The Equivalent of Fourier Holography at the Nanoscale

Abstract

The paper deals with optical holographic interferometry at the nanometric scale. The observed objects are sodium chloride nanocrystals. The object illumination is done through the use of evanescent wavefronts. The observed crystals become self-luminous objects producing pseudo-non-diffracting wavefronts. The wavefronts emerging from the crystals are the result of electromagnetic resonances of the crystals. A microscope is utilized to register the wavefronts generated by the crystals. A 6 μm spherical particle made of polystyrene acts as a relay lens to collect the wavefronts that are recorded by a monochromatic CCD and a color camera attached to the microscope. The structure of the recorded images is determined through Fourier transform analysis. It is shown that the recorded images are lens holograms formed by the interference of the wavefronts generated by the crystals. Fourier transform algorithms and edge detection algorithms are utilized to obtain the dimensions of the crystals. The power of Gabor’s idea when he invented holography is again proven in this study. If the problem of super-resolution is viewed from the point of view of the Theory of Communications, the fact that one can register both amplitude and phase of a signal of a self-luminous object provides the means of reaching spatial resolutions with average standard deviation of ±3 nm using helium–neon laser illumination of λ = 632.8 nm. The resolution that can be achieved depends on the structure of the observed material, in the present case ±5d, where d is the distance of the atomic planes of the NaCl. With improvements in the hardware and software higher resolutions may be feasible.

Appendices

Appendix 1

Semi-classical model to explain the observed electromagnetic resonance phenomena

To provide supporting evidence to the phenomena utilized in this paper to observe objects at the nanoscale, a brief outline of the theoretical foundations is presented in this appendix. There are several areas of research both theoretical and experimental in Optics connected with the subject matter of this paper. A large amount of papers were published in the last ten years on the subject of solutions of the Maxwell equations beyond the known classical solutions. To mention just a few examples: super-resolution, that is obtaining images with resolutions well beyond the Rayleigh limit; propagation of pseudo-non-diffracting wave packets in space and through optical devices (for example, Bessel beams); quantum lithography which uses non classical properties of photons to achieve super-resolution; plasmonics that utilizes properties of the free electrons in metals to get super-resolution; the so called “left handed” materials, that also produce images that are super-resolution images. Many of these areas have developed independently, and have been applied to different physical systems. The utilized approaches to deal with them vary from simple heuristic developments to very sophisticated quantum mechanics arguments. All these different fields have a common physical underpinning, and a mathematics coming from the theory of vibrations of solids. Every oscillator, whether a mass on a spring, a violin string, or a Fabry–Perot cavity, has some common properties that arise from the mathematics of vibrating systems and the solutions of the differential equations that govern the vibratory motions. In what follows some of these common properties will be utilized. Although simple, the following model illustrates the point made above in this section without getting into the very complex subject of the solution of a quantum resonator under conditions that are not yet fully understood. Furthermore, the model is in the same line of thought utilized in [22].

The observed diffraction of light that produces orders 0 and ±1, described in the section “Description of the formation of the images” of the main paper, is similar to the typical of the acousto-optic effect denominated Bragg’s diffraction in analogy to X-ray diffraction where a similar process takes place. The acousto-optic effect is similar to the photoelastic effect, in the sense that the permittivity ɛ of a material is changed due to strain components ɛ _ij caused by the propagation of an acoustic wave. The diffraction grating moves with the velocity of the sound wave in the medium. The light going through the transparent material is diffracted. This diffraction pattern corresponds with a conventional diffraction grating pattern at angles θ _n from the original direction, and is given by:

$${\text{sin}}\,\theta _n {\text{ = }}n\frac{\lambda }{\Lambda }$$

(A1)

where λ is the wavelength of the optical wave, Λ is the wavelength of the acoustic wave and n is the order. From the quantum mechanics point of view this process can be described as the collision between a phonon of wavelength Λ and a photon of wavelength λ. A similar effect occurs with the interaction between near infrared photons and visible light photons [23].

As was said before, the acousto-optic effect is due to the propagation of mechanical waves in a transparent medium; in the present case, NaCl nanocrystals. The mechanical waves are generated by mechanical resonance effects caused by the electromagnetic field of the evanescent waves. The electromagnetic field causes atomic planes to vibrate. At the same time, the electromagnetic resonance generates light by resonance of the electronic layers of the crystals.

