Beyond the Rayleigh Limit in Optical Lithography

CHAPTER

8

Beyond the Rayleigh Limit in Optical Lithography Mohammad Al-Amria,b , Zeyang Liaob and M. Suhail Zubairyb a The National Center for Mathematics and Physics, KACST, P.O. Box 6086, Riyadh 11442, Saudi Arabia b Institute of Quantum Science and Engineering (IQSE) and Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA

Contents

1. 2.

3. 4.

5.

6.

7.

Introduction Classical Photolithography and the Diffraction Limit 2.1 Mask-Based Photolithography 2.2 Classical Interferometric Lithography Classical Multi-Photon Lithography Quantum Interferometric Optical Lithography 4.1 Entanglement Helps to Break the Diffraction Limit 4.2 A Proof-of-Principle Experiment for Quantum Interferometric Photolithography Subwavelength Interferometric Lithography Via Classical Light 5.1 Nonlinear Interferometric Optical Lithography by Controlling the Phase 5.2 Subwavelength Lithography by Coherent Control of Classical Light Pulses 5.3 Subwavelength Lithography Via Correlating Wave Vector and Frequency Resonant Subwavelength Lithography Via Dark State 6.1 Three-Level  Type System 6.2 Generalization to 2× System 6.3 Generalization to N ×  System 6.4 Some Concerns 6.5 Experimental Demonstration of This Scheme Subwavelength Photolithography Via Rabi Oscillations

410 413 413 415 416 418 418 421 423 424 426 430 440 440 442 443 446 447 450

Advances in Atomic, Molecular, and Optical Physics, Volume 61, Copyright © 2012 Elsevier Inc. ISSN 1049-250X, http://dx.doi.org/10.1016/B978-0-12-396482-3.00002-8. All rights reserved.

409

410

Mohammad Al-Amri et al.

7.1 7.2

Achieving the Subwavelength Pattern Arbitrary Subwavelength Patterns in a Macroscopic Area 7.3 Potential Realizations 8. Summary and Outlook Acknowledgments References

Abstract

452 456 458 460 461 462

It is well-known that traditional optical lithography is restricted by the Rayleigh limit such that the smallest feature that can be generated is restricted to half the wavelength of the light source. Thus light beams with shorter and shorter wavelength have been applied to print smaller and smaller circuit images. However, when it comes to the extreme ultraviolet or X-ray, severe problems can emerge. In the past 10–15 years, several novel optical lithography schemes have been illustrated to overcome the diffraction limit. In this article, we review these schemes and explain their basic principles with possible experimental realizations.

1. INTRODUCTION Computer is one of the most important inventions in the last century. The performance of the computer chips has increased dramatically over the past few decades, and meanwhile the size of the integrated circuits reduced roughly following the famous Moore’s law. Optical lithography has been the most important driving force for these changes (Alfred, 2001; Chris, 2007; Harry, 2005; Lu & Lipson, 2010; Mack, 2007; Seisyan, 2011). In fact, almost any electronic equipment that uses processors or memory to work, such as cellular phone, digital photo cameras or automobiles, is the beneficiary of the optical lithography. About 20 years ago, the smallest features printed with optical lithography were twice the wavelength used to print them. Today the industry is pressing toward the need for much smaller resolution. However, there is a diffraction limit that restricts the smallest patterns we can print to about half of the wavelength of light source (Abbe, 1873; Brueck et al., 1998; Rayleigh, 1879). Therefore, to make the chip smaller, we should switch to shorter working wavelength. For example, we quote an interesting statement by one of the experts (Rothschild, 2010): Without some invention that significantly changes the way optical lithography is practiced, a next-generation lithography technology—such as extreme ultraviolet lithography or

Beyond the Rayleigh Limit in Optical Lithography

411

electron projection lithography—will be required to extend the roadmap to 45 nm node and beyond. Nowadays, the working laser can operate in deep ultraviolet (DUV 190– 250 nm) (Chiu & Shaw, 1997; Marconi & Wachulak, 2010; Taylor et al., 1998). Using the immersion lithography technology, the half-pitch nodes currently obtained with 193 nm light are 45 nm and 32 nm (Ivan & Scaiano, 2010). New tricks such as double exposure lithography (DEL) or double patterning lithography (DPL) are possible to extend the resolution by a factor of 2 (Lee et al., 2008). However, these technologies are not possible without the development of new material with nonlinear response to the exposure dose. While we switch to shorter wavelength, two major problem arise (Williams et al., 2006): First, the traditional lens and the air absorb the light significantly. We need to invent new materials for the lens which is hard to come, and the system should work in a vacuum system that can be very expensive. Second, the bandgap of SiO2 is about 9 eV. When the wavelength of the light is close to or smaller than 138 nm, it will cause adverse charging in the SiO2 layer and destroy the substrate. This motivates us to go beyond the Rayleigh limit and study ways to overcome the diffraction limit. The diffraction limit not only affects the photolithography, but also plays an important role in the imaging system. The resolution of a far-field optical microscopy is also limited by the diffraction limit. In the past few decades, a number of schemes have been developed to improve the resolution of the microscopy. To get a better resolution, people have to switch to shorter wavelength (e.g., electron microscope and X-ray lithography, Rudenberg & Rudenberg, 2010; Spille & Feder, 1977; Vieu et al., 2000) which is usually invasive to the system. While confocal microscopy introduced optical sectioning and can get a better resolution than the conventional ones, it did not overcome the diffraction limit (Diaspro, 2010). Near-field scanning microscopy can obtain optical imaging with sub-diffraction resolution (Alkaisi et al., 2001; Betzig & Trautman, 1992; Binnig & Quate, 1986; Dryakhlushin et al., 2005; Ono & Esashi, 1998 ). These techniques are however surface bound and are thus limited in terms of applications. Two-photon fluorescence microscopy was first developed to achieve a higher resolution than classical one-photon fluorescence microscopy (Denk et al., 1990; Denk & Svoboda, 1997; Hell, 1994; Helmchen & Denk, 2005; Strickler & Webb, 1991a, 1991b). Stimulated-emission-depletion fluorescence microscopy (STEP) was then developed by Hell and Wichmann (1994) and the related concepts such as ground-state depletion (GSD) are then developed (Hell, 2007; Hell & Kroug, 1995). A number of experiments have also been carried out (Donnert et al., 2006; Klar & Hell, 1999; Klar et al., 2000; Rittweger et al., 2009). Some other techniques such as Spatially Structured Illumination Microscopy (SSIM) (Gustafsson, 2005), Photoactivated

412

Mohammad Al-Amri et al.

Localization Microscopy (PALM) (Betzig et al., 2006), Stochastic Optical Reconstruction Microscopy (STORM) (Zhuang, 2009) are also invented to achieve super resolution. Dark state (Agarwal & Kapale, 2006; Gorshkov et al., 2008; Kiffner et al., 2008; Yavuz & Proite, 2007) and resonance fluorescence (Chang et al., 2006; Chang et al., 2006; Macovei et al., 2007; Mompart et al., 2009; Sun et al., 2011; Qamar et al., 2000) are also employed to localize the atoms with a subwavelength resolution. In photolithography, several schemes have been proposed to break the diffraction limit in the past two decades. In 1992, Wu et al. pointed out that two-photon excitation in laser scanning photolithography can allow exposure of patterns not possible with conventional one-photon direct writing (Wu et al., 1992). Unfortunately, this scheme is based on point-by-point scanning which has limited applications. In fact the ordinary two-photon absorption can only achieve a sharper peak but not improve the spatial resolution. In 1999, Yablonovitch and Vrijen illustrated how to suppress the normal resolution term and get a super-resolution image based on two-photon absorption (Yablonovitch & Vrijen, 1999). The visibility of this scheme is reduced due to a constant background. In 2000, Boto et al. showed that quantum entanglement can successfully eliminate the normal resolution term without a constant background (Boto et al., 2000). After that several papers showed that quantum entanglement is not necessary to break the diffraction limit but the nonlinear response of the recording medium (Bentley & Boyd, 2004; Hemmer et al., 2006; Peér et al., 2004). In 2008, Kiffner et al. came out with a novel idea that subwavelength resolution can be achieved by preparing the system in a position dependent trapping state (Kiffner et al., 2008). In 2010, Liao et al. discovered that coherent Rabi oscillations can lead to subwavelength resolution (Liao et al., 2010). This method does not require quantum entanglement or multi-photon absorber but just quantum coherence of the medium. In addition to these schemes, there are some near-field subwavelength lithography methods such as photolithography based on surface plasmon (Liu et al., 2005; Luo & Ishihara, 2004a, 2004b; Martin, 2003; Schuller et al., 2010; Srituravanich et al., 2004; Xie et al., 2011), which will not be covered in this article. In this article, we mainly focus on optical lithography. In Section 2, we begin by introducing traditional photolithography, the interferometric photolithography and the diffraction limit. In Section 3, we introduce the classical multi-photon lithography. In Section 4, we show how to achieve subwavelength resolution via quantum entanglement. In Section 5, we will show quantum entanglement can be mimicked by carefully manipulating the classical light. In Section 6, we discuss a novel idea to break the diffraction limit based on dark state. In Section 7, we illustrate our novel and simple method based on coherent Rabi oscillations. Finally we present the summary and the outlook.

413

Beyond the Rayleigh Limit in Optical Lithography

2. CLASSICAL PHOTOLITHOGRAPHY AND THE DIFFRACTION LIMIT 2.1 Mask-Based Photolithography Mask-based photolithography is commonly used to print the circuit image in the industry nowadays (Brueck, 2005; Sheats & Smith, 1998). The typical setup is shown in Figure 1a. The light projects the image of the pattern on the mask onto the photoresist. Some places are bright while others are dark in the photoresist. The photoresist changes its solubility at the place where it is shined by the light. There are two types of photoresist: positive and negative (see Figure 1b). In a positive-tone photoresist, areas of the material that are exposed to light are removed after development. While in a negative-tone photoresist, the areas exposed by the light remain behind. After development, the pattern is printed onto the photoresist. The SiO2 layer is then etched at the place without the protection of the photoresist. The minimum feature size that a projection system can print is approximately given by the Rayleigh criterion (Abbe, 1873; Brueck et al. 1998; Rayleigh, 1879): λ , (1) CD = 0.61 NA where λ is the wavelength of the light used and NA is the numerical aperture of the lens as seen from the wafer (Figure 2). In most areas of optics, the numerical aperture is defined by NA = n sin θ where n is the index of

a

b

Light from laser source Photolithographic Process Photoresist Si Substrate

Condenser lens

Mask or Reticle

SiO2

Coating

Exposure Negative

Projection lens

Positive

Transfer Wafer

TAMU & KACST

Strip Figure 1

(a) Mask-based photolithography. (b) Positive and negative photoresist.

414

Mohammad Al-Amri et al.

Object Plane

θ

Figure 2

Numerical aperture of a lens.

refraction of the medium in which the lens is working (1.0 for air, 1.33 for pure water, and up to 1.56 for oils), and θ is the half-angle of the maximum cone of light that can enter or exit the lens. Equation (1) is also called the diffraction limit, i.e., for a working wavelength and the numerical aperture, this is the resolution limit. The diffraction limit can be explained by the loss of high spatial frequencies due to the evanescent wave (Alkaisi et al., 1999; Brueck, 2005; Neice, 2010). According to the Fourier optics, the electric field on the imaging plane is the summation of varies frequencies components emitted from the object plane: ε(x, y, z) =

  σ

kx

ky

εσ (kx , ky )eikx x+iky y+ikz z dkx dky ,

(2)

where σ is the polarization, z is the propagation direction, and kx2 +ky2 +kz2 = n2 ω2 /c2 (n is the refractive index of the medium, ω is the angular frequency of the light, and c is the speed of the light). For the high spatial frequency such that kx2 + ky2 > n2 ω2 /c2 , kz is pure imaginary which means that this component decays in the propagation direction. This corresponds to the evanescent wave and such waves can not reach the imaging plane in the far field.The highest spatial frequency that can reach the imaging plane is k =

kx2 + ky2 = nω/c, which corresponds to a resolution of 2π/k =

2πc/nω (Pendry, 2000). This corresponds to the maximal resolution for the field to be equal to λ/n. The corresponding maximal resolution for the intensity is therefore equal to λ/2n. Considering the aperture of the lens (see Figure 2), the maximal transverse wave vector that can reach the image plane is k = k sin θ = nω sin θ/c, and thus the maximum resolution is λ/2n sin θ which is the well known diffraction limit.

Beyond the Rayleigh Limit in Optical Lithography

415

2.2 Classical Interferometric Lithography Interferometric lithography is a new kind of mask-less lithography, which uses conventional laser light (Brueck et al., 1998; Menon et al., 2005a, 2005b; Pau et al., 2001). In interferometric lithography, two coherent plane waves of laser radiation converge from the apposite directions and hit a surface with θ being the angle between the direction of propagation and the substrate surface (see Figure 3). The wave vectors of the two light beams are: k1 = k( − cos θ, − sin θ), k2 = k(cos θ, − sin θ) where k = 2nπ/λ0 with n being the refractive index of the medium and λ0 being the wavelength of the light in the vacuum. The two coherent waves form a standing wave in the substrate plane. Consequently, the fringes pattern on the substrate is determined by the in-plane wave vectors k1 and k2 . The intensity distribution at the focal plane and along the x direction is:  2   I(x) ∝ expik1 ·x + expik2 ·x  = 2(1 + cos 2kx cos θ).

(3)

According to the criterion by Rayleigh (1879) the minimal resolvable feature size x occurs at a spacing corresponding to the distance between two intensity maxima. From Equation (3), we obtain 2k x cos θ = 2π, which leads to the following formula for the maximum resolution: x =

λ0 , 2n cos θ

(4)

thus the resolution is wavelength dependent and has its maximum value when x is minimum, and that happens when cos θ = 1. This leads to the classical diffraction limit x = λ0 /2n, where we consider the grazing limit θ = 0. λ

k1

k2 θ

λ

θ Substrate

λ 2 cos θ Figure 3 The scheme of interferometric lithography with classical light. Two counter-propagating plane waves interfere on a photosensitive substrate.

416

Mohammad Al-Amri et al.

Figure 4 Spatial distribution of excitation near the focus of a diffraction limited laser beam for one-photon absorption (a) and two-photon absorption (b). Figure reprinted with permission from Wu et al. (1992). Copyright 1992 by SPIE.

