Spectral anomalies by superposition of polychromatic Gaussian beam and Gaussian vortex beam Panfeng Ding,1 Jixiong Pu,1 Junhong Weng,2 and Pin Han3,* 1
3
College of Information Science & Engineering, Huaqiao University, Xiamen, Fujian 361021, China 2 Department of Electrical Engineering, Tunghai University, Taichung, 40704, Taiwan Graduate Institute of Precision Engineering, National Chung Hsing University, 250 Kuo Kuang Road, Taichung 402, Taiwan *[email protected]
Abstract: We study the spectral property of a composite field superposed by a polychromatic Gaussian beam and a polychromatic Gaussian beam with an embedded mth-order vortex. It is shown that, in the overall spectral shift distribution, there exist m small areas where sharp spectral anomaly takes place, similarly and respectively, which are related with the ratio of the respective amplitudes of the two composite beams and the relative phase between them. Detailed investigation reveals that, for each small area, there exists a “main line”, along which spectral switch can be observed. ©2014 Optical Society of America OCIS codes: (300.6170) Spectra; (050.4865) Optical vortices; (070.7345) Wave propagation.
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#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20193
18. P. Han, “Spectral switches for a circular aperture with a variable wedge,” J. Opt. Soc. Am. A 26(3), 473–479 (2009). 19. P. Han, “All optical spectral switches,” Opt. Lett. 37(12), 2319–2321 (2012). 20. M. Born and E. Wolf, Principles of Optics, 7th ed, (Cambridge University, 1999).
1. Introduction The existence of optical vortex or phase singularity, which is closely related to singular optics, is usually indicated by a specific point of an optical field where the intensity is zero and thus the phase becomes undefined [1,2]. The study of singular optics has developed into a new branch of optics in the past decades and most of the researches concerning with vortex deal with monochromatic wave even at present [3–5]. In 2002 the work in singular optics was extended from monochromatic light to spatially fully coherent and polychromatic light [6–8]. It was found that, when a converging, spatially fully coherent and polychromatic wave is diffracted at an aperture, anomalous spectral changes take place near a phase singularity of some special spectral component in the far field; i.e. the spectrum is red-shifted at some points, blue-shifted at others, and split into two lines elsewhere. The prediction was confirmed in Popescu’s experiment by implying a spatially fully coherent and polychromatic Gaussian beam, diffracted by an aperture lens in [9]. During the research of spectral changes, a new effect was discovered in diffracted partially coherent and polychromatic light, referred to as a spectral switch [10–12]. It was found by numerical research that the spectral switch may also occur in a type of experiment such as the Young’s two-slit setup illuminated by partially coherent light [13,14], which was verified by Kandpal’s experiments [15,16]. And the relation between the spectral anomalies and the spectral switches has been discussed in [17,18]. The spectral switch can also be produced by completely coherent light and can be controlled through Kerr effect [19]. However, except for the special setup of diffraction, it is interesting to find that spectral anomaly and spectral switch can also occur in the superposition of two polychromatic beams with and without inherent vortex. In this paper, we will focus discussion on the spectral property of a composite beam, superposed by a polychromatic Gaussian beam and a polychromatic Gaussian vortex beam with the same polarization. The details of the distribution of spectral shift and location of spectral switch will be investigated. 2. Theory We consider a composite optical field, superposed by a polychromatic Gaussian beam and a polychromatic Gaussian beam carrying a mth-order vortex with the same polarization direction. The corresponding composite electric field in the original plane can be expressed as E ( ρ , ϕ , z = 0) = E ( ρ , ϕ , z = 0) + E ( ρ , ϕ , z = 0) G GV 2 2 0 EG ( ρ , ϕ , z = 0) = EG 0 exp − ρ σ E (ω ) exp[iφ0 ] m 2 2 0 EGV ( ρ , ϕ , z = 0) = EGV 0 ( ρ σ ) exp − ρ σ exp(imϕ ) E (ω )
(1)
where ρ and ϕ are the transverse radial and azimuthal coordinates in the source plane respectively. EG 0 and EGV 0 are the characteristic amplitudes of two component beams. σ represents the beam size. φ0 is the relative phase between the two beams. E 0 (ω ) is related to the original spectrum, which is assumed to have a Gaussian profile, centered at frequency of ω0 and rms width Γ (ω − ω0 ) 2 E 0 (ω )= S 0 (ω ) = exp − 4Γ 2
(2)
Based on paraxial approximation, the propagated complex light amplitude takes the form [20]
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20194
E (r ,θ , ω , z ) = −
iω ωz ik E ( ρ , ϕ , ω , z = 0) exp (r − ρ) 2 dρ exp i 2π zc c 2z
(3)
r and θ are the transverse radial and azimuthal coordinates in the observation plane. From Eqs. (1-3) the composite field of each frequency component during propagation can be expressed as
1 − i z zR ω z (1 − i z z R )r 2 − E (r , θ , ω , z ) = E 0 (ω ) exp i 2 2 2 2 2 1 + z z R c (1 + z z R )σ m m r 1 − i z zR × EGV 0 exp(imθ ) + EG 0 exp(iφ0 ) 2 2 σ 1 + z zR
