Numerical aperture invariant focus shaping using spirally polarized beams

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Optics Communications 281 (2008) 1924–1928 www.elsevier.com/locate/optcom

Numerical aperture invariant focus shaping using spirally polarized beams Bing Hao *, James Leger Department of Electrical and Computer Engineering, University of Minnesota, 200 Union Street SE, Minneapolis, MN 55455, United States Received 25 April 2007; received in revised form 14 November 2007; accepted 23 November 2007

Abstract The concept of spiral polarization is proposed as an extension of the generalized cylindrical vector beam. The focusing properties of this spatially variant polarization under high NA are studied. It can be shown that with one such polarization, the focus maintains a flattop intensity shape independent of NA from NA = 0.82 up to NA = 0.95. Ó 2007 Elsevier B.V. All rights reserved. PACS: 42.25.Ja; 42.25.p; 42.15.Eq Keywords: Polarization; Laser beam shaping; Image formation theory; Optical confinement and manipulation; Cylindrical vector beams

1. Introduction Beam shaping using spatially variant polarization has been of an increasing research interest in recent years. In dealing with two-dimensional configurations, radial and azimuthal polarizations are usually chosen as the polarization basis states due to symmetry considerations. Using this basis, the generalized cylindrical vector beam [1] has been proposed as a linear combination of radial and azimuthal components as shown in Fig. 1. Mathematically, the generalized cylindrical vector beam can be expressed as [1] ~ Eðr; /Þ ¼ P ½cos u0 e~r þ sin u0 e~/ , where e~r and e~/ are the radial and azimuthal basis polarizations, respectively. The relative strength of the radial and azimuthal components is determined by the rotation angle u0 from the radial direction. This generalized cylindrical vector beam is shown to be quite useful in beam shaping [1] under high numerical aperture (NA), which is due to the special focusing properties of radial and azimuthal polarization. (Here NA is defined as *

Corresponding author. E-mail address: [email protected] (B. Hao).

0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.11.093

NA ¼ sin h, where focusing in the air is assumed.) For a given (high) NA, the focal field intensity distribution of the radially polarized light has a peak at the center, contributed by the strong z-component [1,2]; the focal pattern of the azimuthally polarized light has a null at the center, with only the transverse component and no z-component in place. So by changing the angle u0 of the generalized cylindrical vector beam, we can balance the z-component and the transverse components to yield a flat-top focus. For example, in Ref. [1], Zhan and Leger have shown that for NA = 0.8, with the rotation angle u0 of 24°, the focal intensity shape is flat-top, as is shown in Fig. 1b. Such uniform irradiance is very important in a variety of applications, such as laser micro machining [3], laser-assisted thermal annealing [4], optical recording [5] etc. Other beam shapes can also be realized by appropriate polarization modulation. For example, the annular-shaped spot resulting from the transverse-only polarization component has certain advantages in materials processing and micro-welding [6,7]. One inconvenient aspect of this polarization beam shaping scheme involves adjusting the angle u0 for different NA’s. Fig. 2 shows the NA dependence of the rotation angle u0 for the generalized cylindrical vector beam to

B. Hao, J. Leger / Optics Communications 281 (2008) 1924–1928

1925

0.14 z−component transverse component total intensity

0.12

ϕ0 intensity (a.u.)

0.1 0.08 0.06 0.04 0.02 0 −3

−2

−1

0

1

2

3

r (wavelength)

Fig. 1. (a) Schematic of generalized cylindrical vector beam. u0 is the rotation angle from the radial direction. (b) The flat-top intensity distribution in the focal plane produced by (a), NA = 0.8 and u0 ¼ 24 .

40

rotation angle (degrees)

35 30 25 20 15 10 5 0

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

numerical aperture

Fig. 2. Rotation angle u0 as a function of numerical aperture to achieve a flat-top beam at focus.

produce a flat-top focal pattern. It can be clearly seen in Fig. 2 that as the NA increases, the characteristic angle u0 that achieves flat-top beam shaping also becomes larger. In practice, the changing of the angle u0 is usually performed by mechanically rotating a wave-plate to some prescribed orientation. In this paper, we propose another polarization beam shaping method based on the use of ‘‘spiral polarization”, which is an extension of the generalized cylindrical vector beam concept. By numerical simulation, we show that one such polarization family is capable of delivering the flat-top focal pattern invariant to NA, from NA = 0.82 up to NA = 0.95.

