## Abstract

In this work, we offer a novel and flexible approach of spectral switches which can be handled more simply by controlling the phase of the diffracted light field of a completely spatially coherent incident beam with spectral profile from a one-dimensional phase step. This scheme has the benefit of easy implementation by simply varying the height of a one-dimensional phase step which causes spectral switches to occur when the step height reaches certain critical values without modulating any properties of the light source. To illustrate this effect, an explicit and analytical expression at an observation point corresponding to the step edge is obtained and some numerical examples are given and examined experimentally. Finally, based on the obtained results, it is shown that this method with the capability of very short response time can be easily applied to information encoding and transmission.

© 2010 Optical Society of America

## 1. Introduction

Wolf has shown theoretically that the normalized spectrum of radiation emitted by an extended source even on propagation in free space will change significantly at all observation points as a function of the propagation distance. However, there is an exception, when the degree of spectral coherence of the light across the source obeys a certain scaling law [1–3], this change does not occur. This effect which deals with coherence-induced spectral changes has been the subject of extensive theoretical and experimental studies [4–8]. Unlike most earlier works, it has been shown that spectral changes are produced when a class of partially coherent light obeying the scaling law are incident on different apertures as secondary sources [9–14]. Also, the diffracted light shows a gradual change of the spectrum for both on-axis and off-axis points on the observation plane as red-shifted or blue-shifted. It further shows a rapid change close to a critical value of a certain parameter which is called spectral switch and has been growing as a new effect in singular optics [15–20]. These effects are in connection with diffraction-induced spectral changes when the incident light-field over an aperture is spatially completely coherent and it has been found that they have a close link to the phase anomaly in the vicinity of phase singularity [21–24]. Over the past two decades, spectral shifts and spectral switches have received a lot of attention for free-space communications since the fluctuation in the light intensity prevents the errors in bits transmission over appreciable distances due to self-similarity of the far-zone spectrum [22]. Spectral switch is now being widely exploited in optical signal processing, optical selective interconnects development, optical computing, information encoding and cryptography. To do so, spectral switch is modulated by different kinds of light sources [25, 26], spectral width of the original spectrum [10], degree of coherence at the input plane [27, 10], various optical elements [28–30], and the propagation distance or diffraction angles.

In this paper, based on the Fresnel diffraction integral, an analytical expression for the diffraction field of a completely spatially coherent incident beam with spectral profile from a one-dimensional (1-D) phase step with variable height and the effect of the phase modulation on the spectral behavior are presented. Numerical analyses are given to illustrate how it is possible to control and tune spectral anomalies at will by controlling the step height. This method provides a precise and very flexible approach with the very short response time for spectral changes. The advantage of this method over the traditional one, as we will see, is that the spectral switches can be handled quite easily and rapidly for information encoding and transmission.

## 2. Analytical expression for the spectral intensity from a 1-D phase step

We consider a 1-D reflective phase step as indicated in Fig. 1. To build a 1-D reflective phase step we can use the two rectangular mirrors with the same reflectance so that the edge of the step is perpendicular to the page. We have shown that, an arbitrary change of the optical path difference in diffraction by a variable 1-D phase step leads to many novel and interesting metrological applications [31]. Also, we have shown that a phase step with constant phase *π* can be produced naturally in external reflection of the TM component of the light field at Brewster’s angle and examined the anomalous behavior of the light field [32]. Our first assumption is that a secondary source can be produced on a 1-D phase step with variable height *h*. Also it is assumed that this source has a uniform spectral distribution and that the intensity distribution is constant. We denote the field at a narrow strip, which acts as a linear source and is parallel to the step edge, at point *M* specified by position vector **x** on the wave front Σ which is momentarily placed on the 1-D phase step by the expression

*A*(

*λ*) is the disturbance amplitude for wavelength

*λ*,

*c*is the speed of light in vacuum and

*k*= 2

*π*/

*λ*is the wave number. When a uniform plane wave impinges on such a phase step, the phase of the wave front undergoes a sharp change, therefore an unfamiliar form of Fresnel diffraction (FD) will occur [33]. Using the Fresnel diffraction integral, the spectral intensity can be calculated at an arbitrary narrow strip on the observation plane at point