(4)
where z R = ωσ 2 2c . It is straightforward that the spectral density of an observation point takes the form S (r ,θ , ω , z ) =
S 0 (ω ) −2r 2 exp 2 2 2 2 2 1 + z zR (1 + z z R )σ m
m
r 1 − i z zR × EGV 0 exp(imθ ) + EG 0 exp(iφ0 ) 2 2 σ 1 + z zR
2
(5)
3. The property of the central frequency component To explore the property of spectral distribution on a certain observation plane, the central frequency component plays an important role, which is related to phenomenon of spectral anomaly. In Eq. (5), if we set ω =ω0 , it refers to the distribution of central frequency component and the locations of the corresponding singularities satisfies
r = γ 1/ m 1 + z 2 σ , ( z = z z , z = ω σ 2 2c, γ = E E ) n n R0 R0 0 G0 GV 0 θ = arc tan( z z R 0 ) + (π + φ0 + 2nπ ) m , (n = 0,1, ⋅⋅⋅, m − 1)
(6)
It is straightforward in Eq. (6) that there are m singly zero points of the central frequency component, equally distributed in the azimuthal direction at the same radial distance. On a certain observation plane, the radius relies on the ratio of the amplitudes of the two composite beams and the angle is related to the relative phase. Figure 1 shows the simulation results of distribution of central frequency component during propagation while m = 2, σ =0.1mm , γ =1 , φ0 =0 , ω0 =1015 s −1 and Γ =0.02ω0 .
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20195
Fig. 1. Distribution of central frequency component during propagation (m = 2). (a) zn = 0.5; (b) zn = 1; (c) zn = 2; (d) zn = 10.
There exist two symmetric singularities (two dark spots) of the central frequency component in Figs. 1(a)-1(d), where the intensity is zero. With increase of propagation distance from zn = 0.5 to zn = 10, the corresponding radial distance and angle of the singularities increase meanwhile, and the angle has limitation while z approaches infinity ( arc tan( z z R 0 ) → π 2 ), which satisfies Eq. (6). Usually spectral anomaly takes place in the vicinity of the singularity of the central frequency component. To provide a clear proof, the overall property of spectral anomaly on a certain observation plane will be firstly investigated. 4. The overall property of Spectral anomalies
Generally, for an observation point, the spectral centre can be characterized by the mean frequency
ω (r ,θ , ω , z )=
ω S (r ,θ , ω , z )d ω S (r ,θ , ω , z )d ω
(7)
To show the property of spectral shift, the relative mean frequency can be employed
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20196
Ω( r , θ , ω , z ) =
ω (r , θ , ω , z ) − ω0 ω0
(8)
While Ω(r , θ , ω , z ) > 0 , blue shift appears; Ω(r , θ , ω , z ) < 0 , red shift appears in stead. On a certain observation plane, Ω(r , θ , ω , z ) is a function of the positional coordinates of all the observation points, which can provide an overall illustration of the property of spectral shift distribution. To show relation of the intensity distribution of central frequency component and the spectral shift distribution. Simulation of the spectral shift distribution is performed in Fig. 2. The parameters are the same as in Fig. 1.
Fig. 2. Distribution of the relative mean frequency of the spectrum (m = 2). (a) zn = 0.5; (b) zn = 1; (c) zn = 2; (d) zn = 10. The dashed line indicates the positions where spectral shift is zero.
In Fig. 2, the dashed line forms a centro-symmetrically closed zone, inside which the spectra are all blue-shifted and outside which, red shifted, so called “zero shift line”. Along the zero shift line, there exist two small symmetric areas where the spectral shifts (including red shift and blue shift) are outstanding and the maximum red shift and maximum blue shift appear
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20197
on each side. In the two small areas, distributions of the relative mean frequency of the spectrum are the same with each other if one of them rotates an angle of π. By comparing Figs. 1(a)-1(d) with Figs. 2(a)-2(d) respectively, it can be found that the two singularities of the central frequency component in Fig. 1 are just located in the corresponding two small areas in Fig. 2. For instance, in Fig. 1(c) the two singularities are positioned at (−0.2mm, 0.1mm) and (0.2mm, −0.1mm); and the two small areas in Fig. 2(c) are located around (−0.2mm, 0.1mm) and (0.2mm, −0.1mm), respectively. The numbers of the small areas in the spectral shift distribution is equal to the topological charge of the composite beam with inherent vortex. And strong spectral anomaly takes place where the central frequency component is weak. This relation is not a special case. We have performed simulations while m is equal to other value and found the same conclusion, which can also be seen in the following discussion. 5. Influence of γ and Φ0 on the property of spectral shift distribution
As for the combined beam, the two parameters γ and φ0 are important, which can influence the property of spectral shift distribution greatly. It is found that the influence induced by φ0 is simple, but the influence induced by γ is a bit complex. 5.1 Influence of Φ0
Figure 3 shows the spectral shift distribution for different values of φ0 and other parameters are the same as in Fig. 2(c).