generalized cylindrical vector beam concept, we let the rotation angle u0 be a function of radius q, which typically has a larger value as q increases. Thus the instantaneous electric vector at various points follows the shape of a spiral line, resulting in the coined name ‘‘spiral polarization”. Fig. 3 shows the schematic plots of the spiral polarization, which starts off almost radial at the center and curves towards azimuthal polarization at the edge. At each point, the polarization can be decomposed into radial and azimuthal basis components and hence preserves radial symmetry. Depending on the specific form of the function u0 ðqÞ, where q denotes the radius, the ‘‘spiral lines” curve in different patterns. The focusing properties of the spirally polarized beam can be analyzed using the Richards and Wolf [8] vectorial diffraction theory to take into account the spatial inhomogeneity vector property. This method is extensively used when considering high NA imaging and focusing in applications such as lithography and optical recording. Furthermore, in dealing with radial and azimuthal polarization or the generalized cylindrical vector beam, this method has been routinely employed and the integral formula can be shown to take a relatively simple form [1,2]. The geometry of the beam shaping problem is shown in Fig. 4. The incident beam bearing the spiral polarization is focused by an aplanatic high numerical aperture lens. The

2. System configuration and numerical simulation Spiral polarization is another kind of spatially variant polarization bearing radial symmetry. By extending the

Fig. 3. Schematic of spiral polarization.

B. Hao, J. Leger / Optics Communications 281 (2008) 1924–1928 High NA lens y

P(r,ϕs ) Spirally polarized beam

ϕs

x

θ z=0

Blocking Aperture

z

f

Fig. 4. Diagram of the focusing configuration. The blocking aperture indicates the changing of NA.

blocking aperture in the diagram forms the entrance pupil of the lens and thereby establishes its numerical aperture. Our aim is to find a specific u0 ðqÞ that will result in flattop focusing that is invariant to numerical aperture. In the formula describing the focal plane intensity distribution, it is very convenient to express u0 ðqÞ as a function of h, where h is the polar angle as shown in Fig. 4 instead of q. For every q, there is a corresponding polar angle h, where NA ¼ sin hmax . Following the same notation as in Ref. [1], the electric field in the focal plane can be expressed as follows: ~ Eðr; us ; zÞ ¼ Er e~r þ Eu e~u þ Ez e~z ;

ð1Þ

where Er , Eu ; and Ez are the three orthogonal components that determine the total intensity distribution and can be written as Z hmax pffiffiffiffiffiffiffiffiffiffi Er ¼ A cos h sin h cos h cos u0 ðhÞJ 1 ðkr sin hÞeikz cos h dh;

increasing NA. Typically, the z-component grows faster than the transverse component as the NA increases. The generalized cylindrical vector beam can then be used to rotate the angle u0 to retard the fast growth of the z-component (see Fig. 2) so that it can be balanced by the transverse component. To achieve the same flat-top pattern in the focal plane, we change the functional form of u0 ðhÞ to attenuate the larger z-component that occurs at large values of h. The contribution of the z-component mainly comes from the edge rays instead of center rays, so generally the polarization starts off radial and gradually changes to azimuthal at the edge to decrease the z-component growth and balance the transverse component. Thus the polarization directions take on the appearance of a spiral line. Typically, the functional form of u0 ðhÞ needs to be found using a numerical method. In our study, a numerical solution of u0 ðhÞ was obtained by modifying a quadratic function. The reason to fit u0 ðhÞ with a quadratic function

1

rotation angle (radians)

1926

0.8

0.6

0.4

0.2

0

Eu ¼ A

ð2Þ

Z

hmax

pffiffiffiffiffiffiffiffiffiffi cos h sin h sin u0 ðhÞJ 1 ðkr sin hÞeikz cos h dh;

Ez ¼ iA

ð3Þ hmax

0

0.2

0.4

0.6

0.8

1

1.2

polar angle θ (radians)

0

Z

0

Fig. 5. Function of u0 ðhÞ to achieve an NA-invariant flat-top focus, from NA = 0.82 to NA = 0.95.

pffiffiffiffiffiffiffiffiffiffi 2 cos h sin h cos u0 ðhÞJ 0 ðkr sin hÞeikz cos h dh;

0

0.06

ð4Þ

0.05

0.04

intensity (a.u.)

In the above formula, hmax is determined by the blocking aperture (also the numerical aperture). J n ðxÞ is a Bessel function of the first kind of order n. The radial symmetry is easily seen from the above expressions because none of the components is dependent on the observation angle us in the focal plane. Obviously, the intensity distribution is determined by the specific form of u0 ðhÞ. To produce a flat-top focus, the z-component must be balanced with the transverse component (see Fig. 1b, for example, for the balancing effect), which means the NA needs to be large enough for the z-component to gain a comparable strength with the transverse component. However, the increased intensity of the z-component with respect to the transverse component does not take on a simple functional form with

0.03

0.02

0.01

0 −3

−2

−1

0

1

2

3

r (wavelength)

Fig. 6. The intensity distribution in the focal plane, for NA = 0.88.