*P*. As can be seen from Fig. 1, the intensity at an observation point

*P*depends on the distance

*P*

_{0}from the step edge and varies with its position. We use this point as the origin of the coordinate system for the spectral intensity calculation at point

*P*. Here only a portion of the wave front that lies in the neighborhood of the step edge contributes to diffraction integral as far as FD is concerned; hence, mirrors of a few millimeters width are quite adequate. We can use a procedure similar to that in [33], and introducing the normalized distance

*v*as we obtain the following expression:

*C*

_{0}and

*S*

_{0}represent the well known Fresnel cosine and sine integrals, respectively, and defined as Here,

*v*

_{0}corresponds to

*x*

_{0}which is associated with the distance between

*P*

_{0}and the step edge and

*ϕ*= 2

*kh*. Now, we define the spectral intensity at point

*P*

*P*

_{0}in the left and right side of the step edge, respectively. Because of the role sin

*ϕ*plays, when

*h*is replaced by −

*h*, the two former sings are interchanged and consequently the observed interference pattern for monochromatic incident beam at wavelength

*λ*will be interchanged with respect to the step edge [Fig. 2]. We can define the source spectrum as and the spectral modifier function as

*P*is expressed as It becomes evident from (9) how the incident light spectrum

*S*

^{(0)}(

*λ*) is modulated at wavelength

*λ*as a result of diffraction from the phase step. Also it shows how we can control spectral shifts by adjusting the height of the step edge.

A close examination of the spectral modifier *M _{P}*(

*x*

_{0},

*h*,

*λ*) reveals interesting features for monochromatic incident beam. One of the most important cases will occur when the observation point on the screen is along the line corresponding to the step edge, that is,

*C*

_{0}=

*S*

_{0}= 0. If so, the modifier function takes the following form:

*h*= (2

*m*+ 1)

*λ*/4,

*M*(0,

_{P}*h*,

*λ*) and

*S*

^{(}

^{P}^{)}(0,

*h*,

*λ*) are zero. Thus, by satisfying the above mentioned condition one can produce line-singularity in the diffracted field at wavelength

*λ*[34]. In other words, Fresnel diffraction from 1-D phase step of height

*h*produces an isolated zero of the spectral component of wavelength

*λ*. The zero for each

*λ*will be in a different step height

*h*and according to (10), the total light intensity on the observation point at a specified height never vanishes.

## 3. Numerical results and analyses

Now we can consider how the observed spectrum is affected by varying the height of the phase step at a fixed *R* numerically. Since the spectral shift is mainly caused by the spectral modifier *M _{P}*(

*x*

_{0},

*h*,

*λ*), the numerical results do not depend critically on our choice of the spectral profile. To obtain the spectral intensity produced by such a secondary source, we consider that the spectrum of the plane incident spatially completely coherent light upon this phase step, consists of a single line of Gaussian profile, with spectral density

*S*

^{(0)}(

*λ*), centered at wavelength

*λ*

_{0}and

*rms*width

*σ*,

*i.e*.

*S*

^{(}

^{P}^{)}(

*x*

_{0},

*h*,

*λ*)along the line corresponding to the step edge takes the following analytical form:

*λ*

_{0}= 543

*nm*and

*σ*= 70

*nm*. Obviously, as can be expected from (11), when the step height starts to change from zero for which normalized height,

*η*=

*h*/(

*λ*

_{0}/4), is zero, the modifier function for a given

*h*and as a function of wavelength falls off from one to zero and as

*λ*grows larger, the modifier function will rise to unity. By the further increasing of

*h*, the modifier function presents more zeros which move toward larger wavelengths. Therefore, for a given step height, the modifier function is an oscillatory function of the wavelength. Also, zeros of this function are not equally spaced in distance and the distance between zeros increases with increasing wavelength. As shown in Fig. 3(a), by increasing

