Focused fields of given power with maximum longitudinal electric field component inside a substrate

Q. Y. van den Berg; S. F. Pereira; H. P. Urbach

doi:10.1364/JOSAA.33.001010

1. INTRODUCTION

When focusing a polarized beam with a diffraction-limited lens of high numerical aperture, refraction of the vector electromagnetic field must be considered, using vectorial diffraction theory proposed first by Ignatovky and later by Richards and Wolf to obtain the vectorial Airy pattern. With a radially polarized incident beam, the strong bending of the light rays causes the longitudinal electric field component in the focal point to become dominant. When focusing in air or in an immersion system, the maximum of the longitudinal field component is attained precisely in the focal point. For the same numerical aperture, the symmetrical squared amplitude distribution of the longitudinal field component is more confined than the elliptically shaped squared modulus of the total electric field of the classical Airy pattern obtained by focusing a linear polarized plane wave.

A small and symmetric focused spot has applications in optical lithography [1], microscopy [2–4], laser writing [5], and optical trapping [6]. In many of these applications, the light is focused through an interface into a medium of a different refractive index, which can have considerable consequences for the distribution of the different components of the focused vectorial field. Annular illumination [7] and shaded-ring filters [3] are commonly used in order to increase the relative strength of the longitudinal field, although this can lead to undesired strong sidelobes and loss of power. A new optimization of the longitudinal field in focus is required to take into account the influence of the susbtrate while preserving the power transmission. This can be considered an extension of the work of Urbach in [8] for the case of a layered interface.

Here, we present a closed form expression for the electric field in the lens pupil, which maximizes the longitudinal electric field component in the focal point inside a single-layer medium for a fixed power. The optimum incident field in the pupil is still radially polarized, but it depends on the position of the distance of the focal point from the interface. The optimum amplitude distribution is a continuous, monotonically increasing function of the radial coordinate in the pupil of which the precise shape depends on the numerical aperture. Furthermore, for focusing behind an interface, there is an additional phase distribution of the pupil field, which depends on the depth of the focal point inside the material.

2. OPTIMIZATION PROBLEM

We proceed by maximizing the longitudinal field component on the optical axis at an arbitrary depth behind the interface and define the pupil field, which attains this goal as the optimum field. We will first derive the angular spectrum of the optimum electric field in the focal region as a generally valid result that does not depend on the diffraction model used to simulate the lens. Then, we determine the optimum pupil field using vectorial diffraction theory proposed first by Ignatovsky and later by Richards and Wolf [9,10].

With respect to Cartesian coordinates $r = (x, y, z)$ , we consider the complex amplitudes of a time–harmonic electromagnetic field $E (r)$ and $H (r)$ . The incident field is propagating in the positive $z$ direction in the half-space $z < 0$ with real permittivity $ε_{1}$ . It is incident on the substrate, which fills the half-space $z > 0$ and has a permittivity $ε_{2}$ with non-negative imaginary part. If the imaginary part is positive, the material the substrate consists of is absorbing. The time dependence of the fields is given by the factor $\exp (- i ω t)$ , with $ω > 0$ , which is omitted throughout the rest of the paper.

The incident field in $z < 0$ can be expanded in plane waves. It is assumed that the sources that generate the field are at a distance that is far enough so that the waves that are evanescent in the positive $z$ direction can be neglected and, hence, only plane waves that propagate in the positive $z$ direction are included. We furthermore assume that only plane waves of which the wave vectors are inside the cone $\sqrt{k_{x}^{2} + k_{y}^{2}} < k_{0} NA \sqrt{ε_{1}}$ are contributing to the field, where NA is the numerical aperture of the lens in vacuum. Hence, the plane wave expansion of the incident field in $z < 0$ is

E (r) = \frac{1}{4 π^{2}} \iint_{\sqrt{k_{x}^{2} + k_{y}^{2}} \leq k_{0} NA \sqrt{ε_{1}}} A^{(1)} (k_{x}, k_{y}) e^{i k^{(1)} \cdot r} d k_{x} d k_{y},