To relate equation (A1) with the vibrations of the sodium chloride crystals the following model is introduced. One can consider a uniaxial crystal made of N _a planes of atoms in the x-direction of a coordinate system. The atomic planes are equidistant and react to each other according to a given law [24, 25]. This is a classical model introduced by Born in 1912 to analyze the NaCl lattice and later utilized by Brillouin to analyze the vibrations of lattices. Although highly simplified, it illustrates the two phenomena that operate simultaneously in the observations made in this paper. There are resonance modes in the electromagnetic field (i.e., different colors of emitted light) that depend on the length of the nanocrystals. At the same time, there are resonances (phonon-photon interaction, photon-photon interaction, near infrared-visible light interaction) that produce the observed diffraction angles. Brillouin [24] designates these two simultaneous events, respectively, as the optical branch and the acoustical branch of the resonant modes. It is possible to show that the eigenvalues of crystal vibrations are given by the condition:

$${\text{2}}\pi a\left( {N_{\text{a}} {\text{ + 1}}} \right)d{\text{ = 2}}\pi aL{\text{ = }}\pi K$$

(A2)

where: “a” is a wave number, a spatial frequency that characterizes the eigen vibrations of the crystals; d is the distance between atomic planes, a very important parameter that defines the eigenvalues of the crystal; L is the length of the resonant medium; K is an integer. The argument is semi-classical: a transition between quantum mechanics and quantum theory. The spatial frequency “a” is not given by a continuous function but by the discrete values provided by the discontinuous series:

$$a{\text{ = }}\frac{K}{{{\text{2}}\left( {N_{\text{a}} {\text{ + 1}}} \right)d}}.$$

(A3)

This model relates the frequency of the emitted light with the eigenmodes of a standing wave produced in the crystal by two waves propagating in opposite directions. It can be shown that these waves are solutions of the Helmholtz equations, the scalar form of the Maxwell equations for the electric and magnetic fields. The connection between the model of the crystal vibrations and the observed phenomena can be established through the following equation:

$${\text{sin }}\theta {\text{ = }}\frac{\lambda }{{{\text{2}}\,n_{\text{r}} \,p_{\text{p}} \,N}}{\text{ = }}\frac{{n\lambda }}{\Lambda }$$

(A4)

where the double equality derives from equation (A1). For the first order (n = 1), it follows:

$$\Lambda {\text{ = 2}}n_{\text{r}} p_{\text{p}} N$$

(A5)

In equation (A4), sinθ is a measured quantity (coming from correlation of data in Fig. 19 of the main part of the paper); λ is determined from the color images (Table 6); the index of refraction n_r (Table 6) is a quantity depending on the value of the wavelength λ that is measured experimentally and is found in the literature pertaining to the properties of NaCl; p _p, as already described in the main part of the paper and then recalled later in this appendix, is measured from the FT of the real part of the FT of the diffraction pattern of the recorded image; N is an integer. Therefore, N can be computed from equation (A4) while Λ can be computed from equation (A5).

Table 6 Determination of N from equation (A4)

The next step is to connect equations (A3–A5) with the model defined by equation (A2). The spatial frequency is equal to:

$$a{\text{ = }}\frac{{{\text{2}}n_{\text{r}} N}}{\lambda }.$$

(A6)

By combining equations (A2) and (A6), by multiplying and dividing the right hand side of (A6) by p _p, and recalling equation (A1), the following relationships can be written:

$$a{\text{ = }}\frac{K}{{{\text{2}}L}}{\text{ = }}\frac{\Lambda }{{\lambda p{}_{\text{p}}}}{\text{ = }}\frac{{\text{1}}}{{p{}_{\text{p}}{\text{sin}}\theta {\kern 1pt} {\kern 1pt} }}.$$

(A7)

Table 6 contains all the necessary data to compute N. By knowing N, Λ can be also computed with equation (A5). Finally, K can be obtained from equation (A7) and from equation (A2).

Table 7 shows the corresponding results and the excellent agreement of the values of K computed from alternative paths. Table 7 shows also the values of Λ computed directly from equation (A1) for n = 1. We have then obtained relationships that provide the connection between the mechanical vibrations, producing the diffraction of the light, and the electromagnetic resonance modes of the crystals.

