Beam shaping of focused partially coherent beams ... - OSA Publishing

Beam shaping of focused partially coherent beams by use of the spatial coherence effect Jixiong Pu, Shojiro Nemoto, and Xiaoyun Liu

We demonstrate that when a partially coherent beam with a Gaussian intensity distribution is focused by a lens, the desired partially coherent flat-topped intensity distribution or doughnut-shaped intensity distribution at the geometrical focus can be generated by choice of appropriate form of spectral degree of coherence. We provide a novel approach to beam shaping of a partially coherent beam and offer new schemes for their potential applications such as material processing, optical therapy, and optical tweezers. © 2004 Optical Society of America OCIS codes: 030.1640, 140.3300.

1. Introduction

Over the past decade, considerable interest has centered on the conversion of a Gaussian intensity distribution into a flat-topped profile or into other kinds of profiles.1–3 This is so because laser beams with flat-topped and other kinds of profiles are desired for various applications, such as applications of laser fusion, laser heat treatment, and optical data processing. Many methods for generating a flat-topped intensity distributions in either the far or the near field have been proposed. Partially coherent light beams have generated interest in recent years.4,5 It has been demonstrated that a laser beam with many transverse modes can be described by a partially coherent beam.6 Moreover, partially coherent beams have been applied to laser fusion, in which a highly coherent beam is transformed into a partially coherent beam,7 to reduce speckle and achieve a smoother focused spot. Nevertheless, most studies of beam shaping have concentrated on shaping of spatially completely coherent beams; little attention has been paid to shaping of partially coherent beams.8 In this paper we study the focusing of particular classes of partially coherent beams with Gaussian

J. Pu 共[email protected]兲 and X. Liu are with the Department of Electronic Science & Technology, Huaqiao University, Quanzhou, Fujian 362021, China. S. Nemoto is with the Graduate School of Systems and Information Engineering, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan. Received 15 December 2003; revised manuscript received 24 May 2004; accepted 1 July 2004. 0003-6935兾04兾285281-06$15.00兾0 © 2004 Optical Society of America

intensity distributions. An expression to represent the intensity distribution of a partially coherent beam at geometrical focus has been derived, from which we can find that the intensity distribution at the geometrical focus is strongly dependent on both the spectral degree of coherence and the intensity distribution of incident partially coherent beams in front of a lens. Accordingly a novel approach to shaping of partially coherent light beams by controlling the spectral degree of coherence is proposed. It is shown that, by choosing the appropriate spectral degree of coherence of partially coherent beams at the lens plane, one can achieve a desired focal spot at the geometrical focus. Specifically, special focal spots, such as a flat-topped spots and doughnut-shaped spots at the geometrical focus, can be generated by choice of the appropriate forms of spectral degree of coherence. 2. Expressions for Intensity Distribution at the Geometrical Focus

Let us consider a partially coherent beam of frequency ␻ focused by a lens of focal length f. The cross-spectral density of the partially coherent beam in front of the lens is given by6,9 W 共0兲共r1⬘, r2⬘, ␻兲 ⫽ 关I 共0兲共r1⬘, ␻兲I 共0兲共r2⬘, ␻兲兴 1兾2␮ 共0兲共r2⬘ ⫺ r1⬘, ␻兲,

(1)

where ␮共0兲共r2⬘ ⫺ r1⬘, ␻兲 is the spectral degree of coherence 共which may be called the degree of spatial coherence兲 that characterizes the spatial coherence of the beam and I 共0兲共r⬘, ␻兲 ⫽ W 共0兲共r⬘, r⬘, ␻兲 ⫽ I 0 exp共⫺r⬘ 2兾2w 02兲 1 October 2004 兾 Vol. 43, No. 28 兾 APPLIED OPTICS

(2) 5281

is the intensity distribution of the partially coherent beam in front of the lens. r1⬘ and r2⬘ are two position vectors in the lens plane. As shown in Eq. 共2兲, the intensity distribution of the beam in front of the lens is assumed to be of Gaussian profile, and the beam radius is equal to w0. I0 is a positive constant that represents the average intensity. According to the propagation law for a partially coherent beam, we get the intensity distribution at the geometrical focus5: I共r, ␻兲 ⫽

