Probing limits on spatial resolution using nonlinear optical effects and nonclassical light Y. Leng,1,3,* D. H. Park,1,3 D. Schmadel,2 V. E. Yun,1,3 W. N. Herman,3 and J. Goldhar1,3 1
Department of Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742, USA 2
Department of Physics, University of Maryland, College Park, Maryland 20742, USA 3
Laboratory for Physical Sciences, College Park, Maryland 20740, USA *Corresponding author:
[email protected]
Received 26 March 2013; revised 18 November 2013; accepted 25 November 2013; posted 26 November 2013 (Doc. ID 187528); published 23 December 2013
Using a simple optical setup to detect and characterize transmission gratings in the far field, we demonstrate that going beyond the diffraction limit is not possible using linear interaction of nonclassical illumination with the target grating. We also confirm that nonlinear optical interactions with the target grating, or with the optical medium around it, do allow improvement in resolution. © 2013 Optical Society of America OCIS codes: (190.1900) Diagnostic applications of nonlinear optics; (270.5290) Photon statistics; (180.4315) Nonlinear microscopy. http://dx.doi.org/10.1364/AO.53.000051
1. Introduction
In this work we consider the problem of obtaining, with the highest possible spatial resolution, an image of arbitrary structure inside of a transparent solid object, where the use of fluorescent dyes, or placement of structured masks in the object plane, is not an option. Our goal is to critically evaluate the feasibility of the use of two types of techniques to improve spatial resolution: multiphoton detection with nonclassical light illumination and nonlinear optical interactions with the viewed object. We chose an experimental setup that is relatively easy to implement experimentally and to analyze theoretically. Extending the spatial resolution of optical microscopy is a problem of great practical importance and has been a subject of research for a long time [1–5]. One approach is described by Goodman [4] as “bandwidth extrapolation.” It involves recording the image and correcting it numerically by full characterization of the linear response of the imaging system and 1559-128X/14/010051-13$15.00/0 © 2014 Optical Society of America
performing deconvolution. Very high signal-to-noise requirements make this approach impractical for many applications. Another concept for improved spatial resolution was introduced by di Francia [6]. Inserting an appropriate mask with radial rings into a parallel optical beam at the lens, although sacrificing efficiency, he was able to focus the beam to a spot size smaller than the one generated by uniform illumination of the lens. This smaller spot size would allow improved resolution in a scanning microscope. The technique is valid, and recently has been used in conjunction with nonlinear optical techniques for improved resolution [7]. In another classical approach introduced by Lukosz and co-workers [8–11], appropriate grating masks are inserted into the imaging systems, effectively increasing the numerical aperture of the imaging system and improving the resolution. We will discuss these techniques and their relationship to our work in Section 6. More recently, there has been significant progress in fluorescence microscopy. Use of structured illumination [12,13], stimulated emission depletion (STED) [14,15], and stochastic optical reconstruction 1 January 2014 / Vol. 53, No. 1 / APPLIED OPTICS
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microscopy (STORM) [16] all resulted in breaking the diffraction limit. Unfortunately, these fluorescence microscopy techniques are not applicable to our problem as stated in the very first paragraph. However, improvement of resolution using intrinsic optical nonlinearity of observed objects, without adding fluorescent dyes, was also recently proposed and demonstrated [7,17–20]. We will comment on these results and the relationship to our observations in Section 6. An intriguing possible approach utilizing nonclassical light to improve the spatial resolution in microscopy has been suggested in a number of publications [21–27]. It has been noted that N photons when acting as one particle can exhibit a de Broglie wavelength that is λ∕N [28]. For the past decade there has been a flurry of activity in research involving N photon states, such as the N00N states [29–32], with possible applications to improved resolution in optical lithography and microscopy. It has been clearly demonstrated [26,29,32] that sub-diffraction-limited two-photon interference patterns can be produced using these states. Experimental demonstrations of application of this approach to lithography have been successfully performed [32], but so far there have been no reported demonstrations of application of this effect to microscopy. From the literature it is not clear if the problem is one of practical difficulty of producing, interfering, and detecting such states, or if there is a fundamental problem with this approach. Resolving this issue is one of the main goals of this work. We consider an optical system that is constrained by a fixed numerical aperture (NA) for illumination of the object and the same numerical aperture for viewing the image. This means that the illuminating beam will produce on the target a spot size no smaller than allowed by the diffraction limit, or if we use a pair of beams separated by the maximum angle allowed by the numerical aperture, it will produce an intensity grating that has a period no smaller than λ∕2 NA. We use this system to test techniques for detecting and characterizing features that are significantly smaller than the diffractionlimited spot, or equivalently, to detect gratings with periods smaller than λ∕2 NA. An image can be considered to be a superposition of spatial Fourier components, each of which represents a grating, and therefore determination of the shortest period grating that can be detected and characterized gives the resolution of the optical system. In the classical regime, as described in Section 3, this gives the well-known Abbe diffraction limit [1,5]. In this work we use simple and robust classical and nonclassical light interferometers, described in Section 2, to investigate the limits of spatial resolution. These interferometers are quite insensitive to environmental perturbations and very practical for work with femtosecond optical pulses. Detailed analysis of operation of the linear interferometer is described in Section 3, which discusses a simple 52
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linear optical technique for determining the optimal spatial resolution of an imaging system by detecting and characterizing optical gratings. In Section 4, we use a modified version of this setup to investigate the possibility of improved resolution with nonclassical illumination. We found that no improvement in resolution was possible with this scheme, as long as the interaction of light with the target was linear. It is not enough to generate and to detect the multiphoton states. There is also need for interaction of the illuminating beam with the target in multiphoton fashion. In Section 5, we consider what happens if the classical measurement is performed in the nonlinear optical regime, simply by turning up the optical power in the operation of the linear interferometer, and we find that we can readily obtain a factor of two improvement in resolution beyond the diffraction limit, which confirms our theoretical model and is consistent with the results of Refs. [7] and [19]. For practical microscopy it is desirable to detect and characterize both amplitude and phase variations of features of the targets, although phase is usually more important. For practical reasons, in this work we limit our investigation to detection and characterization of phase gratings. They are less likely to be damaged by intense pulses in nonlinear measurements, and less intensity is highly desirable in preserving nonclassical states of light. 2. Classical and Nonclassical Interferometers
In Fig. 1, we show simple and robust classical and nonclassical light interferometers for investigating the limits of spatial resolution. These interferometers are quite insensitive to environmental perturbations and very practical for work with femtosecond
Fig. 1. (a) Classical coherent light imaged grating interferometer. (b) Nonclassical interferometer of the type used in Ref. [36]. (c) Imaged grating squeezed vacuum interferometer used in this work, which is equivalent to (b).