Fig. 3. Distribution of the relative mean frequency of the spectrum (m = 2, zn = 2) (a) φ0 = π 2 ; (b) φ0 =π . The dashed line indicates the positions where spectral shift is zero.
In Fig. 2(c), parameter φ0 is 0; while it changes to π/2 and to π, distribution of spectral shift turns angles of π/4 and π/2 clockwise in Fig. 3(a) and 3(b), respectively. This phenomenon agrees with the rule in Eq. (6) for the central frequency component and it also satisfies Eq. (5).
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20198
5.2 Influence of γ
Unlike φ0 , γ can influence the overall property of distribution of spectral shift on a certain observation plane. We choose zn = 2, m = 2 and perform simulation of distribution of spectral shift for different values of γ . The representative results are shown in Fig. 4.
Fig. 4. Distribution of the relative mean frequency of the spectrum (m = 2,zn = 2). (a) γ = 0.5; (b) γ = 0.63; (c) γ = 0.65; (d) γ = 0.67; (e) γ = 1.2; (f) γ = 1.48; (g) γ = 1.5; (h) γ = 1.52; (i) γ = 2. The dashed line indicates the positions where spectral shift is zero.
In Fig. 4, while γ is 0.5, there are three closed zero shift lines, two small inside one big. The observed spectra of the points inside the two small closed zero shift lines are red-shifted; and for points outside of the big zero shift line, red-shifted too; else, all blue-shifted. With increase of γ to 0.63, the two small zero shift lines both tend to approach the big one. While
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20199
γ increases to 0.65, the two small zero shift lines touch the big one. Meanwhile they are merged into a single crossed line and two single cross points appear. It should be pointed out that the two crossed points are also the connection points, at which the inside red-shift zone touch the outside red-shift zone. With continuous increase of γ to 0.67 and to a bit bigger, the merged crossed-line tends to separate at the two single cross points respectively and tends to become smooth; meanwhile all the red-shift zone is connected and forms a whole. However, while γ increases to 1.48, the smooth line tends to attract itself on both sides at two centro-symmetric parts respectively; with continuous increase of γ to 1.5, the smooth line touches itself at two centro-symmetric points and becomes crossed, for which the whole blue-shift zone is divided into three parts, connected at the two single cross points. While γ increases to 1.52, separation takes place at the two single cross points again. Finally with further increase of γ , three apart closed zero shift lines appear. Meanwhile, three apart blue-shift zones are formed, of which the middle one is like a segment of band. Then we consider the similarity in Figs. 4(a)-4(i). Although γ changes from 0.5 to 2 and it results in strong influence on the spectral shift distribution, there are always two small areas where the spectral shifts (including red shift and blue shift) are outstanding in all the figs, located in the same angle as in Fig. 2(c). An obvious difference between them is that with increase of γ , the two small areas tend to move away from the center, which agrees with conclusion in Eq. (6). Discussion above refers to the case while m = 2, if we change the topological charge of the vortex embedded in one composite beam, similar phenomenon can be found and Fig. 5 shows the case while m = 5 and other parameters are the as in Fig. 4
Fig. 5. Distribution of the relative mean frequency of the spectrum (m = 5, zn = 2). (a) γ = 2.5; (b) γ = 2.73; (c) γ = 2.8; (d) γ = 4.5; (e) γ = 4.7; (f) γ = 5. The dashed line indicates the positions where spectral shift is zero.
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20200
While m changes to 5 in Fig. 5(a), there are six closed zero shift lines, five small inside one big. The observed spectra of the points inside the five small closed zero shift lines are red-shifted; and for the points outside of the big zero shift line, red-shifted too; else, all blue-shifted. With increase of γ , the five small zero shift lines all tend to approach the big one and they are merged into a single crossed line and five single cross points appear in Fig. 5(b). Just as in Figs. 4(d)-4(f), with continuous increase of γ , the merged crossed-line separate at the five cross points and tends to become smooth; then it attract itself on both sides at five centro-symmetric parts respectively. Similarly it becomes into a crossed line with five cross points in Fig. 5(e), as in Fig. 4(e). With further increase of γ , separation occurs at the five cross points and gradually six apart closed zero shift lines are formed, of which five lines are centro-symmetrically distributed around the middle zero shift line (and the middle line engages the shape of pentagon). It also should be pointed out that, despite the strong influence of γ on the spectral shift distribution in Figs. 5(a)-5(f), there are always five small areas where the spectral shifts (including red shift and blue shift) are outstanding in all the figs, located in the same angle. Similarly, the five small areas tend to move away from the center, which agrees with conclusion in Eq. (6). 6. Spectral switches
Distribution of spectral shift can only provide a rough and overall illustration of the property of spectral anomaly. To explore the exact position of the spectral switch, the details of the spectral property in the areas where both red shifts and blue shifts are outstanding should be deeply investigated. Since the property of spectral shift distribution is similar in the small areas, we choose the left one in Fig. 2(c) as an example. Figure 6 presents the details of spectral shift distribution in the left small area in Fig. 2(c), enclosed by a red circle. (The corresponding singularity of the central frequency component is located at (−0.2mm, 0.1mm)).