It is important to note that, in this analysis above, we have made no mention of the atomic structure of the photoresist. It is implicitly assumed that the atomic system is initially in the ground state and the ionizing rate is proportional to the intensity of the light incident on the photoresist. Thus the atomic response of the photoresist is considered to be linear.

3. CLASSICAL MULTI-PHOTON LITHOGRAPHY The concept of two-photon excitation was first described by Goeppert– Mayer (1931) in her doctoral dissertation, and first observed in cesium vapor using laser excitation by Abella (1962). Two-photon excitation is found to be able to improve the resolution in the fluorescence microscopy (Denk et al., 1990; Strickler & Webb, 1991b) and increase the data capacity of the storage (Strickler & Webb, 1991a). Such a process depends quadratically on the photon intensity I of the incident beam (Goeppert-Mayer, 1931; McClain, 1974), i.e., the excitation rate, or the number of photon being absorbed per unit time, W = δI 2 , where δ is the absorption cross section, typically of the order of 10−58 m4 s/photon. The quadratic dependence of the intensity improves the resolution. In Figure 4, we show the onephoton and two-photon excitation rate of a Gaussian shape pulse. From the figure, it is clear that the two-photon process provides a sharper feature. Similar idea is proposed to improve the resolution of photolithography (Kawata et al., 2001; Wu et al., 1992). By scanning the focal volume in a programmed 3D pattern through a thick positive photoresist, it is possible to produce patterns with high aspect ratio trenches and multilayered undercut. However, the point-by-point scanning is time-consuming which limits its applications.

Beyond the Rayleigh Limit in Optical Lithography

417

Figure 5 (a) The ordinary intensity fringe pattern, proportional to (1 + cos 2kx ), produced by converging rays as in Figure 3. (b) The intensity squared fringe pattern (3/2 + 2 cos 2kx + 1/2 cos 4kx ), which consists of a normal-resolution spatial harmonic at 2k , a super-resolution spatial harmonic at 4k , and a constant term. (c) Here (3/2 + 1/2 cos 4kx ), where the normal-resolution spatial harmonic at 2k was canceled. The super-resolution component at 4k remains, on a constant background. (d) By employing a photoresist with a sharp atomic transition the constant background is eliminated, leaving a pure super-resolution image. Figure reprinted with permission from Yablonovitch and Vrijen (1999). Copyright 1999 by SPIE.

Can we generate a super-resolution image by two-photon absorption in one step? Let us look at the two-photon process of a standing field. The excitation rate is proportional to (1 + cos 2κx)2 =

3 + 2 cos 2κx + cos 4κx, 2

(5)

where κ = k cos θ. Comparing the fringe pattern with the one-photon process, we see that the pattern of two-photon absorption is a mixture of a normal-resolution image represented by cos 2κx term and a superresolution image represented by cos 4κx. Indeed, cos 4κx, represents a doubling of the spatial resolution over the one-photon process. The cos 2κx term destroys this super resolution. From Figure 5a and b, we see that the two-photon process does have a sharper peak, but the period of the structure is the same as that for one-photon process. The resolution of the whole pattern does not improve!

418

Mohammad Al-Amri et al.

The second term in Equation (5) comes from the absorption of photons from different paths, i.e., one photon from the left and other photon from the right. If we can eliminate this term, we can obtain a super-resolution image. Yablonovitch and Vrijen (1999) showed that the normal resolution term can be entirely suppressed, using a classical frequency modulation scheme, where simultaneous absorption of a pair of photons is accompanied by a twofold spatial-resolution enhancement. Their scheme is shown in Figure 6. The incident rays on one edge of the lens have frequency ω0 , while rays on the other edge consist of two frequencies (i.e., ω1 = ω0 + δ, ω2 = ω0 − δ). Fringes resulting from the interference of rays from opposite edge oscillate rapidly at the different frequency δ and the normal-resolution image is washed away, forming a constant background. Provided that the frequencies 2ω0 and ω1 + ω2 are coherently related, the super-resolution image is stationary. The resulting fringes are indicated in Figure 5c. The constant background can be entirely eliminated in principle. If the atomic transition at 2ω0 is sufficiently sharp, the background two-photon transitions of the frequency combinations ω0 + ω1 , ω0 + ω2 , 2ω1 , and 2ω2 do not occur. In this case, the background should vanish and leave only the double frequency component (Figure 5d).

4. QUANTUM INTERFEROMETRIC OPTICAL LITHOGRAPHY 4.1 Entanglement Helps to Break the Diffraction Limit Quantum entanglement can effectively eliminate the normal resolution term in Equation (5) and achieve sub-diffraction limited pattern (Boto et al., 2000; Lee & Lee, 2008; Williams et al., 2006). The system is described schematically in Figure 7, where two photon beams are incident on a symmetric, loss-less beam splitter BS at ports A and B. The output beams get reflected off by a mirror pair into the substrate. The two beams get converged on the imaging plane. The photoresist consists of two-photon absorber. Now, let us first look at what happens when we start with the input state | 11 = |1A 1B , i.e., one photon from the upper arm and one photon from the lower arm, see Figure 7. Interference effect upon passage through a symmetric, loss-less beam splitter can cause the product state √ |1A 1B  to become the quantum entanglement state (|2C 0D  + |0C 2D )/ 2. Hence, after the beam splitter, the two photons emerge either both in the upper arm C or both in the lower arm D, but never one photon in each arm. Absorption of one photon from upper path and one photon from lower path never happens. The normal resolution term is therefore completely washed away and leaves only the super-resolution term. This is the basic principle of this scheme.

Beyond the Rayleigh Limit in Optical Lithography

419

ω1+ ω2

ω0+ ω0

Lens

ω + ω2

2ω0

1

θ

θ

Fringe Pattern Figure 6 The fringe pattern produced by two-photon excitation of a photoresist, in which the incident rays on opposite sides of the lens are separated into distinct frequency grouping. Figure reprinted with permission from Yablonovitch and Vrijen (1999). Copyright 1999 by SPIE.

Two photons are incident on the photoresist in such a way that both of them are either in the upper path or in the lower path. The deposition rate is then proportional to  2  2ikx  + e−2ikx  = 2(1 + cos 4kx). e

(6)

420

Mohammad Al-Amri et al.

Mirror

Substrate Α

C BS D

Β

θ

Mirror Figure 7 Interferometric lithography setup where two photon beams hitting a beam splitter at ports A and B, and then get reflected by two mirrors. The two photon beams get interfere on the substrate.

We can see clearly that the slowly oscillating term cos (2kx) has been completely eliminated and we are left with only cos (4kx) terms which gives the resolution of λ/4, half of the usual diffraction limit. The essential physics is simple: The two photons incident on a two-photon absorbing substrate corresponds to an effective doubling of the frequency or reducing the effective wavelength by a factor of 2, thus giving rise to a λ/4 resolution. This result can be generalized by considering the entangled state at ports C and D in Figure 7 to be in the so-called NOON state, i.e., | (N) = √1 ( |NC |0D + |0C |ND ). The deposition rate on an N-photon absorbing 2 substrate is then proportional to N = 1+cos 2Nkx, with resolution λ0 /2N. The generation of NOON state is not simple. This is a maximally entangled photon number state and it was suggested (Boto et al., 2000) that this state can be created using optical components such as parametric down converters, particularly for the case when N = 2. The schemes for the generation of NOON states with higher values of N in a deterministic manner remains a challenge. This proposed method for subwavelength lithography was generalized further to arbitrary patterns in one and two dimensions (Kok et al., 2001). This requires arbitrary entangled Fock states of the form | (N, M) = √1 (|NC MD  + |MC ND ). The application of these ideas 2 to quantum imaging lithography is discussed by Shih (2007).

Beyond the Rayleigh Limit in Optical Lithography

421

Besides the obvious difficulty of generating higher order NOON states, this method for subwavelength lithography suffers from some other serious problems as well (Agarwal et al., 2001). On the one hand, one needs to produce weak light field to contain only two photons per mode, and at the same time, use this particular field to excite two-photon absorption which requires a strong field. When one photon is localized, the momentum of the other photon becomes completely delocalized, and thus this photon can end up anywhere (Tsang, 2007). Hence, the general usefulness of this method is quite limited. The limitations of this N-photon entanglement technique are summarized in the following quote by Anisimov and Dowling (2009): Nearly ten years ago, a quantum optical approach to imaging quantum imaging, which uses path-entangled states was suggested as a solution for this requirement of ever-shorter wavelengths and a way to beat the classical limit (Abella, 1962). This turned out to be easier said than done. While quantum lithography is viewed as one of the killer apps for the nascent field of quantum imaging, the bugaboo in its implementation has been the continued lack of the right kind of multi-photon photoresists that would operate at the low flux levels required for a real proof-of-principle experiment. 4.2 A Proof-of-Principle Experiment for Quantum Interferometric Photolithography A proof-of-principle experiment was reported (D’Angelo et al., 2001) utilizing the entangled photon pairs in a Young double-slit experiment. A similar experiment in the context of measuring the de Broglie wavelength of two-photon wave packets in a Young double-slit experiment was also reported by Fonseca et al. (1999). The schematic setup of the actual experiment is illustrated in Figure 8. However, in order to explain the essential physics of the subwavelength lithography scheme based on entangled light we first look at the setup shown in Figure 9. Here the two slits are placed symmetrically on the left and right sides of the entangled photon source. Region V at the center of Figure 9a is the place where an entangled photon pair can be generated such that the photons of the same pair propagate horizontally and in opposite direction. They are indicated by straight lines as seen in the figure. There is one photon counting detector on each side. These two detectors scan symmetrically in x direction for the arrival of the photon pair, and register the coincident “clicks." The joint detection counting rate is sin c2 (2β) which has a narrower feature than the classical pattern by a factor of 2 (here β = πa sin θ/λ, a is the width of the slit and θ is the scattering angle) (Pittman et al., 1995; Strekalov et al., 1995). This proof-of-principle

422

Mohammad Al-Amri et al.

Figure 8 Schematic of the experimental setup. The 458 nm line of an argon ion laser is used to pump a 5 mm BBO crystal that produce pairs of orthogonally polarized signal and idler photons, which emerges collinearly. The pump is separated from the signal–idler pair by a mirror M that reflects the pump while transmits the signal–idler pair. A cutoff filter F is used to suppress the pump further. The signal–idler beam passes through a double slit, which is placed close to the output side of the crystal, and is reflected by two mirrors, M1 and M2 , onto a pinhole P followed by a polarizing beam splitter PBS. The signal and idler photons are separated by PBS and are detected by the photon counting detectors D1 and D2 , respectively. The output pulses of each detector are sent to a coincidence counting circuit for the signal–idler joint detection. Figure reprinted with permission from D’Angelo et al. (2001). Copyright 2001 by the American Physical Society.

experimental setup can indicate that entanglement can break the one-slit diffraction limit. If we replace the single slit in the above setup with a double slit, see Figure 9b, we can observe the two-photon interference corresponding to the case when N = 2. The entangled photon pairs will either pass through the upper two slits or through the lower two slits. In this situation, we can find that the double-slit two-photon spatial interference pattern has a higher modulation frequency than the classical double-slit interference pattern. In the actual setup shown in Figure 8, the two-photon state N = 2 is generated via spontaneous parametric down conversion (SPDC). In the experiment, the double slit must be placed as close as possible to the output surface of the BBO crystal. In this case, the two entangled photons travel through the same slit and we can erase the first-order interference. Figure 10 shows the experimental results, where Figure 10a shows the distribution of coincident detection events versus the rotation angle θ of mirror M1 . In Figure 10b we present the first-order interference-diffraction pattern of a classical light by the same double slit in the same experimental setup. When comparing the two-photon interference-diffraction pattern result with that of the first-order interference-diffraction pattern of a classical light, one can

Beyond the Rayleigh Limit in Optical Lithography

423

Figure 9 Schematic of a two-photon diffraction-interference thought experiment. Detectors D1 , D2 perform the joint coincidence detection for entangled photon pair “signal and idler” which are represented by the right and left sides of the sketch. Scheme with a single slit in (a) and with a double slit in (b). Figure reprinted with permission from D’Angelo et al. (2001). Copyright 2001 by the American Physical Society.

see that the two-photon interference-diffraction pattern has a higher spatial interference modulation period and a narrower diffraction pattern width.

5. SUBWAVELENGTH INTERFEROMETRIC LITHOGRAPHY VIA CLASSICAL LIGHT The practical difficulties of the experiment presented in the previous section have stimulated several elegant high resolution schemes without entanglement. In these schemes the resolution enhancement can result solely from the properties of the N-photon absorption process if we can somehow eliminate the low spatial frequency components while retaining the high frequency components (Bentley & Boyd, 2004; Boyd & Bentley, 2006a, 2006b; Chang et al., 2006; Chang et al., 2006; Hammer et al., 2006; Kawabe et al., 2007; Peér et al., 2004). In the following subsections, we introduce these subwavelength lithography schemes.

424

Mohammad Al-Amri et al.

Figure 10 (a) Experimental measurement of the coincidences for the twophoton double-slit interference-diffraction pattern. (b) Measurement of the interference-diffraction pattern for classical light in the same experimental setup. Figure reprinted with permission from D’Angelo et al. (2001). Copyright 2001 by the American Physical Society.

5.1 Nonlinear Interferometric Optical Lithography by Controlling the Phase Bentley and Boyd (2004) pointed out that super-resolution of quantum lithography results primarily from the nonlinear response of the recording medium and not from quantum features of the light field. The basic principle of their proposal is shown in Figure 11. The key feature of this setup is that one component of the light beam is shifted by a phase and this phase is incremented by a fixed amount for the successive laser pulse. The successive mth laser pulse will have relative phase given by

φm = 2πm/M,

(7)

Beyond the Rayleigh Limit in Optical Lithography

425

Mirror Δφm N−photon absorber

BS θ

Mirror Figure 11 Sketch of the used technique, where intense laser pulse is separated into two equal beams at a beam splitter BS. One of them gets phase shifted, while the other is not. The two beams are brought together on a recording medium that functions by means of N-photon absorption. Figure reprinted with permission from Bentley and Boyd (2004). Copyright 2004 by the Optical Society of America.

where M is the total number of the pulse. The deposition rate of the Nphoton absorber is I(N, M) =

M 

(Em E∗m )N ,

(8)

m=1

where Em = eikx cos θ + e−ikx cos θ ei φm .