H (r) = \frac{1}{4 π^{2}} \frac{1}{k_{0}} \sqrt{\frac{ε_{0}}{μ_{0}}} \iint_{\sqrt{k_{x}^{2} + k_{y}^{2}} \leq k_{0} NA \sqrt{ε_{1}}} k^{(1)} \times A^{(1)} (k_{x}, k_{y}) e^{i k^{(1)} \cdot r} d k_{x} d k_{y},

with

k = (k_{x}, k_{y}, k_{z}^{(1)})

being the wave vector with

k_{z}^{(1)} > 0

because the waves are propagating in the positive

z

direction, and

k_{0} = 2 π / λ_{0}

the wavenumber in vacuum,

NA = \sin θ_{\max}

, where

θ_{\max}

is the maximum angle between the positive

z

direction and the wave vectors in the first medium. We will define the vectors of a basis in

k

space for medium 1 and 2 labeled by the superscript j, respectively:

k^{(j)} = k_{∥} \cos φ \hat{x} + k_{∥} \sin φ \hat{y} + \sqrt{k_{0}^{2} ε_{j} - k_{∥}^{2}} \hat{z}, {\hat{p}}^{(j)} = \frac{k_{z}^{(j)}}{| k^{(j)} |} \cos φ \hat{x} + \frac{k_{z}^{(j)}}{| k^{(j)} |} \sin φ \hat{y} - \frac{k_{∥}}{| k^{(j)} |} \hat{z}, \hat{s} = - \sin φ \hat{x} + \cos φ \hat{y},

where

k_{∥} = \sqrt{k_{x}^{2} + k_{y}^{2}}

is defined as the transverse wavenumber. Every pair

(k_{∥}, φ)

with

0 \leq k_{∥} \leq k_{0} NA \sqrt{ε_{1}}

and

0 \leq φ \leq 2 π

corresponds to a wave vector. Because the electric field amplitudes of the plane waves satisfy

A^{(1)} (k_{x}, k_{y}) \cdot k^{(1)} = 0

, the vector

A^{(1)} (k_{x}, k_{y})

has only a

\hat{p}

and

\hat{s}

component:

A^{(1)} (k_{x}, k_{y}) = A_{p}^{(1)} (k_{x}, k_{y}) {\hat{p}}^{(1)} + A_{s}^{(1)} (k_{x}, k_{y}) \hat{s} .

Note that when the substrate is absorbing, $k_{z}^{(2)}$ is complex and, hence, a $p$ -polarized wave is elliptically polarized. An $s$ -polarized wave, however, is always linearly polarized.

From Eq. (3), it becomes clear that the $\hat{s}$ components of the plane waves do not contribute to the longitudinal field in the focal spot; hence, the optimum field consists only of the $\hat{p}$ component.

The time-averaged incident power flow in the positive $z$ direction through any $z = constant$ plane between the lens and the interface can be computed with the Poynting vector $\frac{1}{2} Re {E (r) \times H {(r)}^{*}} \cdot \hat{z}$ and Plancherel’s theorem:

P (A_{p}^{(1)}) = \frac{1}{8 π^{2} k_{0}} \sqrt{\frac{ε_{0}}{μ_{0}}} \iint_{k_{∥} \leq k_{0} NA \sqrt{ε_{1}}} {| A_{p}^{(1)} (k_{x}, k_{y}) |}^{2} k_{z}^{(1)} d k_{x} d k_{y} .

The presence of the interface affects the power flow due to reflection in the negative $z$ direction, as well as possible absorption in the second medium.

For the field transmitted in $z > 0$ , a similar plane wave expansion holds as Eqs. (1) and (2), except that the permittivity $ε_{1}$ must be replaced by $ε_{2}$ . We write $A_{p}^{(2)}$ instead of $A_{p}^{(1)}$ in $z > 0$ :

E (r) = \frac{1}{4 π^{2}} \iint_{k_{∥} \leq k_{0} NA \sqrt{ε_{1}}} A_{p}^{(2)} (k_{x}, k_{y}) {\hat{p}}^{(2)} e^{i k^{(2)} \cdot r} d k_{x} d k_{y} .