Fig. 6. Details of the distribution of the relative mean frequency of the spectrum in the vicinity of the left singularity of the central frequency component in Fig. 2(c). The dashed line indicates the positions where spectral shift is zero. The white line is the “main line”.
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20201
From Fig. 6 it can be found that, in the vicinity of the singularity of the central frequency component, the contour of the blue shift and the red shift are similar and the maximum width of the blue shift and red shift is 0.0194, 0.0204 respectively, close to each other. According to the connotation of spectral switch, it should be located at the point where there are two peaks with the same level in the curve of spectral distribution and by each side of which different peak prevails remarkably. In Fig. 6, the contour of spectral shift is just like the contour of electric potential. The gradient line is like field line, which is tangential to the contour of spectral shift. Among all the gradient lines, there is one line which connects the maximum blue-shift point and the maximum red-shift point directly, nearly a straight line. It is most important, here we call “main line”, as shown in white color in Fig. 6. Along this special main line, the spectral shift will show sharpest transition and the spectral switch can be detected. It should be emphasized that the “main line” is different from the concept of “critical direction” in [8]. Main line is a transverse line, along which only one spectral switch can be observed; however, critical direction refers to a spatial angle, along which all the points are “spectral-switch-points”. By careful calculation, it is fount that the spectral switch is at the point (0.000094mm, 0.1mm), so called spectral switch point. It is interesting to discover that in the vicinity of the spectral switch point, the spectra shows different tendency of variation in different direction. For simplification, the spectral switch point is regarded as the origin of coordinates and the “main line” is taken as “x” axis. In other words, we use coordinates (r', θ') to denote the relative position of the observed point to the spectral switch point. Figure 7 shows the spectral distribution at different observation points in the direction θ' = 0 (π)
Fig. 7. Normalized spectral distribution of different observation points in the direction θ' = 0 (π). (a).r' = 0.0005mm, θ' = π; (b) r' = 0.0004mm, θ' = π; (c) r' = 0.0002mm, θ' = π; (d) r' = 0mm; (e) r' = 0.0002mm, θ' = 0; (f) r' = 0.0004mm, θ' = 0. The red curves indicate the original spectra S0(ω) and the blue curves indicate the spectra S(ω).
At the point (r' = 0.0005mm, θ' = π), the observed spectrum has two peaks and the right prevails, which means the existence of blue shift; while the observed point moves to (r' = 0.0004mm, θ' = π) and to (r' = 0.0002mm, θ' = π), the left peak tends to level up gradually. Finally it reaches the same level with the right peak at the point (r' = 0mm) and spectral switch appears. With continuous move of the observed point to (r' = 0.0002mm, θ' = 0) and to (r' =
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20202
0.0004mm, θ' = 0), the right peak levels down gradually, which indicates the existence of red shift. It should be addressed that from Figs. 7(a)-7(f), the move distance of the observation point is very small, but the consequent change on spectral property is sharp. Unlike the spectral property along “main line”, the spectra in the direction θ' = 3π/2 (π/2) show absolutely different characteristic, as shown in Fig. 8
Fig. 8. Normalized spectral distribution of different observation points in the direction θ' = 3π/2(π/2). (a).r' = 0.008mm, θ' = 3π/2; (b) r' = 0.005mm, θ' = 3π/2; (c) r' = 0.002mm, θ' = 3π/2; (d) r' = 0mm; (e) r' = 0.002mm, θ' = π/2; (f) r' = 0.005mm, θ' = π/2. The red curves indicate the original spectra S0(ω) and the blue curves indicate the spectra S(ω).