*h*, the last zero of the modifier function comes near the region enclosed by the source spectrum’s function and causes

*S*

^{(}

^{P}^{)}(0,

*h*,

*λ*) to change and its peak start to shift toward red wavelengths. When

*h*increases to a large enough value, a dip appears on the left side of the diffracted spectrum and the single spectral line splits into two asymmetric lines [Fig. 3(b)]. Due to the asymmetric behavior of the modifier function around its zeros, the first critical value (first spectral switch) of the normalized height will occur at

*η*

_{c+1}= +0.966276 rather than

*η*

_{c+1}= +1. In this case,

*M*(0,

_{P}*h*,

*λ*) falls to zero around

*λ*=

*λ*

_{0}. Then the center of the line is suppressed and

*S*

^{(}

^{P}^{)}(0,

*h*,

*λ*) is divided into two nearly equal and similar peaks [Fig. 3(c)]. This corresponds to a phase difference

*π*between the interfering wavelets and therefore a complete destructive interference for

*λ*

_{0}. Spectral changes are most rapid in the vicinity of phase singularity. The spectral shift shows a sudden change from red shift to blue shift and the diffracted field undergoes a spectral switch at observation point for the wavelength

*λ*

_{0}. A further increase in

*η*results in an increasing dominance of the blue-shifted spectral line over the red-shifted one and the dip shifts to the right side of the diffracted spectrum [Fig. 3(d)].

Consequently, the spectral shift at some observation point can be changed by controlling the step height and thus the phase of the diffracted light and the spectrum’s peak can be controlled accordingly. If the step height still grows larger, the zeros of *S*^{(}^{P}^{)} (0,*h*, *λ*) increase, what gives the location of spectral gaps. Then the greater the number of the zeros, the more narrowly the lobes accompanied by a major lobe. When *η* reaches *λ*_{0}/2, the major central lobe peaks exactly at *λ* = *λ*_{0}. This is true for the cases in which *η* = *mλ*_{0}/2, [Fig. 3(e)]. Although, for 0 < *η* < *η*_{c+1}, the position of the peak of the major lobe may be considered as a good criterion for the spectral shift, for *η*_{c+1} < *η* the spectral shift is determined by the relative strength of the spectral band or minor lobes rather than the position of the major central peak. Therefore, in general, the spectral anomaly can be characterized by the mean wavelength, which is defined as [23]:

*η*

_{c+2}= + 2.91979 rather than

*η*

_{c+2}= +3.

Let *λ _{m}* represent the wavelength at which the spectrum of the diffracted field takes its maximum, then if we differentiate (13) relative to

*λ*to obtain its extremum points, we will obtain a transcendental equation for

*λ*as

_{ext}*rms*width of the incident spectrum to the extremum points in terms of

*λ*and from its roots, we will choose its maximum value as

_{ext}*λ*. Now let us analyze this expression and study some of its consequences. As can be expected and verified by (15),

_{m}*σ*and

*h*are the most important parameters because of the role they play in the spectral behaviour. Accordingly, for a given

*h*, the observed shift of the diffracted spectrum toward red or blue wavelengths and the distance between the two equal peaks during the spectral switch would increase with increasing

*σ*and vice versa. Hence, for a broad band spectrum the observed shift is larger than the shift observed for a narrow one. Similarly, for a given

*σ*when

*h*increases to a large enough value, the number of zeros of

*M*(0,

_{P}*h*,

*λ*), in the region of the source spectrum’s function would increase to more than one and consequently the spectrum is split into more than two peaks. Therefore, with increasing lobes the distance between the two equal peaks during the spectral switch would decrease.