We have

A_{p}^{(2)} (k_{x}, k_{y}) = \frac{ε_{1}}{ε_{2}} \frac{| k^{(2)} |}{| k^{(1)} |} t_{p} (k_{∥}) A_{p}^{(1)} (k_{x}, k_{y}),

where

t_{p}

is the Fresnel coefficient for

p

-polarization, given following the convention of Born and Wolf as [11]

t_{p} (k_{∥}) = \frac{2 k_{z}^{(1)} ε_{2}}{k_{z}^{(1)} ε_{2} + k_{z}^{(2)} ε_{1}},

where the subscripts 1 and 2 denote the medium in which the quantity is measured, and the dependence of

t_{p} (k_{∥})

on

ε_{1}

and

ε_{2}

is not explicitly written. Therefore, for

z > 0

,

E (r) = \frac{1}{4 π^{2}} \frac{ε_{1}}{ε_{2}} \iint_{k_{∥} \leq k_{0} NA \sqrt{ε_{1}}} \frac{| k^{(2)} |}{| k^{(1)} |} t_{p} (k_{∥}) A_{p}^{(1)} (k_{x}, k_{y}) \cdot {\hat{p}}^{(2)} e^{i k^{(2)} \cdot r} d k_{x} d k_{y},

and, in particular, for

z > 0

,

E_{z} (0,0, z) = - \frac{1}{4 π^{2}} \frac{ε_{1}}{ε_{2}} \iint_{k_{∥} \leq k_{0} NA \sqrt{ε_{1}}} t_{p} (k_{∥}) A_{p}^{(1)} (k_{x}, k_{y}) \cdot \frac{k_{∥}}{| k^{(1)} |} e^{i k_{z}^{(2)} \cdot z} d k_{x} d k_{y} .

In this paper, we always place the geometrical focus of the lens at the origin $r = 0$ , i.e., in the interface. At a fixed depth $z_{0} > 0$ , we will consider $E_{z} (0, 0, z_{0})$ as a functional of $A_{p}^{(1)} (k_{x}, k_{y})$ . One may assume that $E_{z} (0,0, z_{0})$ is positive real because this assumption amounts to stating that the phase of the $z$ component of the electric field is zero at time $t = 0$ , and this can always be achieved by a shift in time. Then, we will find the plane wave amplitudes $A_{p}^{(1)} (k_{x}, k_{y})$ , which maximize the axial longitudinal field amplitude in the point $(0,0, z_{0})$ under the constraint that the mean power flow $P_{0}$ through the exit pupil be held constant, i.e.,

{\begin{cases} maximize E_{z} (0, 0, z_{0}) under the constraints \\ P (A_{p}^{(1)}) = P_{0}, Im E_{z} (0, 0, z_{0}) = 0 \end{cases} .

Solving Eq. (11) can be done with the Lagrange multiplier rule,

δ E_{z} (0, 0, z_{0}) - Λ δ P = 0,

where

Λ

denotes the Lagrange multiplier. From Eq. (12) follows a system of linear equations for the real and imaginary parts of the plane wave amplitudes through which they can be expressed in the Lagrange multiplier:

A_{p}^{(1)} (k_{∥}) = - \frac{1}{Λ} k_{0} \sqrt{\frac{μ_{0}}{ε_{0}}} \frac{ε_{1}}{ε_{2}^{*}} \frac{k_{∥}}{| k^{(1)} | k_{z}^{(1)}} {(t_{p} (k_{∥}) e^{i k_{z}^{(2)} z_{0}})}^{*} .

The Lagrange multiplier is obtained by substituting Eq. (13) into the power constraint:

Λ = \frac{1}{2 \sqrt{π}} \sqrt{\frac{k_{0}}{P_{0}}} {(\frac{μ_{0}}{ε_{0}})}^{1 / 4} \frac{ε_{1}}{| ε_{2} |} \cdot {[\int_{0}^{k_{o} NA \sqrt{ε_{1}}} \frac{k_{∥}^{3}}{{| k^{(1)} |}^{2}} \frac{1}{k_{z}^{(1)}} {| t_{p} (k_{∥}) |}^{2} e^{- 2 z_{0} Im {k_{z}^{(2)}}} d k_{∥}]}^{1 / 2} .