(9)

If the phase shifts φm were not introduced, the resulting deposition rate is simply I(N, M) = [1 + cos (2kx cos θ)]N which generate the same spatial period as one-photon interference pattern, but with a sharper features. However, if the phase is introduced as appearing in Equation (9), the slowly varying terms can be averaged out, leaving only a spatial component cos (2Mkx sin θ), and possibly harmonics of this component if N is at least twice as large as M. Therefore, the pattern generated in this way can be M times better than that allowed by the normal interferometric lithography. To achieve the optimum minimum feature size, we can take θ as 0◦ , which is the situation of two beams striking the recording plate near the grazing incidence. The visibility of this pattern is given

426

Mohammad Al-Amri et al.

by

 AN,M + AMHo  , V= AN,0 + AMHe

(10)

where AN,M =

2N! (N − M)!(N + M)!

(M = 0),

AN,0 =

2N! , 2(N!)2

(11)

where AN,M is the desired component, AN,0 the dc component of the deposition pattern, AMHo the odd harmonics, and AMHe the even harmonics of the desired frequency. A proof-of-principle experiment was conducted to demonstrate the resolution enhancement in this scheme (Figure 12). In this experiment, the properties of an N-photon absorber are simulated by an Nth-harmonic generator followed by a CCD camera. This meant that the desired harmonic is recorded by the CCD and any other light was spectrally filtered out. The repeated M laser shots data is collected and summed by a computer. Another technical point is that the prism is mounted on a micrometercontrolled translation stage, which is used as the phase shifter. When N = M = 1, we obtain the usual interference pattern that would be recorded on a linear absorber (see Figure 13a). As seen in Figure 13b and c, the fringes become narrow but the spacing between them remains constant as N is increased. When M  2, the resolution is enhanced. This is seen in Figure 13d and e where the resolution is enhanced by a factor of 2 when M = 2, while the resolution enhancement is three times when M = 3. Moreover, when N = M = 3, the data shows a fringe spacing that is one-third of that for N = 1. The fringe visibility is reduced as predicted by the theory. The conclusion is that this technique that uses classical light can lead to an improvement in the resolution. The procedure is quite straightforward to implement. However, the primary drawback of this technique is the lack of suitable N-photon absorbing media, especially for large N. Also, for increasing M, each individual phase shift becomes smaller, which requires greater phase resolution. 5.2 Subwavelength Lithography by Coherent Control of Classical Light Pulses In another subwavelength resolution scheme Peér et al. (2004) showed that super resolution can be achieved by coherent control of the classical light pulses without entanglement. The basic idea is as follows: If, in a N-photon absorbing medium, the excitation lifetime is much longer than the pulse duration, the two excitations induced by two pulses can still interfere to each other even if they do not overlap in time. Moreover, because two excitation pulses do not overlap in time, the cross absorption terms

Beyond the Rayleigh Limit in Optical Lithography

427

Figure 12 The experimental setup. Here, the output of a picosecond Nd:YAG laser is directed onto a thin plate beam splitter. The transmitted component propagates to a right-angle prism, where it is translated and reflected back to the beam splitter, while the reflected component from the beam splitter propagates to a plane mirror and is reflected back at an angle such that it will overlap the translated component in the detection plane. Figure reprinted with permission from Bentley and Boyd (2004). Copyright 2004 by the Optical Society of America.

are eliminated. For two exciting pulses (E1 and E2 ), the intensity of the excitation is  2   N I(x) ∝ EN 1 (x, ωA ) + E2 (x, ωA ) ,

(12)

where x is the spatial coordinate. We note that the mixed terms such as p N−p are absent and the resolution is enhanced by a factor N. E1 E2 To achieve a desired narrow lithography spot, we should shape the spatial phase fronts of the excitation pulses at the focus such that they interfere constructively in the center of the spot and destructively near the

428

Mohammad Al-Amri et al.

Figure 13 Measured intensity distributions for (a) M = N = 1, (b) M = 1, N = 2, (c) M = 1, N = 3, (d) M = N = 2, (e) M = 2, N = 3, and (f) M = N = 3. Note that the first three patterns have the same period (because M = 1) but that the fringes become sharper with increasing N. Note also the doubling of the fundamental frequency in (d) and (e) and the tripling of the frequency in (f). Figure reprinted with permission from Bentley and Boyd (2004). Copyright 2004 by the Optical Society of America.

edges. For example, we should search for M pulse fields such that M 2     N I(xf ) =  Ek (xf ) ,  

(13)

k=1

where I(xf ) is the desired spot. Figure 14 is a schematic setup where a glass plate delays a segment of the pulse with respect to the other. If the delay between the pulses is tuned correctly, it will lead to the desired constructive interference at the focal spot while destructive interference at the edge. A Fourier equivalent of the segmentation scheme was suggested for doubling the resolution with two-photon absorption (Korobkin & Yablonovitch, 2002; Yablonovitch & Vrijen, 1999). While this scheme works for two-photon case, generalization to the N-photon case is not straightforward and its implementation is more complicated. An experiment is conducted to prove the resolution enhancement of this scheme. The experimental setup is presented in Figure 15 which utilize the technique in Figure 14. The pulses (100 fs around 778 nm) are emitted from a Ti:Sapphire laser. The cylindrical telescope weakly focuses the laser into the Rb cell. The delay line introduces a relative delay between the two halves of the pulse. The cylindrical lens in front of the cell is used to tightly focuses the beam in the perpendicular dimension in order to increase the

Beyond the Rayleigh Limit in Optical Lithography

429

Figure 14 Schematic setup for generating quantum interference in lithographic medium. A glass plate delays half of a planar pulse with respect to the other half. As a result, the nonlinear medium at the focus is excited by two instant pulses. The delay between the pulses can be fine tuned by a small tilt of the glass, in order to generate two-photon interference. Figure reprinted with permission from Bentley and Boyd (2004). Copyright 2004 by the Optical Society of America.

Figure 15 Experimental configuration and relevant level diagram for atomic Rb. Figure reprinted with permission from Peér et al. (2004). Copyright 2004 by the Optical Society of America.

signal. The right part of Figure 15 is the energy level diagram of Rb in which the 5S–5D two-photon transition centered at 778 nm was used in this experiment. To avoid the one-photon excitation of the intermediate 5P level (at about 780 nm), a pulse shaper is placed in front of the cylindrical telescope to block the resonant frequency at 780 nm. Moreover, a π phase shift to frequencies above and below the resonance can be induced by the pulse shaper to maximize the two-photon excitation (Dudovich et al., 2001). Finally, the two-photon excitation can be detected by imaging the resulting fluorescence at 420 nm onto an enhanced CCD camera. In Figure 16, results are shown where double resolution is observed. Two CCD images of dark spot (destructive interference in the center of the spot) and the corresponding transverse line cross sections are shown

430

Mohammad Al-Amri et al.

Figure 16 Experimental results. (a) Images and transverse cross sections of “dark spots” (destructive at the center) for a short relative delay (crosses: data, gray line: theoretical fit) and a long relative delay (circles: data, line: theoretical fit), demonstrating the double resolution of two-photon interference compared to one-photon interference. (b) is the corresponding two-photon “bright spot” as compared to the diffraction limited one-photon spot (dashed). Figure reprinted with permission from Peér et al. (2004). Copyright 2004 by the Optical Society of America.

in Figure 16a. Here, two distinct cases should be noticed. First, a regular one-photon interference is observed when the delay was tuned shorter than the coherence length of the pulse. Second, two-photon interference is observed when the delay was tuned far beyond the coherence length. We can see that the dark spot in the second case is about half of that of first case. In Figure 16b “bright spot" (constructive at the center) is observed in the two-photon interference regime. Here we also see that the central lobe is about twofold narrower than the one-photon case. This experiment verified a scheme for sub-diffraction limit that relies on the quantum nature of the lithographic material and not of the exciting field. In order for this method to be practical, a nonlinear lithographic material with a narrow excitation line is required. However, similar to previous scheme, it suffers from the low efficiency of N-photon absorption and the high intensity requirement, especially for large N. This is a significant obstacle in making use of this scheme in real lithography. 5.3 Subwavelength Lithography Via Correlating Wave Vector and Frequency Hemmer et al. (2006) showed that the diffraction limit can be broken by correlating wave vector and frequency in a narrow band, multi-photon detection process that uses Doppleron-type resonances. The motivation for this work comes from two points of view. First, the early theoretical

Beyond the Rayleigh Limit in Optical Lithography

431

suggestion (Berman & Ziegler, 1977; Haroche & Hartmann, 1972; Kyröla & Stenholm, 1977), and experimental observation (Freund et al., 1975; Reid & Oka, 1977), of directional multi-photon resonances, called “Doppleron,” in saturated absorption spectroscopy. Second, the recently initiated research (Herkommer et al., 1997; Qamar et al., 2000; Zubairy et al.„ 2002), where an atom (or molecule) can be localized to subwavelength precision based on the conditional detection of fluorescence photons as the atom passes through a standing-wave field. The basic idea of this scheme is shown in Figure 17. Two counterpropagating plane waves consisting of signal frequencies ν± interfere on a photosensitive substrate. The drive fields ω± assist a directional resonance for pairs of signal photons, i.e., ωab = 2ν± − ω± .

(14)

If the detection bandwidth is narrow, the ± channels will realize distinct resonances (Mollow, 1968). The atoms will absorb two photons from the left beam or from the right beam, but never one photon from each beam, which is similar to the path-number entanglement in quantum field lithography. As a consequence, the one-photon interference term will be suppressed and keep only the pure two-photon interference term which has a resolution half of the diffraction limit.

5.3.1 Illustrative Calculation for the Case N = 2 Doppleron-type resonances can be observed in a two-level system, provided that the one-photon detunings and field strengths dominate the linewidths in a saturated absorption process (Pritchard & Gould, 1985).

ω− ν Δ1+

ν

ω

ω θ

θ

ν−

λ

Δ1−

ν Δ2+

λ/

θ

ν−

ω−

ν− Δ2−

Figure 17 Subwavelength interference with classical light. Two counter-propagating plane waves consisting of signal frequencies ν± interfere on a photosensitive substrate. The drive fields ω± assist a directional resonance for pairs of signal photons. Figure reprinted with permission from Hemmer et al. (2006). Copyright 2006 by the American Physical Society.

432

Mohammad Al-Amri et al.

The theory of photoelectron counting has been developed in the semiclassical (Mandel et al., 1964) and quantum field (Scully & Lamb, 1969) regimes. In Figure 17, two signal frequencies and two drive frequencies are used to complete three-photon resonance for each direction. The intermediate levels cj are off-resonant by detunings 1± = ωc1 b −ν± and 2± = ωac2 −ν± . Assuming that the signal Rabi frequency S is the same for the two transitions, the interaction Hamiltonian in the rotating-wave approximation (S , D ν± , ω0 ) is given by   HI = S |c1  b|ei 1± t + |a c2 |ei 2± t + h.c.   (15) + D |c1  c2 |ei( 1± + 2± )t + h.c. , where the second term was written make use of Equation (14). Next we derive the Schrödinger equations for the state amplitudes. For large onephoton detunings j±  S , D , the intermediate levels cj can be adiabatically eliminated by setting the time derivatives of the slowly varying amplitudes, c˜ j = cj exp ( − i j± t), to zero. This furnishes an effective coupling between levels a and b: i˙a −

2S 2 D a=− S b, 2± 1± 2±

(16)

and similarly with a ↔ b and 1 ↔ 2. Apart from dispersive phase shifts, the effective coupling is thus described by a three-photon Rabi frequency, eff = (2S D )/( 1± 2± ). In the usual perturbative regime, 1/ j± t 1/eff , the rate of excitation from b to a is given to lowest order by a thirdorder Fermi Golden rule:     2  2 D   R(3) = 2π  S (17)  δ(ωab + ω± − 2ν± ).  1± 2±  This gives the effective rate of two-photon absorption of the signal field ν± when assisted by the drive field ω± . The application to subwavelength interference proceeds as follows: – As the ± channels realize distinct resonances, the atoms will absorb two photons from one signal beam or the other, but never one photon from each beam. – As a consequence, the spatial period of the fringes will carry the twophoton wavelength, which is one-half the wavelength of each photon, the same as achieved by a quantum, entangled state of the form |2, 0+ |0, 2.

Beyond the Rayleigh Limit in Optical Lithography

433

The net electric field as seen by the atoms on the surface of the photoresist consist of two pairs of counter-propagating signal fields (of same intensity) as well as the normally incident drive fields (see Figure 17) can be written as:



E(x, t) = E S ei(k+ x−ν+ t) + ei(k− x−ν− t) + ED e−iω+ t + e−iω− t + c.c., (18) where k± = ±(ν± /c) cos θ. Hence, the third-order excitation rate of the atoms takes the general form  2  t3  t2  d  t R (x, t) ∝ dt3 dt2 dt1 a|HI (x, t3 )HI (x, t2 )HI (x, t1 )|b ,  dt 0 0 0 (19) where the interaction Hamiltonian is given in Equation (15). Under conditions of three-photon resonance, the leading contributions to the above integral will comprise exactly the two channels for the frequency-selective excitation shown in Figure 17, whose rates were calculated in Equation (17). One ends up with the only two significant terms in the field product where the same beam, + or −, contributes twice: (3)

R(3) (x, t) ∝ (3) r± (t)

 =



t 0

dt3



t3 0

dt2

0

 d  i2k+ x (3) (3) 2 r+ (t) + ei2k− x r− (t) ; e dt

t2

(20)







dt1 ES ei 1± t1 ED e−i( 1± + 2± )t2 ES ei 2± t3 ,

(21) where the dipole moments have been suppressed. If the one-photon detunings are large, j±  ν+ − ν− , then the excitation amplitudes r± (t) are approximately equal, and the single beam, two-photon spatial frequencies 2k± make up the interference pattern, i.e., the inter-beam cross terms exp[i(k+ + k− )x] are absent because they are out of three-photon resonance. Here one can see that the two-beam semiclassical lithography exactly simulates quantum field lithography (Boto et al., 2000) with unlimited spatial coherence. Moreover, the visibility is only limited by the small difference in excitation amplitudes of the two channels in Equation (20).