optical pulses. As shown in Fig. 1(a), a phase grating G1 is illuminated by a laser beam to produce two beams that are synchronized and coherent [33,34]. Additional beams produced by the grating can be easily suppressed with beam blocks. The two beams then propagate through two lenses that reimage the grating G1 onto a target grating G2. With proper choice of magnification, the period of the image of G1 can be matched to the period of grating G2. The two main output beams after G2 act as the output arms of a Mach–Zehnder interferometer. As discussed in detail in Section 3, this setup is used to detect and characterize the target gratings. A number of research groups [31,32,35] have demonstrated that spatial frequency modulation in twophoton detection, which is proportional to g2 (degree of second-order correlation), is twice as high as allowed by the diffraction limit with coherent beams, and this can be observed by interfering the spontaneous outputs of a parametric amplifier. The high frequency spatial modulation persists even for high parametric amplifier gains when the beams contain a large number of photons [36]. A typical experimental setup for generation of the two-photon interference involves production of correlated signal-idler beams, which are then combined in an interferometer schematically illustrated in Fig. 1(b). Spatial modulation in two-photon detection can be observed in the plane perpendicular to the second beam splitter. Aligning and maintaining stable operation in this setup is a serious technical challenge. We have a more practical scheme, shown in Fig. 1(c), to generate the same nonclassical interference pattern. The 400 nm pump beam is split into two pump beams by the phase grating G1. The first lens collimates and sends the beams parallel to each other through a BBO crystal aligned for type-I phase matched second harmonic generation from 400 to 800 nm, which is the same as for degenerate parametric gain [37]. Two squeezed vacuum beams that are copropagating with the pump beams are generated by the parametric downconversion process [38]. However, also generated at the crystal will be nondegenerate signal and idler beams, which need to be eliminated by properly placed apertures. The intense 400 nm pump beams are eliminated by prisms, which are not shown in this schematic. The squeezed vacuum beams are imaged to overlap at the target grating G2 , which plays the role of the second beam splitter from Fig. 1(b). This nonclassical imaged grating interferometer guarantees the synchronization of femtosecond pulses and makes it relatively easy to observe quantum interference effects, which show spatial variations in the second-order correlation g2 that have smaller spatial period than the classically allowed intensity interference patterns. Aligning the crystal and apertures can be a tricky experimental procedure since we are dealing with very low intensity light. We have developed a reliable procedure for this alignment. The grating G1 is
fabricated on a substrate next to another phase grating with precisely two times larger period. Simple translation of the substrate allows us to switch between the gratings. We first send an alignment beam at 800 nm through the system and use the larger period grating. The crystal is then tilted for optimal second harmonic generation, and two apertures are aligned on the 800 nm beam. The path of 400 nm beams generated by the crystal is carefully marked with apertures. Not shown on the schematic above (see Fig. 6 for more details) are additional prisms that separate the 400 and 800 nm beams. The 800 nm beams are then imaged on the grating exactly as in the interferometer in Fig. 1(a). The entire alignment can be tested by observing linear interference at 800 nm, with a proper target grating G2. For the actual experiment we switch to the smaller period grating and the 400 nm beam. We make sure that after the crystal this beam follows the same path as the 400 nm beam that was created in the crystal during the alignment. The squeezed vacuum beams are thus automatically aligned to interfere at the target grating. In Section 4, we describe characterization of two-photon interference between the two squeezed vacuum beams, and attempt to use it to improve the resolution of the optical system. 3. Linear Classical Optical Characterization of a Grating
A classical coherent single optical beam can be used to detect the presence of a particular period grating if we observe a non-zeroth-order diffracted beam. However, determination of the location of the grating minima, or phase of the modulation, is essential for image reconstruction from Fourier components. To obtain the phase measurement, we need to use at least one other coherent optical beam. Detection and full characterization of a grating is performed using an interferometer of the type shown in Fig. 1(a). The output part of the interferometer and the intensity interference pattern are shown in Fig. 2. Two monochromatic beams illuminate a grating at angles θ adjusted (by varying the magnification of the optical system) so that the interference pattern has the same periodicity as the grating. This way the light scattered by the grating from one beam can interfere with the other beam increasing the sensitivity of detection and providing information about the phase of the scattered light. We can use beams of
Fig. 2. Schematic of a setup for characterization of grating G2 . 1 January 2014 / Vol. 53, No. 1 / APPLIED OPTICS
