1. Introduction

In recent years, there have been increasing interests in the points of the optical field at which the intensity has zero value, because the phase at the points becomes undetermined and the structure of the field in the neighborhood of such points has a rather complex structure, for example, exhibiting dislocations and optical vortices [1–4]. Studies of the phenomenon associated with phase singularities have gradually developed into a new branch of physical optics, referred as singular optics [4]. While most of the publications concerning with singular optics dealt with monochromatic wave, Wolf and his coworkers showed that drastic spectral changes take place in the vicinity of intensity zeros in the focal region of polychromatic scalar field [5,6], and the experimental studies of the singular behaviors around such points were also carried out [7]. More recently, it has been demonstrated that the spectral anomalies also occur in the vicinity of the dark rings of a Fraunhofer diffraction pattern [8]. The spectral anomalies seem to have a close link to the phenomena of spectral switches, which was found several years ago [9–12], and the link between the spectral anomalies and the spectral switches has been discussed in Ref. [13].

Young’s interference experiment is one of the most fundamental experiments of all physics, which is widely applied to physical optics and quantum optics. In this paper we investigate the spectral behaviors in the interference field of Young’s double-slit interference experiments illuminated by spatially completely coherent polychromatic light or partially coherent polychromatic light. It will be shown that the spectral anomalies also take place at positions located in both the near field and the far field, associated with the dark fringes for a fixed frequency. The evolution of the spectral anomalies from the near field to the far field will be investigated. The potential applications of the spectral anomalies in information encoding and information transmission in free space will be studied.

2 The spectral anomalies in the interference field

2.1 Young’s double-slit illuminated by polychromatic spatially completely coherent light

Let us first consider a spatially completely coherent light, with spectral density S ⁽⁰⁾(ω), is incident upon double slits, as indicated in Fig. 1. The two slits have the identical width, and the inner distance and the outer distance are 2b and 2a, respectively. We define the parameter ε = b/a(1 > ε ≥ 0), for representing the width of the two slits.

It is assumed that the spectrum of the incident spatially completely coherent light consists of a single line of Gaussian profile, centered at frequency ω ₀ and rms width Γ, i.e.,

The spectrum in the interference field can be expressed as

here

here $E_0\left(\omega \right)=\sqrt{S^{\left(0\right)}\left(\omega \right)},z_N=\frac{z}{\frac{a^2}{{\lambda }_0}},u_N=\frac{u}{a}$ , and λ ₀ is the central wavelength of the original spectrum. Equation (3) holds if the observation distance away from the plane containing the double-slit satisfies $z^3\gg \frac{\pi }{4\lambda }{\left[u-x\right]}_{max}^4$ . We can also found from Eq. (3) that the zone z_N ≫ π in the interference field can be viewed as the far field.

 

Fig. 1. Notation relating to Young’s double-slit interference experiment

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The mean frequency is plotted as a function of the observation position (u and z) in Fig. 2, for ω ₀ = 10¹⁵ s^-1 and Γ = 0.01ω ₀. The color in the Fig. 2 is more red and more blue as the spectrum is more redshifted and blueshifted, respectively. As shown in Fig. 2, there exist some positions at which the spectrum exhibits significant change, that is, at one side the spectrum is redshifted, but becomes blueshifted at the other side. It is also shown that the spectral anomalies take place in both the near field and the far field. This means that the spectrum in the interference field is redshifted at some place, and at some other place is blueshifted, and between these places the spectrum is split into two lines [6]. It has been known that this critical position, at which the spectrum exhibits drastic changes, exists just at the neighborhood of the singular point for the frequency ω ₀ [5,6].

 

Fig. 2 (color). Color-coded plot of the relative mean frequency ω̅(u_N,z_N) of the spectrum in the interference field as a function of u_N and z_N . The color is more red or blue as the spectrum is more redshifted or blueshifted, respectively. The parameters are chosen as ε = 0.95 ω ₀ = 10¹⁵s^-1 and Γ = 0.0ω ₀.