5.3.2 Generalization to N Photons We now turn to a multi-photon resonance. The schematics for the system are given in Figure 18. Two bunches of signal fields counter-propagate along the substrate (θ = π/2) and a drive field is incident normally. Either bunch of fields together   with the drive field can excite the multi-photon transition from level b to |a. The photons from the signal fields of frequencies νn± are absorbed and the photons of the drive field with frequency ω0

434

Mohammad Al-Amri et al.

a N+

N-

1+

1-

0

0

2Nsin

b

a c3 c1

N

3

1

0 2 1

0 4

c2

c2N-2

(2N-2)

2

b

Figure 18 (a) The scheme of interferometric lithography. Two bunches of signal fields counter-propagate (θ = π/2) and the drive field incidents normally. (b) The level structure of the substrate atom. Either bunch of fields together with the drive field satisfies the multi-photon resonance. n± is the detuning of intermediate level cn . Figure reprinted with permission from Sun et al. (2007). Copyright 2007 by the American Physical Society.

are emitted. The N signal photons satisfy a frequency summation resonance condition N 

νn± = ωab + (N − 1)ω0 = Nν0 ,

(22)

n=1

such that the N-photon wave vector, Nν0 /c = 2π/(λ0 /N) is the same for both bunches. We further require that any interchange of photons between bunches, νn+ ↔ νn − , results in a loss of resonance. Therefore only two resonant processes make up the interference. The electric field on the surface is: E(x, t) =

N 

ESn+ ei(kn+ x−νn+t) + ESn− ei(kn− x−νn− t) + ED e−iω0 t + c.c.,

n=1

where kn± = ±(νn± /c). In the level structure of the substrate, the intermediate levels cj are off-resonant by detunings 1± = ωc1 b − ν1± , 2± = ν1± − ω0 − ωc2 b , . . . , (2N−2)± = ωac2N−2 − νN± .

Beyond the Rayleigh Limit in Optical Lithography

435

Under the conditions of multi-photon resonance, the leading contributions to the multi-photon excitation rate come from the two resonant processes, i.e.,  2 Nν x  d  i Nν0 x (2N−1) (2N−1) −i c0 (2N−1)  c R (x, t) ∝ r+ (t) + e r− (t) . (23) e dt  If the one-photon detunings are large and ESn± are suitably chosen, the excitation amplitudes r± (t) can be made approximately equal with a phase difference. Factoring them out we find that the remaining expression looks like the interference of single photon absorption with k = Nν0 /c. So the exposure pattern are fringes with distance λ0 /2N. The semiclassical scheme to multiple beams can be generalized as seen in Figure 19 for N = 2. Each point on the slit plane is associated with two complementary frequencies, ν1k and ν2k , that satisfy a sum frequency resonance achieved through opposing spatial chirps created using inverted prisms. Then, photon pairs from a single spatial point on the slit plane will be absorbed collinearly (i.e., same wave vector) in the focal plane. This simulates the multi-mode state vector |2, 0, . . . , 0 + · · · + |0, . . . , 0, 2. As shown for quantum field lithography (D’Angelo et al., 2001), this would achieve subwavelength resolution not only in the carrier fringe (double-slit interference), but also in the envelope (single slit diffraction). The ratio of the

ν

ν

ω

ν

ω

ν

Figure 19 Subwavelength diffraction for classical light. Two laser pulses are given opposite spatial chirps using inverted prisms, and the resulting beams are combined by a beam splitter (BS) to illuminate the slit plane with a position-dependent frequency doublet, such that ν1k + ν2k = const. This creates a correlation between wave vector and frequency pairs in the focal plane of lens L3, and writes a two-photon pattern onto the Doppleron substrate in both carrier and envelope. Figure reprinted with permission from Hemmer et al. (2006). Copyright 2006 by the American Physical Society.

436

Mohammad Al-Amri et al.

pulse bandwidth to detector bandwidth determines the effective number of wave vectors constituting the diffraction pattern, or equivalently, the number of discrete partitions of the slit apertures. We note that the above semiclassical approach can be adapted to imaging, for example, two lenses in an f − f − f − f configuration. Here, one introduces a correlation between wave vector and frequency in the focal plane of the first lens, after the light has passed through the object, i.e., once the angular spectrum of the light is prescribed by the diffracting apertures. This can be accomplished by using a filter array in the focal plane that selects the desired spatial chirp from a broadband input. Using a dual filter array, one can associate a frequency pair (ν1k , ν2k ) with each wave vector such that the sum frequency is fixed: ν1k + ν2k = const. The result is a subdiffraction image spot (airy disk) created on the substrate in the image plane when vignetting due to the lens apertures is taken into account. As in the diffraction scheme, the bandwidth of the multi-photon process effectively discretizes the angular spectrum on the substrate, which in turn determines the resolution needed for the filter array in this imaging scheme. A concern was expressed by Cho (2006) that this scheme may generate only a tight pattern of parallel lines. We now turn to the possibility of generating arbitrary subwavelength patterns using this scheme.

5.3.3 Generation of Arbitrary Patterns Here we discuss the procedures to obtain arbitrary patterns in both one and two dimensions (Sun et al., 2007). The pattern is described by a truncated Fourier series and the scheme is based on multiple exposures. In order to enable subwavelength resolution, we need to have the fundamental frequency much larger than the signal frequencies. This can be done by modifying the resonance condition.

(a) Arbitrary 1D Pattern According to Equation (22), the fundamental frequency ν0 is the average of νn± . This means that the fringes can be subwavelength, however, for arbitrary pattern we need more harmonic components. Another point to note is that ν0 is also limited by the level separation ωab . To remove these limitations, we change the resonance condition to nN 

νn± = ωab + (nN − 1)ωN = Nν0 .

(24)

n=1

This is the key equation. The main difference is that the number of signal fields involved in a multi-photon resonance changes from N into nN . As far as the frequency summation equals Nν0 , there is no requirement

Beyond the Rayleigh Limit in Optical Lithography

437

this process must absorb N photons. The fundamental frequency ν0 can be much larger than νn± to obtain a subwavelength resolution. ν0 can even be much larger than ωab , which is a unique property due to the Dopplerontype resonances of this scheme (Hemmer et al., 2006; Sun et al., 2007). The drive frequency is also changed from ω0 to ωN , which can be different for each N. The arbitrary photon number nN and the drive frequency ωN provides more freedom to choose the fields. In order to write an arbitrary 1D pattern, we start with a high frequency ν0 . For any N (=1, 2, 3, …) we can always find some suitable nN and ωN to achieve the multi-photon wave vector Nν0 /c. Two bunches of signal fields grazing from +x and −x directions give the excitation rate (2nN −1)

R

 2 Nν x  d  i Nν0 x (2nN −1) −i c0 (2nN −1)  c (x, t) ∝ r+ (t) + e r− (t) . e  dt

(25)

For each N we can use the above method to make an exposure. After multiple exposures we get fringes corresponding to N = 1, 2, 3, . . . , Nmax. The final pattern is

P(x) =

N max  tN  N=1

=

N max  N=1

=Q+

0

R(2nN −1) (x, t)dt

2  2  Nν0 x Nν x   (2nN −1)   i c −i c0 iθN  cN r (tN ) e +e e  N max  N=0

2Nν0 x 2Nν0 x + bN sin , aN cos c c

(26)

where cN is the ratio coefficient in R(2nN −1) . Here P(x) is a truncated Fourier series with a penalty deposition Q. Such a series can approximate any 1D pattern if enough components are included. The coefficients and phases of each component can be controlled by ESn± and tN .

(b) Arbitrary 2D Pattern Next, we would like to see how this method can be applied to generate arbitrary pattern in two dimensions. Such a generalization still relies on multiple exposures, using two bunches of signal fields each time. However, the direction of these two bunches is not limited to the x axis. In this approach, we firstsend in the two bunches from the opposite directions ±(N xˆ + Myˆ )/ N 2 + M2 and make an exposure. The sum

438

Mohammad Al-Amri et al.

frequency of either bunch should satisfy n√

N 2 +M2 

νn± =



N 2 + M2 ν0 .

(27)

n=1

Like the 1D case, we have the multi-photon excitation rate  2 Nν x+Mν y  d  i Nν0 x+Mν0 y (2n−1) −i 0 c 0 (2n−1)  c r (t) + e r (t) e + −   . dt (28) ˆ Then change the directions of these two bunches into ±(N x − M yˆ )/  N 2 + M2 and make another exposure. Note the field coefficients and phases for the twoexposures could be different. For each nonzero (N, M) pair that satisfies N 2 + M2  Nmax we make two exposures like this. If N or M = 0 then only one exposure is needed. From Equation (21) we find the final exposure pattern   2ν0 (Nx + My) P(x, y) = aNM cos c √ (2n√

R

N 2 +M2

−1)

(x, t) ∝

0<

N 2 +M 2 Nmax

 2ν0 (Nx + My) A + a2NM + b2NM c 2ν0 (Nx − My) 2ν0 (Nx − My) + dNM sin +cNM cos c c    2 + d2NM = Q + + cNM √

+ bNM sin

0

N 2 +M 2 Nmax

 2ν0 My 2ν0 Nx cos × (aNM + cNM ) cos c c 2ν0 My 2ν0 Nx +( − aNM + cNM ) sin sin c c 2ν0 My 2ν0 Nx cos + (bNM + dNM ) sin c c  2ν0 My 2ν0 Nx +(bNM − dNM ) cos sin . c c

(29)

This is a truncated 2D Fourier series with a penalty deposition. It can approximate any two-dimensional pattern in principle. As an example, we consider the test function  2ν y h if − π2 < 2νc0 x , c0 < π2 , F(x, y) = (30) 0 elsewhere.

Beyond the Rayleigh Limit in Optical Lithography

439

Figure 20 Approximation of a 2D pattern using direction variation and multiple exposures. Here the penalty deposition is already subtracted to get the truncated Fourier series. The upper limit is Nmax = 10. Figure reprinted with permission from Sun et al. (2007). Copyright 2007 by the American Physical Society.

For the upper limit Nmax = 10 we get the truncated Fourier series P(x) − Q as shown in Figure 20. Except for the abrupt ramp and the corners of the hat, the error to the test function is within ±0.1h, which is acceptable considering the number of components included. The penalty deposition Q = 2.15h. Q and h are in arbitrary unit. This unit has to be chosen carefully to ensure the photoresist threshold dose falls between Q and Q + h (Levinson, 2001). For the places with exposure dose close to Q, the photoresist only has a small loss after the development. While for the places with exposure dose close to Q + h, the photoresist is completely removed. The scheme due to Sun et al. (2007) has many advantages: It requires neither superposition of entangled Fock states, nor broadband sensitive substrate. It can approximate any 2D pattern in principle. In all the interferometric schemes discussed so far, we require N-photon absorbers. However, whereas the earlier schemes require the signal frequency summation equal to the level separation of the substrate, in the present scheme the summation can be much larger than the level separation. The way to get larger summation is by increasing νn± or add another signal field. As a result, higher fundamental frequency can be obtained which basically means smaller pattern, and more Fourier components. This is the main advantage of this scheme. The main limitation of this scheme is the same as before: Due to multiphoton absorption, we require highly intense fields which may make the experimental realization quite difficult. With these difficulty in mind, we now turn to possible schemes for subwavelength lithography that require resonant atom field interaction.

440

Mohammad Al-Amri et al.

6. RESONANT SUBWAVELENGTH LITHOGRAPHY VIA DARK STATE In 2008, Kiffner et al. presented an alternative and novel scheme for resonant subwavelength lithography without the requirement of an N-photon absorption process (Kiffner et al., 2008). This scheme relied on the phenomenon of coherent population trapping (CPT) (Arimondo, 1996; Scully & Zubairy, 1997). Atoms are prepared in a position dependent state, the subwavelength spatial distribution coming from the phase shifted standing wave patterns in a multi-level resonant atom-field system. 6.1 Three-Level  Type System It is known that CPT occurs in a three-level  type system as shown in Figure 21a. The two ground states are represented by |b1  and |b2 , which are resonantly coupled to the excited state |a1  by laser fields with Rabi frequencies R1 and S1 , respectively. In such configuration, we can get the dark state once the system is optically pumped into a coherent

a

b a1

1

1

b2

2

2

1

1

b1

a2

a1

b3

b2

b1

c

aN a2

a1

N

1

b1

2

1

b2

2

b3

bN

N

b N+1

Figure 21 Considered level schemes of the substrate. The ground states |bn  and |bn+1  are resonantly coupled to the excited state |an  via Rabi frequencies Rn and Sn , respectively. Each excited state |an  decays to the ground states |bn  and |bn+1  by spontaneous emission. (a) Single  system. In (b), a sequence of two  systems is displayed. (c) General level scheme with N excited and N + 1 ground states as a sequence of N -type systems. Figure reprinted with permission from Kiffner et al. (2008). Copyright 2008 by the American Physical Society.

Beyond the Rayleigh Limit in Optical Lithography

441

superposition of the two ground states which is then decoupled from the applied light fields. The dark state is given by  (31) |D  = (S1 |b1  − R1 |b2 )/ |S1 |2 + |R1 |2 . In this scheme, R1 and S1 represent standing waves in the z direction with wave number k0 = 2π/λ0 that are formed by plane waves incident on the substrate, see Figure 22a. We can see from Equation (32) that the population of these two ground states in |D  depend very much on the ratio of the Rabi frequencies R1 and S1 . If the standing waves corresponding to R1

Figure 22 (a) The standing wave patterns R1 and S1 are formed by two plane waves Xi , Yi with wavelength λi . The period of each intensity pattern is given by λ0 /2, where λ0 = λi / cos θi , and θi is chosen such that the effective wavelength in the substrate plane is equal to λ0 for both R1 and S1 . Subfigures (b) and (c) correspond to the  system shown in Figure 21b. Part (b) illustrates the intensity profiles of the standing waves R1 and S1 according to Equation (31). Note that |R1 |2 and |S1 |2 are not drawn to scale. The solid line in (c) shows the population of state |b1  corresponding to R1 = (0 /10) cos (k0 z) and S1 = 0 sin (k0 z).