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equal or different intensities. The phase and amplitude of the scattered light are determined by observing the modulation of the output as a function of the relative phase between the beams, which translates the interference fringes relative to the grating. Moving the grating with respect to the stationary interference pattern provides the same information. The transmission gratings used in our experimental work consist of an array of rectangular crosssection ridges fabricated lithographically on fused silica substrates. Gratings with repetition period Λg varying from 2 to 256 μm were fabricated on the same substrate. The first grating in the imaged grating interferometer (G1 ) was designed to produce a phase modulation of π rad, resulting in suppression of the zeroth-order diffraction beam and approximately 90% of the incident power going into the 1 orders. The target grating (G2 ) was typically fabricated with a π∕2 phase shift, and, for small angles of incidence, it transmitted ∼43% into the zeroth order, 22% into the 1 orders, and 2% or less into each higher order. We can model this thin phase grating by a complex transmission function [4,39]
Each term in the above expression corresponds to a beam in the far field. We are considering gratings that in the near field contain a large number of periods, and therefore in the far field produce welldefined separate beams that can be modeled by δ functions. We monitor the fields E1out and E2out , which correspond to beams propagating in the same direction as the input beams. For a linear system we can write the input–output relationship as
m
∞ X
An einK g x−x0 A−n e−inK g x−x0 ;
(1)
n1
where K g 2π∕Λg is the fundamental spatial frequency of the grating, m is the peak-to-peak modulation, and An J n m∕2 are the integer Bessel function coefficients in the Jacobi–Anger expansion. We choose an input field consisting of two beams, which can be approximated by plane waves as Ein x; z E1 e
ikx xkz z
E2 e
i−kx xkz z
;
(2)
with kx 2π∕λ sin θ. For the simplest case, we have taken the polarization to be in the y direction, perpendicular to the plane of the intersecting beams. For beams with a relative phase shift ϕ and equal intensities, E1;2 E0 e∓iϕ∕2 . The spatial frequency of the intensity interference pattern of these two beams is K 2kx. Varying ϕ translates the interference pattern across the grating by distance xd , where Kxd ϕ. The electric field right after the grating at z 0 is given by Eout x TxEin x; 0 E1 A0 ei 2 x E2 A0 e−i 2 x ∞ X K E1 An e−inK g x0 ei 2 nK g x K
n1
E1 A−n einK g x0 ei
∞ X
K 2 −nK g
E2 An e−inK g x0 ei
n1
x
−K2 nK g x
54
e
K inK g x0 i − 2 −nK g x
E2 A−n e
K
:
APPLIED OPTICS / Vol. 53, No. 1 / 1 January 2014
(3)
out
τ11 τ21
τ12 τ22
E1 E2
:
(4)
in
The elements of the transfer matrix are obtained by inspection of Eq. (3). For an arbitrary K, there is no overlap between any of the beams and the matrix is diagonal. For the special case of K K g, there are two additional terms having the same direction of propagation as the input beams, and we get
Tx ≈ ei 2 sin K g x−x0 ≈ A0
E1 E2
τ11
τ12
τ21
τ22
A0
A1 e−iK g x0
A−1 eiK g x0
A0
:
(5)
We also note that all other noninterfering beams are outside the numerical aperture of the detection system. Since A1 −A−1 , the intensities of the two detected beams are given by I 1 jE0 j2 jA0 A1 e−iK g x0 eiϕ j2 jE0 j2 A20 A21 2A0 A1 cos K g xd − x0 ;
(6)
and I 2 jE0 j2 jA0 A−1 eiK g x0 e−iϕ j2 jE0 j2 A20 A21 − 2A0 A1 cos K g xd − x0 :
(7)
As we vary ϕ, or x0, and translate the optical interference pattern across the grating, modulation of the two beams at the grating’s spatial frequency can be readily observed and used to calculate A1 ∕A0 , which is the strength of the fundamental Fourier component of the grating. The value of x0 here specifies the position of the grating. In this work we will concentrate on analysis and detection of phase gratings, although it is straightforward to perform a similar analysis for amplitude gratings. (There we find that the modulation intensities of the two beams are in phase. This actually makes the detection of modulation more difficult because we cannot use balanced detection.) As discussed above, only a grating with the correct period, K g K, produces modulation of the probe beams as we vary ϕ. This approach was successfully used for precise measurement of nonlinear transient gratings in order to determine the phase angle of the complex third-order nonlinearity χ 3 [40]. The largest spatial frequency grating that can be observed by
0.16
this method is equal to the largest value of K of the illuminating beams, which is determined by the maximum allowed angle θ in this imaging system. We have 4π sin θmax ; λ
(8)
and the smallest observable features in the image have the spatial extent comparable to the spatial period Λg of this grating, which is Λg
2π λ λ 0 ; K max 2 sin θmax 2 NA
(9)
where λ0 is the optical wavelength in vacuum, and NA n sin θmax is the numerical aperture. This is equivalent to the Abbe diffraction limit. The imaged grating interferometer shown in Fig. 1(a) was used to perform the preliminary grating characterization measurements in the linear regime. The laser beam consists of 150 fs pulses at 800 nm from a Ti:sapphire laser system. The beam propagates through a phase grating G1 with 256 μm period, fabricated lithographically on a 1 mm thick fused silica substrate. After the grating, the zeroth-order beam is suppressed and a lens combines the 1 order beams to project an image of G1 onto the test grating G2 . Keeping the target grating at the same axial position and by translating the lenses and the first grating, the demagnification is adjusted so that the intensity interference pattern has a period of 10 μm. Figure 3 shows the outputs of the two detectors when we use a phase grating with a period of 10 μm as the test grating G2 . This is the smallest period grating that can be measured using the available numerical aperture of 0.04. We see a very strong intensity interference pattern as the test grating is laterally translated. The test grating can be characterized by measuring the amplitude and the relative phase of the two sinusoidal signals. From this we can find the real and imaginary parts of the complex transmission function of the grating. In this case, we see that the two signals are oscillating approximately 180 deg out of phase, which according to Eqs. (6) and (7) is expected for a pure phase grating. We will next consider what improvement in resolution is possible if we use nonclassical illumination and nonlinear optical effects to probe the grating.
