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Optics Communications 267 (2006) 24–31

Tunable spectral switching in the far field with a chirpedcosh-Gaussian pulse

Soumendu Jana, S. Konar *

Department of Applied Physics, Birla Institute of Technology, Mesra, Ranchi 835215, India

Received 6 February 2006; received in revised form 22 May 2006; accepted 7 June 2006

Abstract

We have shown the possibility of tunable spectral switching in the far field with an apertured chirped cosh-Gaussian pulse by varyingits cosh parameter (X). We have also shown that the switching frequency and both blue and red shifted frequencies can be also tuned byvarying X. Possible application of our results in optical space communications has been highlighted.� 2006 Elsevier B.V. All rights reserved.

Keywords: Far-field power spectrum; Cosh-Gaussian pulse; Spectral switching; Critical angle of diffraction; Blue shift; Red shift

1. Introduction

During last few years, a new branch of physical opticscalled singular optics [1,2] has attracted considerableattention. The phase is indeterminate at zero intensitypoints of a monochromatic wave field, these points arecalled phase singularity. Different types of optical phasesingularities were described and classified long back byNye and Berry [3] and Wright [4] and was experimentallyobserved in Kr+ laser [5] and laser speckle fields [6]. Atthe vicinity of these singular points several interestingoptical phenomena viz. wave front dislocation, opticalvortices [3,7], spectral switching [8–10] may occur. Theresearch interest in the field of singular optics, which ismainly theory driven until recently is manifold. Amongthem spectral shifting, anomalous spectral behaviourand hence spectral switching have attracted spectacularattention. Wolf pointed out that a partially coherentlight when propagating even through free space, owingto the correlation of the source, undergoes spectral shiftunless the degree of spectral coherence obeys certain scal-

0030-4018/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2006.06.013

* Corresponding author. Tel.: +91 6512276274; fax: +91 6512275401.E-mail address: swakonar@yahoo.com (S. Konar).

ing laws [11–13]. Later on it has been proved that evenobeying the scaling laws, spectral changes may occurwhen a partially coherent light propagates through anaperture [14,15]. Initially, major attention regardingsingular optics was confined to monochromatic light.However, recent investigations by several workers haveextended the domain of singular optics to the polychro-matic field. The research interest on phase singularitywith polychromatic light may be divided into two majorcategories, i.e. fully coherent and partially coherent. In alandmark paper, Gbur et al. have showed in 2002 thatspectral switching is possible for spatially fully coherentpolychromatic light [8,9]. They theoretically showed thatremarkable spectral changes take place in the neighbour-hood of phase singularities near the focus of a converg-ing spatially fully coherent polychromatic wave diffractedthrough an aperture. Their prediction for spectral switch-ing has been experimentally verified by Popescu et al. inthe same year [10]. These spectral changes are proved tobe the characteristic features of polychromatic field nearphase singularity [16]. For these fully coherent fields thespectral shifts are due to diffraction. Apart from thisfully coherent light, phase singularity for partially coher-ent light has also been investigated [17–24]. Spectralshifts of partially coherent light are induced by coherence

S. Jana, S. Konar / Optics Communications 267 (2006) 24–31 25

and/or diffraction. Pu et al. have shown that rapid spec-tral change i.e. spectral switching may occur for partiallycoherent light. In this case the switching is observed atcertain critical lengths along the axis of the aperturethrough which the light is diffracted [17,18]. Subse-quently, above prediction was experimentally verified byKandpal et al. [19,20].

