spectral switches for a circular aperture with a variable wedge

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Pin Han Vol. 26, No. 3 /March 2009/J. Opt. Soc. Am. A 473

Spectral switches for a circular aperture witha variable wedge

Pin Han

Institute of Precision Engineering, National Chung Hsing University, 250 Kuo Kuang Road, Taichung 402, Taiwan([email protected])

Received November 11, 2008; revised December 12, 2008; accepted December 13, 2008;posted December 16, 2008 (Doc. ID 103969); published February 5, 2009

The far-field spectral anomalies of a short Gaussian pulse incident on a circular aperture with a variable wedgeare theoretically studied, and some numerical examples are given to illustrate these effects. It is shown thatsome anomalous behaviors (such as red shift or blue shift for the spectral peak of the diffracted pulse) can befound under different conditions. Also, the important phenomenon called “spectral switches” is presented,which can be controlled by varying the angle of the wedge part in the circular aperture. Its potential applica-tion in information encoding and transmission for free-space communications is proposed, and this scheme hasthe benefit of easy implementation compared with other previous methods. © 2009 Optical Society of America

OCIS codes: 260.1960, 300.6170.

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. INTRODUCTIONingular optics [1], which is the study of the structure ofave fields in the neighborhood of singular points where

he field amplitude has zero value and the phase is thusot determined, has been developing into a rich field [2]nd has gained more interest during the past few years.round these singular points, many different kinds of be-avior can exist (e.g., wavefront dislocations, optical vor-ices [3], spectral switches, spectra distortions [4–14],tc.).

Most earlier publications used monochromatic wavess light sources [1–3] to discuss these singular behaviors.n more recent papers, it has been found that polychro-atic light [6,10] or a broadband short pulse [7,8] can also

xhibit drastic spectrum changes. These spectral changesere then analyzed and shown to be related to the spatial

orrelation of the light sources [4] (correlation-inducedpectral changes) and/or to the limiting aperture [6–10]aperture-induced spectral changes), and the latter is alsoeferred to as “aperture dispersion” [7,8,15,16]. It waslso pointed out that the lack of spatial coherence tends toessen diffraction-induced spectral changes [5]. Sincehen, there are many works in which different kinds ofight sources (e.g., Bessel–Gauss beam [13], Gaussian–chell model [11], Gaussian pulse [7,8], etc.) are used orarious optical elements (e.g. single slit [12], double slit10], circular aperture [15], gratings [16], etc.) are em-loyed to study the spectral anomalies in the near field orar field; meanwhile, some of them are experimentallyerified [17–19].

In this work, a circular aperture with a variable wedges used to illustrate the spectral changes for a Gaussianhort pulse in the far field. The theoretic analysis andnalytical expression is obtained first, and the numericalxamples that follow indicate the red shift or blue shift of

diffracted spectrum’s peak. The important “spectralwitches” phenomenon is investigated in detail, and weill show how to control it by adjusting the wedge angle

1084-7529/09/030473-7/$15.00 © 2

f the circular aperture. This scheme can be applied to in-ormation encoding and transmission, and it is easier toerform than in other previous works that modulate theroperties of the light source to fulfill this task.

. THEORYonsider an incoming wave with a spectral scalar field,��p� ,��, incident on a circular aperture (radius a) with a

ariable wedge portion expanding to angle � radian, ashow in Figs. 1(a) and 1(b). The diffraction field U�p ,�� onhe observation plane can be obtained using the Fresneliffraction integral [20]:

U�p,�� =1

j� ��

U��p�,��exp�j�r/c�

r��d��, �1�

here �� is the obliquity factor, � is the wavelength, � ishe angular frequency, c is the velocity of the light wave,nd r is the distance from point p��x� ,y� ,0� on the aper-ure plane to point p�x ,y ,z� on the observation (image)lane. The 2-D Cartesian coordinate systems used for theperture plane and the observation plane are �x� ,y�� andx ,y� coordinates, respectively. �� is the aperture functionnd d�� is the integration of ��, as shown in Fig. 1(a).ince it is easier to analyze this aperture with a cylindri-al coordinate, we adopt the coordinate systems �� ,��nd �� ,�, respectively, for the aperture plane and the ob-ervation plane. The relationships between Cartesian co-rdinates and cylindrical coordinates are as follows: x�� cos �, y�=�� sin �, x=� cos , y=� sin . It is alsooted that Eq. (1) is usually used for a monochromatic in-ident field, U��p� , t�=U��p� ,��e−j�t, with a single fre-uency � and the constant complex amplitude U��p� ,��;owever, it is also applicable for a polychromatic field orroadband pulse [21], which can be superposed withonochromatic fields via the Fourier integral.