B. Hao, J. Leger / Optics Communications 281 (2008) 1924–1928

is to put more weight on the edge rays (than the center rays) to control the z-component. It produces the desired flat-top pattern for lower NA’s, but as h becomes large (i.e. for large values of NA), the z-component in the focal plane associated with the quadratic function cannot keep up with the corresponding transverse component and destroys the flat-top shape. This suggests that the initial quadratic function must roll off beyond some specific h. We then used another quadratic function to simulate the roll-off. By evaluating the integrals (2)–(4) as h grows larger, we found the specific roll-off point and determined the specific functional forms of the two quadratic functions. After some numerical optimization (by changing the coefficients related to the quadratics), we found an analytical form of u0 ðhÞ that yields the flat-top focus independent of NA, from NA = 0.82 up to NA = 0.95. The expression of u0 ðhÞ is the splicing of two quadratic forms and can be written as

u0 ðhÞ ¼

in the low NA region (where h is small), it does not affect the overall z-component very much as long as the high NA region is included. Fig. 7 shows one such example. In this particular case, the dotted curve takes a quadratic form, while the solid curve has a ‘‘plateau” in the low NA region, but has very similar values as the dotted one in the high NA region. They both achieve NA invariant beam shaping with the same NA range (in this case from NA = 0.82 to NA = 0.9). Another reason for the non-unique property is that the function of u0 ðhÞ is not required to be smooth or continuous. This is because the intensity distribution in the focal plane is determined by the integrals (2)–(4), where the smoothness or continuity of u0 ðhÞ is not necessary. This provides the possibility of different u0 ðhÞ yielding the same result after integration, hence destroys the uniqueness of the functional form. Of course if the desired NA application range changes, u0 ðhÞ will take some different forms.

0 6 h 6 sin1 ð0:86Þ

0:8769h2 2

1:8917ðh  1:2532Þ þ 1:0297

ð5Þ

sin1 ð0:86Þ 6 h 6 sin1 ð0:95Þ

where both h and u0 are in radians. The coefficients of the function are found by numerical simulations. Fig. 5 shows the function graphically. Note that it is continuous and piecewise smooth. For this specific functional form, the flat-top intensity in the focal plane is readily demonstrated. Fig. 6 shows one example of the flat-top focus for NA = 0.88. The top remains flat, from NA = 0.82 to NA = 0.95, while the width of the focal spot decreases as the NA increases. 3. Discussion Note that the functional form of u0 ðhÞ to achieve NAindependent beam shaping needs to be selected carefully by simulation. However, the functional form of u0 ðhÞ is by no means unique. There are two reasons for this. One is that the effect of different polarizations on spot shape is only significant at high numerical apertures where there are significant differences in the value of the z-component of polarization at the focused spot. The choice of polarization states at low NA’s has negligible effect on beam shape, and thus many different polarization functions are possible. As a numerical example, the z-component intensity of a focused radially polarized beam is only about 50% of that of the transverse component at NA = 0.5; while at NA = 0.8, the z-component intensity is about 1.8 times that of the transverse component for radially polarized light. To be able to obtain a flat-top focus, the z-component needs to be greater than the transverse component, as is shown in Fig. 1b, which requires high NA. On the other hand, if the NA is low, the relative intensity of the z-component is not very significant. So if u0 ðhÞ changes

So a reasonable design procedure would be to first fix the NA range required by the application and then develop the appropriate functional relationship by simulation. If an analytical function is required by the application, we can approximate the numerical functional form using standard curve-fitting techniques. We have chosen to use loworder polynomials to perform a piece-wise fit. In this paper, the NA range chosen (from 0.82 to 0.95) in Fig. 5 covers a large number of the high NA systems encountered in practice (for example [3–5]). The above discussion applies to plane wave illumination (uniform amplitude and phase in the pupil plane). Note

1

rotation angle (radians)

(

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0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

1.2

polar angle θ (radians)

Fig. 7. Two different curves of u0 ðhÞ to achieve NA independent focus from NA = 0.82 to NA = 0.9.

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B. Hao, J. Leger / Optics Communications 281 (2008) 1924–1928

that this technique of beam shaping can also be applied to some other forms of apodization [9,10]. One example is the Gaussian illumination, which is generated by most laser systems. With Gaussian illumination and the same blocking aperture, the ‘‘effective” NA becomes smaller. To achieve the same balancing effect between the z-component and the transverse component, the required numerical aperture is higher, which means the range of numerical apertures that produce flat-top beams is shifted to larger values. The specific functional form of u0 ðhÞ can again be determined by simulation to take into account the intensity profile of the Gaussian illumination. One concern regarding this method is how one can generate the required polarization. There are at least two possibilities in practice. One is to generate radially polarized light first using any one of a variety of methods [11–14]. The polarization could then be rotated locally using a spatial light modulator, to provide spatially variant retardance. Another method is to use form birefringence [15] to directly obtain the desired spiral polarization. 4. Conclusion In this paper, spiral polarization is proposed as an extension to the well-known cylindrical vector beam. The focusing properties have been studied and according to numerical simulation, the flat-top shaped focus is achieved independent of NA over an extended range. This selfadjustment with NA is the unique property of the spiral polarization. The technique may find application under circumstances where very small focused spots with flat-tops

are desired, and the exact numerical aperture is variable or unknown.

Acknowledgement The authors would like to thank Cymer Inc. for its generous support of this work.

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[6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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