Results of investigations as elucidated above have gen-erated renewed interest in the field. The polarization sin-gularity as well as the Lissajous singularities has alsobeen investigated for two-colour vector field [25]. Earlier,Palma et al. have widely studied the spectral shifts forGaussian–Schell model (GSM) beams, which are par-tially coherent in nature, due to propagation throughboth free space and an aperture lens [26,27]. Recently,Lu et al. have shown that the spectral switching forGSM beams propagating through an aperture lens isdependent on the truncation parameter, coherenceparameter and Fresnel number [28]. Similar investigationwith a multimode laser (using Hermite–Gaussian beams)instead of a fundamental mode laser also shows spectralswitching [29]. The diffraction of GSM beam through anastigmatic lens has also showed spectral switching in thevicinity of the intensity minimum in the geometric focalplane [30]. However, astigmatism has an effect on thecritical position, spectral minimum and the transitionheight of spectral switches. This study explains some dis-crepancy between experimental results of spectral shiftand theoretical prediction where aberration effect hasbeen neglected. The Fraunhofer diffraction of spatiallycoherent polychromatic beam through single slit alsoexhibits drastic spectral change near the dark lines ofthe diffraction pattern. This has been proved both theo-retically and experimentally [31]. Young’s double slitexperiment for both partial coherent [32] and fully coher-ent light [33] are other notable investigations along thisdirection. Both investigations confirm spectral switchingat the near zone of dark fringes. Though most investiga-tions on the anomalous spectral behaviour have beencarried out for light beams, ultra-short pulses [34–36]also have drawn attention. For example, spectral behav-iour as well as the spectral switching of diffracted chirpedGaussian pulses in near field [37] and far field have beenstudied recently [38–40]. An important issue pertaining tospectral switching is whether such switching could bemade tunable as well. Recently in an elegant investiga-tion, cosh-Gaussian beam has been introduced in theatmospheric optical communication links [41]. In thiscommunication, we present an investigation to highlightthe possibility of tunable spectral switching in the farfield of an aperture with a chirped cosh-Gaussian pulseby varying the cosh parameter of the pulse. This seemsto be promising because in addition to the chirp, thecosh parameter should provide another significant toolin controlling spectral switching and also spectral shift.Since, generating such pulses does not seem to bedifficult with the available optoelectronic technology,

proposed controllable spectral switching may find appli-cations in free space optical communications and signalprocessing. Particularly, in free space optical communica-tion systems different wavelengths of wavelength divisionmultiplexed (WDM) signals can be switched at differentangles. Moreover, since these angles can be varied byvarying cosh-parameter of the pulse, there by giving anopportunity to communicate signals of a particular wave-length to receivers located at different angles from thesource plane.

The organization of the paper is as follows. In Section 2a theoretical model has been developed to describe the farfield power spectrum of the pulse. Section 3 contains resultsand discussions. In this section we have discussed both onand off-axis spectral behaviour. Spectral switching and itscontrolling have been discussed in detail. A brief conclusionis added in Section 4.

2. Theoretical model

We consider a cosh-Gaussian laser pulse which is inci-dent on a rectangular aperture at z = 0 plane. The widthsof the aperture along x and y directions are 2a and 2b,respectively. The initial field of the incident pulse can beexpressed as

Eðx0; y0; 0; tÞ ¼ exp � x20 þ y2

0

w20

� �UðtÞ; ð1Þ

where w0 is the beam waist, U(t) is the temporal profile ofthe pulse which has the form

UðtÞ ¼ exp �ð1þ iCÞt2

2T 2

� �coshðXtÞ expð�ix0tÞ; ð2Þ

where T is the pulse duration, C is the chirp parameterand x0 is the central frequency of the pulse. It can be eas-ily shown that a cosh-Gaussian laser pulse is actuallycomposed of two identical in phase decentred Gaussianpulses. Therefore, such a pulse can be created by splittinga Gaussian pulse and allowing one of them to travelthrough an optical delay line. These two pulses can berecombined to create a cosh-Gaussian pulse whose cosh-Gaussian parameter X can be varied by varying the lengthof the delay line.

The field given by Eq. (1) can be described in the space-frequency domain by Fourier transformation as

Eðx0; y0; 0;xÞ ¼1ffiffiffiffiffiffi2pp

Z 1

�1Eðx0; y0; 0; tÞ expðixtÞ dt

¼ exp � x20 þ y2

0

w20

� �F ðxÞ; ð3Þ

where

F ðxÞ ¼ 1ffiffiffiffiffiffi2pp

Z 1

�1UðtÞ expðixtÞ dt ð4Þ

26 S. Jana, S. Konar / Optics Communications 267 (2006) 24–31

represents the Fourier spectrum of the pulse at the incidentplane and x is the frequency. From Eqs. (2)–(4) we get theinitial field at the plane z = 0 as

Eðx0;y0;0;xÞ

¼ 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT 2ð1� iCÞ

1þC2

sexp

T 2ð1� iCÞ2ð1þC2Þ

fX2�ðx�x0Þ2g� �

� expðCþ iÞT 2Xðx�x0Þ

1þC2

� ��

þexp �ðCþ iÞT 2Xðx�x0Þ1þC2

� ��exp �x2

0þ y20

w20

� �: ð5Þ

The initial power spectrum at the incident plane z = 0 iseasily obtained from Eqs. (2) and (4) as