The magnitude of the plane wave amplitudes is independent on the location $z_{0}$ when medium 2 is nonabsorbing. There is, however, an additional phase modulation that is nonzero for all $z_{0} \neq 0$ . The integral in Eq. (14) cannot be computed analytically due to the Fresnel coefficients and must be evaluated numerically.

After substituting Eq. (13) into Eq. (9), the incident electromagnetic field in a point $r$ with cylindrical coordinates $(ρ, φ, z)$ and $z > 0$ can be expressed as

E (r) = - \frac{i}{2 π Λ} \sqrt{\frac{μ_{0}}{ε_{0}}} \frac{ε_{1}^{2}}{{| ε_{2} |}^{2}} [g_{1}^{2,1} (ρ, z) \hat{ρ} + i g_{0}^{3,0} (ρ, z) \hat{z}]

and

H (r) = - \frac{i}{2 π Λ} \frac{ε_{1}^{2}}{ε_{2}^{*}} k_{0} g_{1}^{2,0} (ρ, z) \hat{φ},

where

\hat{ρ}

,

\hat{φ}

, and

\hat{z}

are unit vectors of the cylindrical coordinate system and where the function

g_{l}^{ν, μ} (ρ, z)

is defined by

g_{l}^{ν, μ} (ρ, z) = \int_{0}^{k_{0} NA \sqrt{ε_{1}}} \frac{k_{0}}{k_{z}^{(1)} {| k^{(1)} |}^{2}} k_{∥}^{ν} {(k_{z}^{(2)})}^{μ} {| t_{p} (k_{∥}) |}^{2} \cdot J_{l} (k_{∥} ρ) e^{i (k_{z}^{(2)} z - {(k_{z}^{(2)})}^{*} z_{0})} d k_{∥} .

In Eq. (17), the special functions $J_{l}$ represent Bessel functions of the first kind and order $l$ .

3. OPTIMUM PUPIL FIELD DISTRIBUTION

Now we will address the problem of how to realize the optimum field as computed in Section 2 by focusing with a high numerical aperture lens. The optimum pupil field is based on a model for high NA focusing derived by Ignatovski and Richards and Wolf [9,10]. Spatial light modulators can then be used to shape the amplitude, phase, and polarization in the pupil. In the model of Ignatovski and Richards and Wolf, the electric field $E^{pupil}$ at a pupil point with polar coordinates $(ρ_{p}, φ_{p})$ is related to the field of the plane wave with transverse wave vector $(k_{∥}, φ)$ such that

ρ_{p} = f \frac{k_{∥}}{k_{0} \sqrt{ε_{1}}}, φ_{p} = φ + π,

where

f

denotes the focal distance. The complex pupil field is given by [12]

E^{pupil} (ρ_{p}, φ_{p}) = \frac{{(ε_{1})}^{1 / 4} \sqrt{k_{0} k_{z}^{(1)}}}{2 π i f} \cdot [A_{p}^{(1)} (k_{∥}, φ) \hat{ρ} + A_{s}^{(1)} (k_{∥}, φ) \hat{φ}] = \frac{i}{2 π Λ} \sqrt{\frac{μ_{0}}{ε_{0}}} k_{0} \frac{ε_{1}}{ε_{2}^{*}} \frac{ρ_{p}}{f^{3 / 2} {(f^{2} - ρ_{p}^{2})}^{1 / 4}} \cdot {(t_{p} (k_{∥}) e^{i k_{0} z_{0} \sqrt{ε_{2} - ε_{1} {(ρ_{p} / f)}^{2}}})}^{*} \hat{ρ} .

The optimum pupil field is radially polarized and has rotational symmetry in amplitude and phase. Its amplitude is a monotonously increasing function of the radial coordinate $ρ_{p} \leq f NA$ . This amplitude distribution depends on the dielectric permittivities on both sides of the interface and on NA but not on the focal plane location if and only if the media are nonabsorbing ( $ε_{2} \in R$ ). The phase does, however, depend on the assumed position of the focal point.

