. As shown in Fig. 6, the spatial spectrum of the filtered beams with pinhole SF and slit SF at the beam quality of 8XDL (0.35mm-1), 10XDL (0.43mm-1) and 12XDL (0.52mm-1) is given. The corresponding near-field intensity distribution of the filtered beams with pinhole SF is displayed in Figs. 7(a)–7(c). Figures 7(d)–7(f) show the subtraction of the near-field distribution for the filtered beams of slit SF and pinhole SF. It can be seen that the maximum intensities in Figs. 7(d)–7(f) are lower than the output beam intensities from the spatial filters by three orders of magnitude, which means that the spatial filtering effect of slit SF is almost the same as that of the pinhole SF. For the beam quality of 8XDL, 10XDL and 12XDL, the differences between the two filtered beams in near-field modulation and near-field contrast are 0.31% and 0.69%, 0.02% and 1.39% and 0.05% and 0, respectively. According to the simulation for the beam quality ranged from 6XDL to 16XDL, the difference between the filtered beams from the two kinds of SFs in near-field contrast decreases basically with the cutoff frequency increasing. The preliminary explanation is that the effective filtering aperture of slit SF in spectral plane is a square region while pinhole SF’s circular. Spectrum always appears with strong center and weak edge, so that the larger the cutoff frequency is, the smaller the difference in filtering effect will be.
Fig. 6. The spatial spectra of the filtered beams. (a), (b) and (c) are the filtered spectra with pinhole SF at the beam quality of 8XDL, 10XDL and 12XDL, respectively. (d), (e) and (f) are the filtered spectra with slit SF at the beam quality of 8XDL, 10XDL and 12XDL, respectively.
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22216
Fig. 7. The near-field distribution of the filtered beams. (a), (b) and (c) are the filtered beams with pinhole SF at the beam quality of 8XDL, 10XDL and 12XDL, respectively. (d), (e) and (f) are the subtraction of the near-field distribution for the filtered beams of slit SF and pinhole SF at the beam quality of 8XDL, 10XDL and 12XDL, respectively.
3. Simulation for NIF system
For large laser systems such as NIF, laser beam has a very high power. In general, the laser is designed to have a large beam aperture due to the laser damage limitation. Moreover, the SF has a long focal length to overcome spherical aberration, etc. According to the practical parameters of TSF in NIF system, the incident beam is an 8th-order super-Gaussian square beam with beam area of 350mm × 350mm, pulse energy of 20kJ, pulse width of 3.5ns, wavelength of 1053nm and cutoff frequency being set at 39XDL (0.096mm−1). For convenience, the cylindrical lenses and spherical lens used in simulation are all spherically curved instead of the practical aspherically curved lenses. Figure 8(a) is the near field of the simulated incident beam, whose near-field modulation is 1.0684 and near-field contrast is 0.0420. Figure 8(b) is the corresponding PSD of the incident beam. For TSF in NIF system, the simulated peak intensity in the focus is 4.3 × 1016 W/cm2, which basically agrees with the 3.5 × 1016W/cm2 in [3]. The intensity on inner edge of pinhole is 4.8 × 1011 W/cm2. The beam filtered with TSF and its PSD are shown in Figs. 8(c) and 8(d), respectively. It can be seen that the beam profile becomes smoother after filtering and the spatial frequencies above 0.096mm−1 are cleared up. The near-field modulation and the near-field contrast are of 1.0205 and 0.0313, respectively. Figure 9 shows the dependence of the peak intensity (dashed line) and the intensities on the inner edge (solid line) of the slits with the focal length of lenses. It can be seen that the peak intensity reaches a maximum near 7 meter focal length because aberrations dominate at short focal lengths, and decreases with the focal length beyond 7 meters. The intensity on the edge of the slit is 6.1 × 108 W/cm2, which is far lower than that of 1010 W/cm2. The beam profile and its PSD filtered with 7-meter-long focal length slit SF are shown in Figs. 8(e) and 8(f), and the near-field modulation and near-filed contrast are 1.0212 and 0.0305, which is better than that of the TSF and can be used in the laser systems. In this way, the overall length of slit SF will be only 28 meters, which is much shorter than the 60-meter-long TSF in NIF system. Besides, the simulated peak intensity in slit SF is 6.3 × 1013 W/cm2, which is lowered by about three orders of magnitude than that in TSF, so that the vacuum degree in SF can be reduced by about one order of magnitude [3].