I0ðxÞ¼ jF ðxÞj2

¼ T 2

2ffiffiffiffiffiffiffiffiffiffiffiffiffi1þC2

p expT 2 X2�ðx�x0Þ2n oð1þC2Þ

24

35

� cosh2T 2XCðx�x0Þ

1þC2

� �þ cos

2T 2Xðx�x0Þ1þC2

� �� �:

ð6Þ

The field described by Eq. (3) satisfies Huygens–Fresnelintegration during its propagation and can be expressedas [38]

Eðx; y; z;xÞ ¼ i

kzexpð�ikzÞ

Z a

�a

Z b

�bEðx0; y0; 0;xÞ

� exp � ik2zfðx� x0Þ2 þ ðy � y0Þ

2g� �

dx0 dy0;

ð7Þ

where k is the wave number related to the wavelengththrough k = 2p/k. Substituting Eq. (5) into Eq. (7) we getthe expression for the field at a point (x,y,z) as

Eðx; y; z;xÞ ¼i8

z0xzx0

1þ i z0xzx0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT 2ð1� iCÞ

1þ C2

sexpð�ikzÞ

� expT 2ð1� iCÞ2ð1þ C2Þ

fX2 � ðx� x0Þ2g� �

� expðC þ iÞT 2Xðx� x0Þ

1þ C2

� ��

þ exp �ðC þ iÞT 2Xðx� x0Þ1þ C2

� ��

� exp �a2

x þ a2y

xx0

2

þ i xzx0z0

� �

1þ z0xzx0

2

2664

3775

� ½erfðC1Þ � erfðC2Þ� � ½erfðC3Þ � erfðC4Þ�;ð8Þ

where

C1 ¼�dx � idx

z0xzx0þ iax

xx0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� i z0x

zx0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ z0x

zx0

2r ;

C2 ¼dx þ idx

z0xzx0þ iax

xx0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� i z0x

zx0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ z0x

zx0

2r ;

C3 ¼�dy � idy

z0xzx0þ iay

xx0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� i z0x

zx0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ z0x

zx0

2r ;

C4 ¼dy þ idy

z0xzx0þ iay

xx0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� i z0x

zx0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ z0x

zx0

2r :

k0 is the central wavelength, z0 ¼pw2

0

k0is the Rayleigh length,

ax ¼ x=zh0

and ay ¼ y=zh0

are, respectively, the normalized angles

of diffraction in x and y directions, h0 ¼ k0

pw0is the diver-

gence angle, dx ¼ aw0

and dy ¼ bw0

are the truncation param-eters along x and y directions, respectively, and ‘‘erf(x)’’stands for error function of the argument. Making use ofEq. (8) we derive the diffraction induced modified powerspectrum as,

Iðax; ay ; z;xÞ ¼ jEðax; ay ; z;xÞj2 ¼ I0ðxÞMðax; ay ; z;xÞ; ð9Þwhere

Mðax;ay ; z;xÞ ¼z0xzx0

2

16 1þ z0xzx0

2� � exp �

2 a2x þ a2

y

xx0

2

1þ z0xzx0

2

264

375

�ferfðC1Þ � erfðC2Þg� ferfðC3Þ � erfðC4Þg� erf C�1

� �� erf C�2

� � �� erf C�3

� �� erf C�4

� � �ð10Þ

and C�1, C�2, C�3 and C�4 are, respectively, the complex conju-gate of C1, C2, C3 and C4. M can be identified as spectralmodifier and arises due to diffraction of the pulse throughthe finite aperture. The modifier depends on frequency, an-gle of diffraction and distance of propagation. The modifierM does not involve cosh parameter X, signifying that themodifier is identical for both Gaussian as well as cosh-Gaussian pulses. This result may be attributed to the factthat the modifier is a geometrical effect which depends onthe aperture and observation point, not on pulse shape.A simplified expression for the modified power spectrumwhich is valid at the far field (i.e. at z0/z� 1) may be easilyderived from Eq. (9) as follows:

Iðax; ay ; z;xÞ ¼ jEðax; ay ; z;xÞj2

¼ I0ðxÞMðax; ay ; z;xÞ; ð11Þ

where

-0.4 -0.2 0 0.2 0.40

0.5

1

I(ω

)

(a)

0.5

1

I(ω

)

(b)

Ω=0

Ω=0.3

Ω=0.6

Ω=0 Ω=0.3

Ω=0.6

S. Jana, S. Konar / Optics Communications 267 (2006) 24–31 27

Mðax; ay ; z;xÞ ¼1

16

z0xzx0

� �2

exp �2 a2x þ a2

y

x2

x20

� �

� erf dx � iaxxx0

� �þ erf dx þ iax

xx0

� �� �2

� erf dy � iayxx0

� �þ erf dy þ iay

xx0

� �� �2

:

ð12Þ

Eqs. (11) and (12) have been used subsequently to investi-gate the far field spectral behaviour of the pulse.

