46 1, 018003 January 2007 Axial intensity distribution of lens axicon illuminated by Gaussian-Schell model beam Yuan Chen Jixiong Pu Xiaoyun Liu Huaqiao University Department of Electronic Science and Technology Quanzhou, Fujian 362021, China Abstract. In this paper, both backward and forward-type lens axicons illuminated by a Gaussian-Schell model GSM beam are studied. A numerical calculation is performed to investigate the axial intensity distribution of the focal segment. It is shown that when a backward-type lens axicon is illuminated with an appropriate beam radius and spatial coherence of the incident GSM beam, one can achieve a focal segment with uniform axial intensity. The rapid oscillation of the axial intensity can be reduced by the decreasing of spatial coherence. This method is very simple and efficient. A forward-type lens axicon illuminated by a GSM beam is also investigated. 2007 Society of Photo-Optical Instrumentation Engineers. DOI: 10.1117/1.2431360 Subject terms: lens axicon; Gaussian-Schell model beam; spherical aberration. Paper 060160R received Mar. 5, 2006; revised manuscript received Jun. 10, 2006; accepted for publication Jun. 14, 2006; published online Jan. 16, 2007. 1 Introduction Axicons, which are elements that produce a longitudinal focal line rather than a point focus, have attracted considerable interest because of their unusual properties and versatility in practical applications. 1 7 A generalized axicon with its phase retardation function given by an appropriate logarithmic expression can achieve uniform axial intensity within the desired range. This kind of axicon can be designed and fabricated by electron beam lithography. 8,9 More recently it has been proved that a lens with spherical aberration is a simple and efficient optical element to generate Bessel fields. 10 In practical applications, the focus segments with uniform optical intensity along the element axial within the specified range are very important; they have been extensively used in alignment, metrology, laser optics, and so on. 11,12 The focusing of completely coherent light by lenses without or with aberration has been studied extensively over a period of several decades. However, less work has been done on focusing partially coherent light. In recent years, partially coherent light has been of interest. 13,14 This is because laser beams with many transverse modes can be described by partially coherent light sources. 15 18 In addition, partially coherent light is closely connected to laser fusion, in which a high coherent light is transformed into a low coherent light by induced spatial incoherence and random phase shifts. 14,16 18 Therefore, it is quite important to investigate the problem of focusing of partially coherence light and to discuss how the spatial coherence affects the distribution of the axial intensity. In this paper, we will investigate a lens axicon illuminated by a Gaussian-Schell model GSM beam. Both backward and forward-type lens axicons are studied. It is shown that when a backward-type lens axicon is illuminated with an appropriate beam radius and spatial coherence of the incident GSM beam, one can achieve a focal segment with 0091-3286/2007/$25.00 2007 SPIE uniform axial intensity. We will show that our approach for generating the focal segment with uniform axial intensity is very simple. 2 Theoretical Formulation As shown in Fig. 1, a lens with negative SA the backwardtype lens axicon is illuminated by a partially coherent light of wavelength. The paraxial focal length of the aberrated lens is f. The inner and outer radii of the aperture are, respectively, R 1 and R 2. For aberration lens coefficient, the function of SA can be described by T = exp ik 4 1 4 2. 1 When is positive, the lens axicon is forward-type; when is negative, it is backward-type. 1 The cross-spectral density function of the incident GSM beam in the z=0 plane is given by w 1, 2,z =0 = I 0 exp 1 2 2 + 2 2 exp 1 2 2 w 0 2 2, 2 where I 0 is a constant. 1 and 2 are two position vectors in the z=0 plane, is the length of spatial coherence. w 0 is the waist of the GSM beam. It is assumed that the waist of Fig. 1 A converging annular lens with negative SA, which forms a backward-type lens, is illuminated by partially coherent light. 018003-1
Fig. 2 The axial intensity of the focal segment when the lens axicon is illuminated by GSM beam of four different beam radii w 0 : a w 0 =5 m, b w 0 =8 mm; c w 0 =11 mm; d w 0. Other parameters are =0.5 mm, f=85.714 mm, =6.667 10 5, and =6328 10 7 mm. the GSM beam is located just at the lens plane, denoted as the z=0 plane. Based on the Collins formula, the cross-spectral density of arbitrary point in the field can be expressed by 19,20 w r 1,r 2,z = 2 z k 2 R 2 R 2 w 1, 2,z =0 exp ik 2B A 1 2 2 2 + D r 1 2 r 2 2 +2 2 r 2 1 r 1 d 1 d 2, where r 1 and r 2 are two position vectors in the field of focus, and A, B, C, D are the matrix elements of the following optical matrix A B C D = 1 z 0 1 1 0 1/f 1 = 1 z/f z 3 1/f 1 4 with AD BC=1. The distribution of axial intensity can be given by I 0,z = W r 1 = r 2 =0,z R 2 exp r 1 2 2 + r 2 where = k z 2 R 2 w 0 2 exp r 1 2 2 + r 2 2 2 exp ikf r 1,r 2 I 0 r 1 r 2 / 2 r 1 r 2 dr 1 dr 2, f r 1,r 2 = r 1 4 r 2 4 + 1 2z 1 2f r 1 2 r 2 2. The particular values of f and depend on the focal segment coordinates and the aperture of the lens axicon 1 = d 2 d 1 4d 1 d 2 R 2 2 R 1 2, f = d 1d 2 R 2 2 R 2 1 d 2 R 2 2 2 d 1 R. 7 1 Based on Eqs. 5 7, we can perform the numerical calculation. In Sec. 3, we give some numerical results that show the effect of SA, the beam radius, and the spatial coherence of the Gaussian-Schell beam on the axial inten- 5 6 018003-2