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22217
Fig. 8. Spatial filtering effect of pinhole and slit SFs. (a) and (b) are the incident beam and its PSD, respectively. (c) and (d) are the filtered beam with TSF in NIF system and its PSD, respectively. (e) and (f) are the filtered beam with slit SF when focal length is at 7 meters and its PSD, respectively.
Fig. 9. The dependence of peak intensity and the intensities on the inner edge of the slits with the focal length of lenses.
Since the output beam quality of slit SF is better than that of the TSF, the cutoff frequency of 39 XDL may not be required. Here we show that larger cutoff frequencies may work in large laser systems. As shown in Table 1, the near field parameters of the filtered beams at the beam quality of 40XDL (0.098mm−1), 41XDL (0.101mm−1), 42XDL (0.103mm−1), 43XDL (0.106mm−1), 44XDL (0.108mm−1) and 45XDL (0.111mm−1) are given. It can be seen that the near-field contrast is still better than that of TSF when the cutoff frequency is of 41XDL. Even at 42XDL, the difference between the two filtered beams with slit SF and TSF in nearfield modulation and near-field contrast is only 0.78% and 1.60%, respectively.
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22218
Table 1. The near-field parameters and corresponding intensities on inner edge of slits of the filtered beams at different cutoff frequencies.
Near-field modulation Near-field contrast Intensities on slits (W/cm2)
39XDL
40XDL
41XDL
42XDL
43XDL
44XDL
45XDL
1.0205
1.0209
1.0263
1.0285
1.0354
1.0368
1.0421
0.0313
0.0307
0.0314
0.0318
0.0327
0.0329
0.0334
6.05 × 108
5.62 × 108
6.03 × 108
5.59 × 108
4.56 × 108
4.62 × 108
4.99 × 108
4. Conclusions
In conclusion, we propose a new-type slit SF with the combination of two cylindrical lenses and a spherical lens. This slit SF has the simple structure and is beneficial to reducing the cost and easing the maintenance of slit SF. The characteristics of this slit SF on image relay, aperture matching and spatial filtering are discussed. In simulation, we used this new-type SF to replace the TSF in NIF system, and found that the focal intensity on the slim drops by about 3 orders of magnitude, which greatly improves the pinhole closure for present large laser systems and could reduce the vacuum degree in SF by about one order of magnitude. Besides, the overall length of SF can be optimized to less than half that of the original one since the focal length in this slit SF is greatly reduced, which is helpful in building and maintaining the large laser systems. Acknowledgments
This work is financially supported by the united foundation between NSFC and Chinese Academy of Engineering Physics under contract Nos. of 11176021 and 11076021, the Natural Science Foundation of China NSFC (91023009, 61108024, 61275140), the Natural Science Foundation of Jiangsu Higher education institutions (10KJA140045, BK2011278, 09KJB140008), the project of the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions, the Research and Innovation Project for College Graduates of Jiangsu Province (CXLX12_0797, CXZZ11_0095).
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22219
Abstract: A new-type slit spatial filter system with three lenses was proposed, in which the focal spot was turned into focal line by adding cylindrical lenses to increase focal area and then lower the focal intensity. Its performances on image relay, aperture matching and spatial filtering are comprehended by detailed theoretical calculations and numerical simulation. According to transmission spatial filter in national ignition facility, we present a replaceable slit spatial filter, which can reduce the overall length of laser system, improve the beam quality and suppress or even avoid the pinhole (slit) closure in the spatial filter. ©2014 Optical Society of America OCIS codes: (070.6110) Spatial filtering; (110.4850) Optical transfer functions; (120.4570) Optical design of instruments.