-0.4 -0.2 0 0.2 0.40

(ω−ω0)/ω

0

Fig. 1. Normalized on-axis power spectrum for unchirped (C = 0)Gaussian and cosh-Gaussian pulses with normalized pulse durationT = 3 and normalized central frequency x0 = 2.36, dx = dy = 0.8. (a)Initial spectrum, (b) modified far field spectrum.

–0.5 0 0.50

0.5

1

I(ω

)

(a)

–0.5 0 0.50

0.5

1

(ω−ω0)/ω

0

I(ω

)

(b)

Ω=0

Ω=0.3 Ω=0.6

Ω=0

Ω=0.3 Ω=0.6

Fig. 2. Normalized on-axis power spectrum for chirped (C = 1) Gaussianand cosh-Gaussian pulses with normalized pulse duration T = 3 andnormalized central frequency x0 = 2.36, dx = dy = 0.8. (a) Near fieldspectrum, (b) modified far field spectrum.

3. Results and discussions

We present our discussion under two subtitles: (i) Onaxis far-field power spectrum, (ii) Off-axis far-field powerspectrum.

3.1. On axis far-field power spectrum

The expression for on-axis far-field power spectrum ormodified power spectrum can be derived by settingax = ay = 0, in Eq. (11) which turns out to be

Ið0;z;xÞ¼ 1

2

z0

z

2 xx0

� �2 T 20ffiffiffiffiffiffiffiffiffiffiffiffiffi

1þC2p exp

T 2fX2�ðx�x0Þ2gð1þC2Þ

" #

� cosh2T 2XCðx�x0Þ

1þC2

� �þ cos

2T 2Xðx�x0Þ1þC2

� �� ��½erfðdxÞ�2½erfðdyÞ�2; ð13Þ

where T, X, x and x0 are normalized as follows: T 0 = T/T0,X 0 = XT0, x 0 = x0T0; T0 is some characteristic time in fem-to seconds. For brevity, we have omitted the prime over T,X, x and x0 in our further discussion. The expression (13)has been used to examine the modified on-axis spectralcharacter of the pulses. For better perception, it is worthmentioning the values of the parameters used for our anal-ysis. These are as follows: dx = dy = 0.8 (unless statedotherwise), T0 � 10�15 s and C = 0,1,2. The central wave-length (k0) of the laser source corresponding to the centralfrequency x0 is 800 nm. One example of such source isTi:Sapphire laser which can generate shortest optical pulses(<5 fs) having spectrum centered at k0 = 800 nm. Figs. 1(a)and (b), respectively, show the on-axis initial and far fieldpower spectrum of unchirped Gaussian and cosh-Gaussianpulses. The initial spectrum is symmetrically distributedaround the central frequency whereas the far-field spec-trum is shifted towards higher frequency region i.e. to-wards the frequency of blue light. This phenomenon isreferred as blue shift. The spectral shift Dx is measuredas the difference between the frequencies correspondingto the spectral maxima of the initial and modified powerspectrum i.e. Dx = (xm � x0)/x0. In absence of chirping,the blue shift is less for cosh-Gaussian pulses in compari-son to the Gaussian one. In fact, the amount of blue shiftdecreases with the increase in cosh parameter X. On the

other hand, the situation is different in case of chirpedpulses. For chirped pulses (C = 1), Figs. 2(a) and (b) are,respectively, the counter part of Figs. 1(a) and (b). Likeunchirped pulses, initial spectrum of chirped pulses is alsosymmetrical. But more prominent blue shift is observed forincreasing X at the far field. The variation of the amount ofthe blue shift (DxB) with X has been depicted in Fig. 3. Forunchirped pulse (C = 0), DxB remains unaffected with thevariation in X. For chirped pulses DxB remains unaffectedfor low value of X, then increases with the increase in thevalue of X. Particularly, at higher value of X, the changeis more significant. Influence of X is more prominent forpulses with large chirping. In order to qualitatively under-stand how pulses of different duration are affected by thecosh parameter X, we have analysed the variation of DxB

with X of unchirped pulses, for different pulse durationT. It has been realized that for unchirped pulses, shorter

0 0.2 0.4 0.62

2.5

3

3.5

4

Ω

ΔωB

C=0

C=1

C=2

Fig. 3. Variation of far field on axis blue shift (DxB) with respect to X. C isa parameter. Normalized pulse duration T = 3 and normalized centralfrequency x0 = 2.36.