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474 J. Opt. Soc. Am. A/Vol. 26, No. 3 /March 2009 Pin Han

In a cylindrical coordinate, the Fourier transform is aourier–Hankel transform, which is defined as [22]

FH�g��,�� = �n=−

�− j�n exp�jnf�2�

��0

��gn��Jn�2�f��d��, �2a�

ith

gn�� =1

2��

−�

�

g��,��exp�− jn��d�. �2b�

In Eqs. (2a) and (2b), the following notations are used:H�g�� ,�� is the Fourier–Hankel transform of the ap-rture function g�� ,��; f� and f are the spatial fre-uency variables; Jn�x� is the Bessel function of the firstind and nth order.From the aperture geometry in Fig. 1(b), the aperture

unction can be rewritten as: g�� ,��=whole circlewedge portion. The Fourier–Hankel transform of thehole circle is

FH�circ��/a�� =a

f�

J1�2�af��, �3�

here circ�� is the circle function which is defined as:irc�� /a�=1 for 0 � a and circ�� /a�=0 for � �a. For

ig. 1. (a) Geometry and notation for an incoming Gaussianulse incident on a wedged circular aperture. (b) Structure of theperture with radius a and wedge angle �.

� � � �

he wedge portion, we first use Eq. (2b) to get

gn�� = circ��/a�� 1

2��

−�/2

�/2

exp�− jn��d��= �circ��/a�

1

n�sin�n�

2 , n � 0

circ��/a��

2�, n = 0 . �4�

ith Eq. (4) into Eq. (2a), the Fourier–Hankel transformf the wedge portion can be found as

FH�wedge portion� = �n=−

except n=0

�− j�n exp�jnf�2

nsin�n�

2 ��

0

a

��Jn�2�f��d��

+�

2�

a

f�

J1�2�af��. �5�

ombining Eqs. (3) and (5), the Fourier–Hankel trans-orm of the aperture function g�� ,�� takes the form of

FH�g��,�� = �1 −�

2� a

f�

J1�2�af�� − 2 �n=−

except n=0

�− j�n

�exp�jnf�1

nsin�n�

2 �0

a

��Jn�2�f��d��.

�6�

ow consider the following time-dependent Gaussianulse with unit amplitude:

u�p�,t� = exp�−1

2� t

�2

+ j�0t� , �7�

here �0 is the pulse central frequency and � is its dura-ion time. The spectral complex amplitude U��p� ,�� cor-esponding to Eq. (7) can be obtained using the Fourierransformation U��p� ,��=�−

u��p� , t�exp�−j�t�dt as

U��p�,�� = �2�� exp −1

2�� − �0��2� . �8�

ith the far-field approximation [20] and substituting Eq.8) into Eq. (1), the diffraction field U�p ,�� at the obser-ation plane can be obtained as

U�p,�� =1

j�Rexp�jkR�U��p�,��FH�g��,��, �9�

here R is the distance for o�p, k=� /c=2� /�, and the lasterm FH�g�� ,�� is the Fourier–Hankel transform of theperture function g�� ,��, as in Eq. (6) with the spatialrequencies f�=� /�R. Substituting Eq. (6) into Eq. (9), weave

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U�f�,,�� =1

jR� �

2�cej�kR��2��exp −1

2�� − �0��2�

� �1 −�

2��

a

f�

J1�2�af�� − 2� �n=−

except n=0

�− j�n exp�jnf�1

nsin�n�

2 ��

0

a

��Jn�2�f��d�� . �10�

To facilitate further discussion, we can limit our obser-ation point p along the positive y axis only, that is, theondition f==� /2 is used. Let us denote this specifiediffracted field as Uy�f� ,��=U�f� ,=� /2 ,��, where theubscript y in Uy�f� ,�� indicates this field is found alonghe positive y axis. Under this condition and using the re-ations 1/�=� /2�c and f�=�� /2�cR=� sin � /2�c, weave