Fig. 3 The axial intensity of the focal segment when the lens axicon is illuminated by GSM beams of five different beam radii w 0 : a w 0 =3 mm, b w 0 =5 mm, c w 0 =7.3 mm, d w 0 =10 mm, e w 0. Other parameters are =6328 10 7 mm, =0.5 mm, R 1 =2.5 mm, R 2 =5 mm, d 1 =200 mm, d 2 =100 mm which results in f=300 mm, = 6.667 10 5. sity distribution of the focal segment. A uniform focal segment can be achieved by the proposed approach. 3 Numerical Results and Discussions This paper illustrates the effect of the beam radius and the spatial coherence of the Gaussian-Schell beam on the axial intensity distribution of the focal segment. Based on this illustration, we can choose the appropriate parameters of GSM sources to achieve uniform axial intensity. First, we discuss the forward-type lens axicons illuminated by GSM beams. The forward-type lens axicon is a lens with positive SA. The simplest case of the forwardtype lens axicon is a doublet consisting of a diverging aberrated lens and a converging perfect lens. 1 The parameters of the lens axicon and the focal segment are chosen as R 1 =2.5 mm, R 2 =5 mm, d 1 =100 mm, d 2 =200 mm which results in f =85.714 mm, = 6.667 10 5. Figure 2 illustrates the axial intensity of the focal segment for GSM beams of four beam radii w 0 =5,8,11mmandw 0, respectively when the spatial coherence length of the GSM beams is equal to 0.5 mm. The smaller the beam radius of the incident GSM beam is, the faster the axial intensity decreases in increments of z. However, when the beam radii of the GSM beams w 0, the axial intensity of the focal segment is still nonuniform. Besides, oscillations exist in the axial intensity along the focal segment. Now we consider the backward-type lens axicons illuminated by a GSM beam. The parameters of the lens axicon and the focal segment are chosen as R 1 =2.5 mm, R 2 =5 mm, d 1 =200 mm, d 2 =100 mm which results in f =300 mm, = 6.667 10 5. The spatial coherence length of the GSM beams remains 0.5 mm. Figure 3 represents the axial intensity of the focal segment when the lens axicon is illuminated by GSM beams of five different beam radii. The axial intensity of the focal segment exhibits rapid oscillation. When w 0 7.3 mm, the axial intensity increases in increments of z, but when w 0 7.3 mm, the axial intensity decreases in increments of z. In Fig. 3 c w 0 =7.3 mm, a focal segment with uniform axial intensity is achieved, which is similar to that achieved by the logarithmic axicon. 1 3 Our approach for generating the focal segment with uniform axial intensity is very simple and highly efficient. For the logarithmic axicon illuminated by a plane wave, we would need to expand a Gaussian beam into a beam with a very large radius and use only the small fraction of light located on the axicon. But in our approach, we need only expand the GSM beam to the appropriate beam radius and use the resultant GSM beam to illuminate the lens axicon. Now we want to discuss how the spatial coherence affects the axial intensity of the focal segment. Figure 4 represents the axial intensity of the focal segment when the lens axicon is illuminated by GSM beams of three different spatial coherence. It is shown that the larger the spatial coherence of the incident GSM beam is, the more rapid is the oscillation of the axial intensity. This kind of oscillation of the axial intensity is due to the interference of the light delivered from the lens axicon. Therefore, we can reduce the undesired oscillation by decreasing the spatial coherence of the incident GSM beam. We had discussed the effect of the beam radius and the spatial coherence of the GSM beam on the axial intensity 018003-3
Fig. 4 The axial intensity of the focal segment when the lens axicon is illuminated by GSM beam of three different spatial coherence : a =0.2 mm, b =0.5 mm; c =0.8 mm. Here w 0 =6.8 mm. Other parameters are the same as in Fig. 3. distribution of focal segment. Based on this, we can choose the appropriate beam radius and the spatial coherence to achieve uniform axial intensity. Figure 5 shows the relationship between the beam radius w 0 and the coherent length to generate uniform axial intensity. When the coherent length 5 mm and w 0 6.8 to 7.8, there exist two coherent lengths at which the uniform axis intensity can be produced. Moreover, when 2 mm, the appropriate beam radius w 0 at which the uniform focal segment is generated increases with the increment of the spatial coherence of the GSM beam and reaches its maximum when equals to 2 mm. Then with the further increment of, the beam radius w 0 decreases. Nevertheless, when 11 mm, the beam radius w 0 nearly remains invariant w 0 =5.2 mm. Based on this point, we can choose w 0 as 5.2 mm and change spatial coherence as you want on the condition that 11 mm to achieve the uniform axial intensity. 