28 S. Jana, S. Konar / Optics Communications 267 (2006) 24–31

the duration of the pulse width (T) larger is the blue shift.However, DxB decreases with increase in X, but this varia-tion is small as stated earlier. Thus for unchirped shortpulses the effect of X is to diminish blue shift. On the otherhand, for chirped pulses DxB remarkably increases with theincrease in X for all higher values of T. To demonstrate theeffect of chirp on spectral shift more prominently, we havedepicted far field power spectrum of the pulse with differentC in Fig. 4 which clearly shows that blue shift is larger forpulses with higher chirp. This in turn is supported byFig. 3.

3.2. Off-axis far-field power spectrum

We now switch over to the off-axis case for which atleast one of the angles of diffraction ax and ay is non zero(i.e. either ax 5 0, ay = 0 or, ax = 0, ay 5 0 or ax 5 0,ay 5 0). We have investigated the far field power spectrumat different angle of diffraction along both x and y direc-tions. Eq. (11) implies that the far field spectrum is theproduct of the initial spectrum and the spectral modifierM. Primarily, it is necessary to examine the nature of the

–1.5 –0.75 0 0.75 1.5 0

0.5

1

(ω − ω0)/ω

0

I(ω

)

C=0

C=1 C=2

Fig. 4. Normalized on-axis far field power spectrum for cosh-Gaussianpulses for different chirps. Normalized pulse duration T = 3 and normal-ized central frequency x0 = 2.36, X = 0.6.

modifier. Fig. 5 shows the spectral modifier at the centralfrequency, with dx = dy = 0.8. This is symmetrical in nat-ure. We have found several minima along both x and y

axes, among which the first two, respectively, occurs atax = ay = 2.255 and at ax = ay = 4.057. These in turn cor-responds to the zero intensity point or singular points.The spectral switching occurring at the neighbourhood ofthese angles is known as first and second order spectralswitching, respectively. We have considered the first orderswitching in our present communication. The spectralmodifier for dx 5 dy, though not shown, is rather asym-metrical but also contains multiple singular points. We firstconsider the case of symmetrical aperture which results insymmetrical spectral modifier (dx = dy = 0.8). Therefore,it is reasonable to confine our discussion along any oneaxis, say, x axis. The off axis far field spectral behaviourcan be investigated again under two different conditions(i) for ay = 0 and (ii) for ay 5 0. First, fixing ay = 0, wekeep on increasing the value of ax. At a certain value ofax the spectrum is divided into two peaks of equal height.This angle is known as critical angle aCx . At an angle smal-ler than aCx the spectrum shifts towards lower frequencyregion, i.e., the spectrum is red shifted and for larger anglethe spectrum is blue shifted. Therefore, by simply changingthe angle of diffraction one can switch over from one col-our to another colour. This phenomenon is known as spec-tral switching and has been pointed out by many authors[8–10,37,38]. A typical spectral switching is shown inFig. 6. The main objective of the present investigation isto control this spectral switching. In order to achieve thisobjective we have examined the influence of X on spectralswitching. We have depicted the variation of critical angleaCx with X for ay = 0 in Fig. 7. From figure it is evident thatby changing the value of X, we can achieve switching at dif-ferent angle. We have found that for unchirped pulses theeffect of X on the variation of aCx is marginal but significantfor chirped pulses, particularly for higher X. It should bepointed out that chirped cosh-Gaussian pulses show

–5 0 50

0.5

1

α x

–50

5

–5

0

5

0

0.5

1

αxαy

–5 0 50

0.5

1

α y

M M

M

Fig. 5. 3D spectral modifier at central frequency (x0) for symmetricalaperture (i.e. dx = dy).