References and links 1.
N. N. Akhmediev, V. I. Korneev, and R. F. Nabiev, “Modulation instability of the ground state of the nonlinear wave equation: optical machine gun,” Opt. Lett. 17(6), 393–395 (1992). 2. J. E. Murray, D. Milam, C. D. Boley, K. G. Estabrook, and J. A. Caird, “Spatial filter pinhole development for the National Ignition Facility,” Appl. Opt. 39(9), 1405–1420 (2000). 3. J. E. Murray, D. Milam, C. D. Boley, K. G. Estabrook, and F. Bonneau, “Spatial filter issues,” Proc. SPIE 3492, 496–503 (1999). 4. J. E. Murray, K. G. Estabrook, D. Milam, W. D. Sell, B. M. Van Wonterghem, M. D. Feit, and A. M. Rubenchik, “Spatial filter issues,” Proc. SPIE 3047, 207–212 (1996). 5. S. Sato, and H. Ashida, “Vacuum-cored hollow waveguide for transmission of high-energy, nanosecond Nd:YAG laser pulses and its application to biological tissue ablation,” Opt. Lett. 25(1), 49–51 (2000). 6. W. Yu, G. Wang, S. Wang, and Y. Li, “Diagnosis and compensation of wavefront distortion of laser beams,” Chin. J. Lasers 10(4), 220–224 (1983). 7. A. C. Erlandson, “Spatial filter for high average power lasers,” US Patent: 8,320,056, 1–15 (2012). 8. A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. A 18(8), 1897–1904 (2001). 9. J. Chen, “Propagation and transformation of flat-topped multi-Gaussian beams in a general nonsymmetrical apertured double-lens system,” J. Opt. Soc. Am. A 24(1), 84–92 (2007). 10. Y. Gao, B. Zhu, D. Liu, and Z. Lin, “Propagation of flat-topped multi-Gaussian beams through a double-lens system with apertures,” Opt. Express 17(15), 12753–12766 (2009). 11. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chaps. 3 and 5.
1. Introduction Since the existence of spatial imperfections, such as dust, scratches and impurities in the optical component and optical paths, intense laser beam will inevitably encounter modulation of spatial frequencies. These modulations will grow in the laser systems and lead to fast nonlinear growth in the beam as well as small-scale self-focusing [1], degrading the laser beam quality and resulting in damages in the laser medium and optical components, which is considered one of the main limitations in designing and building high-power laser systems, especially in inertial confinement fusion lasers that have higher intensities. Traditional spatial filter (SF) consists of two convex lenses and a pinhole placed at their common focal plane. The pinhole is used to improve output beam quality by clearing off the middle and high spatial frequencies. However, the intensity in the focal spot in large laser system is so powerful that efficient cutoff radius of pinhole is always irradiated by intense laser and then results in pinhole closure [2]. Besides, high vacuum environment in SF system [3] is required
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22211
to avoid air breakdown caused by high intensity [4,5]. Therefore, it is of great significance for large laser system development if the intensity of focal spot in SF could be significantly dropped down and keeps the efficient filtering effects. Fortunately, Slit SF is able to meet these requirements, which can be obtained by adding cylindrical lenses in SF. But the study on slit SF is still rare so far. In 1983, W. Yu, G. Wang, S. Wang and Y. Li [6] in Shanghai Institute of Optics and Fine Mechanics used double slits to filter holographic image of astigmatic beam. But the relevant characteristics of the slits about spatial filtering were not mentioned, and high power laser was not involved either. In 2012, A. C. Erlandson at the Lawrence Livermore National Laboratory applied for a patent [7] about slit SF, which used four cylindrical lenses to replace the original spherical lenses in the pinhole SF. This method turns the original focal spot into a focal line to enlarge the focal area, which is beneficial in