442

Mohammad Al-Amri et al.

and S1 are phase shifted with respect to each other, then the ratio R1 /S1 becomes position dependent. Hence the population in any ground level (|b1  or |b2 ) can be made position dependent. The question we address is whether a subwavelength population distribution can be obtained in one of the ground levels (say |b1 ). Let us look at the case that the two standing waves are phase shifted by π/2, i.e., R1 = 0 cos (k0 z), S1 = 0 sin (k0 z). (32) The populations of |b1  and |b2  in |D  are then given by |S1 |2 = [1 − cos (2k0 z)]/2, |R1 |2 + |S1 |2 |R1 |2 = [1 + cos (2k0 z)]/2. | b2 |D |2 = |R1 |2 + |S1 |2

| b1 |D |2 =

(33a) (33b)

The two ground states populations show the same spatial modulation as the intensity profiles of the standing waves corresponding to S1 and R1 , respectively. It is important to note that the populations do not depend on the maximal Rabi frequency |0|, but rather on the ratio of the Rabi frequencies R1 and S1 . Here the atomic population in (say level |b1 ) is modulated with spatial frequency 2k0 giving a resolution of λ0 /2 which gives the same result as the Rayleigh limit. This same limit is obtained by assuming a linear response of a two-level atomic system. Here, in the three-level atomic system, we have recovered the same limit but with very different physics. A question is whether we can obtain subwavelength resolution beyond Rayleigh limit using the dark state physics used here. We will address this question in the following sections. Before moving to the more complicated system, we point out another interesting feature in the present three-level system. For unequal amplitudes of the Rabi frequencies of the two standing-wave fields, a single very narrow spatial structure at a controllable position within a range of λ/2 can be generated. For example, if we choose

R1 = (0 /10) cos (k0 z),

S1 = 0 sin (k0 z),

(34)

the population in the ground state |b1  is shown in Figure 22c. Here the population in level |b1  is unity everywhere except at those point where S1 = 0. This can be used to write desired structures point by point. 6.2 Generalization to 2× System The above result for a single  type three-level system can be generalized to 2 ×  or so-called M system, see Figure 21b. The generalization of the

Beyond the Rayleigh Limit in Optical Lithography

443

dark state Equation (32) for this system is given by Zubairy et al. (2002)  |D2×  = (S1 S2 |b1  − R1 S2 |b2  + R1 R2 |b3 )/ C2 ,

(35)

where S1,2 and R1,2 are the driving fields, and C2 is the normalization constant and given by: C2 =

3 n−1  

|Rk |2

n=1 k=1

2 

|Sj |2 .

(36)

j=n

The probability to find the system in state |b1  is proportional to |S1 S2 |2 . This involves the product of the fields S1 and S2 . If both S1 and S2 have a sinusoidal oscillation behavior with respect to position, i.e., S1 ∼ sin (k0 z) and S2 ∼ sin (k0 z + φ), we obtain |S1 S2 |2 ∼ [cos (φ) − cos (2k0 z + φ)]2 .

(37)

It is crucial to choose the relative phase shift of the two standing waves as φ = π/2, in order to get: |S1 S2 |2 ∼ [1 − cos (4k0 z)]/2.

(38)

The population oscillations with wave number 4k0 are obtained, while the contribution with wave number 2k0 has been canceled, see Figure 23c. The spatial resolution is half of the classical limit! However, we should note that for the moment we have neglected the normalization constant C2 in Equation (36) which is also position dependent. Here, unlike previous schemes by Yablonovitch and Vrijen (1999), Bentley and Boyd (2004), Peér et al. (2004), and Hemmer et al. (2006) where one needs high light field intensities, the scheme can work at very low laser intensities. This happens because there is no need for nonlinear transition amplitudes between different states but rather one exploits the nonlinear dependence of the ground state population probabilities on the Rabi frequencies, which only depends on relative field strengths. 6.3 Generalization to N ×  System In this section we extend the generalization to level schemes with an N ×  structure, see Figure 21c. In the interaction picture and in rotating-wave approximation, the interaction Hamiltonian of the N ×  system takes the form: N    HN× =  Rn |an  bn | + Sn |an  bn+1 | + H.c. , (39) n=1

444

Mohammad Al-Amri et al.

Figure 23 (a) Each standing wave pattern Rn , Sn is formed by two plane waves Xi , Yi with wavelength λi . The period of each intensity pattern is given by λ0 /2, where λ0 = λi / cos θi , and θi is chosen such that the effective wavelength in the substrate plane is equal to λ0 for all Rn and Sn . Subfigures (b) and (c) correspond to the M system shown in Figure 21b. Part (b) illustrates the intensity profiles of the standing waves Rn and Sn (n = 1, 2) according to Equation (36). Note that |Rn |2 and |Sn |2 are not drawn to scale. The solid line in (c) shows the population of state |b1  corresponding to Equations (37) and (39) with η = 1/20. It varies with wave number 4k0 . The dotted line is the corresponding result with nonzero ground state decoherence rates γcoh . We set γcoh = γ , where γ is the full decay rate on the |an  ↔ |bn±1  transition. Figure reprinted with permission from Kiffner et al. (2008). Copyright 2008 by the American Physical Society.

where H.c. denotes the Hermitian conjugate. One key assumption is that the resonance frequencies of the various transitions are sufficiently distinct such that the Rabi frequencies Rn and Sn can be chosen individually. The dynamics of the atomic density operator can be described by a master equation ∂t  = −i[Hint , ]/ + Lγ ,

(40)

Beyond the Rayleigh Limit in Optical Lithography

445

where Lγ is just the spontaneous emission of the N excited states to the ground states. We assume that each excited state |an , in this system, decays to both ground states |bn  and |bn+1 . One needs to pay attention to the case of steady state in order not to have singular cases. The reason is that the system, in this case, depends very much on the initial condition. This can be avoided by setting either all Rabi frequencies Rn or all Sn (1  n  N) to be different from zero at any point in space. The steady state of the system is then evaluated by Matsko et al. (2003) to be |DN×  = √

N+1 n−1 N   1  ( − 1)n+1 Rk Sj |bn , CN n=1 k=1

where CN =

N+1  n−1  n=1 k=1

|Rk |2

N 

(41)

j=n

|Sj |2

(42)

j=n

  is the normalization constant and we set 0k=1 = N j=N+1 = 1. Thus as in the case of the single  system the atoms are optically pumped into a dark state |DN× . From now on, we suppose that the atoms have reached this steady state. As discussed in the previous subwavelength schemes, the main key point here is to have a product of N sinusoidal waves with wave number k0 to display spatial oscillations with wave number Nk0 only. However, the question is what about all other harmonics with wave number nk0 with n  N? The answer simply is that they can be canceled with a suitable choice of the relative phase shifts of the standing waves. This property is described by the trigonometric identities N 

sin[k0 z + (n − 1)π/N] =

n=1 N 

sin (Nk0 z) , 2N−1

sin[k0 z + (2n − 1)π/(2N)] =

n=1

cos (Nk0 z) . 2N−1

(43a)

(43b)

It is straightforward to see the applicability of these identities to this system. For this, we notice that the coefficient of |b1  in the expansion of the dark state in Equation (42) is proportional to the product of all Rabi  frequencies Sn , i.e., b1 |DN×  ∼ N n=1 Sn . Similarly, the matrix element  R involves the product of all Rabi frequencies Rn .

bN+1 |DN×  ∼ N n=1 n If we choose the position dependence of Rn and Sn according to

Sn (z) = Sn sin[k0 z + (n − 1)π/N], Rn (z) = Rn sin[k0 z + (2n − 1)π/(2N)],

(44a) (44b)

446

Mohammad Al-Amri et al.

it follows from Equations (42) and (43) that we have | b1 |DN× |2 = A1 [1 − cos (2Nk0 z)]/2, 2

| bN+1 |DN× | = AN+1 [1 + cos (2Nk0 z)]/2.

(45a) (45b)

This equation is the main result of this scheme and it shows that the population of ground state is modulated by a spatial frequency N times of the classical diffraction limit. We again need to note that the amplitudes  N 2 N−1 ) and A 2 N−1 ) also depend A1 = N N+1 = n=1 |Sn | /(CN 4 n=1 |Rn | /(CN 4 on position which will be discussed in next subsection. 6.4 Some Concerns In the previous subsections, we mentioned that the normalization constant is position dependent. This may, in general lead to undesirable spatial oscillations for the atomic population. In order to have a high lithography contrast, a full population oscillation amplitude is required. In principle, it is best to set the amplitudes A1 and AN+1 to be equal to unity, and that can be obtained if the parameters Rn and Sn in Equation (42) are chosen according to |R1 | = |SN | = η0 ,

0 < η 1,

|RN | = |S1 | = |Rn | = |Sn | = 0 ,

(46a) 1 < n < N,

(46b)

where 0 is an arbitrary positive Rabi frequency. Thus, the laser fields driving the outermost ground states should be much weaker than all other fields. In this case, the amplitudes A1 and AN+1 are then given by

(47) A1 = AN+1 = 1/ 1 + η2 fN (z) , where the function fN is independent of η. As η is much smaller than unity, A1 ≈ AN+1 ≈ 1 which is what we want. We also note that the population of the remaining ground state (1 < n  N) are suppressed by a factor of η2 . The second concern for this scheme is that CPT mechanism relies very much on the preservation of the ground state coherence in order to evolve into the stationary dark state. The question then is: Is this scheme still valid for large ground state decoherence rate? The answer is yes, even a large ground state decoherence rate γcoh does not affect the applicability of this scheme as one can see in Figure 21c, the dotted curve coincides with the solid line curve. Here dotted curve shows the result for the same parameters as the solid line, but with γcoh set equal to the population decay rate γ on the dipole-allowed transitions from the excited state to the ground states. The third concern is that, in all our calculations, we assume Rn and Sn have the same wave numbers k0 , but we also require that the frequencies

Beyond the Rayleigh Limit in Optical Lithography

447

of the fields Rn and Sn are distinct for individual addressing. These two assumptions seems to be inconsistent. However, both conditions can be met by choosing the appropriate incident angles θi , such that k0 = ki cos θi . Despite of that, for the case of N ×  systems with larger N, it can be a challenging task. However, Kiffner et al. (2008) estimate the influence of small wave vector mismatch and find that the scheme also works for that mismatch. In this proposed scheme, a desired 2D final pattern can be achieved via multiple exposure with different harmonics based on a Fourier decomposition as discussed before. We would need a medium that supports the generation of oscillations with maximal wave number 2Nk0 , where all smaller wave numbers 2nk0 with 0 < n  N can be generated by appropriately modifying the incident angle θ (Bentley & Boyd, 2004). The required harmonics can also be generated without changing θ using different n ×  (n  N) subsystems of the same full level structure.

6.5 Experimental Demonstration of this Scheme Here we discuss an experiment where the scheme presented above based on dark state can be used to obtain sub-diffraction imaging. The experiment approach of Li et al. (2008) is based on coherent population trapping in Rb vapor. The experimental setup is schematically shown in Figure 24. Before describing the setup, we briefly discuss the three-level  system that was used, see the inset in Figure 24. Probe and drive fields are applied to the three-level  atoms which, in this case, are 87 Rb atoms with |a = |52 P1/2 , F = 1, m = 0, |b = |52 S1/2 , F = 2, m = −1, and |c = |52 S1/2, F = 2, m = +1. The system evolves to the dark state which is given by  |D = (p |c − d |b)/ 2p + 2d , where d is the Rabi frequency of the drive field, whereas p is of the probe filed. Usually, d  p . When drive field is nonzero, the dark state practically is |b and the medium is transparent to the probe beam. However, if the drive field is zero, then the dark state is |c, and the probe beam can be absorbed at these positions. Therefore, the intensity profile of the transmitted probe beam is modulated by the spatial intensity of the drive beam. An external cavity diode laser is sent through a polarization-preserving single-mode optical fiber. The output laser is vertically polarized and split into two beams: drive and probe. The probe beam carries a small portion of the laser intensity, and its polarization is rotated to be horizontal. The drive beam is split into two beams that cross at a small angle, using a Mach–Zehnder interferometer, see the dashed square in Figure 24. This trick is used to generate a double-peak spatial distribution for the drive field. Figure 24a shows neatly a two-peak interference pattern of crossing

448

Mohammad Al-Amri et al.

Figure 24 Experimental schematic. λ/2: half-wave plate; λ/4: quarter-wave plate; L1, L2, L3: lenses; Mach–Zehnder interferometer MZ; piezoelectric transducer PZT; polarizing beam splitter PBS, photodiode PD; CCD camera CCD. Image (a) is the spatial intensity distribution of the drive field. Image (b) is the beam profile of the parallel probe beam without the lens L1. Image (c) is the beam profile of the diffraction limited probe beam with the lens L1. The inset is the energy diagram of the three-level  Rb atom. Figure reprinted with permission from Li et al. (2008). Copyright 2008 by the American Physical Society.

beams. Now, the probe and drive beams combine on a polarizing beam splitter. This leads the probe field and the interference pattern of the drive field to be overlapped in a Rb cell. Just before the cell, a quarter-wave plate converts probe and drive beams into left and right circularly polarized beams. The output beam is directed through Rb cell which is filled with 87 Rb and has a length of 4 cm. Magnetic shield is applied to isolate the cell from the environmental magnetic fields. Inside the cell a solenoid provides an adjustable, longitude magnetic field. The cell is installed in an oven that heats the cell to reach an atomic density of 1012 cm3 . The laser is tuned to the D1 line of 87 Rb at the transition 52 S1/2, F = 2 → 52 P1/2 , F = 1. After passing through the cell, the probe and drive beams are converted back to linear polarizations by another quarter-wave plate and then get separated by a polarizing beam splitter. A photodiode PD is used to monitor the power of transmitted probe field, while the spatial intensity distribution of probe field is recorded by an imaging system. This system is consisting of the lens L3 and a CCD camera. Two experiments have been done. In the first experiment, the lenses L1 and L2 are removed. Figure 25a shows the spatial intensity distribution of

Beyond the Rayleigh Limit in Optical Lithography

449

Figure 25 The results of the first experiment, the lenses L1 and L2 are not used. Image (a) shows snap shot of the intensity distribution of the drive field in the Rb cell. While (b) shows the intensity distribution of the transmitted probe beam. Figure (c) and (d) are the corresponding intensity profiles. Figure reprinted with permission from Li et al. (2008). Copyright 2008 by the American Physical Society.

the drive beam, while Figure 25b shows the intensity distribution of the transmitted probe beam. Both beams have the same spacing between two peaks, but the probe intensity distribution has sharper peaks than the drive intensity. Figure 25c and d are the horizontal cross sections of the drive and the transmitted probe distributions. In the drive intensity profile, the width (FWHM) of the peaks of the drive intensity is 0.4 mm (Figure 25c), while it is 0.1 mm for the transmitted probe intensity (Figure 25d). The finesse (ratio of spacing between peaks to the width of peaks) of the transmitted probe intensity distribution is smaller than that of the drive intensity distribution by a factor of 4. In the second experiment, the lenses L1 and L2 are used. A parallel probe beam with a diameter of 1.4 mm is focused by the lens L1. The focal length of lens L1 is 750 mm, and the beam size at the waist has a diffraction limited size 0.5 mm. The lens L2 is used to make the drive beam smaller in the Rb cell, where the pattern of drive field is spatially overlapped with the waist of the probe beam. Figure 26 shows the experimental result where in (a) the drive field still has a double-peak intensity distribution. Again,

450

Mohammad Al-Amri et al.