At the point (r' = 0.008mm, θ' = 3π/2), the observed spectrum has only one peak; while at the point (r' = 0.005mm, θ' = 3π/2), the peak tends to split, slightly. With continuous move of the observation point to (r' = 0.002mm, θ' = 3π/2), the spectral intensity at the center of the curve levels down and two discrete peaks appear. At the switch point, the center of the curve approaches zero. If the observation point moves to (r' = 0.002mm, θ' = π/2) and to (r' = 0.005mm, θ' = π/2), the center of the curve levels up and the two peaks tend to merge gradually until finally a center peak is formed. It should be pointed out that the two peaks in Figs. 8(c) and 8(e) are not at the same level, and the right prevails slightly, which means that there is no corresponding spectral switch. Except for the different characteristic of spectral property between Fig. 7 and Fig. 8 described above, another difference is also straightforward: from Figs. 7(a)-7(f), movement of the observed point is 0.0009mm, which is big enough to show the outstanding spectral variation; however the observed point moves 0.013mm in Figs. 8(a)-8(f) to show the corresponding spectral variation. It reveals that in the vicinity of the spectral switch point, the main line is most sensitive where spectral property varies sharply. This conclusion agrees with the discussion on the contour of spectral shift in Fig. 6. As for other direction in the vicinity of spectral switch point, the spectral property can be by the large estimated. 7. Conclusion
In conclusion, the properties of spectral anomaly and spectral switch have been studied, for a composite field, superposed by a polychromatic Gaussian beam and a polychromatic Gaussian beam carrying a mth-order vortex. It is found that, in the overall spectral shift distribution, there
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20203
are m small areas where there exist maximum shift width (including both red shift and blue shift), and the number of the small areas is always equal to the topological charge of the vortex of one composite beam. The spectral property of the small areas is similar to each other, and their locations are dependent on the ratio of the respective amplitudes of the two composite beams and the corresponding relative phase. During the detailed investigation of the property in one small area for an instance, the sharp variance of spectral distribution is found along the so called “main line”, which means that all m spectral switches can be explored in the same way. The research introduces a new way to generate spectral switches, of which the number and the location can be controlled by changing the parameters of the combined beam. Acknowledgments
This research was supported by the NSC of ROC under contract no. NSC 101-2221-E-005-062-MY3; the National Natural Science Foundations of China under Grant Nos. 61307001 and 61178015; the National Natural Science Foundation of Fujian Province under Grant No.2013J05094; and the National Chung Hsing University, Taiwan.
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20204
3
College of Information Science & Engineering, Huaqiao University, Xiamen, Fujian 361021, China 2 Department of Electrical Engineering, Tunghai University, Taichung, 40704, Taiwan Graduate Institute of Precision Engineering, National Chung Hsing University, 250 Kuo Kuang Road, Taichung 402, Taiwan *[email protected]
Abstract: We study the spectral property of a composite field superposed by a polychromatic Gaussian beam and a polychromatic Gaussian beam with an embedded mth-order vortex. It is shown that, in the overall spectral shift distribution, there exist m small areas where sharp spectral anomaly takes place, similarly and respectively, which are related with the ratio of the respective amplitudes of the two composite beams and the relative phase between them. Detailed investigation reveals that, for each small area, there exists a “main line”, along which spectral switch can be observed. ©2014 Optical Society of America OCIS codes: (300.6170) Spectra; (050.4865) Optical vortices; (070.7345) Wave propagation.
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#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20193
18. P. Han, “Spectral switches for a circular aperture with a variable wedge,” J. Opt. Soc. Am. A 26(3), 473–479 (2009). 19. P. Han, “All optical spectral switches,” Opt. Lett. 37(12), 2319–2321 (2012). 20. M. Born and E. Wolf, Principles of Optics, 7th ed, (Cambridge University, 1999).
1. Introduction The existence of optical vortex or phase singularity, which is closely related to singular optics, is usually indicated by a specific point of an optical field where the intensity is zero and thus the phase becomes undefined [1,2]. The study of singular optics has developed into a new branch of optics in the past decades and most of the researches concerning with vortex deal with monochromatic wave even at present [3–5]. In 2002 the work in singular optics was extended from monochromatic light to spatially fully coherent and polychromatic light [6–8]. It was found that, when a converging, spatially fully coherent and polychromatic wave is diffracted at an aperture, anomalous spectral changes take place near a phase singularity of some special spectral component in the far field; i.e. the spectrum is red-shifted at some points, blue-shifted at others, and split into two lines elsewhere. The prediction was confirmed in Popescu’s experiment by implying a spatially fully coherent and polychromatic Gaussian beam, diffracted by an aperture lens in [9]. During the research of spectral changes, a new effect was discovered in