solving of pinhole closure since the lowering of focal intensity. However, this slit SF will raise the cost and make it more difficult to align and maintain due to the complex angular adjustment raised by cylindrical lenses and slit apertures. It will be helpful for the practical application of slit SF with fewer lenses in slit SF or using spherical lenses instead of cylindrical lenses as far as possible. In this article, we propose a new-type slit SF with the combination of two onedimensional cylindrical lenses and a spherical lens. The characteristics of this new-type filter on image relaying, aperture matching and spatial filtering are comprehended by both theoretical calculations and numerical simulation. Besides, we use this slit SF model to simulate national ignition facility (NIF) system, and better performances were shown compared to the original transmission spatial filter (TSF) in NIF system. 2. Simulation and the characteristics of slit SF The slit SF system includes two cylindrical lenses, a spherical lens and two slit apertures placed in front of and after the common focal planes, as shown in Fig. 1. In the slit SF, the original focal spot is transformed into two independent and orthogonal focal lines, as shown in Fig. 2. Focal intensity is significantly reduced with the larger focal areas, and the spatial frequencies aside the focal lines are filtered with the slits placed in the focal planes. Therefore, the pinhole closure in pinhole SF can be effectively eased.
Fig. 1. Schematic diagram of slit SF with the combination of two cylindrical lenses and a spherical lens.
Since the introduction of cylindrical lenses, the positions of the front (back) focal planes in x- and y-directions will get apart from each other, as shown in Fig. 3, where Y-FFP means the front focal plane in y-direction, and X-BFP means the back focal plane in x-direction. In order to make understand the image relaying function of slit SF and find out the front (back) focal plane of the filtering system, theoretical analysis will be adopted based on RayleighSommerfeld diffraction integral and lens transformation.
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22212
Fig. 2. Focal lines at the focal planes. (a)3-dimensional view and (b) vertical view of spatial spectrum in plane of slit-I. (c) 3-dimensional view and (d) vertical view of spatial spectrum in plane of slit-II.
Fig. 3. The marked positions of focal planes.
A super-Gaussian square beam expressed in the form of a similar multi-Gaussian function [8–10] is incident on the silt SF system. The incident plane is chosen before cylindrical lens-I by a distance of z1, and the output plane is chosen behind cylindrical lens-II by a distance of z2. Both the distances z1 and z2 are undefined constants. The incident beam used here is a multi-Gaussian function: R
U i ( x, y ) =
( x − mW ) 2 R ( y − nW ) 2 exp − W2 W2 n =− R
exp −
m =− R
R
R
exp(−m ) exp(−n 2
m =− R
2
,
(1)
)
n =− R
where R is the order of multi-Gaussian function, W is waist radius of each Gaussian component in multi-Gaussian function. According to Rayleigh-Sommerfeld diffraction integral and lens transformation theory [11], the output laser beam can be obtained as: U o ( x, y ) =
exp(ikS ) R
R
exp(−m ) exp(−n
m =− R
2
2
)
−ikW 2 N (2 z1 − ikW 2 ) + 2 z2 N
(2)
n =− R
ik ( x + NmW ) ik ( y + NnW ) × exp 2 exp 2 , 2 2 m =− R N (2 z1 − ikW ) + 2 z2 n =− R N (2 z1 − ikW ) + 2 z2 R
2
R
2
where S is the whole optical path of the system, and k is wave number. At the condition of N2·z1 + z2 = 0, Eq. (2) can be converted into:
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22213
( x N + mW ) 2 R ( y N + nW ) 2 − exp exp − W2 W2 m =− R n =− R , U o ( x, y ) = R R 2 2 N exp(−m ) exp(− n ) R
m =− R
(3)
n =− R