Table 7 Determination of values of K and Λ

There is a geometrical representation illustrating the physical meaning of the quantities that appear in the adopted model (see Fig. 20). The wavelength of the crystal oscillating in the mechanical fundamental mode is twice the length of the crystal: λ _v = 2L. Let us call λ _v the fundamental mode wavelength of the mechanical vibrations. The quantity K hence becomes:

$$K{\text{ = }}\frac{{\lambda _{\text{v}} \,\Lambda }}{{\lambda p{}_{\text{p}}}}.$$

(A8)

Fig. 20

(Color online) Schematic representation of the vibrating mode of the nanocrystal of length 86 nm. Two waves propagate in opposite directions and form a standing wave. The red dots correspond to the nodes that do not change in time; p _p is the distance between nodes and antinodes (the latter are indicated by red circles)

Utilizing information provided by the color images, it is possible to know the wavelength λ of the resonant modes of the different crystals. In Fig. 21, the values of λ vs. twice the length of the particle obtained experimentally are plotted along with the values predicted by the Born–Brillouin model [24, 25]. Another relationship between the mechanical vibrations and the light emitted by the crystals come through the following relationship,

$$\lambda {\text{ = KN}}_{\text{ $ \lambda $ }} p_{\text{p}} {\text{sin}}\,\theta .$$

(A9)

Fig. 21

(Color online) Wavelength λ of resonant modes as a function of λ _v (i.e., twice the crystal length)

In equation (A9), $N_{\text{ $ \lambda $ }} {\text{ = }}\frac{\lambda }{{\lambda _{\text{v}} }}$ is the ratio between the wavelength of the emitted light and the wavelength of the mechanical vibrations. Equation (A9) has been plotted in Fig. 21.

Figure 22 shows that the resonance wavelength values determined in this study agree well with data reported in literature for sodium chloride crystals in the bulk [26]. In order to be consistent with the notation used in [26], in the figure, the wavelength of light emitted by each nanocrystal has been replaced by the corresponding energy value ħω expressed in eV: ħ is the reduced Planck’s constant and ω is the angular frequency.

Fig. 22

(Color online) Reflectance energy spectrum of NaCl crystals. Dotted lines correspond to resonance modes observed in the present experiment for the different nanocrystals (“1” corresponds to L = 54 nm; “2” corresponds to L = 55 nm, “3” corresponds to L = 86 nm; “4” corresponds to L = 120 nm; “I” corresponds to the extrapolated point in Fig. 21). The reflectance is directly connected with the emission of light in the resonant mode of the crystal

The luminescence of sodium chloride was observed in the early 1960s and in the current literature there are many publications that confirm the above mentioned reference. It is worth noting that actual resonance wavelengths observed for nanocrystals are slightly different from their counterpart for the bulk material [27]. The extrapolated point has been added in Figs. 21 and 22 in order to illustrate the minimum possible size of a resonant crystal.

The values of p _p defined in equation (A4) can be determined by utilizing the procedure already described in the section of the main paper “Analysis of the observed images”. The FT of the real part of the FT of the image is computed. To take the FT of the real part of the diffraction pattern corresponds to taking the cosine transform of the FT of the original image. This operation puts back the information from the frequency space into the physical space. Since we have a distribution of intensities as shown in Fig. 3 (upper left corner) of the main paper, the obtained spatial distribution corresponds to the three overlapping orders 0 and ±1 (approximately, three rectangles of uniform intensity). These orders can be clearly identified in the FT pattern of the real part of the image FT. By filtering these orders, one can estimate the value of p _p which corresponds to half of the shift between orders 0 and ±1.

Table 8 gives the values of shift measured in the recorded images using edge detection and the corresponding values 2*p _p evaluated from Fourier transforms. In view of the smallness of the measured quantities, the results show a good agreement. The average standard deviation is ±2.7 nm which is consistent with the errors found on nanocrystal dimensions. Referring these data to the dimension d of the elementary cell, it can be seen that the average standard deviation is about ±5d.