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It is found from Eq. (3) that the spectrum in the interference field is generally different from that of the incident light, and it varies with the observation position (u and z). It is also found that the spectral anomaly may take place at some position. The spectral anomaly can be characterized by the mean frequency for different observation position, which is defined as [6]

It is found from Fig. 2 that when z_N > 9 , the spectral anomaly shows the characteristic of regularity for a fixed propagation angle. This means that, along a fixed propagation direction (θ_c ≈ u_N / z_N), the spectrum is always redshifted at the angle a little smaller than θ_c, and always blueshifted at the angle a little larger than θ_c. This special characteristic may be used in the information transmission in free space. We will, in this paper, present approaches for realizing the information transmission in free space.

Now we want to investigate the variation of the spectral anomalies at the fixed point of the interference fields with some parameters. In Fig. 3, we plot the relative mean frequency at the point z_N = 15 and u_N = 3.82 as a function of the spectral width Γ of the incident light. We notice that when Γ < 0.066ω ₀, the spectrum at this point shows redshifted, but when Γ > 0.066ω ₀, the spectrum is blueshifted. Figure 4 plots the relative mean frequency near the observed point (z_N = 15 and u_N = 3.82), with color marks, in the cases of Γ = 0.01ω ₀ (Fig. 4(a)) and Γ = 0.2ω ₀ (Fig. 4(b)), respectively. It is seen that when Γ = 0.01ω ₀ the spectrum at this point (u_N = 3.82) is redshifted, and becomes blueshifted when the spectral width Γ of the incident light is Γ = 0.2ω ₀. The reason for this spectral property may be attributed to that spectral shift at some specific point is dependent upon the spectral width Γ .

2.2 Young’s double-slits illuminated by polychromatic partially coherent light

Now we consider Young’s double-slit illuminated by partially coherent light, whose cross-spectral density in front of the two slits is given by

where S ⁽⁰⁾(ω) is the spectrum of the incident light, as expressed by Eq.(1), and

is the rms spatial correlation distance of the incident partially coherent light, and σ ₀ is the rms spatial correlation distance at the central frequency ω ₀.

Based on the propagation law for the cross-spectral density, the spectrum in the interference field can be expressed as

 

Fig. 3 The relative mean frequency as a function of the spectral width (Γ) of the incident light. The parameters are chosen as ε = 0.95 , ω ₀ = 10¹⁵s^-1 , z_N = 15 and u_N = 3.82 .

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Fig. 4. The spectral shifts are represented by the color marks. The more red or more blue indicates the spectrum more redshifted or more bluedshifted. (a) Γ = 0.01ω ₀ (b) Γ = 0.2ω ₀ . Other parameters are ε = 0.95 , ω ₀ = 10¹⁵s^-1, z_N = 15 . The arrows indicate the positions at which the spectral detector is placed for measuring the spectra. v_N = v/a.

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where ∆₀ = σ ₀ / a is the normalized spatial correlation distance for the central frequency and the integral is performed over the plane containing the two slits. It has been found that the change of the coherence gives rise to the drastic change of the spectrum of the interference field. This property can also be used for information encoding and transmission. In Fig. 5, we plot the relative mean frequency near the observation point (z_N = 15 and u_N = 3.82) as a function of the coherence ∆₀. It is seen that when the coherence is low, the spectrum is blueshifted, and turns into the redshifted when ∆₀ is larger than ten. It has been known that the spectrum in the interference is strongly dependent on the coherence. This indicates that the change of the coherence gives rise to re-distribution of the different frequency component, so that results in the drastic spectral shift at some specific point.

 

Fig. 5 The relative mean frequency as a function of the relative coherence ∆₀ . Γ = 0.01ω ₀ (solid curve), Γ = 0.02ω ₀ (dashed curve) The parameters are chosen as ε = 0.95 , ω ₀ = 10¹⁵s^-1, z_N = 15 and u_N = 3.82 .

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3. The potential applications of the spectral anomalies in information encoding and transmission

As shown in Section 2, the spectral anomalies take place in the interference field. Such spectral anomalies may have applications in information encoding and free-space communications, in which the blueshift could be associated with a bit of information such as a “1”, and the redshift could be associated with “0”. In Fig. 6, the information “1” is encoded by the blueshift; and the information “0” is encoded by the redshift. As the spectral width of the incident spatially completely coherent light is Γ = 0.2ω ₀ , the blueshift occurs in the observation point (z_N = 15 and u_N = 3.82), this means an information “1” is received; when Γ = 0.01ω ₀, the redshift takes place, that is, an information “0” is received.