diffracted partially coherent and polychromatic light, referred to as a spectral switch [10–12]. It was found by numerical research that the spectral switch may also occur in a type of experiment such as the Young’s two-slit setup illuminated by partially coherent light [13,14], which was verified by Kandpal’s experiments [15,16]. And the relation between the spectral anomalies and the spectral switches has been discussed in [17,18]. The spectral switch can also be produced by completely coherent light and can be controlled through Kerr effect [19]. However, except for the special setup of diffraction, it is interesting to find that spectral anomaly and spectral switch can also occur in the superposition of two polychromatic beams with and without inherent vortex. In this paper, we will focus discussion on the spectral property of a composite beam, superposed by a polychromatic Gaussian beam and a polychromatic Gaussian vortex beam with the same polarization. The details of the distribution of spectral shift and location of spectral switch will be investigated. 2. Theory We consider a composite optical field, superposed by a polychromatic Gaussian beam and a polychromatic Gaussian beam carrying a mth-order vortex with the same polarization direction. The corresponding composite electric field in the original plane can be expressed as E ( ρ , ϕ , z = 0) = E ( ρ , ϕ , z = 0) + E ( ρ , ϕ , z = 0) G GV 2 2 0 EG ( ρ , ϕ , z = 0) = EG 0 exp − ρ σ E (ω ) exp[iφ0 ] m 2 2 0 EGV ( ρ , ϕ , z = 0) = EGV 0 ( ρ σ ) exp − ρ σ exp(imϕ ) E (ω )
(1)
where ρ and ϕ are the transverse radial and azimuthal coordinates in the source plane respectively. EG 0 and EGV 0 are the characteristic amplitudes of two component beams. σ represents the beam size. φ0 is the relative phase between the two beams. E 0 (ω ) is related to the original spectrum, which is assumed to have a Gaussian profile, centered at frequency of ω0 and rms width Γ (ω − ω0 ) 2 E 0 (ω )= S 0 (ω ) = exp − 4Γ 2
(2)
Based on paraxial approximation, the propagated complex light amplitude takes the form [20]
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20194
E (r ,θ , ω , z ) = −
iω ωz ik E ( ρ , ϕ , ω , z = 0) exp (r − ρ) 2 dρ exp i 2π zc c 2z
(3)
r and θ are the transverse radial and azimuthal coordinates in the observation plane. From Eqs. (1-3) the composite field of each frequency component during propagation can be expressed as
1 − i z zR ω z (1 − i z z R )r 2 − E (r , θ , ω , z ) = E 0 (ω ) exp i 2 2 2 2 2 1 + z z R c (1 + z z R )σ m m r 1 − i z zR × EGV 0 exp(imθ ) + EG 0 exp(iφ0 ) 2 2 σ 1 + z zR
(4)
where z R = ωσ 2 2c . It is straightforward that the spectral density of an observation point takes the form S (r ,θ , ω , z ) =
S 0 (ω ) −2r 2 exp 2 2 2 2 2 1 + z zR (1 + z z R )σ m
m
r 1 − i z zR × EGV 0 exp(imθ ) + EG 0 exp(iφ0 ) 2 2 σ 1 + z zR
2
(5)
3. The property of the central frequency component To explore the property of spectral distribution on a certain observation plane, the central frequency component plays an important role, which is related to phenomenon of spectral anomaly. In Eq. (5), if we set ω =ω0 , it refers to the distribution of central frequency component and the locations of the corresponding singularities satisfies
r = γ 1/ m 1 + z 2 σ , ( z = z z , z = ω σ 2 2c, γ = E E ) n n R0 R0 0 G0 GV 0 θ = arc tan( z z R 0 ) + (π + φ0 + 2nπ ) m , (n = 0,1, ⋅⋅⋅, m − 1)
(6)
It is straightforward in Eq. (6) that there are m singly zero points of the central frequency component, equally distributed in the azimuthal direction at the same radial distance. On a certain observation plane, the radius relies on the ratio of the amplitudes of the two composite beams and the angle is related to the relative phase. Figure 1 shows the simulation results of distribution of central frequency component during propagation while m = 2, σ =0.1mm , γ =1 , φ0 =0 , ω0 =1015 s −1 and Γ =0.02ω0 .
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20195
Fig. 1. Distribution of central frequency component during propagation (m = 2). (a) zn = 0.5; (b) zn = 1; (c) zn = 2; (d) zn = 10.
There exist two symmetric singularities (two dark spots) of the central frequency component in Figs. 1(a)-1(d), where the intensity is zero. With increase of propagation distance from zn = 0.5 to zn = 10, the corresponding radial distance and angle of the singularities increase meanwhile, and the angle has limitation while z approaches infinity ( arc tan( z z R 0 ) → π 2 ), which satisfies Eq. (6). Usually spectral anomaly takes place in the vicinity of the singularity of the central frequency component. To provide a clear proof, the overall property of spectral anomaly on a certain observation plane will be firstly investigated. 4. The overall property of Spectral anomalies
Generally, for an observation point, the spectral centre can be characterized by the mean frequency
ω (r ,θ , ω , z )=
ω S (r ,θ , ω , z )d ω S (r ,θ , ω , z )d ω
(7)
To show the property of spectral shift, the relative mean frequency can be employed
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20196
Ω( r , θ , ω , z ) =
ω (r , θ , ω , z ) − ω0 ω0
(8)
While Ω(r , θ , ω , z ) > 0 , blue shift appears; Ω(r , θ , ω , z ) < 0 , red shift appears in stead. On a certain observation plane, Ω(r , θ , ω , z ) is a function of the positional coordinates of all the observation points, which can provide an overall illustration of the property of spectral shift distribution. To show relation of the intensity distribution of central frequency component and the spectral shift distribution. Simulation of the spectral shift distribution is performed in Fig. 2. The parameters are the same as in Fig. 1.