When magnification N is a unity, the form of Eq. (3) is similar to Eq. (1). Since the range of x and y are symmetrical with respect to the coordinate origin, it takes no effect on Eq. (3). If magnification N is not a unity, the amplitude item in Eq. (3) will decrease by N times (intensity will decrease by N 2 times) in the situation of beam expansion of N times, and the distribution of laser beam will increase by N times in both x- and y-directions (beam area will increase by N2 times), which coincides with the principle of optical transmission that the product of intensity and beam area is constant. So it meets the requirements of image relaying and aperture matching. The condition N2·z1 + z2 = 0 can be further converted into z1 = z2 = 0, which implies the front and back focal planes of the system are just in the positions of cylindrical lens-I (marked with S-FFP in Fig. 3) and cylindrical lens-II (marked with S-BFP in Fig. 3), respectively. For conventional pinhole SF, the output beam can be similar to the input beam at the condition of:
z1 N 2 − z2 = ( N − 1) Nf ,
(4)
z1 N 2 − N 2 f = z2 − Nf ,
(5)
or in another form of:
where f is the focal length of the first lens in pinhole SF. It can be seen from Eq. (4) that z1 must be equal to z2 when N is a unity. Set the both sides of Eq. (5) to zero, and then the solutions can be obtained: z1 = f ,
(6)
z2 = Nf ,
(7)
Equations (6) and (7) show the positions of the front and back focal plane of pinhole SF, which is consistent with the image relay characteristics of the conventional pinhole SF.
Fig. 4. Simulation of image relaying. In situation of 1 time aperture matching, (a), (b) and (c) are the incident beam, the relayed beam with pinhole SF and the relayed beam with slit SF, respectively. In situation of 4 times aperture matching, (d), (e) and (f) are the incident beam, the relayed beam with pinhole SF and the relayed beam with slit SF, respectively.
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22214
We choose a 20th-order super-Gaussian square beam with the aperture of 20mm × 20mm to demonstrate the image relay process. As shown in Fig. 4(a), the incident beam is modulated by a crisscross with the modulation depth of 10% and the width of 1mm. The simulation shows that the cross images relayed by pinhole SF and slit SF are both complete and distortionless, as shown in Figs. 4(b) and 4(c), respectively. Near-field modulation and near-field contrast are used herein for evaluating the image relay effects of pinhole SF and slit SF. The near-field modulation is defined as the ratio of peak intensity to average intensity, and the near-field contrast is the ratio of root-mean-square value of intensity distribution to the average, as shown: C=
1 I avg
n
(I t =1
i
− I avg ) 2 / n .
(8)
where n is the number of sampling points, Ii is the intensity of point i, and Iavg is the averaged intensity over the beam. The difference between the two output beams in near-field modulation and near-field contrast is only 0.38% and 0.15%, respectively, which means that the image relaying function of the slit SF is equivalent to that of pinhole SF. In situation of 4 times aperture matching, it can be seen that the relayed cross images are still complete and distortionless, as shown in Figs. 4(e) and 4(f), but the difference between the two output beams in near-field modulation and near-field contrast becomes to 1.64% and 0, respectively. So the image relaying function can be still satisfied even in situation of multi-times aperture matching.
Fig. 5. Simulation of spatial filtering. (a) and (b) are the incident beam and its spatial spectrum, respectively. The spatial spectrum is 3-dimensional and shown in side view, and the insertion in (b) is the spatial spectrum shown in vertical view. (c) and (d) are the filtered beam with pinhole SF and its spatial spectrum, and (e) and (f) are the filtered beam with slit SF and its spatial spectrum, respectively.