Uy��,�� =�2��a2

2�cjRej�kR��exp −

1

2�� − �0��2� ,

��2� − ��J1�a sin��/c�

a sin��/c � − 2 �n=−

except n=0

1

nsin�n�

2 ��

0

a

��Jn�a sin��

c d�� , �11�

here sin �=� /R and � is the angle between o�o and o�ps indicated in Fig. 1(a). The spectral intensity Iy�� ,�� ofhe diffraction field, through the equality Iy�� ,��Uy�� ,��2=Uy�� ,��Uy�� ,��*, is

Iy��,�� = A exp�− �� − �0��2��2��2� − ��

��J1�a sin��/c�

a sin��/c � − 2 �n=−

except n=0

1

nsin�n�

2 ��

0

a


c d��2

� A G�� M��,��,

�12�

here A=�2a4 /2�c2R2, G��=exp�−��−�0��2� is the nor-alized incident spectrum, and

M��,�� = �2��2� − ��J1�a sin��/c�

a sin��/c � − 2

� �n=−

except n=0

1

nsin�n�

2 �0

a


c d��2

s called the modifier function. As indicated in Eq. (12),his modifier function illustrates how the incident spec-rum G�� is modified (or modulated) as a result of dif-raction at the aperture. Eq. (12) is the main result of thisork, and it is used to give some numerical results below

hat characterize spectral anomalies (e.g., red shift, bluehift, spectral switches) in different cases.

. NUMERICAL RESULTSe first discuss how the spectral intensity distribution

hanges at different observation position, along the posi-ive y axis (or equivalently, the angle �) in Subsection 3.A.n Subsection 3.B, the spectrum changes influenced bydjusting the angle of the wedge portion (the parameter �)re investigated, and the method applying it to informa-ion encoding and transmission in free space is suggested.

. Spectral Intensity Distribution at Differentbservation Positionset us first examine the spectral changes at different ob-ervation angles � for a fixed wedge with �=� /3. We willtart from the optical axis and move along the positive yxis with the increasing value of � to examine the changesf the diffracted spectrum at different angles.

. Spectral Intensity Distribution When �=0 (on Axis)hen the observation point p is exactly at the center o of

he observation plane, we are under the condition �=0.ubstituting sin��=0 into Eq. (12) and using the follow-

ng equalities: limx→0 J1�x� /x=1/2, J0�0�=1, and Jn�0�=0or n�0, the diffracted spectrum at �=0 is

Iy�0,�� = A� G�� 2, �13�

here A�= �5� /6�A. Thus it is found that G�� now isodified by a simple function M��=0,��=�2. Figure 2

lots the resultant normalized spectrum Iy�0,�� for twoifferent values of �=1/�0�=0.3,0.5, respectively. As indi-ated in the figure, the peak of normalized diffracted spec-rum Iy�0,�� is always blue-shifted, and the amount ofhift increases as the bandwidth � rises. The maximum ofhe spectral intensity is at �max I=1/2�1+ �1+ �2��2�1/2��0,nd the amount of shift is ��=�max I−�0= �1/2��1�2��2�1/2−1��0. This behavior, in which the incident spec-

rum G�� is modified by �2 at �=0, can also be found inther works with different aperture structure [7,8,16].