Fig. 2. Distribution of the relative mean frequency of the spectrum (m = 2). (a) zn = 0.5; (b) zn = 1; (c) zn = 2; (d) zn = 10. The dashed line indicates the positions where spectral shift is zero.
In Fig. 2, the dashed line forms a centro-symmetrically closed zone, inside which the spectra are all blue-shifted and outside which, red shifted, so called “zero shift line”. Along the zero shift line, there exist two small symmetric areas where the spectral shifts (including red shift and blue shift) are outstanding and the maximum red shift and maximum blue shift appear
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20197
on each side. In the two small areas, distributions of the relative mean frequency of the spectrum are the same with each other if one of them rotates an angle of π. By comparing Figs. 1(a)-1(d) with Figs. 2(a)-2(d) respectively, it can be found that the two singularities of the central frequency component in Fig. 1 are just located in the corresponding two small areas in Fig. 2. For instance, in Fig. 1(c) the two singularities are positioned at (−0.2mm, 0.1mm) and (0.2mm, −0.1mm); and the two small areas in Fig. 2(c) are located around (−0.2mm, 0.1mm) and (0.2mm, −0.1mm), respectively. The numbers of the small areas in the spectral shift distribution is equal to the topological charge of the composite beam with inherent vortex. And strong spectral anomaly takes place where the central frequency component is weak. This relation is not a special case. We have performed simulations while m is equal to other value and found the same conclusion, which can also be seen in the following discussion. 5. Influence of γ and Φ0 on the property of spectral shift distribution
As for the combined beam, the two parameters γ and φ0 are important, which can influence the property of spectral shift distribution greatly. It is found that the influence induced by φ0 is simple, but the influence induced by γ is a bit complex. 5.1 Influence of Φ0
Figure 3 shows the spectral shift distribution for different values of φ0 and other parameters are the same as in Fig. 2(c).
Fig. 3. Distribution of the relative mean frequency of the spectrum (m = 2, zn = 2) (a) φ0 = π 2 ; (b) φ0 =π . The dashed line indicates the positions where spectral shift is zero.
In Fig. 2(c), parameter φ0 is 0; while it changes to π/2 and to π, distribution of spectral shift turns angles of π/4 and π/2 clockwise in Fig. 3(a) and 3(b), respectively. This phenomenon agrees with the rule in Eq. (6) for the central frequency component and it also satisfies Eq. (5).
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20198
5.2 Influence of γ
Unlike φ0 , γ can influence the overall property of distribution of spectral shift on a certain observation plane. We choose zn = 2, m = 2 and perform simulation of distribution of spectral shift for different values of γ . The representative results are shown in Fig. 4.
Fig. 4. Distribution of the relative mean frequency of the spectrum (m = 2,zn = 2). (a) γ = 0.5; (b) γ = 0.63; (c) γ = 0.65; (d) γ = 0.67; (e) γ = 1.2; (f) γ = 1.48; (g) γ = 1.5; (h) γ = 1.52; (i) γ = 2. The dashed line indicates the positions where spectral shift is zero.
In Fig. 4, while γ is 0.5, there are three closed zero shift lines, two small inside one big. The observed spectra of the points inside the two small closed zero shift lines are red-shifted; and for points outside of the big zero shift line, red-shifted too; else, all blue-shifted. With increase of γ to 0.63, the two small zero shift lines both tend to approach the big one. While
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20199
γ increases to 0.65, the two small zero shift lines touch the big one. Meanwhile they are merged into a single crossed line and two single cross points appear. It should be pointed out that the two crossed points are also the connection points, at which the inside red-shift zone touch the outside red-shift zone. With continuous increase of γ to 0.67 and to a bit bigger, the merged crossed-line tends to separate at the two single cross points respectively and tends to become smooth; meanwhile all the red-shift zone is connected and forms a whole. However, while γ increases to 1.48, the smooth line tends to attract itself on both sides at two centro-symmetric parts respectively; with continuous increase of γ to 1.5, the smooth line touches itself at two centro-symmetric points and becomes crossed, for which the whole blue-shift zone is divided into three parts, connected at the two single cross points. While γ increases to 1.52, separation takes place at the two single cross points again. Finally with further increase of γ , three apart closed zero shift lines appear. Meanwhile, three apart blue-shift zones are formed, of which the middle one is like a segment of band. Then we consider the similarity in Figs. 4(a)-4(i). Although γ changes from 0.5 to 2 and it results in strong influence on the spectral shift distribution, there are always two small areas where the spectral shifts (including red shift and blue shift) are outstanding in all the figs, located in the same angle as in Fig. 2(c). An obvious difference between them is that with increase of γ , the two small areas tend to move away from the center, which agrees with conclusion in Eq. (6). Discussion above refers to the case while m = 2, if we change the topological charge of the vortex embedded in one composite beam, similar phenomenon can be found and Fig. 5 shows the case while m = 5 and other parameters are the as in Fig. 4
Fig. 5. Distribution of the relative mean frequency of the spectrum (m = 5, zn = 2). (a) γ = 2.5; (b) γ = 2.73; (c) γ = 2.8; (d) γ = 4.5; (e) γ = 4.7; (f) γ = 5. The dashed line indicates the positions where spectral shift is zero.