–1 0 10

0.5

1

I(ω

)

(a)

–1 0 10

0.5

1

I(ω

)

(ω −ω0)/ω

0

(b)

–1 0 10

0.5

1

(ω −ω0)/ω

0

I(ω

)(c)

(ω −ω0)/ω

0

Fig. 6. Demonstration of a typical switching phenomenon. (a) Spectrumat critical angle (aCx ¼ 2:315), (b) blue shift (ax ¼ 2:415 > aCx ), (c) red shift(ax ¼ 2:215 < aCx ).

0 0.3 0.62

3

4

5

Ω

α cx

C=2

C=1.5

C=1

C=0

Fig. 7. Variation of critical angle aCx with X. ay = 0, dx = dy = 0.8 andT = 3.

0 0.3 0.6

1

2

(a)

ωs

0 0.3 0.6 0.9

1

2

(b)

Ω

ωs

C=0

C=1

C=1.5

C=2

C=0

C=1

C=1.5

C=2

Fig. 8. Variation of switching frequency xs with X. dx = dy = 0.8 andT = 3. (a) ay = 0, (b) ay = 1.

-0.5 0 0.50

0.5

1

(ω−ω0)/ω

0

I(ω

)

Ω=0 Ω=0.2

Ω=0.3

Ω=0.4

Fig. 9. Blue shifted power spectrum of unchirped pulses for different X.ay = 0, dx = dy = 0.8 and T = 3.

S. Jana, S. Konar / Optics Communications 267 (2006) 24–31 29

spectral switching at larger angle of diffraction in compar-ison to chirped Gaussian pulses if cosh parameter X > 0.3.At this stage it would be worth considering an examplewith finite ay. Therefore, we investigate the spectral behav-iour for ay = 1, the spectral behaviour is identical to that ofay = 0, however the switching for chirped pulses occurs atgreater critical angle in comparison to the case of ay = 0.Thus, with the increase in the asymmetry in modifier theswitching angle increases. The variation of switching fre-quency xs with X for chirped and unchirped pulses forsymmetrical modifier has been depicted in Fig. 8. Forunchirped pulses switching frequency is virtually invariantwith X but decreases with the increase in the value of X forchirped pulses. For higher value of X and chirp switchingfrequency is comparatively low. The variations of switch-ing frequency with X for ay = 0 and ay = 1 are identical,however for ay = 1 we get switching at greater X forC = 1 and 1.5. These findings are important because bycontrolling X we can tune the spectral switching frequency.

We now investigate the possibility of tuning red andblue shifts, respectively, at angles slightly less than orgreater than critical angles. In Fig. 9, we have depicted blueshifts at an angle ax ¼ 2:415 > aCx and ay = 0 for differentX. The frequency corresponding to the peak point of theblue shifted spectrum is referred as the blue shifted fre-quency (xB). We have plotted the variation of xB with Xin Fig. 10. It is found that, irrespective of chirping xB

decreases with the increase in X. For unchirped pulses thisvariation is small and fairly linear. On the other hand forchirped pulses, xB significantly decreases at higher X andC. Since switching frequencies as well as blue shift fre-quency are tunable, it is expected that red shift frequencymay be also tunable which has been verified. The variationof red shift frequency with X has been depicted in Figs.11(a) and (b) for ay = 0 and 1, respectively. For both cases,we found that unlike blue shift frequency, red shift fre-quency of unchirped pulses increases with increase in X.However, for chirped pulses red shift frequency decreaseswith the increase in X. Therefore, switching frequency, blue

0 0.2 0.42

2.5

ωB

(a)

0 0.2 0.4 0.6 0.8 11

2

(b)

Ω

ωB

C=0 C=1

C=1.5 C=2

C=1

C=0

C=1.5 C=2

Fig. 10. Variation of the blue shift frequency (xB) with X. dx = dy = 0.8and T = 3. (a) ay = 0, (b) ay = 1.

0 0.2 0.4 0.61

2

(a)

ωR

0 0.3 0.6 0.9

1

2(b)

Ω

ωR

C=0

C=1

C=1.5 C=2

C=0

C=1

C=1.5

C=2

Fig. 11. Variation of red shift frequency (xR) with X. dx = dy = 0.8 andT = 3. (a) ay = 0, (b) ay = 1.

0 1 22

3

4

αy

αC

x

C=0, Ω=0

C=0, Ω=0.4

C=1, Ω=0.4

C=1, Ω=0

Fig. 12. Variation of aCx with respect to ay for different combinations of C

and X. dx = dy = 0.8 and T = 3.