. Spectral Intensity Distribution When ��0 (offxis)hen the observation point p is not on the optical axis

��0�, we can plot the behavior of spectral intensityith Eq. (12). Since in Eq. (12) the resultant spectrum�� ,�� is the product of the incident spectrum G�� and
y

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he modifier function M�� ,��, the behavior of Iy�� ,�� cane discussed according to the different properties of�� ,�� as follows. Usually it can be characterized by ex-

mining the value or slope of M�� ,�� at �=�0 as ex-lained below.(a) Smooth movement of spectrum’s peak. Let the detec-

ion angle begin from sin��=1.0�10−4 for typical param-ter values �0=3�1015 rad/s, a=1.0 mm, �=� /3, �=0.5,nd these values are used in all of the following numericalxamples unless specified otherwise. The resultant spec-rum is shown in Fig. 3(a). As indicated in the figure, theiffracted spectrum Iy�� ,�� is blue-shifted due to the in-reasing function (positive slope) of M�� ,�� at �=�0. For

bigger angle at sin��=1.67�10−4, now M�� ,�� iseaked at central frequency �0 as shown in Fig. 3(b). Be-ause G�� is a symmetric function of � and also has itsaximum value at �=�0, the peak of diffracted spectrum

y�� ,�� also occurs at �0, which means that the peak ofy�� ,�� is unshifted. For the next angle at sin��=2.510−4, the diffracted spectrum Iy�� ,�� is red-shifted due

o the decreasing function (negative slope) of M�� ,�� at=�0, as indicated in Fig. 3(c). It can be seen from Fig. 3

hat when the angle is increased from case (a) to case (c),he shift of the spectrum’s peak moves smoothly from bluehift, to no shift, to red shift.

(b) Sudden change of spectrum’s peak (spectral switch).or the next angle at sin��=3.06�10−4, as shown in Fig.(a), the main peak that has the maximum spectrumalue (left peak, as indicated by the letter L) is still red-hifted, but the right peak is larger compared with thene in Fig. 3(c). The solid dot(s) on the top of the plots in-icate the position of the spectrum maximum. If the anglencreases a little bit to sin��=3.16�10−4, the spectrumas two peaks with equal height. For a little larger anglealue at sin��=3.26�10−4, the right peak is now theain peak. Thus we can find a fast change from red shift

o right shift near the neighborhood of some criticalngles, and this phenomenon is called the spectral switch.

ig. 2. Normalized spectral intensity of Iy�0,��2 G�� for twoifferent bandwidths, �=1/ ��0��=0.3 and �=0.5. The spectrum islways blue-shifted. As the bandwidth � increases, the amount ofhe peak’s shift increases. (For all the spectral curves in the fol-owing figures, each curve is normalized to its maximum value.)

he author has pointed out that the necessary conditionor the appearance of spectral switches lies in some kindf oscillatory behavior associated with the modifier func-ion [23]. In this case, we can find from Fig. 4(b) that

ig. 3. Normalized spectra for G�� (dotted curve), M�� ,��dashed curve), Iy�� ,�� (solid curve) at different angles. (a)in��=1.0�10−4. (b) sin��=1.67�10−4. (c) sin��=2.5�10−4.The same curve types are used consistently in all the followinggures.)

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hen the minimum (zero) of M�� ,�� falls around �=�0,ts oscillatory property can redistribute the incident spec-rum into two peaks with equal magnitude and conse-uently causes the spectral switch. The small circles onhe horizontal axis indicate the frequency where M�� ,��

ig. 4. Normalized spectra for G�� (dotted curve), M�� ,��dashed curve), Iy�� ,�� (solid curve) at different angles. (a)in��=3.06�10−4. (b) sin��=3.16�10−4. (c) sin��=3.26�10−4.he small solid dot(s) in the plots indicates the position of theaximum of the spectrum and the capital letter L (R) indicates

he left (right) peak. The small circles on the horizontal axis in-icate the frequency where M�� ,�� and I�� ,�� have zeromplitude.

nd I�� ,�� have zero amplitude and thus the phase is sin-ular at that frequency.

For a better illustration of spectral behavior from thebove discussion, the normalized frequency-shift concepts used and defined as

� = ��p − �0�/�0, �14�

here �p is the frequency at which the spectrum of theiffracted spectrum peaks. This quantity is plotted as aunction of sin�� in Fig. 5, and the angles indicated on theaxis from �1 to �6 marked with “�” correspond to the se-

ected angles for Figs 3(a)–3(c) and Figs. 4(a)–4(c), respec-ively. It is obvious from this plot that the shift is continu-us, and at �2 [see Fig. 3(b)] there is no shift for thepectrum’s peak ��=0�. Then at �5 [see Fig. 4(b)] there is

discontinuous jump for � (spectral switch) from redhift [Fig. 4(a)] to blue shift [Fig. 4(c)].

