1. Introduction

Apodizing techniques have been developed in a number of optical imaging and processing systems to optimize the balance of power between the primary lobe and side-lobes of a propagating wavefront. For instance, apodization filters have been used in astronomy to improve discrimination of faint objects near much brighter sources by suppressing the side-lobe component in the aerial image of the bright source [1, 2]. In a similar fashion, apodization has been used in remote sensing to minimize radiance errors from diffraction off of cloud cover [3]. Apodizing filters can be used to improve signal-to-noise levels and depth discrimination in confocal and fluorescent spectroscopy [4, 5], to control the beam width of laser diodes in laser printers [6], to measure low-angle scattering of optical surfaces [7], or as a means to improve the lithographic depth of focus in optical lithography [8–10]. While the underlying principle is the same, i.e., that of controlling the magnitude of side-lobes induced by diffraction through a hard aperture by spatially modulating the transmission through the aperture, a number of specific transmission functions have been designed and implemented [1, 2, 6, 7, 11–13]. Among these, a Gaussian transmission function has proven to be both common and versatile [2, 6, 7, 12], according to which the transmission of a given aperture, such as the pupil, is a Gaussian or inverted Gaussian with predetermined width and amplitude. The fabrication of these filters commonly requires a complex thin-film deposition process. One approach is to approximate the Gaussian transmittance with a binary representation [12].

Another alternative is to introduce a semi-transparent medium between two spherical surfaces of lenses that are in near contact. In fact, a dye-containing solution was used in Ref. 7 for the preparation in this manner of a Gaussian apodizer operating at 633 nm.

In this paper we report on the design, assembly, and use of a Gaussian apodizer, which is achieved by filling the gap between two optical surfaces with a suitably absorbing gas. This apodizer has certain unique desirable properties. The degree of apodization is dynamically variable over a wide range, on the scale of minutes, controlled by varying the pressure and flow rate of the gas. Such capability can have many applications. For instance, in photolithography a variable apodizer could be dynamically tuned to optimize depth of focus, central-lobe width, and contrast for certain geometries or when switching between different photomasks or different illumination modes. Furthermore, the concept is applicable to wavelengths from the vacuum-UV to the infrared, with essentially the same setup. In addition, it may be especially useful when used with high-power lasers, which can damage more conventional thin-film or solid apodizers.

2. Apodization at 157 nm

Our demonstration has been carried out at 157 nm, where we constructed an instrument to measure the low-angle scatter of optical surfaces, such as the scatter that can lead to flare in optical lithographic systems. The choice of absorber gas is determined primarily by the wavelength. At 157 nm oxygen is suitable, since it is absorptive below ~185 nm. The Gaussian apodization is obtained by flowing oxygen at controlled partial pressures between the opposing spherical surfaces of two custom meniscus lenses. Under the paraxial approximation, the optical path difference between the elements scales quadratically with the radial distance, r, from the optical axis. Filling the space between the mated lens faces with an absorptive medium with absorption coefficient k_gas results in a Gaussian intensity transmission factor for the apodizing filter, P(r):

where R is the radius of curvature of the mated lens faces. At 157 nm, k_gas of molecular oxygen is 150 atm^-1cm^-1 (Ref. [14]).

The optical system that includes the apodizer is shown in Fig. 1. It uses a pulsed 157-nm fluorine excimer laser (Lambda Physik LPF 220) operating at 200 Hz. Since apodization is most efficient with a spatially coherent source, the multimode output of the laser was first spatially filtered with the aid of two pinholes fabricated in 25-μm-thick tungsten membranes. The pinholes were 400 μm and 75 μm in diameter, and separated by 600 μm. Based on application of the van Cittert–Zernike Theorem [15], incoherent light emanating from the first, 400-μm-diameter pinhole gives rise to approximately 75-μm zones of coherence after propagating 600 μm. Therefore, the second pinhole is matched to transmit a highly coherent beam.

 

Fig. 1. Schematic drawing of the experimental test-bed.