Above results indicate that if the spectral width can be changed flexibly, the information transmission is realizable. Unfortunately this job is not easy. In the following, another scheme for information transmission by modulating spatial coherence of the incident light is presented.

 

Fig. 6. Illustration for the information (data) encoding and information transmission by controlling the spectral width of the incident light. The blueshift (B, in short) could be associated with a bit of information such as a “1”, and the redshift (R, in short) could be associated with “0”. The observation point is at z_N = 15 and u_N = 3.82. Γ is the spectral width of the incident spatial completely coherent light.

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Fig. 7. Illustration for the information (data) encoding and information transmission by modulating spatial coherence of the incident light. The blueshift (B, in short) could be associated with a bit of information such as a “1”, and the redshift (R, in short) could be associated with “0”. The observation point is at z_N = 15 and u_N = 3.82 . ∆₀ is the normalized spatial correlation distance.

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It is also shown in Fig. 7 that the spectral shift at some observation point of the interference field can be changed by controlling the spatial coherence of the incident light. It is shown that when the coherence is low (for example, ∆₀ = 3), the spectral shift at the observation point (z_N = 15 and u_N = 3.82) is blueshifted, but when the coherence is changed to be ∆₀ = 20, the spectral shift turns into the redshift. In this sense, the information can be transferred in free space. Moreover, several approaches for modulating spatial coherence are available now [14,15]. A flexible approach for modulating spatial coherence with the responding time of the order of 1 μs has been proposed by electronically synthesized holographic grating [14]. Therefore information transmission system can be constructed by modulating the spatial coherence of the incident laser beams with spectral width Γ . The transmitted data are sent to the observation point (for example, z_N = 15 and u_N = 3.82) by modulating the spatial coherence of the laser beam; and a spectral detector located at the observation point receives the spectral shifts with time. If the time interval of the bits is longer than 1 μs , and the measuring for the spectral shifts is fast enough, the detected spectral shifts with time represent a set of data delivered from the double-slit. This new proposal for information transmission in free space by spectral shifts may have some advantages over traditional communication system, because in the new scheme, the information is encoded by spectral shifts, the fluctuation in the light intensity does not cause the errors in bits transmission. Moreover, due to the symmetry of the spectral shifts in the interference field along the axis u_N = 0 (see Fig. 2), the information can be transmitted to two symmetrical points, at which the amount of spectral shifts is the same.

4 Conclusions and discussion

It has been shown that the spectral anomaly takes place in the interference field, both in the near field and in the far field, when the double-slit is illuminated by polychromatic completely coherent light or polychromatic partially coherent light. The spectrum in the interference field is redshifted in some position, and in some other position is blueshifted, and at the point between these positions the spectrum is split into two lines. The spectral anomalies just occur near the singular points for the frequency ω ₀. Such a spectral anomaly may have applications in free-space communications, in which the blueshift could be associated with a bit of information such as a “1”, and the redshift could be associated with “0”.

As described in Section 3, for the fixed observation point (for example, z_N = 15 and u_N = 3.82) where a spectral detector is placed, the variation of the spectral shifts with the spectral width of the original spectrum is studied. It is found that the spectral shifts may change from the redshift to the blueshift as the spectral width of the original spectrum is changed. This characteristic may be used to information transmission, if we define the blueshift or redshift as a bit of information “1” or “0”. The detected spectral shifts in the fixed observation point represent a set of information delivered from the plane containing the double-slit.

The influence of the spatial coherence on the spectral anomalies is also investigated. Similar interesting phenomenon is found, i.e., as the coherence changes, the spectral shifts in some observation place (for example, z_N = 15 and u_N = 3.82) may change from the redshift to the blueshift. This specific property can also be used for information transmission, that is, due to the dependence of the spectral shifts (at the observation place) on the coherence of the incident partially coherent light, the information can be transferred from the plane containing the double-slit to the fixed observation point by modulating the coherence of the incident partially coherent light.