To evaluate the filtering effects, the 20th-order super-Gaussian beam with the aperture of 20mm × 20mm is used to analyze the filtering characteristics of slit SF and pinhole SF. The
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22215
near-field distribution of the incident beam is shown in Fig. 5(a), and the corresponding spatial spectrum in side view and vertical view is shown in Fig. 5(b). The Focal length of lenses used in both SF systems are 1m, both the width of slit and diameter of pinhole are 0.73mm, and the corresponding cutoff frequency is of 0.35mm−1 or 8 times diffraction limitation (DL). The near-field distribution of the beams filtered with pinhole and slit SFs are shown in Figs. 5(c) and 5(e), respectively. Figures 5(d) and 5(f) are the spatial spectra of the beams filtered with pinhole and slit SFs, respectively. The results show that the frequencies beyond 0.35mm−1 of the beam output from slit SF are cleaned off, which is in accordance with that of the pinhole SF. As shown in Fig. 6, the spatial spectrum of the filtered beams with pinhole SF and slit SF at the beam quality of 8XDL (0.35mm-1), 10XDL (0.43mm-1) and 12XDL (0.52mm-1) is given. The corresponding near-field intensity distribution of the filtered beams with pinhole SF is displayed in Figs. 7(a)–7(c). Figures 7(d)–7(f) show the subtraction of the near-field distribution for the filtered beams of slit SF and pinhole SF. It can be seen that the maximum intensities in Figs. 7(d)–7(f) are lower than the output beam intensities from the spatial filters by three orders of magnitude, which means that the spatial filtering effect of slit SF is almost the same as that of the pinhole SF. For the beam quality of 8XDL, 10XDL and 12XDL, the differences between the two filtered beams in near-field modulation and near-field contrast are 0.31% and 0.69%, 0.02% and 1.39% and 0.05% and 0, respectively. According to the simulation for the beam quality ranged from 6XDL to 16XDL, the difference between the filtered beams from the two kinds of SFs in near-field contrast decreases basically with the cutoff frequency increasing. The preliminary explanation is that the effective filtering aperture of slit SF in spectral plane is a square region while pinhole SF’s circular. Spectrum always appears with strong center and weak edge, so that the larger the cutoff frequency is, the smaller the difference in filtering effect will be.
Fig. 6. The spatial spectra of the filtered beams. (a), (b) and (c) are the filtered spectra with pinhole SF at the beam quality of 8XDL, 10XDL and 12XDL, respectively. (d), (e) and (f) are the filtered spectra with slit SF at the beam quality of 8XDL, 10XDL and 12XDL, respectively.
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22216
Fig. 7. The near-field distribution of the filtered beams. (a), (b) and (c) are the filtered beams with pinhole SF at the beam quality of 8XDL, 10XDL and 12XDL, respectively. (d), (e) and (f) are the subtraction of the near-field distribution for the filtered beams of slit SF and pinhole SF at the beam quality of 8XDL, 10XDL and 12XDL, respectively.
3. Simulation for NIF system
For large laser systems such as NIF, laser beam has a very high power. In general, the laser is designed to have a large beam aperture due to the laser damage limitation. Moreover, the SF has a long focal length to overcome spherical aberration, etc. According to the practical parameters of TSF in NIF system, the incident beam is an 8th-order super-Gaussian square beam with beam area of 350mm × 350mm, pulse energy of 20kJ, pulse width of 3.5ns, wavelength of 1053nm and cutoff frequency being set at 39XDL (0.096mm−1). For convenience, the cylindrical lenses and spherical lens used in simulation are all spherically curved instead of the practical aspherically curved lenses. Figure 8(a) is the near field of the simulated incident beam, whose near-field modulation is 1.0684 and near-field contrast is 0.0420. Figure 8(b) is the corresponding PSD of the incident beam. For TSF in NIF system, the simulated peak intensity in the focus is 4.3 × 1016 W/cm2, which basically agrees with the 3.5 × 1016W/cm2 in [3]. The intensity on inner edge of pinhole is 4.8 × 1011 W/cm2. The beam filtered with TSF and its PSD are shown in Figs. 8(c) and 8(d), respectively. It can be seen that the beam profile becomes smoother after filtering and the spatial frequencies above 0.096mm−1 are cleared up. The near-field modulation and the near-field contrast are of 1.0205 and 0.0313, respectively. Figure 9 shows the dependence of the peak intensity (dashed line) and the intensities on the inner edge (solid line) of the slits with the focal length of lenses. It can be seen that the peak intensity reaches a maximum near 7 meter focal length because aberrations dominate at short focal lengths, and decreases with the focal length beyond 7 meters. The intensity on the edge of the slit is 6.1 × 108 W/cm2, which is far lower than that of 1010 W/cm2. The beam profile and its PSD filtered with 7-meter-long focal length slit SF are shown in Figs. 8(e) and 8(f), and the near-field modulation and near-filed contrast are 1.0212 and 0.0305, which is better than that of the TSF and can be used in the laser systems. In this way, the overall length of slit SF will be only 28 meters, which is much shorter than the 60-meter-long TSF in NIF system. Besides, the simulated peak intensity in slit SF is 6.3 × 1013 W/cm2, which is lowered by about three orders of magnitude than that in TSF, so that the vacuum degree in SF can be reduced by about one order of magnitude [3].