Voltage (V)
K max 2kx
0.12
0.08
0.04
0
0
10
20
30
Grating Position (µm) Fig. 3. Plot of the outputs of two detectors at low intensity for a 10 μm period test grating G2 .
doubled Ti:sapphire pump beam at 400 nm propagates through a transmission phase grating designed for zero diffraction order suppression. After the grating, the 1 orders contain approximately 90% of the incident power. A lens collimates the two diffracted beams, which then propagate through a 5 mm thick BBO crystal with type-I phase matching for degenerate parametric downconversion. The two pump beams generate collinear 800 nm squeezed vacuum beams, which are selected by two properly placed apertures. The two prisms shown in Fig. 4 are used to eliminate the 400 nm light from the detection system. The second lens images the squeezed vacuum beams on an overlap region where the grating to be tested is placed. In order to characterize the nonclassical illumination, we used a long focal length lens to achieve imaging magnification of 10× and a 10 μm wide slit in the region where the two beams overlapped, as shown in Fig. 4. The slit was scanned in the transverse direction. Light after the slit is split into two beams using another zeroth-order suppressing grating that acts as a beam splitter. The beams are recollimated by a lens and sent to photomultipliers (Hamamatsu H10721-20). The laser was operated at 1 kHz, and the signals from the PMTs were digitized by a fast digital oscilloscope (Agilent Infiniium DSO9104A) and stored in a computer. Typically we averaged more than 4096 shots in order to monitor the intensity and calculate the correlation functions. For strong signals containing many photons, the degree of second-order correlation g2
4. Illumination with Nonclassical Light
As discussed in Section 1, there has been significant interest in utilizing multiphoton interference with nonclassical light for improving the resolution of both optical lithography and microscopy [21–24,30–32,35,36]. Our simple and robust nonclassical light interferometer, shown in Fig. 1(c) and with more detail in Fig. 4, produces the type of illumination comparable with that in Ref. [35], and here we consider its utility for testing improvement in resolution. A frequency
Fig. 4. Experimental setup for generation and characterization of two-photon spatial interference pattern. 1 January 2014 / Vol. 53, No. 1 / APPLIED OPTICS
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[36,41] was calculated simply by taking the ratio of the average of signal squared to the square of the average signal. The spatial variation of g2 that is expected in this experiment can be calculated theoretically. The correlation properties of nonclassical light beams can be treated in the Schrödinger picture for two-photon states [42], which are good approximations to weak squeezed vacuum. Dealing with larger number states becomes much more complicated. However, it is straightforward to calculate the first- and second-order degrees of correlation of the light beams transmitted through the slit in the Heisenberg picture [41,43]. We can treat the slit as a linear multiport device [44], as illustrated in Fig. 5. The input modes are represented by annihilation operators aˆ n , which, in the Heisenberg picture, propagate in space very much like the plane waves of classical fields. All the input modes, except for the two corresponding to our squeezed vacuum beams, are in the vacuum state so that the input state can be written as jΨin i jζi1 jζi2
Y
j0in ;
τ02 τ01 eiϕxd . K is the spatial frequency of the interference fringe K∕2 2πn∕λ sin θ, where θ is the angle of incidence of the beams. First, we consider the variation of intensity of transmitted beams as a function of slit position. Calculating the number of photons in the mode b0 , we get hΨin jbˆ †0 bˆ 0 jΨin i 2 hζj1 hζjbˆ †0 bˆ 0 jζi1 jζi2 jτ01 j2 jτ02 j2 hn0 i:
(13)
This is as expected, because the relative optical phase between the two squeezed vacuum beams is random, and therefore at the target there is no first-order interference and the intensity is constant. For the two-photon detection rate, after considerable algebra, using identities from Ref. [41] and assuming jτ01 j2 jτ02 j2 , we obtain hΨin jbˆ †0 bˆ †0 bˆ 0 bˆ 0 jΨin i 2jτ01 j4 hn0 i5hn0 i 1 hn0 i 1 cos 2ϕ:
(10)
(14)
n>2
where the squeezed vacuum state is described by a complex parameter ζ seiϕ [41]. The parameter s is related to the number of photons in a single pulse in a squeezed vacuum mode by [41] ˆ hn0 i sinh2 s: hζjaˆ † ajζi
(11)
The annihilation operators for the output modes bˆ m can be written as bˆ m
X
τmn aˆ n
n1;2
X
ρmk cˆ k ;
(12)
k1;2…
which is a linear superposition of the input states aˆ n and the noise operators cˆ n , which describe the loss channels. ˆ We are P interested in the operator b0 τ01 aˆ 1 τ02 aˆ 2 k ρ0k cˆ k , because the only relevant nonzero coefficients of τ0n are τ01 and τ02 . Just like in classical optics, these coefficients are equal in magnitude and their relative phase is ϕxd Kxd . We will write
bˆm
aˆ1 bˆ0
aˆ2 aˆn Fig. 5. Propagation of quantum mechanical operators through a single slit in the Heisenberg picture corresponding to the experimental setup in Fig. 4. 56
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The Langevin noise operators do not affect the result because we are calculating the expectation value of normally ordered operators. The degree of secondorder correlation is given by g2 bb
hbˆ †0 bˆ †0 bˆ 0 bˆ 0 i hbˆ † bˆ 0 i2 0
1 2 hn0 i5hn0 i
1 − hn0 i 1 cos 2ϕ : hn0 i2
(15)