0 0.5 1 1.50

4

8

δx

αC

x

Ω=0.4, C=2

Ω=0, C=2

Ω=0, C=0 Ω=0.4, C=0

Fig. 13. Variation of aCx with dx for dy = 0.8, T = 3 and ay = 0.

0 0.15 0.3 0.450.6

0.9

1.2

1.5

Ω

δ xmax

αy=0,C=0

αy=1,C=0

αy=0,C=2

αy=1,C=2

Fig. 14. Variation of dxmax with X for both chirped and unchirped pulses.ay = 0, ay = 1 and T = 3.

30 S. Jana, S. Konar / Optics Communications 267 (2006) 24–31

shift frequency and red shift frequency are significantlytunable which is the most important finding of this paper.

The critical angle aCx for switching depends on the dif-fraction angle ay. The influence of ay on aCx has beendepicted in Fig. 12. It is evident from figure that increasein ay, increases aCx irrespective to chirp and X. Further-more, in absence of chirp, cosh-Gaussian pulses showswitching at lower critical angle in comparison to that forGaussian one, but the opposite behaviour is observed incase of chirped pulses.

We now turn our attention to examine the influence ofthe aperture dimension on switching. The variation of aCx

with dx for fixed dy (=0.8) has been demonstrated inFig. 13. When ay = 0, for unchirped pulses aCx decreasesrapidly with the increase in dx. However, for chirped pulsesaCx first decreases with the increase in dx, then increases.Higher the value of X and C this tendency is moreprominent. It is evident from figure that for fixed dx andC, cosh-Gaussian pulses (X = 0.4) exhibit switching at lar-ger critical angle in comparison to the Gaussian pulses. For

ay = 1, the variation of aCx is more or less similar to that ofunchirped pulses. But in presence of chirp (C = 2), switch-ing is observed at higher aCx for cosh-Gaussian pulses(X = 0.4) in comparison to the Gaussian pulses.

One important issue is the value of the maximum dimen-sion of the aperture above which switching is not possible.

S. Jana, S. Konar / Optics Communications 267 (2006) 24–31 31

For fixed X and dy there is a maximum value of dx (i.e.dxmax ) upto which spectral switching is possible. Presentinvestigation shows that this dxmax is dependent on X andC. Fig. 14 shows the variation of dxmax with X for bothchirped and unchirped pulses. Broadly speaking the dxmax

decreases with the increase in X. dxmax is greater for unchir-ped pulses in comparison to chirped pulses. Chirp alsoinfluences dxmax , particularly it decreases with the increasein C. For both ay = 0 and 1, Gaussian pulses show greaterdxmax in comparison to cosh-Gaussian pulses.

4. Conclusion

In conclusion, we have elucidated the possibility of tun-able spectral switching and spectral shifts. We have shownthat this goal may be achieved by controlling the so-calledcosh-parameter. In order to explain our findings, we havecompared the spectral behaviour of cosh-Gaussian pulseswith that of Gaussian pulses. An analytical expression forthe far field power spectrum of the pulse has been derivedby employing Fourier transformation method and far fieldapproximation. Detailed numerical simulations were per-formed to explore both the on-axis and off-axis spectralbehaviours. It has been shown that the on-axis far fieldpower spectrum is blue shifted and the amount of the shiftincreases with the increase in cosh parameter (X) forchirped pulses. For the case of unchirped pulses, theamount of blue shift decreases with increasing X at sucha slower rate that it seems to be invariant in comparisonto the case of chirped pulses. The phenomenon of spectralswitching which occurs for off-axis position has been dis-cussed in detail. For chirped pulses the critical angle forswitching significantly increases with the increase in X. Incontrary, X has marginal effect on critical angle in case ofunchirped one. We have shown that switching frequency,blue shift frequency as well as the red shift frequency canbe tuned by varying X. The influence of aperture dimensionon the switching phenomena has also been examined. Asfar as the application part is concerned, we have pointedout a method of generating cosh-Gaussian pulses. The tun-able spectral switching is important because of its potentialapplication in optical space communication and signal pro-cessing. Particularly, it may be very useful as a switchingtool in space communication.

Acknowledgements

We are thankful to anonymous referees for insightfulcomments and valuable suggestions. The authors thank

Prof. H.C. Pande, Prof. S.K. Mukherjee and Prof. P.K.Barhai for their encouragement and moral support.

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