It should be noticed that in practice, there exists a maximum value for |*η*_{c±n}| due to the limitation on the longitudinal or temporal coherent length *l*. As indicated in Fig. 4, with the increase of |*η*| (for both *h* and –*h*), the critical positions of all spectral switches are distributed in the closed and nearly regularly spaced intervals in which the difference interval is about 2 and repeated nearly periodically. That is,

[To characterize the spectral changes clearly, we define the normalized wavelength shift as

*η*| and the sudden spectral changes take place in the vicinity of critical heights. It is seen that at

*h*

_{c+n<}=

*h*

_{c+n}–

*δh*and

_{n}*h*

_{c+n>}=

*h*

_{c+n}+

*δh*′

*the spectrum is the most red- and blue-shifted, respectively.*

_{n}Among all the critical values, only *η*_{c±1} is of major practical importance. Since as can be seen from Fig. 4, at *h*_{c±1}, *δλ* achieves its maximum value (in our example 2.6 *σ*), and the amount of change in the step height between the most red shift and the most blue shift, that is, Δ*h*_{1} = *h*_{c+1>} – *h*_{c+1<}, is the least relative to the others and would be of the order of one nanometer.

## 4. Potential application in information encoding and transmission

Let the shift of the diffracted spectrum toward blue shift be associated with a bit of information such as say “1”, and the red shift be associated with “0”. Therefore, as we change the height of the step to one of the assumed values, one of the desired shifts will be produced. For now, suppose that two photodiode detectors equipped with multilayer bandpass filters are located at a distance *R* from the step edge corresponding to the step edge, each of which is designed to record one of the spectral shifts. In our example, a multilayer bandpass filter has a center wavelength of 435 *nm* and the second has a center wavelength of 620 *nm*, each of which permits isolation of wavelength interval of a few nanometers or less in width. As a result, for blue shift as an output, one of the spectral detectors is “on” which corresponds to bit “1”, and for red shift another spectral detector is “on” which corresponds to bit “0”. Then, the detected spectral shifts in the fixed observation point represent a set of information delivered from the secondary source with time. The above results indicate that if the height of the step can be changed flexibly, the information transmission is realizable. Therefore, an information transmission system can be constructed by modulating the phase of the diffracted light. Such a capability with spectral anomalies in the vicinity of singular points has some potential applications in information encoding and transmission, optical interconnects and communications.This can be done more easily and more rapidly through controllable phase modulations than through any means suggested by previous works. To change the height of the phase step, one of the two mirrors is displaced by using a calibrated bundle of piezoelectric elements and the amount of displacement is controlled by a stabilized power supply. This is depicted graphically in Fig. 5. In this figure, it is assumed that the height is measured from *h* = 0, but, practically it would be better to set up and measure these two heights from the first critical height |*h*_{c±1}|.

## 5. Experiments

We performed two kinds of experiments with two sources. To see how this is done in the first experiment, consider Fig. 6, which shows the setup arrangement. The first light source used to provide a completely coherent light field, was a He-Ne laser with the emitted output at 543 *nm*. By means of a converging system, a laser beam was focused on a pinhole (PH) with a diameter of 0.020 *mm*. It produced symmetric diffraction pattern consisting of a bright central disk known as the Airy disk which fell on a 1-D reflective phase step after collimation and passed through a cubed beam splitter (CBS). The diffracted light was taken by a CCD camera which terminated to an image analyzer (IA). A laser beam typically has a spectral line with the Gaussian shape, which after interaction with a pinhole will obtain a complicated line shape. In our experiment the variation of intensity at the center of the Airy disk within a rectangle region with dimensions 5 × 10 *mm*, which we are interested in, was ignorable. By using a calibrated piezoelectric element (PE) as a precise micropositioner we can move one of the mirrors which produces a dynamic phase step with desired height. In this experiment the step height was tuned at a height of *λ*_{0}/4 (first-order critical height). Figure 7, shows the interference pattern after diffraction from the phase step for a monochromatic laser beam. To study the effect of the derived spectral modifier experimentally, the following experiment was designed and carried out (Fig. 8). A Tungsten-Halogen lamp as a white-light source (S) with maximum wavelength, *λ _{m}*=4700