. Spectral Switches Control by Varying the Wedgeortionow we can consider how the spectrum is affected byarying the extended angle � of the wedge at a fixed �. As-uming that sin��=3.16�10−4 is picked, the normalizedy�� ,�� with respect to �=� /2 ,� /3 ,� /4 are plotted inigs. 6(a)–6(c), respectively. It is found that the spectralwitch can be found again, and this time it is controlled byarying �.

The spectral switches previously have been utilized innformation encoding and transmission in free space [10],nd in that work the modulations for rms bandwidth orpatial coherence of the incident field are used to controlhe switches, which is not an easy task to perform. Fromhe above analysis, we propose another scheme to achievepectral switches by simply adjusting the wedge portionf a circular aperture, while the properties of the incidentight source do not need any changes. Referring to Fig. 7,t is assumed that there is a set of data as shown in the

ig. 5. Plot of the normalized frequency shift � as a function ofin�� for the parameter values �0=3�1015 rad/s, a=1 mm, �� /3, and �=0.5. The six angles indicated on the x axis from �1

o �6 marked with “�” correspond to the picked angles for Figs(a)–3(c) and Figs. 4(a)–4(c), respectively. It is found that at �2here is no shift for spectrum’s peak [�=0 as in Fig. 3(b)], and at

there is a discontinuous jump (spectral switch) as in Fig. 4(b).
5

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rst row that needs to be transmitted to position p, whichakes angle � from the optical axis o�o. The blue shift

nd red shift are designated as a bit of “1” or “0”, respec-ively (the notations B and R are used to indicate the bluehift and red shift in the row under the data row). Thusy properly adjusting �, the blue shift or red shift of thepectrum’s peak can be controlled accordingly, as indi-

ig. 6. Normalized spectra for G�� (dotted curve) and Iy�� ,��solid curve) under different wedge angles at sin��=3.16�10−4.a) �=� /2, (b) �=� /3, (c) �=� /4.

ated in the bottom row of Fig. 7. From the numerical cal-ulation, it is found that the amount of blue shift �� /2� and red shift ��=� /4� are �=0.41 and �=−0.32, re-pectively. This method is direct and easier to implement,nd no incident light source’s properties need to be modu-ated to control the spectral switches. Note that in someorks [9,10] the concept of relative mean frequency issed to indicate the spectral shift; however, we found thathe whole spectrum profile and peak frequency shift � areore suitable for describing the spectral switches because

hey are easy to be measured with a spectrometer [17] oronochromator [18] as in some experiments that justified

he existence of spectral switches.

. CONCLUSIONSn this paper the far-field spectral characteristics of aaussian pulse incident on a circular aperture with aariable wedge are presented. The analytical expressionor spectral intensity distribution is derived, and someumerical examples indicate the blue or red shift for thepectrum peak. Also, how the diffracted spectrum changesor different detection angles and for various wedgengles is illustrated by the numerical results and plots.he modifier function’s oscillatory behavior leads to theppearance of spectral switches, as pointed out by the au-hor. We also suggest another method with which thepectral switches can be controlled by varying the wedge’sngle, and it can be applied to information encoding andransmission in free space. This scheme has the benefit ofasy implementation without modulating any propertiesf the light source.

CKNOWLEDGMENTShis study was supported by the National Chung Hsingniversity, Taiwan. The author also thanks his colleagues

or thoughtful comments and useful suggestions. Thisork was also supported by the National Science Councilf Taiwan (NSCT) under contract NSC 97-2622-E-005-04-CC3 and in part by the Ministry of Education,aiwan under the Aim for the Top University and Eliteesearch Center Development (ATU) Plan.

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ig. 7. Illustration for the data encoding and information trans-ission by controlling the wedge angle. The blue shift (B) is as-

ociated with a bit of information such as “1” and the red shift (R)s associated with a bit of “0.”

1

1

1

1

1

1

1

1

1

1

2

2

22


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spectral switches for a circular aperture with a variable wedge

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