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The second pinhole is imaged onto the detector plane by the combination of a CaF₂ diverging lens with a 157-nm focal length of -36.2 mm and two spherical meniscus lenses, each having surface radii of curvature of 31.25 mm and 29.04 mm. The meniscus lenses are fabricated in fluorine-doped fused silica, which is sufficiently transparent at 157 nm [16] for this proof-of-concept design, and whose refractive index at 157 nm is 1.68. The 157-nm focal length of each meniscus is 400 mm. The meniscus lenses are mounted in a vapor cell allowing oxygen gas to flow between them. To distribute oxygen uniformly through the cell, three separate inlets and outlets are used. All optical elements are contained in aluminum enclosures interconnected with aluminum tubes and purged with boil-off nitrogen, so as to eliminate the chance of photoinduced contamination.

Although the use of an absorptive gas allows a simple delivery system to be constructed, the gas phase dynamics under laser radiation must be considered when determining the appropriate flow rate. In the present situation, oxygen readily dissociates under irradiation and forms ozone (O₃), which at 157 nm is approximately three times more transparent [14], and atomic oxygen in its ground state, O³P, which is transparent at 157 nm. Thus, in general the value of k_gas in Eq. (1) will depend on the steady-state concentrations of all three species of oxygen. These concentrations in turn will depend on the various chemical and photochemical rates, and on the residence time of the gas. The temporal evolution of molecular oxygen, ozone, and atomic oxygen is predicted using the Chapman reaction model, which was developed to simulate the behavior of ozone in the upper atmosphere. Ozone forms over a characteristic time scale ${\tau }_{O_{\mathit{3}}}$ given by [17]:

where k_b is the rate constant for the formation of ozone from atomic and molecular oxygen through a three-body collision, k_d is the destruction rate constant of ozone by its reaction with atomic oxygen, j_a and j_c are the dissociation rates of oxygen and ozone by 157 nm radiation respectively, and n_m is the total number density of gaseous species, including possibly carrier gases in addition to oxygen. Using the parameters summarized in Table 1, and for the average intensities of our low-angle scatterometer (less than 1 mW/cm²), we find that ozone forms on time scales longer than minutes. Note that in Table 1 we use the average intensity, I, even though the laser is a pulsed source, since no significant nonlinearities are expected even at the highest peak intensities of our setup, ~0.5 kW/cm².

Table 1. Rate Constants Used in Eq. (2), at 157 nm [14, 17, 18]

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As mentioned above, the steady-state concentration depends on the residence time as well. Indeed, in our case it is dominated by it. With the volume between the meniscus lenses being under 1 cm³, and the flow rates ~1 liter/min, the residence time is <100 msec. This is at least several orders of magnitude smaller thant ${\tau }_{O_{\mathit{3}}}$, and therefore the ozone concentration (and the atomic oxygen concentration) under our experimental conditions is negligible.

The efficiency of the apodizer is measured experimentally by spatial mapping of the intensity profile of the beam, both at its center and in the wings, where the diffracted signal is orders of magnitude lower than in the primary lobe. This task is carried out with a profiler (Fig. 1) based on a photomultiplier tube (Perkin Elmer channel model 910), which incorporates a CsI photocathode that is solar-blind above 230 nm. The detector is mounted on a precision x-y stage, allowing two-dimensional profiles to be performed. The photomultiplier linearity is determined by comparing its output with that of a pyrodetector standard. In performing measurements over the seven orders of magnitude required by our application, a set of attenuators are cycled into the beam-line. These attenuators are fabricated by depositing a thin-film of platinum on CaF₂. They have transmission of 16% or 40%, depending on the thickness of the film, and are calibrated individually in-situ using a pyrodetector. The appropriate combination of attenuators is chosen, in order to maintain a suitable signal level on the photomultiplier. A portion of the input beam is also split off by a CaF₂ window preceding the attenuators and is detected by a pyrodetector for reference measurements.