Figure 26 The results of the experiment with the diffraction limited probe beam, the lenses L1 and L2 are used. Images (a) and (b) show the image of the intensity distribution of the drive field and the intensity distribution of the transmitted probe field, respectfully, in the Rb cell. Curves (c) and (d) are the corresponding profiles. Figure reprinted with permission from Li et al. (2008). Copyright 2008 by the American Physical Society.

(b) shows similar double-peak intensity distribution for the transmitted probe beam. Figure 26c and d are the horizontal cross sections of the drive and transmitted probe profiles respectively. The width of the peaks in the drive beam is 165 µm, while it is 93 µm for transmitted probe beam. The structure created within the diffraction limit, for the probe beam, has a size five times smaller than that of the diffraction limited size 500 µm. Thus, this experiment successfully demonstrate that the diffraction limit can be broken using the dark state physics.

7. SUBWAVELENGTH PHOTOLITHOGRAPHY VIA RABI OSCILLATIONS So far all the schemes for overcoming the Rayleigh limit are based on multiphoton absorption or multi-level multi-beam systems. In 2010, Liao et al. presented a novel and simple scheme for subwavelength lithography based

Beyond the Rayleigh Limit in Optical Lithography

Pulse laser 2

Pulse laser 1

τ12

451

a

b

Figure 27 Schematics for the proposed lithographic scheme. In the first step a laser pulse induces the Rabi oscillation between the ground state and the excited state of the molecules in the photoresist. Then a second laser pulse is applied to dissociate the atoms in the excited state, cutting the chemical bound of the molecules. The molecules change its solubility in the photoresist developer and the required pattern can then be formed in the photoresist. Figure reprinted with permission from Liao et al. (2010). Copyright 2010 by the American Physical Society.

on Rabi oscillations (Liao et al., 2010). This method is similar to the traditional photolithography but adding a critical step before dissociating the chemical bound of the photoresist. The subwavelength pattern is achieved by inducing the multi-Rabi-oscillation between the ground state and one intermediate state. In Figure 27, the molecules are simplified as a three-level system. In the traditional optical lithography only one light beam is used to dissociate the molecules. Here we sequentially turn on two different frequencies of lights. The first light pulse induces Rabi oscillations between the ground state and the intermediate excited state. Then the second light only dissociates the molecules in the excited states but not those in the ground state. Initially, the valence electrons of the chemical bond are in the ground state |b. The first light has frequency ν1 which is resonant with the energy difference ωab between |a and |b. The atoms or molecules undergo Rabi oscillations between state |a and |b. At some predetermined time, the molecules occupy the excited state |a with spatially modulated probabilities. The second light pulse of frequency ν2 dissociates the molecules which are in the excited state |a. The dissociation of the molecules cuts the chemical bond and changes the chemical properties of the photoresist. We can then use photoresist developer to wash out the dissociated molecules or undissociated molecules (Wayne & Wayne, 1996). The resulting patterns of the photoresist should then depend on the spatial distribution of the excited state induced by the first light pulse. If the spatial modulation of the probability to find the molecules at excited state has subwavelength pattern, then the resulting patterns of the photoresist is also subwavelength. We next show that our method can potentially lead to subwavelength patterns of almost arbitrary accuracy, much easier than any other proposed methods.

452

Mohammad Al-Amri et al.

7.1 Achieving the Subwavelength Pattern The first step is very critical in order to achieve the subwavelength pattern. We illustrate it in more detail and show how to prepare the molecules in a subwavelength position dependent state. Two beams of light from opposite directions are incident on the photoresist and they form a standing wave on the surface of the photoresist. The standing light field interacts with the molecules in the photoresist, for which we consider two kinds of light sources: Continuous wave and a Gaussian pulse.

7.1.1 Continuous Wave Analysis For simplicity, we first consider the continuous wave with frequency resonant to the two atomic levels. The standing electric field on the surface is E(r, t) = E0 cos (ν1 t)eik·r + E0 cos (ν1 t)e−i(k·r+2φ) = 2E e−iφ cos (kx cos θ + φ) cos (ν t) 0

1

(48)

in which E0 is the field amplitude, ν1 is the frequency, θ is the angle between the incident light and the surface, and 2φ is the phase difference of the two beams. Considering the dipole interaction between the electric field and the atoms, and for the resonant case where ν1 = ω, the probability for the atoms to be in the excited state |a at time T is:   1 − cos R T cos (kx cos θ + φ) , (49) Pa (x, T) = 2 where we assume that the atoms are initially in the ground state, R = (2|℘ba |E0 )/ is the Rabi frequency at the peak electric intensity, and |℘ba | is the amplitude of the electric dipole moment. From this equation, we can see that the probability in the excited state is spatially dependent and the shape depends on the field area R T. As the molecules that are in the excited state are dissociated, the spatial pattern also depends on the field area. We now look at the spatial pattern in more detail. For simplicity, we choose θ = 0 and φ = 0 which does not change the overall properties. Then we have Pa (x, T) = (1 − cos[R T cos (kx)])/2 which is a double cosine function and we can calculate the positions of the valleys and the peaks. First, we note that the usual Rayleigh limit is obtained in the linear approximation corresponding to R T 1. In this case Pa (x, T) ≈ α(1 + cos (2kx)) with α = (R T)2 /8 leading to a resolution of λ/2. Next we look at the situation where we are not restricted by the linear approximation and various Rabi oscillations during the interaction time T are allowed. When cos (kx) = 2mπ/R T, the probability Pa (x, T) is 0 which corresponds to the valleys, and when cos (kx) = (2m + 1)π/R T, the probability is 1 which corresponds to the peaks, where m = 0, ±1, ±2, . . .. It is

Beyond the Rayleigh Limit in Optical Lithography

453

Figure 28 Subwavelength patterns generated by fields of different R T . (a) R T = π ; (b) R T = 2π ; (c) R T = 3π ; (d) R T = 4π . The solid line is when the decay is not included whereas the green dashed line shows the results with γ = ωab /1000. Figure reprinted with permission from Liao et al. (2010). Copyright 2010 by the American Physical Society.

readily seen that when R T = π, there are two valleys (x = λ/4, 3λ/4) and three peaks (x = 0, λ/2, λ) within one wavelength (Figure 28a), which gives the same result as the classical interference lithography. However, when R T  2π, more valleys and peaks appear and the classical limitation is broken. For example, when R T = 2π, there are five valleys (x = 0, λ/4, λ/2, 3λ/4, λ) and four peaks (x = λ/6, λ/3, 2λ/3, 5λ/6) within one wavelength (Figure 28b). When R T becomes larger, the pattern becomes smaller (Figure 28c and d). Therefore it is, in principle, possible to achieve arbitrarily smaller subwavelength patterns by using stronger field or lengthening the interaction time to induce more Rabi oscillations. The physics behind the subwavelength pattern is the nonlinearity associated with the Rabi oscillations. If the field is intense enough then the Rabi oscillations are induced that help to modulate the population in level |a, thus leading to the subwavelength oscillations for the population. For example, when R T = π, one photon is absorbed and we are in the linear regime. The corresponding resolution is the same as that obtained in the classical lithography (Figure 28a). When R t = 2π, one photon is absorbed

454

Mohammad Al-Amri et al.

then another photon is emitted, leading to a full Rabi cycle. The resulting resolution is half of the classical case (Figure 28b). And so on. In order to see clearly the advantage of the present method over any previous method for precision lithography, we refer to Equation (50). In the case when φ = π/2, and kx cos θ 1, Equation (50) reduces to 1 − cos (2keff x) , 2

(50)

λeff = λ/(R T cos θ).

(51)

Pa = where keff = R Tk cos θ or

Thus, a large number of Rabi oscillations in the interaction time can lead to an arbitrarily small effective wavelength. Therefore a novel feature of our scheme is that it should be possible to generate a nano-scale pattern using a microwave field. For example, if two sublevels of a system have energy difference of about 3 GHz and the coherence time is of the order of 1 s, we can use a microwave pulse with wavelength 10 cm and pulse duration 0.1 s to induce the Rabi oscillations between these two levels. If R = 0.1 GHz, the resolution could be of the order 10 nm.

7.1.2 Gaussian Pulse Analysis So far we considered the light field to be a continuous wave. However, in practical applications we usually use laser pulses instead. Our study shows that the result of the pulses is similar to that of the continuous wave. Two beams of Gaussian pulses with the same frequency ν1 , same maximal amplitude E0 and same full width at half maximum of the intensity √ tFWHM = 2 ln 2σ are incident on the photoresist from opposite directions with angle θ, and σ is the width of the pulse. They then form a standing electric field described by 

t2 E(x, t) = 2E0 exp − 2 2σ

 cos (kx cos θ + φ) cos (ν1 t),

(52)

where φ is the phase difference between these two pulses. The electric field couples to the molecules in the photoresist. If ν1 is resonant to the two energy levels |a and |b, the electric field drives Rabi oscillations between these two levels. The Rabi frequency is R (x, t) = 2|℘ba |E0 exp 2 (− 2σt 2 ) cos (kx cos θ + φ)/. According to the Area theorem, the upper-level

Beyond the Rayleigh Limit in Optical Lithography

455

probability after the pulse is Pa (x) 

1 − cos[

∞

−∞ R (x, t)dt]

2 √ 1 − cos[2 2π σ |℘ba|E0 cos (kx cos θ + φ)] = 2 1 − cos[0 t0 cos (kx cos θ + φ)] , = 2

(53)

where 0 = 2|℘ba |E0 / is the maximal Rabi frequency and we define t0 =  π t . From the equation, we see that the pattern generated by 2 ln 2 FWHM the Gaussian pulse is the same as that of the continuous wave, but just replace T by t0 . For example, when 0 t0 = 2π, one Rabi cycle is driven and the pattern has a resolution of λ/4 (Figure 29b) which is the same as Figure 28b. When 0 t0 = 4π, two Rabi cycles are driven and the resolution is λ/8 (Figure 29b) which is the same as Figure 28d. In the real system, decoherence time is an important factor we should consider. When the pulse time exceed the decoherence time, the visibility reduces dramatically. Usually the dephasing time is much smaller than the decay time. Therefore here we only consider the effect of dephasing time. Numerical simulation shows that when tFWHM = τ/2 (where τ is the dephasing time), the visibility is reduced to about 80% but the total

Figure 29 (a) The Gaussian pulse. The red dash line is the amplitude profile and the thick dark  line is the intensity profile; (b) The pattern produced by the Gaussian pulse π  t when 2 ln 2 0 FWHM = 2π ; (c) The pattern produced by the Gaussian pulse when  π  t 2 ln 2 0 FWHM = 4π . The solid line is the result without the decoherence while the green dashed line shows the results with tFWHM = τ/2. Figure reprinted with permission from Liao et al. (2010). Copyright 2010 by the American Physical Society.

456

Mohammad Al-Amri et al.

patterns are almost the same as the result without decoherence. Therefore, if tFWHM  τ , our scheme still works well. 7.2 Arbitrary Subwavelength Patterns in a Macroscopic Area In the previous section we have shown how to achieve a simple subwavelength pattern via coherent Rabi oscillations. For any practical applications we should produce more complicated patterns (Kok et al., 2001; Pau et al., 2001; Sun et al., 2007). In the following, we will discuss how to produce arbitrary subwavelength patterns in a macroscopic area. For one-dimensional case, any functions in the range L can be expanded as a Fourier series:

 ∞  a0  2nπx 2nπx f (x) = + an cos + bn sin . (54) 2 L L n=1

For the components with periods L/n larger than optical wavelength λ, we just use the traditional way, i.e., shine two dissociative lasers with frequency large enough to dissociate the molecules directly and they form a standing wave correspond to the component and with strength related to the Fourier coefficient. For the components with L/n < λ, we apply our subwavelength scheme to realize them. We shine two phase locked pulses with amplitude E0 from angle θ to form a standing wave and the third one with amplitude E1 from the right angle to form a constant background. The resulting electric field is E(x, t) = [2E0 cos (kx cos θ + φ) + 2 E1 ] exp (− 2σt 2 ) cos (ν1 t). When nπ −   kx cos θ  nπ +  (n is an integer and  is a small number),   t2 (55) E(x, t)  ±[2E0 kx cos θ + E1 ] exp − 2 cos (ν1 t), 2σ where φ is set to be 90◦ . Then the Rabi frequency is 2|℘ab |E1 2|℘ab | 2|℘ab |E0 k cos θ [E0 cos (kx cos θ + φ) + E1 ] ≈ x+ .    (56) The Rabi frequency is approximately a linear function of the position, and the gradient of intensity is approximately a constant in the region (nπ − )/k cos θ  x  (nπ + )/k cos θ. Then the pattern produced in this linear region is 1 − cos (Ax + B) Pa (x, T)  , (57) 2   where A = 2π/ ln 20 tFWHM k cos θ and B = 2π/ ln 21 tFWHM . The coefficients A and B can be controlled by the field strength and the pulse time. The effective wavelength  λeff = λ/( 2π/ ln 20 tFWHM cos θ) (58) R (x) =

Beyond the Rayleigh Limit in Optical Lithography

457

can be arbitrary small by using stronger field or longer pulse time. We note that ignoring the constant background 1/2, when B = 0, the pattern is a cosine function; when B = π/2, the pattern is a sine function. For example, if we want to produce sine pattern with λ/5 resolution in a large region, we can do it in two steps (Figure 30): First, we etch the pattern in the linear region as shown in Figure 30a. We then shift the standing wave by a phase π/2 such that the linear region shifts by a distance of λ/2. This allows us to write the sine pattern in the remaining region (Figure 30b) thus leading to the resulting sine pattern in the entire region as shown in Figure 30c. The peak power for E0 is about 15 MW/cm2 (cos (θ) = 1/4, |℘ab | = 10 D, tFWHM = 1 ps) (Becker et al., 1988) and the peak power for E1 is about 0.37 MW/cm2 . For larger resolution, the peak power should increase. For example, to reach λ/10 resolution, the peak power for E0 is about 60 MW/cm2 and the peak power for E1 is about 0.37 MW/cm2 . In addition, for the Fourier coefficients an and bn , we can control the strength and time of the dissociation pulse to control the dissociation rate or we can use different wavelengths with different absorption rates. We can also generalize our method to two-dimensional patterns. Arbitrary 2D periodic function with f (x + λ, y + λ) = f (x, y) can be simulated by the truncated Fourier series:    2π(mx + ny) 2π(mx − ny) f (x, y) = + bmn cos amn cos λ λ m=0 n=0     2π(mx + ny) 2π(mx − ny) + dmn sin +cmn sin λ λ M  N  

a



b

c

Figure 30 A proposed scheme to print a sine pattern in an arbitrary large region. Figure reprinted with permission from Liao et al. (2010). Copyright 2010 by the American Physical Society.