When focusing behind the interface in the second medium $(z_{0} > 0)$ , a $ρ_{p}$ -dependent phase modulation is introduced. In absorbing media, the exponential factor also contains additional amplitude modulation through the imaginary part of $ε_{2}$ , which will make the optimum pupil amplitude become $z_{0}$ dependent. Figure 1 shows the pupil field amplitudes for several NA and the same total power with $NA = 0.5$ , $NA = 0.8$ and $NA = 0.95$ , $ε_{1} = 1$ and $ε_{2} = 1.5$ . As a function of radius, these amplitude distributions appear to be similar, although the angular distribution of these incident fields is significantly different for different values of NA. Nevertheless, the dependency of the optimum pupil field on NA is much weaker than when focusing in air [8]. Particularly for far off-axis rays (close to $ρ_{p} / pupil radius = 1$ ), the Fresnel transmission coefficients in Eq. (19) have low values, thus slightly flattening the field amplitude in this regime. The choice of $ε_{1} = 1$ was made throughout this paper in order to demonstrate the largest possible effect of the interface. When the difference in permittivity before and after the interface becomes smaller, the influence of the interface is reduced and the system tends toward an immersion system. Other parameters are chosen in a way that they are representative for the applications mentioned in the introduction, but, since we derived closed expressions, they can easily be changed using Eqs. (17) and (19).

Fig. 1. Electric field amplitude in pupil plane for $NA = {0.5, 0.8, 0.95}$ with $ε_{1} = 1$ and $ε_{2} = 1.5$ and same total power.

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Figure 2 shows the pupil field amplitudes for several $ε_{2}$ and the same total power with $ε_{2} = 1$ , $ε_{2} = 1.5$ and $ε_{2} = 2.5$ , with $NA = 0.9$ and $ε_{1} = 1$ . The total incident power is held constant. The pupil amplitudes in Fig. 1 and 2 are independent of $z_{0}$ because $ε_{2}$ is real.

Fig. 2. Electric field amplitude in pupil plane for $ε_{2} = {1, 1.5, 2.5}$ with $ε_{1} = 1$ and $NA = 0.9$ and same total power.

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When absorption is included, it simultaneously affects the amplitude and phase of the pupil field. To demonstrate the effect on amplitude, Fig. 3 displays the optimum pupil amplitude for a medium with ( $ε_{2} = 1.3 + i \cdot 0.1$ , $ε_{2} = 1.3 + i \cdot 0.2$ ) and without absorption ( $ε_{2} = 1.3$ ). The field is maximized at a depth of $z_{0} = λ_{2} / 2 = λ_{0} / 2 \cdot Re {\sqrt{ε_{2}}}$ . Furthermore, $NA = 0.9$ , $ε_{1} = 1$ . Introducing absorption in medium 2 further reduces the optimum pupil field amplitude at large pupil radii. This effect can be understood, as light rays passing through the edge of the lens travel a greater distance to reach the focal spot, as compared with rays closer to the optical axis, and would be more strongly attenuated along the way.

Fig. 3. Electric field amplitude in pupil plane with and without absorption through $ε_{2} = {1.3, 1.3 + i \cdot 0.1, 1.3 + i \cdot 0.2}$ with $ε_{1} = 1$ at $z_{0} = λ_{2} / 2$ and $NA = 0.9$ and same total power.

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Finally, the phase front of the optimum pupil field is generally nonuniform unless $z_{0} = Im {ε_{2}} = 0$ . Figure 4 shows some phase distributions of the optimum pupil fields, which maximize the longitudinal field in focus at different depths $z_{0}$ . Other conditions chosen for this example are $NA = 0.9$ , $ε_{1} = 1$ , and $ε_{2} = 1.5 + i \cdot 0.1$ . Note that $z_{0}$ is measured in wavelengths units in medium 2.

Fig. 4. Optimum pupil phase distribution for $NA = 0.9$ , $ε_{1} = 1$ , $ε_{2} = 1.5 + i \cdot 0.1$ and different positions $z_{0}$ measured in wavelength units.