To test the operation of the apodizing cell, intensity measurements were first performed directly after the apodizing cell. The cell is flood illuminated by locating the diverging lens at the back focal plane of the lens combination in the apodizer cell. The intensity pattern of the resulting collimated beam is mapped by placing the scanner 100 mm from the meniscus lenses, and is shown in Fig. 2 for three concentrations of O₂ and for the control case of unapodized beam, when transparent N₂ flows through the cell. The dotted lines are theoretical curves, based on multiplying the Airy disk distribution emanating from the 75 µm pinhole by the Gaussian transmission factor given in Eq. (1). The introduction of oxygen clearly shows a marked sharpening of the central lobe and complete elimination of the side lobes. The experimental intensity profiles agree very well with the predicted values at the three different oxygen levels when k_gas is that of oxygen rather than of ozone, in agreement with chemical kinetics analysis above. The higher intensity levels in the tails with apodization are attributed primarily to multiple reflections off the uncoated optics. Ray tracing simulations indicate these beams come to a focus approximately half-way between the detector and lens combination before rapidly diverging, thus manifesting themselves as diffuse scattering. An additional feature of the data shown in Fig. 2 is a reduction in the peak intensity. This is due to a small gap between the meniscus lenses, which was maintained to prevent contact damage during sealing the backsides of the lenses. A curve fitting of the peak intensity vs. oxygen pressure yields a gap of 70±15 µm. We did not attempt to confirm the spacing independently, but it is reasonable given our procedure to assemble the apodizing cell.

 

Fig. 2. Intensity map through a flood-exposed apodizer cell at different levels of oxygen. Each solid trace represents a separate x or y directional scan. The dashed curves represent the theoretical intensity maps. In order to match the position of the diffraction nulls, the Airy distribution for a 65-µm, rather than 75-µm pinhole has been used in computing the theoret ical curves.

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Fig. 3. Intensity map in the image plane with 1 atm N₂ and 2 atm O₂ flowing through the apodizing cell. The dashed curves represent the theoretical profiles.

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Similar profiling measurements were performed in the image plane after repositioning the divergent lens to produce a 1:1 coherent imaging system, so that the 75-µm pinhole produces a geometric image of 75 µm in the absence of aberrations. This is the configuration for using the apodizer as part of a low-angle scatterometer. In this arrangement, a 4-mm-diameter field stop is placed right before the sample being measured, to limit the degree of scattered light from the lens entering the detector plane. Figure 3 shows the experimental results compared with the predicted curves. For clarity, only one line scan (x direction) is shown, and the traces have been normalized to unity peak levels. The simulated results incorporate the primary through fourth order spherical aberrations caused by the two spherical meniscus lenses. Due to the high curvature of the optics, the effect of the spherical aberration is pronounced and can be clearly seen in the unapodized case by the broadened first side-lobe. With apodization, the aberration is suppressed, as a smaller effective portion of the lens is used. Figure 3 also shows a ~ 10x reduction in the relative magnitude of the diffraction-induced first side-lobe, at ~±2 mm, as a clear result of apodization.

Another effect noticeable in Figure 3 also shows that there is a discrepancy between the experimentally achieved intensity profile and the simulated curves, that becomes significant at normalized intensity levels of ~ 10^-4 and below. This effect may be caused by higher order aberrations from lens fabrication imperfections not independently characterized and incorporated in the simulations and by scattering that passes through the field stop.

3. Summary

We propose a new concept of gas-based variable apodizers for producing a smoothly varying transmission filter. It is demonstrated at 157 nm using oxygen as the absorbing gas, for the purpose of quantifying the low-angle scatter of optical surfaces. It is, however, extendible to other wavelengths as well, by selecting suitable gases. As discussed in detail in this paper with respect to 157 nm, the main criterion for the gas is that it be absorptive at reasonable pressures. Other considerations include stabilization of the photochemically formed species and prevention of photochemically induced contamination of the optical surfaces. For instance, an apodizer operating at 193 nm could be based on O₃, N₂O, NO₂, or NH₃, which have absorption coefficients of 13, 2.5,19.3, and 100–400 atm^-1cm^-1, respectively [14], and whose decomposition products are not expected to leave deposits on the lenses.