Note that for weak squeezed vacuum described by hn0 i → 0, the modulation depth goes to unity, as expected, since the detected state (in Schrödinger’s picture) is alternating between j1i1 j1i2 and the p N00N state 1∕ 2j0i1 j2i2 j2i1 j0i2 . As the average number of photons per pulse increases, the depth of modulation decreases, but it does not vanish. Even for a very large number of photons, g2 oscillates between 2 and 3, resulting in fringe visibility of 20%, just as in Ref. [32]. It is interesting to compare this result with the case of interfering classical coherent beams. In that case the intensity is modulated by the phase angle ϕ and the degree of second-order correlation is constant and equal to one. For the case of interfering two squeezed vacuum beams, on the other hand, we find that the intensity is constant and the degree of second-order correlation is modulated at twice the spatial frequency (phase angle is 2ϕ). Figure 6(a) shows the experimental measurements for intensity and the degree of second-order correlation for coherent illumination. For a physical grating with a period of 64 μm, a classical intensity pattern with a period of 64 μm is clearly seen using coherent illumination in Fig. 6(a). As expected, the degree of
3
Intensity or g(2)
Intensity or g(2)
1 0.8 0.6 0.4
2
1
0.2 0
0
100
200
0
0
100
200
Grating Position (µm)
Grating Position (µm)
(a)
(b)
Fig. 6. (a) Interference pattern with coherent light; the intensity (lower trace) is modulated and the degree of second-order correlation (upper trace) remains constant. (b) With squeezed vacuum, the degree of second-order correlation is modulated and the intensity is not. The spatial period of modulation differs by a factor of two.
second-order correlation g2 is constant and equal to one. With squeezed vacuum, for the same imaging system, we see in Fig. 6(b) no modulation in the intensity and a 32 μm period pattern in g2 for the same 64 μm physical grating. We have a reasonably good agreement with the theory. Under ideal conditions the modulation of g2 should be between 2 and 3 [28,31]. We believe that the reason for reduced modulation depth is due to the presence of more than one spatial mode in each squeezed vacuum beam and imperfect overlap of modes. This can be corrected with proper choice of nonlinear crystal thickness and spatial filtering. Also, reducing pump intensity at the nonlinear crystal should result in increased visibility of modulation fringes (and drastically degrade the signal-to-noise ratio), by reducing the background due to contributions from different N00N states constituting the squeezed vacuum beams. The results shown in Fig. 6(b) clearly illustrate that our imaged grating interferometer is capable of creating sub-diffraction-limited spatial patterns, and, in conjunction with a two-photon absorber, it could be used for high-resolution lithography. The intriguing question is whether these patterns could be used for microscopy to probe sub-diffraction-limited features. The interfering squeezed vacuum beams can be used to illuminate and to characterize a target grating, as illustrated in Fig. 7. This target grating replaces the slit in the apparatus shown in Fig. 4. We can readily calculate the intensities and correlations of the output signals after the grating. Again we consider the problem in the Heisenberg picture with two incoming squeezed vacuum beams as the input state and the grating modeled as a multiport device. With squeezed vacuum inputs, the input state is the same as in Eq. (10), and the output operators bˆ n can be written as a linear combination of the input operators aˆ n . Experimentally, we monitor the intensity and correlation only of beams b1 and b2 . In general,
bˆ 1 τ11 aˆ 1 τ12 aˆ 2
X
τ1n0 aˆ n0
n0 >2
X
τ1m cˆ m ;
(16)
m
but only the first two terms contribute to the calculated intensity and to g2 . In the Heisenberg picture, these coupling coefficients, which depend on the input angles of the beams and the period of the physical grating, are exactly the same as for coherent classical light [41,44] and can be obtained from Eqs. (3) and (4). In the special case in which the grating has spatial frequency of K g (the highest period grating observable with classical linear optics) the coefficients are given by Eq. (5). With these coefficients we can calculate the intensities and second-order correlation functions for the output beams. We find that g2 b1 b1 is described by the same expression as Eq. (15), viz., g2 b1 b1
hbˆ †1 bˆ †1 bˆ 1 bˆ 1 i hbˆ † bˆ 1 i2 1
1 2 hn0 i5hn0 i
1 − hn0 i 1 cos2ϕ ; hn0 i2
(17)
while the cross correlation is given by g2 b1 b2
hbˆ †1 bˆ †2 bˆ 1 bˆ 2 i hbˆ † bˆ 1 ihbˆ † bˆ 2 i 1
2
1 2 hn0 i3hn0 i
1 hn0 i 1 cos2ϕ ; hn0 i2
(18)
where hn0 i is the number of photons in each incident beam, and ϕ is the phase between complex transmission coefficients τ11 and τ12 . This functional dependence is observed in our experimental data shown in Fig. 8(b), and is typical of two-mode squeezed light interference patterns [32]. The light illuminating the target grating has modulation in g2 at the spatial frequency of 2 K 1 January 2014 / Vol. 53, No. 1 / APPLIED OPTICS
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bˆ3
aˆ4 aˆ1
bˆ2
aˆ2
bˆ1 aˆ3
period grating, as seen in Fig. 8(c), no modulation is observed. This is consistent with our theoretical prediction. We conclude that, unfortunately, the use of two-photon interference with nonclassical light in this experiment does not lead to improvement in spatial resolution. As mentioned earlier, it is well known that a multiphoton state can manifest a de Broglie wavelength [28,29] that is shorter than the optical wavelength, which motivated the proposed use of such states for potential use in microscopy. However, it is not enough to be able to create and to detect this multiphoton state. The state needs to have a multiphoton interaction with the target in order to take advantage of the shorter wavelength. Multiphoton interaction is intrinsically a nonlinear process, and the limitations of linear optics no longer apply. As we show in the following section, when there is nonlinear interaction with the target, even with coherent light it is relatively easy to obtain spatial resolution beyond the Abbe limit.