冉 冊 兰兰 册 k 2␲f

2

冋

W 共0兲共r1⬘, r2⬘, ␻兲exp

ik r 䡠 共r1⬘ f

⫺ r2⬘兲 dr1⬘dr2⬘,

(3)

where k ⫽ ␻兾c is a wave number associated with angular frequency ␻. r is the position vector in the plane of the geometrical focus. It was found from Eqs. 共1兲 and 共3兲 that the intensity distribution 关I共r, ␻兲兴 at the geometrical focus is dependent on both the beam profile 关I共0兲共r⬘, ␻兲兴 and the spectral degree of coherence 关i.e., ␮共0兲共⌬r⬘, ␻兲, where ⌬r⬘ ⫽ r2⬘ ⫺ r1⬘兴 of the partially coherent beam in front of the lens. This dependence of the intensity distribution 关I共r, ␻兲兴 at the geometrical focus on the spectral degree of coherence may provide a flexible approach to achieving the desired intensity profile at the geometrical focus. For a given intensity profile in front of the lens, the variation of the spectral degree of coherence, ␮共0兲共⌬r⬘, ␻兲, of the incident beam may result in a large change of the intensity profiles at the geometrical focus. One may expect that different spectral degrees of coherence of the incident partially coherent beam may result in different focused intensity distributions at the geometrical focus. In Section 3 we give some specific examples to illustrate this novel approach to beam shaping of partially coherent beams. 3. Shaping of Partially Coherent Beams

As indicated in Section 2, when a partially coherent beam is focused by a lens the intensity distribution at the geometrical focus is strongly dependent on the intensity distribution and on the spatial coherence of the incident partially coherent beam in front of the lens. Therefore we can present a novel method for generating the desired intensity distribution at the geometrical focus by choosing suitable forms of the spatial coherence of the incident partially coherent beam. First, it is assumed that the partially coherent beam is just a Gaussian Schell-model 共GSM兲 beam. The intensity distribution and the spectral degree of coherence of a GSM beam are given, respectively, by Eq. 共2兲 and by the following equation5: ␮ 共0兲共⌬r⬘, ␻兲 ⫽ exp关⫺共⌬r⬘兲 2兾2␴ 2兴,

(4)

where ␴ is the effective coherence width of the GSM beam in front of the lens. As shown above, both the 5282

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Fig. 1. 共a兲 Spectral degree of coherence of the partially coherent beam at the lens plane and 共b兲 intensity distribution at the geometrical focus when the incident partially coherent beam is a GSM beam of ␣ ⫽ ␴兾w0 ⫽ 0.3 共dashed curve兲 and ␣ ⫽ ␴兾w0 ⫽ 1 共solid curve兲. The Fresnel number of the GSM beam is Nw ⫽ w02兾␭f ⫽ 100.

spectral degree of coherence and the intensity distribution of the GSM beam are of Gaussian distribution. Parameter ␣ ⫽ ␴兾w0 can be used to characterize the global coherence of the GSM beam. In Figure 1 we plot the intensity distribution at the geometrical focus and the corresponding spectral degree of coherence when the incident partially coherent light beam is a GSM beam with ␣ ⫽ ␴兾w0 ⫽ 0.3 共dashed curves兲, or ␣ ⫽ ␴兾w0 ⫽ 1 共solid curves兲. In Fig. 1 the Fresnel number of the GSM beam has been chosen as Nw ⫽ w02兾␭f ⫽ 100, and the aperture of the lens is assumed to be much larger than Nw, so the diffraction effect of the lens can be neglected.10 共This applies also to Figs. 2– 6 below兲. It is readily found that the intensity distribution at the geometrical focus is just, as expected, of a Gaussian distribution; and the lower coherence 共i.e., the lower value of ␣兲 results in the larger focal spot.9 To demonstrate further the effect of the spatial coherence of the partially coherent beam on the intensity distribution at the geometrical focus, we

choose the spectral degree of coherence of the partially coherent beam in front of the lens as follows9: ␮ 共0兲共⌬r⬘, ␻兲 ⫽