458

Mohammad Al-Amri et al.





  2π cos (θ) (mx + ny) π m2 + n2  ≈ + amn cos cos cos (θ) λ 2 m2 + n2 m=0 n=0    2π cos (θ) (mx − ny) π m2 + n2  cos + +bmn cos cos (θ) λ 2 m2 + n2    2π cos (θ) (mx + ny) π m2 + n2  cos +cmn sin + cos (θ) λ 2 m2 + n2    2π cos (θ) (mx − ny) π m2 + n2 cos (59) +dmn sin  + cos (θ) λ 2 m2 + n2 M  N 

in which θ is near 90◦ . In the practical application, we should realize each Fourier component one by one. For the first and third components  in Equation (6) we shine the pulses from directions (mxˆ + nˆy)/ m2 + n2 while for the  we shine the pulses from directions  other two components 2 2 (mxˆ − nˆy)/ m + n and 0 t0 = m2 + n2 /cos (θ). Besides, due to the constant 1/2 appears in Equation (3), there is an additional penalty deposition Q which depends on the Fourier coefficients. For example, applying the numerical simulation we print characters “TAMU-KACST" within one wavelength (Figure 31). In the simulation, we take θ = 80◦ and M = N = 15. Q = 0.24h where h is the height of the pattern. We have a total of 15×15×4 = 900 components and each component needs 4 pulses (three for standing wave and one for dissociation). Therefore we need 3600 pulses in total. Each component takes about 1 ms and the whole process takes about 1 s. In our example with the region λ × λ, the required maximal power is about 200 MW/cm2 for a pulse duration of t0 = 5 ps. 7.3 Potential Realizations The scheme shown in Figure 27 is a simplified model. In the following we introduce two possible realizations of our scheme in two different systems. The first one is in the organic molecular photochemistry. The typical state energy diagram for the chemical bound is shown in Figure 32 (Turro et al., 2009). Here S0 and S1 are the ground singlet state and the first excited singlet state, respectively and T1 is the first excited triplet state. KF is the fluorescence decay rate from S1 to S0 ; KP is the phosphorescence decay rate from T1 to S0 ; while KST is the intersystem crossing rate from S1 to T1 . To induce Rabi oscillation, the system should be kept coherently. Therefore, the decoherence time is an important parameter in our scheme. The typical decoherence time τ is about 1∼5 ps at room temperature (Fischer & Laubereau, 1975). To realize our subwavelength scheme, the requirements for these parameters are tFWHM  τ and KST  KF  KP . For tFWHM  τ , the system keeps coherent. For KST  KF , intersystem

Beyond the Rayleigh Limit in Optical Lithography

459

Figure 31 A 2D pattern “TAMU-KACST” printed within one wavelength using the present method. Parameters are M = N = 15, θ = 80◦ . Figure reprinted with permission from Liao et al. (2010). Copyright 2010 by the American Physical Society.

S1

a

k ST

T1 kF kP

b

S0

Figure 32 The schematics for the state energy diagram for molecular organic photochemistry. Figure reprinted with permission from Liao et al. (2010). Copyright 2010 by the American Physical Society.

crossing from S1 to T1 dominates, which means that most of the molecules at S1 will transfer to T1 instead of decaying to S0 . As the transition from T1 to S0 is spin forbidden, the lifetime (or phosphorescence time) of T1 is long. Within the phosphorescence time, we shine the second pulse to dissociate the molecules in state T1 . Indeed, the requirements can be satisfied in some real systems. Usually, the time scale for KF : 105 –109 Hz; KST : 105 –1011 Hz; KP : 10−2 –103 Hz. The Rabi frequency can be chosen as 1012 –1014 Hz. One example is 1-Bromonaphthalene (Turro et al., 2009) for which KF ∼ 106 Hz, KST ∼ 109 Hz, KP ∼ 30 Hz. The lifetime of the intermediate state T1 is about

460

Mohammad Al-Amri et al.

30 ms which is long enough for us to shine the second pulse. It is worthwhile to mention that the dipole–dipole interaction or exchange interaction may induce energy transfer between neighboring molecules which limits the resolution in our scheme (Turro et al., 2009). However these effects can be ignored for the following reasons. The dipole–dipole energy transfer rate is of the order of fluorescence rate when the distance between two molecules is in the range of 1–5 nm. However, as we require KST to be much larger than KF , the intersystem crossing to T1 occurs in times shorter than that required for the dipole–dipole energy transfer to the neighboring molecules. Also when the molecules are in the triplet state the dipole– dipole energy transfer between the two molecules is forbidden. Therefore the energy transfer due to the dipole–dipole interaction can be ignored in our scheme. While the triplet–triplet energy transfer is allowed by the electron exchange interaction, it can only happen at a distance within 1 nm which is about the size of the molecules. Usually we cannot reach such small patterns in the photoresist lithography. The second possible realization is to generate a nanopattern using a microwave. For example, the solid state system such as the NV-diamond has a long dephasing time. The ground triplet state is split into two sublevels (ms = 0 and ms = ±1). The energy difference between these two sublevels are about 2.9 GHz, which corresponds to a microwave with wavelength of about 0.1 m. The dephasing time at room temperature can reach 1.8 ms (Balasubramanian et al., 2009). Let tFWHM = 1 ms and R = 0.1 GHz, then we can reach a resolution of about 300 nm. At the low temperature, the dephasing time can be even larger, and the pattern can be smaller.

8. SUMMARY AND OUTLOOK The diffraction limit is one of the major obstacles for the resolution of optical microscope and the current photolithography techniques. Researchers have been struggling to increase the numerical aperture to improve the resolution, but until now the improvement is not significant. Although the working wavelength is reduced to print finer pattern, the light source, lens and the photoresist working for high energy photons are hard to find. Atomic and electron beam lithography are possible candidates of the nanometer lithography, but they are restricted by the secondary electron scattering and low throughput problem. Therefore, it is very interesting and useful if we can somehow overcome the diffraction limit. In the last two decades several ways to go beyond the diffraction limit have been illustrated. Two-photon process and its generalization to multiphoton process are first studied to increase the resolution of microscope and later illustrated to shrink the pattern of photolithography. The

Beyond the Rayleigh Limit in Optical Lithography

461

principle of this method is straightforward, but the requirements of extremely high laser intensity and low efficiency are the main concerns. Photon entanglement successfully suppress the normal resolution term and keep only the super-resolution term in the absorption rate. How to produce ultrapure NOON quantum entanglement state is the biggest challenge of this scheme. Actually, quantum entanglement is not required to suppress the normal resolution term. If we can carefully control the light source by either controlling the phase relationship between pulses or matching the wave vector and frequency with the material energy levels in a narrow band, we can achieve similar result of quantum entanglement. However, multi-photon absorber is required to generate higher harmonic patterns which also requires extremely high laser intensity and also subject to low visibility. Spatial dependent dark state is a novel idea to produce subwavelength resolution either in the microscopy or in the photolithography. This scheme does not require quantum entanglement or multi-photon absorber, but it requires additional levels and beams for higher harmonic generation. Subwavelength resolution can be also simply achieved by inducing Rabi oscillations between two energy levels in photoresist. The advantages of this scheme are that it does not require quantum entanglement, multi-photon absorber, multi-level and multi-beam. Moreover, it is also very straightforward to produce higher harmonic components in which we just need stronger pulse or longer pulse time. The resolution limitation of this scheme is mainly due to the relaxation time of the material. To achieve a higher resolution, we should find a material which has a relatively long relaxation time. Neither quantum entanglement nor multi-photon absorber is required for subwavelength photolithography, but nonlinearity is somehow involved in every scheme invented to break the diffraction limit until now. Every scheme has its own advantages and disadvantages. In the near future, we should find a suitable light source and a suitable material that match all the requirements of one of the promising schemes. For industry applications, effective way to generate arbitrary 2D pattern and throughput are also important issues that we should consider in the future.

ACKNOWLEDGMENTS We would like to thank many colleagues with whom we discussed the subject matter of this article over the years. In particular we thank Joerg Evers, Phil Hemmer, Martin Kiffner. Ashok Mutukrishnan, Marlan Scully, and Qingqing Sun with whom we collaborated on different aspects of subwavelength lithography. This work is supported by a grant from the King Abdul Aziz City for Science and Technology (KACST). The research of MSZ is supported by NPRP grant 08-043-1-011 by the Qatar National Research Fund (QNRF).

462

Mohammad Al-Amri et al.

REFERENCES Abbe, E. (1873). Beitr¨age zur Theorie des Mikroskops und der mikroskopischen. Archiv für Mikroskopische Anatomie, 9, 413. Abella, I. D. (1962). Optical double-photon absorption in cesium vapor. Physical Review Letters, 9, 453. Agarwal, G. S., Boyd, R. W., Nagasako, E. M., & Bentley, S. J. (2001). Comment on quantum interferometric optical lithography: Exploiting entanglement to beat the diffraction limit. Physical Review Letters, 86, 1389. Agarwal, G. S., & Kapale, K. T. (2006). Subwavelength atom localization via coherent population trapping. Journal of Physics B: Atomic, Molecular and Optical Physics, 39, 3437. Alfred, K-K. W. (2001). Resolution enhancement techniques in optical lithography. SPIE Press. Alkaisi, M. M., Blaikie, R. J., & McNab, S. J. (2001). Nanolithography in the evanescent near field. Advanced Materials, 13, 877. Alkaisi, M. M., Blaikie, R. J., McNab, S. J., Cheung, R., & Cumming, D. R. S. (1999). Subdiffraction-limited patterning using evanescent near-field optical lithography. Applied Physics Letters, 75, 3560. Anisimov, P. M., & Dowling, J. P. (2009). Super resolution with superposition. Physics, 2, 52. Arimondo, E. (1996). Coherent population trapping in laser spectroscopy. Progress in Optics, 35, 259. Balasubramanian, G., Neumann, P., Twitchen, D., Markham, M., Kolesov, R., Mizuochi, N., et al. (2009). Ultralong spin coherence time in isotopically engineered diamond. Nature Materials, 8, 383. Becker, P. C., Fork, R. L., Brito Cruz, C. H., Gordon, J. P., & Shank, C. V. (1988). Optical stark effect in organic dyes probed with optical pulses of 6-fs duration. Physical Review Letters, 60, 2462. Bentley, S. J., & Boyd, R. W. (2004). Nonlinear optical lithography with ultra-high sub-Rayleigh resolution. Optics Express, 12, 5735. Berman, P. R., & Ziegler, J. (1977). Generalized dressed-atom approach to atom-strong-field interactions—Application to the theory of lasers and Bloch–Siegert shifts. Physical Review A, 15, 2042. Betzig, E., Patterson, G. H., Sougrat, R., Lindwasser, O. W., Scott Olenych, S., Bonifacino, J. S., et al. (2006). Imaging intracellular fluorescent proteins at nanometer resolution. Science, 313, 1642. Betzig, E., & Trautman, J. K. (1992). Near-field optics: Microscopy, spectroscopy, and surface modification beyond the diffraction limit. Science, 257, 189. Binnig, G., & Quate, C. F. (1986). Atomic force microscope. Physical Review Letters, 56, 930. Boto, A., Kok, P., Abrams, D., Braunstein, S., Williams, C. P., Dowling, J. P. (2000). Quantum interferometric optical lithography: Exploiting entanglement to beat the diffraction limit. Physical Review Letters, 85, 2733. Boyd, R. W., & Bentley, S. J. (2006). Recent progress in quantum and nonlinear optical lithography. Journal of Modern Optics, 53, 713. Boyd, R. W., & Bentley, S. J. (2006). Recent progress in quantum and nonlinear optical lithography. Erratum Journal of Modern Optics, 53, 1529. Brueck, S. R. J. (2005). Optical and interferometric lithography—Nanotechnology enablers. Proceedings of the IEEE, 93, 1704. Brueck, S. R. J., Zaidi, S. H., Chen, X., & Zhang, Z. (1998). Interferometric lithography: From periodic arrays to arbitrary patterns. Microelectronic Engineering, 41–42, 145. Chang, J., Evers, J., Scully, M. O., & Zubairy, M. S. (2006). Measurement of the separation between atoms beyond diffraction limit. Physical Review A, 73, 031803(R). Chang, H. J., Shin, H., O’Sullivan-Hale, M. N., & Boyd, R. W. (2006). Implementation of subRayleigh-resolution lithography using an N-photon absorber. Journal of Modern Optics, 53, 2271. Chiu, G. L. -T., & Shaw, J. M. (1997). Optical lithography: Introduction. IBM Journal of Research and Development, 41, 3. Cho, A. (2006). A new way to beat the limits on shrinking transistors?. Science, 312, 672. Chris, A. M. (2007). Fundamental principles of optical lithography: The science of microfabrication. John Wiley & Sons.