bˆ4
Fig. 7. Propagation of the beams through a grating can be modeled as a multiport device.
as shown in Fig. 6(b). It is intriguing to try to detect a physical grating with K g 2K 4kx . As discussed in Section 3, for this case the classical coefficients are given by
τ12 τ22
τ11 τ21
A0 0
0 : A0
(19)
5. Nonlinear Optical Characterization of a Grating
According to the results from Section 3, this grating cannot be observed with coherent light. For this grating, the output intensities in the experimental setup shown in Figs. 1(c) and 2 are not affected by the lateral translation of the grating or variation of ϕ, the relative phase shift between the two beams. Unfortunately, in the Heisenberg picture, the coupling coefficients in Eq. (16) are the same for both the classical and the quantum mechanical treatments. This leads us to conclude that there is no modulation of g2 with variation of ϕ, or equivalently, translation of the grating. Figure 8 shows the experimental results. Experimental grating characterization with coherent beams is shown in Fig. 8(a). The setup works as expected in the linear mode with a 32 μm period grating. For the same period grating, when we switched to squeezed vacuum beams we observed, as predicted by Eqs. (17) and (18), a modulation at twice as high a spatial frequency, as shown in Fig. 8(b). When we replace the test grating with a factor of two shorter
0.8 0.6 0.4 0.2 0
20
40
60
Intensity or g(2)
Intensity or g(2)
Intensity or g(2)
1.5
3
1
0
Next we consider coherent light beam propagation through a grating structure that has a period that is half of that observable with linear optics. We are performing experiments with low average power and very high peak power laser pulses in high-quality fused silica, which has negligible linear absorption at 800 nm. In this regime, thermal effects are negligible and the only nonlinearities that can be observed [45] are nonlinear index of refraction effects and catastrophic damage due to optical breakdown. We include the optical nonlinearities in the medium right after the grating. The grating can be modeled with a transmission function as described in Eq. (1). As shown in Fig. 9, beam #1 and beam #2 overlap on the target grating structure and produce interference fringes with a period twice that of the target grating structure. The diffracted beams #3 through #6 have diffraction angles larger than the angle between two beams (they are outside the allowed numerical aperture) and do not overlap with any
2 1 0
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Fig. 8. (a) Interaction of coherent light with 32 μm period target grating; intensity in each detector (lower traces) is modulated and the degree of second-order correlation (upper trace) remains constant. (b) Interaction of squeezed vacuum beams with 32 μm grating. The 2 2 degree of second-order correlation (upper two traces are g2 11 ,g22 , and the middle one is cross correlation g12 ) is modulated with twice the spatial frequency (period of 16 μm). The lowest two traces are the intensity measured by the two detectors, and there is no modulation. (c) There is no observable modulation in the degree of second-order correlation for grating G2 with Λg 16 μm interacting with squeezed vacuum beams. 58
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of the original beams. There is no interference between probe beams and diffracted beams in the linear region, and translating the grating laterally does not produce any observable modulation in probe beam powers. However, at high intensity, this grating can be readily observed due to nonlinear effects, as we discuss below. In order to explain this effect we need to include the analysis of nonlinear propagation for all beams in the fused silica substrate right after the grating. The two incident probe beams have transverse k − vector components with spatial frequencies kx : k⃗ 1 kx xˆ k1z zˆ ;
k1z
q k2 − k2x .
(20)
The interaction with a physical grating of spatial frequency 2K 4kx produces additional beams #3 and #4 at 3kx , and #5 and #6 at 5kx with q k2 − 3kx 2 ; q k2 − 5kx 2 :
k⃗ 3 3kx xˆ k3z zˆ ;
k3z
k⃗ 5 5kx xˆ k5z zˆ ;
k5z
(21)
These diffracted beams are not observable if the numerical aperture is limited to only kx beams. However, the nonlinear interaction, among all the beams as described below, results in change of intensity of the two original probe beam directions, kx . Therefore we can observe modulation of the photodiode outputs as the interference pattern is translated. In order to simulate the interaction of probing light with the target grating and the nonlinear medium that is right after it, we model the electric field at z 0 (right after the target grating) as Kg Ex; z 0 E0 cos x Tx: 2
Since the contribution from higher diffraction orders is negligible, we can express the electric field at z 0 using the 0th and 1∕ − 1 diffraction orders. There are six terms, Kg Kg Ex; z 0 E0 A1 ei 2 xiK g xd A−1 e−i 2 x−iK g xd A3 ei A5 ei
3K g 2 x−iK g xd 5K g 2 xiK g xd
Fig. 9. Schematic showing optical beams used for calculation of nonlinear grating characterization.