Table 8 Comparison of shifts measured from images or evaluated with FT

Appendix 2

Optimization procedure for determining nanocrystal dimensions

The extraction of thickness information from the phase equation

$$\Delta \Phi = \frac{{{\text{2}}\pi }}{p}\left( {n_{\text{c}} - n_{\text{s}} } \right)t$$

(B1)

presents the following problems: (1) determination of the pitch of the order containing thickness information; (2) computation of $\Delta \Phi$; (3) evaluation of the difference between the index of refraction of the nanocrystal and the index of refraction of the saline solution.

There are several possible frequencies that can provide the phase difference produced by the optical path of the rays through the prism. The order must be present in the region where the observed prism is located. Furthermore, the order must be such that the influence of neighbor frequencies is a minimum.

The difference of the index of refraction involves several critical points. The index of refraction is a function of the color. Each nanocrystal generates light of a different color which depends on the dimensions of the nanocrystal. There is an additional effect that must be considered: the index of refraction of the saline solution is also a function of the wavelength of light. For those reasons, to make an accurate determination of the thickness using equation (B1), one must utilize a process of optimization. Let us formulate the problem in the most general form and include the main variables that influence the outcome. There is a prism of length L that is the length in the plane of the image. Let b be the other dimension in the plane and t the thickness. From theoretical information on the crystal size ratio it is known that:

$$r_{\text{1}} \,{\text{ = }}\,\frac{L}{b}$$

(B2)

$$r_{\text{2}} {\text{ = }}\frac{b}{t}$$

(B3)

where equation (B2) relates b with L while equation (B3) connects t with b.

In equation (B1), the index of refraction of the saline solution is a function of the wavelength of the light and of the concentration C _c (i.e., the content in milligrams of sodium chloride per milliliters of water). That is:

$$n_{\text{s}} \,{\text{ = }}f_{\text{1}} \left( {\lambda {\text{,}}C_{\text{c}} } \right).$$

(B4)

There is a function f ₂ that relates L with the wavelength λ:

$$\lambda {\text{ = }}f_{\text{2}} \left( L \right)$$

(B5)

where this function was obtained from the data analysis.

The function f ₁ can be split in two functions that must be satisfied. Firstly, the index of refraction of the solution depends on the wavelength λ:

$$n_{\text{s}} {\text{ = }}\,f_{\text{3}} \left( \lambda \right).$$

(B6)

Secondly, for a given λ there is the relationship:

$$n_{\text{s}} {\text{ = }}\,f_{\text{4}} \,\left( {C{}_{\text{c}}} \right).$$

(B7)

Finally, main geometric dimensions of the nanocrystals must be integer multiples of the elementary cell size d (i.e., d = 0.573 nm at room temperature):

$$\left\{ \begin{aligned}& L{\text{ = }}\,N_{\text{1}} \,d \\& b{\text{ = }}\,N_{\text{2}} \,d \\& t{\text{ = }}\,N_{\text{3}} \,d \\ \end{aligned} \right.$$

(B8)

where N ₁, N ₂ and N ₃ are three integer multipliers.

In summary, the problem is to find the three values L, b, t constrained to satisfy equations (B1) through (B8). The function f ₁(λ,C _c) can be found through the functions f ₂(L), f ₃(λ) and f ₄(C _c) where f ₂ was obtained from the data analysis while f ₃ and f ₄ can be obtained from literature. The optimization process was carried out in a discrete fashion since there is only a limited set of combinations of geometric dimensions L, b, t that correspond to theoretical size ratios of crystals.

Figure 23 shows the index of refraction evaluated at the different wavelengths for the nanocrystals and the saline solution. As expected, the trends exhibited by these two quantities are similar. The computed values of n _s ranged from 1.396 to 1.399 and were slightly higher (about 3%) than the reference value of 1.36, listed in Table 1 of the main paper, which was found to provide consistent results in previous investigations [1–3] and which resulted from an empirical function that gives the index of refraction of the solution as a function of the concentration. The reason for those slight differences mentioned above is that the index of refraction n _s depends on the wavelength of the light going through the saline solution and hence on the resonant mode developed in each nanocrystal.

Fig. 23

(Color online) Index of refraction of nanocrystals and saline solution in function of the wavelength of light

Figure 24 shows the difference of nanocrystal and saline solution refraction index versus the wavelength of the emitted light. It can be seen that this difference oscillates by about 1.5% around the average value of 0.16 and decreases for the smaller crystals.

Fig. 24

(Color online) Difference between nanocrystals and saline solution refraction indices at different wavelengths of the light