4 Conclusions In this paper, we have investigated both backward and forward-type lens axicons illuminated by GSM beams. The effect of beam radius and spatial coherence of the incident GSM beam on the axial intensity distribution of the focal segment is discussed in detail. It is demonstrated that, when a backward-type lens axicon is illuminated with an appropriate beam radius and spatial coherence of the incident Fig. 5 The relationship between the beam radius w 0 and the coherent length when the axial intensity distribution of the focal segment is uniform. Here, R 1 =2.5 mm, R 2 =5 mm, d 1 =200 mm, d 2 =100 mm which results in f=300 mm, = 6.667 10 5. GSM beam, one can achieve a focal segment with uniform axial intensity. Additionally, the rapid oscillation of the axial intensity can be reduced by the decreasing of spatial coherence. For a forward-type lens axicon illuminated by GSM beams, the decrease of the axial intensity with increasing z is faster than that generated by the lens axicon illuminated by a plane wave. The smaller the beam radius of the incident GSM beam is, the faster the axial intensity decreases with the increasing of z. Our approach to generate the focal segment with uniform axial intensity is very simple. Acknowledgments This research was supported by the National Natural Science Foundation of China Grant No. 60477041. References 1. J. Pu, H. Zhang, and S. Nemoto, Lens axicons illuminated by Gaussian beams for generation of uniform-axial intensity Bessel fields, Opt. Eng. 39 3, 803 806 2000. 2. J. Sochaki, S. Bara, Z. Jaroszewicz, and A. Kolodziejczyk, Phase retardation of the uniform-intensity axilens, Opt. Lett. 17, 7 9 1992. 3. J. Sochaki, Z. Jaroszewicz, L. R. Staronski, and A. Kolodziejczyk, Annular-aperature logarithmic axicon, J. Opt. Soc. Am. A 10, 1765 1768 1993. 4. M. Honkanen and J. Turunen, Tandem systems for efficient generation of uniform-axial-intensity Bessel fields, Opt. Commun. 154, 368 375 1998. 5. A. T. Friberg, Stationary-phase analysis of generalized axicons, J. Opt. Soc. Am. A 13, 743 750 1996. 6. N. Davidson, A. A. Friesem, and E. Hasman, Holographic axilens: High resolution and long fical depth, Opt. Lett. 16, 523 525 1991. 7. S. Y. Phpov and A. T. Friberg, Linear axicons in partially coherent light, Opt. Eng. 34, 2567 2573 1995. 8. S. Y. Phpov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography, Opt. Commun. 154, 359 367 1998. 9. S. Y. Phpov and A. T. Friberg, Apodization of generalized axicons to produce uniform axial line image, Pure Appl. Opt. 7, 537 548 1998. 10. Z. Jaroszewicz and J. Morales, Lens axicons: Systems composed of a diverging aberrated lens and a perfect converging lens, J. Opt. Soc. Am. A 15, 2383 2390 1998. 11. V. Jarutis, R. Paskauskas, and A. Stabinbis, Focusing of Laguerre- Gaussian beams by axicon, Opt. Commun. 184, 105 112 2000. 12. A. Thaning, A. T. Friberg, and Z. Jaroszewicz, Synthesis of diffractive axicons for partially coherent light based on asymptotic wave theory, Opt. Lett. 26 21, 1648 1650 2001. 13. S. Y. Phpov and A. T. Friberg, Design of diffractive axicons for partially coherent light, Opt. Lett. 23, 1639 1641 1998. 14. A. T. Friberg and J. Turunen, Imaging of Gaussian Shell model sources, J. Opt. Soc. Am. A 5, 713 720 1988. 018003-4
15. J. Pu, S. Nemoto, and X. Liu, Beam shaping of focused partially coherent beams by use of the spatial coherence effect, Appl. Opt. 43 28, 5281 5286 2004. 16. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Random phasing of high-power lasers for uniform target acceleration and plasma instability suppression, Phys. Rev. Lett. 53, 1057 1060 1984. 17. R. Gase, The multimode laser radiation as a Gaussian Shell model beam, J. Mod. Opt. 38, 1107 1116 1991. 18. J. Turunen, E. Tervonen, and A. T. Friberg, Acousto-optic control and modulation of optical coherence by electronically synthesized holographic gratings, J. Appl. Phys. 67, 49 59 1990. 19. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, UK 1995. 20. S. A. Collins Lens-systems diffraction integral written in terms of matrixoptics, J. Opt. Soc. Am. 60, 1168 1177 1970. Yuan Chen obtained a BS degree in physics at Huaqiao University, and she is now pursuing a MS degree at the same university. Her research topics are beam shaping, propagation, and focusing of laser beams. Jixiong Pu obtained a PhD at the University of Tsukuba Japan. Heisnowaprofessor of physics at Huaqiao University. He is interested in the areas of propagation and focusing of laser beams and nonlinear optics. Xiaoyun Liu received a MS degree in physical electronic in 2006 from Hauqiao University, China. She is intersted in the areas of propagation and focusing of laser beams. 018003-5