3K g 2 x−iK g xd
A−5 e−i
5K g 2 x−iK g xd
;
(23)
where A1 A−1 a0 and A3 A−3 A5 A−5 a1 . In the slowly varying envelope approximation, the electromagnetic wave equation describing diffraction and nonlinear interaction can be written as ∂ 1 Ex; z ∇2 Ex; z − ik0 n2 jEx; zj2 Ex; z: ∂z 2ik0 ⊥ (24) Here the first term is diffraction and the second term is the contribution from the third-order nonlinear optical process. There are 27 terms contributing to the power modulation in probe beam #1. Combining similar terms, we have d E 2in2 k0 dz0 1 jE1 j2 2 2 2 2 2 jE−1 j jE3 j jE−3 j jE5 j jE−5 j E1 2 0
0
E3 E1 E−1 eiΔ31 z E3 E−3 E−5 eiΔ51 z E23 E5 iΔ31 Δ35 z0 e 2 0 0 E5 E−1 E−5 e2iΔ51 z E−1 E3 E5 eiΔ53 z
E2−1 iΔ13 z E e 2 −3
0
E3 E−3 E−1 e2iΔ31 z 0
(22)
A−3 e−i
0
E−1 E−3 E−5 eiΔ35 z E−3 E1 E5 eiΔ31 Δ51 z ;
(25)
where Δ31 k3z − k1z −Δ13 , Δ35 k3z − k5z −Δ53 , and Δ15 k1z − k5z −Δ51 . We integrate this equation and calculate the intensity of beam #1 as a function of propagation distance z with initial conditions given by Eq. (22). We find that there is modulation at the spatial frequency of K g as we vary xd. The maximum modulation depth is calculated for comparison and is reported with the experiment described below. This improvement in resolution is based on increase of the effective numerical aperture of the detection system, and a similar effect has been reported in Ref. [19]. Figure 10(a) shows the schematic of the experimental setup of the robust nonlinear interferometer for the sub-diffraction-limited grating structure diagnostic. A half wave plate and a thin-film polarizer provide a convenient and reliable way to vary the incident laser beam power. As in linear grating characterization, the grating G1 has a period of 256 μm, and the optical system has a demagnification factor of 1 January 2014 / Vol. 53, No. 1 / APPLIED OPTICS
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(e)
Fig. 10. (a) Schematic of the experimental setup of the robust nonlinear interferometer for the sub-diffraction-limited grating structure characterization (G2 with a period of 5 μm). (b) Photodiode outputs from the two probing beams. (c) and (d) are the Fourier frequency components of the individual photodiode output. (e) Fourier frequency components of the difference of the two photodiode outputs.
12.8. The intensity modulation generated from the interference of two probing beams has a period of 10 μm. The grating G2 has a period of 5 μm. In the linear optical regime, scanning G2 will not produce any modulation in photodiode outputs. After increasing the incident laser power density to 200 GW∕cm2 , periodic modulation is observed in both photodiode outputs as grating G2 moves laterally at a step size of 0.2 μm. The results at ∼900 GW∕cm2 are shown in Fig. 10(b); the vertical axis is between 0.85 and 0.93 V, and the modulation depth is about 4%. Figures 10(c) and 10(d) show the Fourier frequency components of the individual photodiode outputs. There are observable Fourier frequency components at spatial frequency 0.2 μm−1 corresponding to 5 μm grating structures, which are much smaller than their DC components. However, we can successfully subtract these DC components from the two photodiode outputs, as clearly illustrated in Fig. 10(e), and the Fourier frequency components of the difference of the two photodiode outputs highlight the peaks at spatial frequency 0.2 μm−1 or 5 μm grating structures. After this subtraction, the DC frequency component is below the noise level. The experimentally measured dependence of modulation on optical intensity is plotted with asterisks in Fig. 11. Within the experimental error, there is a linear variation. When the intensity of the beam was increased above 900 GW∕cm2 , because of selffocusing, the beam breaks up and this limits the magnitude of the observable modulation. According to theory, because of phase mismatch in the nonlinear interaction, the modulation depth oscillates with propagation distance exhibiting a period of approximately 100 μm (the thickness of the substrate is 60
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1 mm). For comparison with the experiment, we calculate the maximum modulation. Although in the experiment we are not able to smoothly vary the propagation distance, the variation of the beam diameter and the location of the beam image effectively had the same effect and allowed us to maximize the signal. The solid line shows the theoretical results using n2 2.5 · 10−7 cm2 ∕GW and a 60 μm spot size. We have a reasonable agreement between the experiment and theory. 6. Discussion
In this work we demonstrated experimentally that a simple imaged grating interferometer can be used to
Fig. 11. Modulation depths versus incident laser intensity. Red asterisks are experiment results and solid line is numerical simulation.