*Å*, at 6165

*K*illuminates a narrow slit of width 0.070 × 12

*mm*, which acts as a linear source and the direction of its length was placed parallel with the step edge to provide a partial coherent field. The transmitted light strikes the surfaces of the two mirrors after being collimated by a cylindrical lens and reflected through a cubed beam splitter (CBS). The width of the slit and the distance between the source slit and the phase step was so that it provided illumination of high spatial coherency. In principle, since a white-light source is capable of producing short longitudinal coherency; we used it to determine the zero of the step height in the first experiment and showed how the spectrum was affected by the spectral modifier at different heights in the second experiment. The diffracted light from 1-D phase step strikes perpendicularly the entrance slit of a monochromator (MC) (1/4

*m*grating monochromator-Oriel Instrument) after passing through CBS. The resolution of MC was 0.1

*nm*, but the data were registered at desired intervals. The width of the entrance slit was 0.12

*mm*(at observation point). The MC’s exit slit faced a Photo Multiplier Tube (PMT) (Type 9558B from Electron Tubes Co.).

In order to compare the experimental results with the theoretical predictions, for every experimental step height *h*, we used the spectral profile of the white-light source and the modifying factors, (11), to simulate the corresponding diffracted spectral profile. The results of the simulation and the experiment are illustrated by the graphs in Fig. 9. Since isolated singularities corresponding to different spectral components at different adjacent heights stop rays of all colors averaging out to white light; some very interesting and beautiful colors with different hues appear for a given height. The colors vary very sharply by variation of the step height. In Fig. (10), we have a color picture recorded by a CCD camera from a spatially coherent beam of white light diffracted from a 1-D phase step of height *h*_{c+1} ≃ 131 *nm*. Based on our scheme, it is plausible to speed up our procedure as fast as possible by a little modification in the experimental arrangement. To do so, we can replace the aforementioned variable phase step by putting two similar small adjacent cubed slice of piezoelectric and instead of using the two mirror, we can deposit a high-reflector coating on one face of each slice to form a reflecting 1-D phase step. The step height can be tuned to about *h* =|*h*_{c+1}| and the dimensions of the two piezoelectric slices can be designed to vary a few nanometers.

The piezoelectric crystal vibrates in several modes over several frequency ranges from the low KHZ range up to MHZ range. A piezoelectric as an electromechanical transducer allows very fast action and very precise controllability [35]. However, the piezoelectric behaviour is restricted by varying degrees of non-linearity, finite speed of the acoustic waves through the crystal, the rising time of the voltage pulse and its response to electrical pulses, to name only some outstanding features. Nevertheless, the step displacement causes the modulated light after diffraction to be delivered typically at a rate that is of the order of 1 *μs* or equivalently, 10^{6} bits per second. It is still possible to double this rate, when the two piezoelectric crystals are driven out of phase with each other by a common stabilized power supply but with half displacement; or even parallel rows of such dynamic phase steps.

## 6. Conclusion

We have studied the spectral shifts and spectral switches of completely coherent wave field by a flexible approach for modulating the phase of the diffracted beam from a 1-D phase step. As was shown, by varying the step height, when it reaches some critical values, *h*_{±n}, we can find the spectral red shift on one side and blue shift on the other side of the critical heights. Also a noticeable change has been detected from red shift (Λ > 0) to blue shift (Λ < 0) near the neighborhood of some critical height at which the spectral switch takes place. This technique is clearly easier to implement and with the response time of the order of 1*μs*, it can be very useful to improve and promote the throughput of information transmission in free space. The utilities it involves are of special interest, since no properties of the incident light source or various optical elements need to be modulated to control the spectral switches. This effect has been illustrated by a number of examples and some of them are experimentally verified for a partially coherent wave field. In effect, our proposal scheme is based on three key techniques which are: (i) to control the phase of the diffracted light by the change of the step height flexibly, (ii) to set the rate at which the spectral shift is delivered from the secondary source to the output ports and (iii) to obtain the desired spectral shift with a given observation point without the need to change the position of detectors.

## Acknowledgments

The corresponding author sincerely acknowledges the technical support provided by the Institute for Advanced Studies in Basic Sciences (IASBS). Special thanks is also due to Dr. A. Fazelimanie from the Dept. of English for proof reading the text.

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