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22217
Fig. 8. Spatial filtering effect of pinhole and slit SFs. (a) and (b) are the incident beam and its PSD, respectively. (c) and (d) are the filtered beam with TSF in NIF system and its PSD, respectively. (e) and (f) are the filtered beam with slit SF when focal length is at 7 meters and its PSD, respectively.
Fig. 9. The dependence of peak intensity and the intensities on the inner edge of the slits with the focal length of lenses.
Since the output beam quality of slit SF is better than that of the TSF, the cutoff frequency of 39 XDL may not be required. Here we show that larger cutoff frequencies may work in large laser systems. As shown in Table 1, the near field parameters of the filtered beams at the beam quality of 40XDL (0.098mm−1), 41XDL (0.101mm−1), 42XDL (0.103mm−1), 43XDL (0.106mm−1), 44XDL (0.108mm−1) and 45XDL (0.111mm−1) are given. It can be seen that the near-field contrast is still better than that of TSF when the cutoff frequency is of 41XDL. Even at 42XDL, the difference between the two filtered beams with slit SF and TSF in nearfield modulation and near-field contrast is only 0.78% and 1.60%, respectively.
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22218
Table 1. The near-field parameters and corresponding intensities on inner edge of slits of the filtered beams at different cutoff frequencies.
Near-field modulation Near-field contrast Intensities on slits (W/cm2)
39XDL
40XDL
41XDL
42XDL
43XDL
44XDL
45XDL
1.0205
1.0209
1.0263
1.0285
1.0354
1.0368
1.0421
0.0313
0.0307
0.0314
0.0318
0.0327
0.0329
0.0334
6.05 × 108
5.62 × 108
6.03 × 108
5.59 × 108
4.56 × 108
4.62 × 108
4.99 × 108
4. Conclusions
In conclusion, we propose a new-type slit SF with the combination of two cylindrical lenses and a spherical lens. This slit SF has the simple structure and is beneficial to reducing the cost and easing the maintenance of slit SF. The characteristics of this slit SF on image relay, aperture matching and spatial filtering are discussed. In simulation, we used this new-type SF to replace the TSF in NIF system, and found that the focal intensity on the slim drops by about 3 orders of magnitude, which greatly improves the pinhole closure for present large laser systems and could reduce the vacuum degree in SF by about one order of magnitude. Besides, the overall length of SF can be optimized to less than half that of the original one since the focal length in this slit SF is greatly reduced, which is helpful in building and maintaining the large laser systems. Acknowledgments
This work is financially supported by the united foundation between NSFC and Chinese Academy of Engineering Physics under contract Nos. of 11176021 and 11076021, the Natural Science Foundation of China NSFC (91023009, 61108024, 61275140), the Natural Science Foundation of Jiangsu Higher education institutions (10KJA140045, BK2011278, 09KJB140008), the project of the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions, the Research and Innovation Project for College Graduates of Jiangsu Province (CXLX12_0797, CXZZ11_0095).
#213923 - $15.00 USD Received 16 Jun 2014; revised 27 Aug 2014; accepted 28 Aug 2014; published 5 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022211 | OPTICS EXPRESS 22219