1 Be sinc共k兩⌬r⬘兩b兲 1 ⫺ ⑀2 ⫺

⑀2 Be sinc共k兩⌬r⬘兩⑀b兲, 1 ⫺ ⑀2

(5)

where Be sinc共u兲 ⫽ 2J1共u兲兾u, J1 is the Bessel function of order unity, and b and ⑀ 共1 ⬎ ⑀ ⱖ 0兲 are two positive constants that represent the spatial coherence properties of the partially coherent beams. Fields with the spectral degree of coherence given by Eq. 共5兲 can be synthesized in the optical system described in Refs. 11 and 12. Some special properties of this kind of partially coherent light have been studied in recent years.11–13 A parameter L៮ ⫽ 3.832兾kb has been defined to characterize the spatial coherence of the partially coherent light in front of the lens.11–13 It was readily found that a higher value of L៮ corresponds to a higher coherence of the light beam at the lens plane. To show the effect of spatial coherence on the intensity distribution at the geometrical focus, we have plotted the intensity distributions and the corresponding spectral degree of coherence, ␮共0兲共⌬r⬘兲 ⬅ ␮共0兲共⌬r⬘, ␻兲, in Figs. 2– 6. It can be seen from Fig. 2共b兲 that, when ⑀ ⫽ 0.0 and w0兾L៮ ⫽ 1.5, the flattopped spot at the geometrical focus, whose intensity distribution can be described as a super-Gaussian function exp关⫺共r兾r0兲14兴 with r0 ⫽ 0.0095w0, is produced. Figure 2共a兲 shows the corresponding spectral degree of coherence. It can also be found from Figs. 3 and 4 that, when w0兾L៮ ⫽ 2 and w0兾L៮ ⫽ 3, the intensity distributions at the geometrical focus are, respectively, super-Gaussian pattern exp关⫺共r兾r0兲18兴 with r0 ⫽ 0.0126w0 and exp关⫺共r兾r0兲32兴 with r0 ⫽ 0.0185w0. These results indicate that different flattopped intensity distributions can be achieved by choice of different spatial coherence. As we know, the larger the value of w0兾L៮ , the lower the coherence of the partially coherent beam at the lens plane. Therefore from Figs. 2– 4 we can readily arrive at the conclusion that the lower the coherence of the partially coherent beam in front of the lens, the higher the order of the super-Gaussian intensity distribution at the geometrical focus, and the larger the focal spot. As discussed above, the spatial coherence of the incident partially coherent beam determines the intensity distribution at the geometrical focus. This characteristic can be explained by examination of Eq. 共3兲. From Eq. 共3兲, it can readily be found that the intensity distribution at the geometrical focus is proportional to the inverse Fourier transformation of cross-spectral density W共0兲共r1⬘, r2⬘, ␻兲 of the partially coherent beam in front of the lens. It is known that both the spectral degree of coherence and the intensity of the incident GSM beam are of Gaussian distribution, and the inverse Fourier transformation of a function of Gaussian distribution is just a Gaussian function. Therefore, when the incident light beam is

Fig. 2. 共a兲 Spectral degree of coherence of the partially coherent beam at the lens plane and 共b兲 intensity distribution at the geometrical focus when the spatially coherent function of the incident partially coherent beam is given by Eq. 共5兲. Nw ⫽ w02兾␭f ⫽ 100, ⑀ ⫽ 0.0, w0兾L៮ ⫽ 1.5, f ⫽ 1 m, and ␭ ⫽ 632.8 nm. The dotted curve represents the super-Gaussian profile exp关⫺共r兾r0兲14兴 with r0 ⫽ 0.0095w0.

a GSM beam, the intensity distribution at the geometrical focus is just Gaussian. For another example in which the spatial coherence of the incident beam is characterized by Eq. 共5兲, we assume that ⑀ ⫽ 0.0, and in this case the spectral degree of coherence of the partially coherent beam in front of the lens can be rewritten as

冉

␮ 共0兲共⌬r⬘, ␻兲 ⫽ Be sinc ⫽

3.832兩⌬r⬘兩 L៮

冊

2J 1共3.832兩⌬r⬘兩兾L៮ 兲 . 3.832兩⌬r⬘兩兾L៮

(6)