Beyond the Rayleigh Limit in Optical Lithography

463

D’Angelo, M., Chekhova, M. V., & Shih, Y. (2001). Two-photon diffraction and quantum lithography. Physical Review Letters, 87, 013602. Denk, W., Strickler, J. H., & Webb, W. W. (1990). Two-photon laser scanning fluorescence microscopy. Science, 248, 73. Denk, W., Svoboda, K. (1997). Photon upmanship: Why multiphoton imaging is more than a gimmick. Neuron, 18, 351. Diaspro, A. (Ed.). (2010). Nanoscopy and multidimensional optical fluorescence microscopy. Boca Raton: CRC Press. Donnert, G., Keller, J., Medda, R., Alexandra Andrei, M., Rizzoli, S. O., Lührmann, R., et al. (2006). Macromolecular-scale resolution in biological fluorescence microscopy. Proceedings of the National Academy of Sciences, 103, 11440. Dryakhlushin, V. F., Klimov, A. Y., Rogov, V. V., & Vostokov, N. V. (2005). Near-field optical lithography method for fabrication of the nanodimensional objects. Applied Surface Science, 248, 200. Dudovich, N., Dayan, B., Gallagher Faeder, S. M., & Silberberg, Y. (2001). Transform-limited pulses are not optimal for resonant multiphoton transitions. Physical Review Letters, 86, 47. Fischer, S. F., & Laubereau, A. (1975). Dephasing processes of molecular vibrations in liquids. Chemical Physics Letters, 35, 6. Fonseca, E. J. S., Monken, C. H., & Pádua, S. (1999). Measurement of the de Broglie wavelength of a multiphoton wave packet. Physical Review Letters, 82, 2868. Freund, S. M., Römheld, M., & Oka, T. (1975). Infrared-radio-frequency two-photon and multiphoton lamb dips for CH3 F. Physical Review Letters, 35, 1497. Goeppert-Mayer, M. (1931). Ueber elementarakte mit zwei quantenspruengen. Annalen der Physik, 9, 273. Gorshkov, A. V., Jiang, L., Greiner, M., Zoller, P., & Lukin, M. D. (2008). Coherent quantum optical control with subwavelength resolution. Physical Review Letters, 100, 093005. Gustafsson, M. G. L. (2005). Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution. Proceedings of the National Academy of Sciences, 102, 13081. Haroche, S., & Hartmann, F. (1972). Theory of saturated-absorption line shapes. Physical Review A, 6, 1280. Harry, J. L. (2005). Principles of lithography. SPIE Press. Hell, S. W. (1994). Improvement of lateral resolution in far-field light microscopy using twophoton excitation with offset beams. Optics Communications, 106, 19. Hell, S. W. (2007). Far-field optical nanoscopy. Science, 316, 1153. Hell, S. W., & Kroug, M. (1995). Ground-state-depletion fluorescence microscopy: A concept for breaking the diffraction resolution limit. Applied Physics B, 60, 495. Hell, S. W., & Wichmann, J. (1994). Breaking the diffraction resolution limit by stimulated emission: Stimulated-emission-depletion fluorescence microscopy. Optics Letters, 19, 780. Helmchen, F., & Denk, W. (2005). Deep tissue two-photon microscopy. Nature Methods, 2, 932. Hemmer, P. R., Muthukrishnan, A., Scully, M. O., & Zubairy, M. S. (2006). Quantum lithography with classical light. Physical Review Letters, 96, 163603. Herkommer, A. M., Schleich, W. P., & Zubairy, M. S. (1997). Autler–Townes microscopy of a single atom. Journal of Modern Optics, 44, 2507. Ivan, M. G., Scaiano, J. C. (2010). Photoimaging and lithographic processes in polymers. In N. S. Allen (Ed.), Photochemistry and photophysics of polymer materials (pp. 479–507). New Jersey: John Wiley & Sons. Kawabe, Y., Fujiwara, H., Okamoto, R., Sasaki, K., & Takeuchi, S. (2007). Quantum interference fringes beating the diffraction limit. Optics Express, 15, 14244. Kawata, S., Sun, H. -B., Tanaka, T., & Takada, K. (2001). Finer features for functional microdevices. Nature (London), 412, 697. Kiffner, M., Evers, J., & Zubairy, M. S. (2008). Resonant interferometric lithography beyond the diffraction limit. Physical Review Letters, 100, 073602. Klar, T. A., & Hell, S. W. (1999). Subdiffraction resolution in far-field fluorescence microscopy. Optics Letters, 24, 954. Klar, T. A., Jakobs, S., Dyba, M., Egner, A., & Hell, S. W. (2000). Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission. Proceedings of the National Academy of Sciences, 97, 8206.

464

Mohammad Al-Amri et al.

Kok, P., Boto, A. N., Abrams, D. S., Williams, C. P., Braunstein, S. L., Dowling, J. P. (2001). Quantum-interferometric optical lithography: Towards arbitrary two-dimensional patterns. Physical Review A, 63, 063407. Korobkin, D. V., & Yablonovitch, E. (2002). Twofold spatial resolution enhancement by twophoton exposure of photographic film. Optical Engineering, 41, 1729. Kyr¨ola, E., & Stenholm, S. (1977). Velocity tuned resonances as multi-Doppleron processes. Optics Communications, 22, 123. Lee, S., Byers, J., Jen, K., Zimmerman, P., Rice, B., Turro, N. J., et al. (2008). An analysis of double exposure lithography options. Proceedings of SPIE, 6924, 69242A. Lee, S. K., & Lee, H. -W. (2008). The role of entanglement in quantum lithography. Journal of the Physical Society of Japan, 77, 124001. Levinson, H. J. (2001). Principles of lithography. Washington: SPIE Press. Liao, Z., Al-Amri, M., & Zubairy, M. S. (2010). Quantum lithography beyond the diffraction limit via Rabi oscillations. Physical Review Letters, 105, 183601. Li, H., Sautenkov, V. A., Kash, M. M., Sokolov, A. V., Welch, G. R., Rostovtsev, Y. V., et al. (2008). Optical imaging beyond the diffraction limit via dark states. Physical Review A, 78, 013803. Liu, Z., Wei, Q. H., & Zhang, X. (2005). Surface plasmon interference nanolithography. Nano Letters, 5, 957. Lu, C., & Lipson, R. H. (2010). Interference lithography: A powerful tool for fabricating periodic structures. Laser and Photonics Reviews, 4, 568. Luo, X. G., & Ishihara, T. (2004a). Subwavelength photolithography based on surface-plasmon polariton resonance. Optics Express, 12, 3055. Luo, X. G., & Ishihara, T. (2004b). Surface plasmon resonant interference nanolithography technique. Applied Physics Letters, 84, 4780. Mack, C. (2007). Fundamental optical principles of lithography: The science of microfabrication. West Sussex, England: John Wiley & Sons. Macovei, M., Evers, J., Keitel, C. H., & Zubairy, M. S. (2007). Localization of atomic ensembles via superfluorescence. Physical Review A, 75, 033801. Mandel, L., Sudarshan, E. C. G., & Wolf, E. (1964). Theory of photoelectric detection of light fluctuation. Proceedings of the Physical Society, 84, 435. Marconi, M. C., & Wachulak, P. W. (2010). Extreme ultraviolet lithography with table top lasers. Progress in Quantum Electronics, 34, 173. Martin, O. J. F. (2003). Surface plasmon illumination scheme for contact lithography beyond the diffraction limit. Microelectronic Engineering, 67, 24. Matsko, A. B., Novikova, I., Zubairy, M. S., & Welch, G. R. (2003). Nonlinear magneto-optical rotation of elliptically polarized light. Physical Review A, 67, 043805. McClain, W. M. (1974). Two-photon molecular spectroscopy. Accounts of Chemical Research, 7, 129. Menon, R., Patel, A., Gil, D., & Smith, H. I. (2005). Maskless lithography. Materials Today, 7021, 26. ISSN:1369 Menon, R., Walsh, M., Galus, M., Chao, D., Patel, A., & Smith, H. I. (2005). Maskless lithography using diffractive-optical arrays. In Frontiers in optics, OSA technical digest series. Optical Society of America (Paper FWU5). Mollow, B. R. (1968). Two-photon absorption and field correlation functions. Physical Review, 175, 1555. Mompart, J., Ahufinger, V., & Birkl, G. (2009). Coherent patterning of matter waves with subwavelength localization. Physical Review A, 79, 053638. Neice, A. (2010). Methods and limitations of subwavelength imaging. Advances in imaging and Electron Physics, 163, 117. Ono, T., & Esashi, M. (1998). Subwavelength pattern transfer by near-field photolithography. Japanese Journal of Applied Physics, Part 1, 37, 6745. Pau, S., Watson, G. P., & Nalamasu, O. (2001). Writing an arbitrary pattern using interference lithography. Journal of Modern Optics, 48, 1211. Peér, A., Dayan, B., Vucelja, M., Silberberg, Y., & Friesem, A. A. (2004). Quantum lithography by coherent control of classical light pulses. Optics Express, 12, 6600. Pendry, J. B. (2000). Negative refraction makes a perfect lens. Physical Review Letters, 85, 3966.

Beyond the Rayleigh Limit in Optical Lithography

465

Pittman, T. B., Shih, Y. H., Strekalov, D. V., & Sergienko, A. V. (1995). Optical imaging by means of two-photon quantum entanglement. Physical Review A, 52, R3429. Pritchard, D. E., & Gould, P. L. (1985). Experimental possibilities for observation of unidirectional momentum transfer to atoms from standing-wave light. Journal of the Optical Society of America B – Optical Physics, 2, 1799. Qamar, S., Zhu, S. -Y., & Zubairy, M. S. (2000). Atom localization via resonance fluorescence. Physical Review A, 61, 063806. Rayleigh, L. (1879). Investigations in optics, with special reference to the spectroscope. Philosophical Magazine, 8, 261. Reid, J., & Oka, T. (1977). Direct observation of velocity-tuned multiphoton processes in the laser cavity. Physical Review Letters, 38, 67. Rittweger, E., Han, K. Y., Irvine, S. E., Eggeling, C., & Hell, S. W. (2009). STED microscopy reveals crystal colour centres with nanometric resolution. Nature Photonics, 3, 144. Rothschild, M. (2010). A roadmap for optical lithography. Optics and Photonics News, 21, 27. Rudenberg, H. G., & Rudenberg, P. G. (2010). Origin and background of the invention of the electron microscope: Commentary and expanded notes on Memoir of Reinhold Rüdenberg. Advances in Imaging and Electron Physics, 160, 207. Schuller, J. A., Barnard, E. S., Cai, W., Jun, Y. C., White, J. S., Brongersma, M. L. (2010). Plasmonics for extreme light concentration and manipulation. Nature Materials, 9, 193. Scully, M. O., & Lamb, W. E. (1969). Quantum theory of an optical maser. III. Theory of photoelectron counting statistics. Physical Review, 179, 368. Scully, M. O., & Zubairy, M. S. (1997). Quantum optics. Cambridge, England: Cambridge University Press. Seisyan, R. P. (2011). Nanolithography in microelectronics: A review. Technical Physics, 56, 1061. Sheats, J. R., & Smith, B. W. (Eds.). (1998). Microlithography science and technology (1st ed.). New York: Marcel Dekker Inc.. Shih, Y. (2007). Quantum imaging. IEEE Journal of Selected Topics in Quantum Electronics, 13, 1016. Spille, E., & Feder, R. (1977). X-ray lithography. Topics in Applied Physics, 22, 35. Srituravanich, W., Fang, N., Sun, C., Luo, Q., & Zhang, X. (2004). Plasmonic nanolithography. Nano Letters, 4, 1085. Strekalov, D. V., Sergienko, A. V., Klyshko, D. N., & Shih, Y. H. (1995). Observation of twophoton “Ghost" interference and diffraction. Physical Review Letters, 74, 3600. Strickler, J. H., & Webb, W. W. (1991). Three-dimensional optical data storage in refractive media by two-photon point excitation. Optics Letters, 16, 1780. Strickler, J. H., & Webb, W. W. (1991). Two-photon excitation in laser scanning fluorescence microscopy. Proceedings of SPIE, 1398, 107. Sun, Q., Al-Amri, M., Scully, M. O., & Zubairy, M. S. (2011). Subwavelength optical microscopy in the far field. Physical Review A, 83, 063818. Sun, Q., Hemmer, P. R., & Zubairy, M. S. (2007). Quantum lithography with classical light: Generation of arbitrary patterns. Physical Review A, 75, 065803. Taylor, J. S., Sommargren, G. E., Sweeney, D. W., & Hudyma, R. M. (1998). Fabrication and testing of optics for EUV projection lithography. Proceedings of SPIE, 3331, 580. Tsang, M. (2007). Relationship between resolution enhancement and multiphoton absorption rate in quantum lithography. Physical Review A, 75, 043813. Turro, N. J., Ramamurthy, V., & Scaino, J. C. (2009). Principles of molecular photochemistry: An introduction. . Sausalito, CA: University Science Books. Vieu, C., Carcenac, F., P’ipin, A., Chen, Y., Mejias, M., Lebib, A., et al. (2000). Electron beam lithography: Resolution limits and applications. Applied Surface Science, 164, 111. Wayne, C. E., & Wayne, R. P. (1996). Photochemistry.. New York: Oxford University Press. Williams, C., Kok, P., Lee, H., & Dowling, J. P. (2006). Quantum lithography: A non-computing application of quantum information. Informatik – Forschung und Entwicklung, 21, 73. Wu, E. S., Strickler, J. H., Harrell, W. R., & Webb, W. W. (1992). Optical/laser microlithography V. Proceedings of SPIE, 1674, 776. Xie, Z., Yu, W., Wang, T., Zhang, H., Fu, Y., Liu, H., et al. (2011). Plasmonic nanolithography: A review. Plasmonic, 6, 565.

466

Mohammad Al-Amri et al.

Yablonovitch, E., & Vrijen, R. B. (1999). Optical projection lithography at half the Rayleigh resolution limit by two-photon exposure. Optical Engineering, 38, 334. Yavuz, D. D., & Proite, N. A. (2007). Nanoscale resolution fluorescence microscopy using electromagnetically induced transparency. Physical Review A, 76, 041802. Zhuang, X. (2009). Nano-imaging with STORM. Nature Photonics, 3, 365. Zubairy, M. S., Matsko, A. B., & Scully, M. O. (2002). Resonant enhancement of high-order optical nonlinearities based on atomic coherence. Physical Review A, 65, 043804.