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To account for values of $z_{0} > 0$ , a defocus term arises in the optimum pupil phase whereas absorption leads to a nonspherical wavefront.

4. VISUALIZATION OF FOCUSED FIELDS

The optimum electromagnetic field in focus is rotationally symmetric around the $z$ axis; hence, both $E (r)$ and $H (r)$ are independent of $φ$ . For a given optical system with $NA = 0.9$ and an air/medium interface with $ε_{1} = 1$ and $ε_{2} = 2.6$ (note that common organic photoresists {Shipley-series [13], MAN-series [14]} have similar dielectric constants in the near-UV range), the focused field amplitudes resulting, which follow from Eq. (15), are plotted for the optimized as well as a uniform-amplitude (top hat) pupil field in Fig. 5 (note that the uniform amplitude field is zero in the center to avoid a polarization singularity). The interface is located such that the longitudinal fields peak at a depth $z_{0} = λ_{2} / 5$ . The longitudinal, transversal, and total focused field amplitudes are denoted by $| E_{z} |$ , $| E_{ρ} |$ , and $| E_{t} |$ , respectively, and the total field peaks at unity.

Fig. 5. Field amplitudes resulting from optimized and uniform amplitude (top hat), radially polarized pupil function for $NA = 0.9$ , $ε_{1} = 1$ , $ε_{2} = 2.6$ , $z_{0} = λ_{2} / 5$ . $| E_{z} |$ , $| E_{ρ} |$ , and $| E_{t} |$ denote longitudinal, transverse, and total field, respectively.

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For the case of an incident field with a top hat amplitude profile, the transverse electric field is the dominant field component. After optimization, the longitudinal field dominates over the transverse field, leading to a narrower total field distribution. This effect becomes stronger when considering the field intensities (squared magnitudes). Interestingly, the transverse field component of the optimized pupil field also becomes narrower, thus further reducing the width of the total field distribution.

Figure 1 seems to suggest that the optimum pupil field amplitude as a function of pupil radius depends only weakly on the numerical aperture. To demonstrate the dependency of the focused fields on NA, Fig. 6 shows the maximum longitudinal field $E_{z}$ , which is positive real by definition, for a focused field maximized on the optical axis at $z_{0} = λ_{2} / 2$ . Taking different values of $ε_{2}$ demonstrates that, as expected, the maximum longitudinal field decreases when either the real or imaginary part of the second dielectric permittivity increases.

Fig. 6. Maximum longitudinal field in the plane $z_{0} = λ_{2} / 2$ as a function of NA, for $ε_{2} = {1.5, 1.5 + i \cdot 0.15, 1.8, 2.0}$ and $ε_{1} = 1$ . Max $E_{z}$ is found on the optical axis.

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5. CONCLUSION

To conclude, we present a closed-form expression for the pupil field, which, when focused through an interface, maximizes the longitudinal electric field component at a selected depth in the substrate for a fixed total power. The optimum pupil field is radially polarized, has rotational symmetry in amplitude and phase, and the field amplitude increases continuously with distance from the optical axis. When focusing into another medium, the phase front shows an extra depth-dependent curvature. In media with absorption, both the phase and amplitude of the optimum pupil field change due to the imaginary part of the respective dielectric constant.

Because the longitudinal field has spatial dimensions below the diffraction limit, this framework can be used to obtain a small and symmetric focused spot inside a possibly absorbing medium at arbitrary depth, which has been implemented in the field of optical lithography [15], and has further applications in microscopy and laser machining. It is also useful for applications that make use of a strong longitudinal field, such as particle trapping [16].

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Focused fields of given power with maximum longitudinal electric field component inside a substrate

Abstract

1. INTRODUCTION

2. OPTIMIZATION PROBLEM

3. OPTIMUM PUPIL FIELD DISTRIBUTION

4. VISUALIZATION OF FOCUSED FIELDS

5. CONCLUSION

REFERENCES

Cited By

Figures (6)

Equations (19)

Journal of the Optical Society of America A