It is well known that the Fourier transformation of a flat-topped function gives a function described by Eq. 共6兲; therefore the inverse Fourier transformation of a function similar to Eq. 共6兲 will produce a function similar to the flat-topped pattern. This is why the intensity distribution of a super-Gaussian distribution 共or a flat-topped intensity distribution; see Figs. 1 October 2004 兾 Vol. 43, No. 28 兾 APPLIED OPTICS

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Fig. 3. 共a兲 Spectral degree of coherence of the partially coherent beam at the lens plane and 共b兲 intensity distribution at the geometrical focus when the spatially coherent function of the incident partially coherent beam is given by Eq. 共5兲. Nw ⫽ w02兾␭f ⫽ 100, ⑀ ⫽ 0.0, w0兾L៮ ⫽ 2, f ⫽ 1 m, and ␭ ⫽ 632.8 nm. The dotted curve represents the super-Gaussian profile exp关⫺共r兾r0兲18兴 with r0 ⫽ 0.0126w0.

2– 4兲 is obtained when the spectral degree of coherence of the incident partially coherent beam is given by Eq. 共6兲. To generate other kinds of intensity distribution at the geometrical focus, one should choose the parameters of the spectral degree of coherence, given by Eq. 共5兲, of the partially coherent beam to be other values. When we choose ⑀ ⫽ 0.4 and w0兾L៮ ⫽ 1, the spectral degree of coherence is as plotted in Fig. 5共a兲. The corresponding partially coherent doughnut-shaped spot at the geometrical focus is generated 共solid curve兲 as plotted in Fig. 5共b兲. The dotted curve in Fig. 5共b兲 is the intensity distribution of the TEM01 mode, I共r兲 ⫽ I0共r兾r0兲2 exp关⫺2共r兾r0兲2兴, with r0 ⫽ 0.006w0. It is shown that the resultant partially coherent doughnut-shaped intensity distribution at the geometrical focus is more tightly concentrated about the axis than the profile of the TEM01 mode, and good similarity is observed near the axis. The doughnut-shaped intensity distribution has many applications; one of the most important is in trapping 5284

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Fig. 4. 共a兲 Spectral degree of coherence of the partially coherent beam at the lens plane and 共b兲 intensity distribution at the geometrical focus when the spatially coherent function of the incident partially coherent beam is given by Eq. 共5兲. Nw ⫽ w02兾␭f ⫽ 100, ⑀ ⫽ 0.0, w0兾L៮ ⫽ 3, f ⫽ 1 m, and ␭ ⫽ 632.8 nm. The dotted curve represents the super-Gaussian profile exp关⫺共r兾r0兲32兴 with r0 ⫽ 0.0185w0.

of microparticles. Ashkin14 investigated the trapping of a dielectric sphere by using TEM01 beam mode. It should be mentioned again that partially coherent doughnut-shaped spots have some advantages over completely coherent doughnut-shaped spots because of their low sensitivity to speckle, etc. Therefore the partially coherent doughnut-shaped spots may be useful in atom optical experiments, such as with atomic lenses, atom switches, and optical tweezers.14,15 As indicated in Fig. 5共b兲, the dip in the doughnutshaped spot is very deep. To achieve a doughnutshaped spot with a shallower dip we choose other parameters of spatial coherence such as ⑀ ⫽ 0.2 and w0兾L៮ ⫽ 1. In this case the corresponding spectral degree of coherence and the resultant focused spot are as shown in Figs. 6共a兲 and 6共b兲, respectively. If we choose other forms of the spectral degree of coherence, we can generate the other desired doughnutshaped spots or other kinds of spot at the geometrical focus, although the intensity distribution 共i.e., the

Fig. 5. 共a兲 Spectral degree of coherence of the partially coherent beam at the lens plane and 共b兲 intensity distribution 共solid curve兲 at the geometrical focus when the spatially coherent function of the incident partially coherent beam is given by Eq. 共5兲. Nw ⫽ w02兾 ␭f ⫽ 100, ⑀ ⫽ 0.4, w0兾L៮ ⫽ 1, f ⫽ 1 m, and ␭ ⫽ 632.8 nm. The dotted curve in 共b兲 is the intensity distribution of TEM01 mode I共r兲 ⫽ I0共r兾r0兲2 exp关⫺2共r兾r0兲2兴 with r0 ⫽ 0.006w0.