probe optical transmission gratings to investigate the possibility of going beyond the diffraction limit using nonclassical light and nonlinear optical effects. Using this interferometer, we observed that the interference of two squeezed vacuum light beams exhibits a spatial degree of second-order correlation that has a period that is a factor of two smaller than the one corresponding to classical intensity interference of two coherent beams. Spatial variation of a classical material property (such as index of refraction) at a particular spatial frequency is directly related to appropriate Fourier components, such as An and A−n of the transmission function given by Eq. (1). Experimental measurement of the fringes, such as those shown in Figs. 3 and 8(a), give the amplitude and phase of these Fourier coefficients and allow reconstruction of the object. In correlation measurements with nonclassical light as in Fig. 8(b), the measurement of modulation of the degree of second-order correlation can also give the same information. But we only observe fringes for those Fourier components of the transmission function that are also observable with classical measurement. The interaction of nonclassical light with a material grating could be treated in the Schrödinger picture, but it is much clearer in the Heisenberg picture, where the annihilation operators evolve just like classical waves. And from the analysis presented in Section 4, it is evident that the modulation of second-order correlation cannot be used for improvement of resolution in microscopy as long as the interaction between the beams and the target grating is linear. The nonlinear interaction of light and matter, for a very low level of illumination, can be significantly stronger for the nonclassical states than for coherent light, but the effect is still negligible under practical conditions with conventional optical materials. It is useful to consider the experimental demonstration of super resolution by Lukosz [8,10] and compare it to our results. A simplified schematic of the experimental setup from Ref. [10] is shown in Fig. 12. Without the masks M and M 0 , the resolution of the imaging system is limited by the aperture at the lens L1 . In Ref. [10], the grating mask M shown in Fig. 12 was placed either before or after the object plane OP. If the mask was placed after OP before the lens L1 , then it collected the rays associated with higher spatial frequencies than normally acceptable by the imaging system. The second mask redirected these rays to the appropriate part of the image plane IP0 . Clearly, this approach is not useful for our problem in which the resolution is limited by available viewing angle of the object. This argument also applies for the mask placed before the OP. It is interesting to consider the case in which the mask is placed exactly at the OP [46]. In this case, as explained by McCutchen [47], “the effect of the stop is similar to that of the local oscillator in a superheterodyne radio receiver. Both serve to shift signal frequencies into the passband, spatial or temporal, of a subsequent filter. In a microscope, however, it
is not very practical to confine the illumination by a stop right next to the object…” In our experiment with nonlinear optics, we also generate a phase grating that has the same effect as mask M in Fig. 12. However, in our case it is a virtual optical element due to the nonlinear index of refraction, and not a real physical grating. Barsi and Fleischer [19] demonstrated increased field of view by propagating the light right after the object through a photorefractive crystal, a highly nonlinear medium. The high spatial frequencies that in a linear system would have escaped from this imaging system experienced nonlinear coupling with beams at low spatial frequencies and created new spatial Fourier components that carried the information to the image plane. This information was then recovered through numerical analysis. This is basically the same physical process that preserves the high spatial frequency information in our experiment, except that we are using the nonlinearity of the substrate material, fused silica. It is important to consider what happens with our technique at smaller grating periods, and appropriately higher numerical apertures. With index of refraction equal to 1.5, for grating periods below half wavelength, the diffracted beams are evanescent waves. Figure 13 shows results of numerical simulations for small period gratings. It is evident from the plot that the effect becomes small, with modulation depth ∼10−4 , but does not vanish. If we consider the illuminating beams incident onto a grating fabricated on a substrate with index of refraction 1.5 from air, then with linear optics the smallest allowed observable grating would be λ0 ∕2 0.4 μm. Using index matching allows us to decrease the observable grating to λ0 ∕3 0.267 μm. The projected limit of this type of nonlinear measurement is still another factor of two below that limit. It will be a challenging task to detect such a small modulation, especially in the presence of classical and shot noise on the optical beams. The common mode pulse-to-pulse fluctuations can be taken care of by balanced detection in the case in which the target is a phase grating. Theoretical analysis shows that the shot noise problem can be alleviated by x
x
OP
f
f
a
M
L1
IP
L2
M’
IP’
Fig. 12. Schematic of experiment setup of Ref. [10] for obtaining super resolution. OP, object plane; IP, image plane; IP0 , another image plane; L1 and L2 , lenses; M and M 0 , grating masks. Green lines illustrate rays for image formation without, and red is with gratings inserted into the optical path. 1 January 2014 / Vol. 53, No. 1 / APPLIED OPTICS
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Fig. 13. Numerical simulation of modulation depth versus period of the target grating.
using squeezed coherent light beams. Of course this could be useful only in systems with extremely low loss. In particular, high signal-to-noise ratios are required in order to detect evanescent waves from very small period gratings, especially if intensity is kept low to prevent optical damage. For example, if photon counting detectors are used, the photon number squeezed probe beams could in principle allow subshot noise detection. In the case of a homodyne detection system, the phase of squeezed light could provide an advantage. We note here that although the nonlinear optical technique described in this work was used to demonstrate the physical feasibility of detecting spatial features below the diffraction limit, it is not necessarily the optimal approach for practical microscopy. For high numerical apertures, there can be significant improvement in signal-to-noise ratio if the nonlinearly generated light is of a wavelength slightly different from the intense illuminating beams, as has been demonstrated recently by Kim et al. [7] and Leng et al. [20]. In summary, we investigated various approaches for improving the spatial resolution beyond the diffraction limit by establishing criteria for detecting high spatial frequency gratings. As long as the interaction of illuminating beams with the grating is linear, we found that illuminating an object with nonclassical light and observing the multiphoton interference pattern cannot lead to improved resolution beyond the classical diffraction limit. This is evident from theory for any illumination scheme, and was experimentally observed in our setup. In the case of nonlinear interaction of the grating with the probe beams (we were using the intensity-dependent index of refraction), we found that a factor of two improvement beyond the diffraction limit is possible. References 1. E. Abbe, “Beiträge zur Theorie des Mikroskops und der Mikroskopischen Wahrnehmum,” Archiv für Mikroskopische Anatomie IX, 413–468 (1873). 2. C. Cremer and B. R. Masters, “Resolution enhancement techniques in microscopy,” Eur. Phys. J. A 38, 281–344 (2013). 62
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