Gaussian distribution兲 of the incident partially coherent beam in front of the lens is unchanged. 4. Conclusions and Discussion

The intensity distribution at geometrical focus when a partially coherent beam is focused by a lens has been investigated. It has been demonstrated that the intensity distribution at the geometrical focus is dependent both on the intensity distribution and on the spectral degree of coherence of the partially coherent beam in front of the lens. Based on this, a novel approach to beam shaping of partially coherent beams can be achieved by choice of the spectral degree of coherence of the incident partially coherent beam. Some special examples are given to prove the usefulness of the approach. For example, partially coherent flat-topped spots or partially coherent doughnut-shaped spots at the geometrical focus can be generated by choice of a suitable spectral degree of coherence. The inverse process of the topic discussed in this

Fig. 6. 共a兲 Spectral degree of coherence of the partially coherent beam at the lens plane and 共b兲 intensity distribution at the geometrical focus when the spatially coherent function of the incident partially coherent beam is given by Eq. 共5兲. Nw ⫽ w02兾␭f ⫽ 100, ⑀ ⫽ 0.2, w0兾L៮ ⫽ 1, f ⫽ 1 m, and ␭ ⫽ 632.8 nm.

paper seems to be more useful; that is, based on the intensity distribution in front of the lens and on the desired intensity distribution at the geometrical focus, we find the corresponding function forms of the spectral degree of coherence. This job seems to be quite complicated, and the precision of reconstructing the desired intensity distribution is poor.16 However, until now only a few specific forms of spatial coherence could be synthesized.8,9,11,12,17,18 In this sense, even if some forms of spatial coherence can be derived for achieving the desired intensity distribution at the geometrical focus, it may be impossible to synthesize specific forms of the spatial coherence, or the efficiency in producing the desired forms of the spatial coherence may be very low. We have demonstrated that the intensity distribution at the geometrical focus is dependent on the intensity distribution and the spatial coherence of a partially coherent beam in front of a lens. Specific forms of partially coherent intensity distributions, such as a flat-topped intensity distribution and doughnut-shaped intensity distribution, can be generated by choice of the spatial coherence as the spe1 October 2004 兾 Vol. 43, No. 28 兾 APPLIED OPTICS

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cific forms. We have demonstrated that the approach given in this paper is an effective and simple method for shaping of a partially coherent beam and offers new schemes for potential applications such as material processing, optical therapy and optical tweezers, etc.

9.

10.

This research was supported by the Fujian Natural Science Foundation of China. References and Notes 1. E. G. Churin, “Diffraction-limited laser beam shaping by use of computer-generated holograms with dislocations,” Opt. Lett. 24, 620 – 622 共1999兲. 2. S. N. Dixit, M. D. Feit, M. D. Perry, and H. T. Powell, “Designing fully continuous phase screens for tailoring focal-plane irradiance profiles,” Opt. Lett. 21, 1715–1717 共1996兲. 3. L. A. Romero and F. M. Dickey, “Lossless laser beam shaping,” J. Opt. Soc. Am. A 13, 751–755 共1996兲. 4. S. Y. Popov and A. T. Friberg, “Design of diffractive axicons for partially coherent light,” Opt. Lett. 23, 1639 –1641 共1998兲. 5. A. T. Friberg and J. Turunen, “Imaging of Gaussian Schellmodel sources,” J. Opt. Soc. Am. A 5, 713–720 共1988兲. 6. R. Gase, “The multimode laser radiation as a Gaussian Schell model beam,” J. Mod. Opt. 38, 1107–1116 共1991兲. 7. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of highpower lasers for uniform target acceleration and plasmainstability suppression,” Phys. Rev. Lett. 53, 1057–1060 共1984兲. 8. J. Turunen, E. Tervonen, and A. T. Friberg, “Acousto-optic control and modulation of optical coherence by electronically

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