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20200
While m changes to 5 in Fig. 5(a), there are six closed zero shift lines, five small inside one big. The observed spectra of the points inside the five small closed zero shift lines are red-shifted; and for the points outside of the big zero shift line, red-shifted too; else, all blue-shifted. With increase of γ , the five small zero shift lines all tend to approach the big one and they are merged into a single crossed line and five single cross points appear in Fig. 5(b). Just as in Figs. 4(d)-4(f), with continuous increase of γ , the merged crossed-line separate at the five cross points and tends to become smooth; then it attract itself on both sides at five centro-symmetric parts respectively. Similarly it becomes into a crossed line with five cross points in Fig. 5(e), as in Fig. 4(e). With further increase of γ , separation occurs at the five cross points and gradually six apart closed zero shift lines are formed, of which five lines are centro-symmetrically distributed around the middle zero shift line (and the middle line engages the shape of pentagon). It also should be pointed out that, despite the strong influence of γ on the spectral shift distribution in Figs. 5(a)-5(f), there are always five small areas where the spectral shifts (including red shift and blue shift) are outstanding in all the figs, located in the same angle. Similarly, the five small areas tend to move away from the center, which agrees with conclusion in Eq. (6). 6. Spectral switches
Distribution of spectral shift can only provide a rough and overall illustration of the property of spectral anomaly. To explore the exact position of the spectral switch, the details of the spectral property in the areas where both red shifts and blue shifts are outstanding should be deeply investigated. Since the property of spectral shift distribution is similar in the small areas, we choose the left one in Fig. 2(c) as an example. Figure 6 presents the details of spectral shift distribution in the left small area in Fig. 2(c), enclosed by a red circle. (The corresponding singularity of the central frequency component is located at (−0.2mm, 0.1mm)).
Fig. 6. Details of the distribution of the relative mean frequency of the spectrum in the vicinity of the left singularity of the central frequency component in Fig. 2(c). The dashed line indicates the positions where spectral shift is zero. The white line is the “main line”.
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20201
From Fig. 6 it can be found that, in the vicinity of the singularity of the central frequency component, the contour of the blue shift and the red shift are similar and the maximum width of the blue shift and red shift is 0.0194, 0.0204 respectively, close to each other. According to the connotation of spectral switch, it should be located at the point where there are two peaks with the same level in the curve of spectral distribution and by each side of which different peak prevails remarkably. In Fig. 6, the contour of spectral shift is just like the contour of electric potential. The gradient line is like field line, which is tangential to the contour of spectral shift. Among all the gradient lines, there is one line which connects the maximum blue-shift point and the maximum red-shift point directly, nearly a straight line. It is most important, here we call “main line”, as shown in white color in Fig. 6. Along this special main line, the spectral shift will show sharpest transition and the spectral switch can be detected. It should be emphasized that the “main line” is different from the concept of “critical direction” in [8]. Main line is a transverse line, along which only one spectral switch can be observed; however, critical direction refers to a spatial angle, along which all the points are “spectral-switch-points”. By careful calculation, it is fount that the spectral switch is at the point (0.000094mm, 0.1mm), so called spectral switch point. It is interesting to discover that in the vicinity of the spectral switch point, the spectra shows different tendency of variation in different direction. For simplification, the spectral switch point is regarded as the origin of coordinates and the “main line” is taken as “x” axis. In other words, we use coordinates (r', θ') to denote the relative position of the observed point to the spectral switch point. Figure 7 shows the spectral distribution at different observation points in the direction θ' = 0 (π)
Fig. 7. Normalized spectral distribution of different observation points in the direction θ' = 0 (π). (a).r' = 0.0005mm, θ' = π; (b) r' = 0.0004mm, θ' = π; (c) r' = 0.0002mm, θ' = π; (d) r' = 0mm; (e) r' = 0.0002mm, θ' = 0; (f) r' = 0.0004mm, θ' = 0. The red curves indicate the original spectra S0(ω) and the blue curves indicate the spectra S(ω).
At the point (r' = 0.0005mm, θ' = π), the observed spectrum has two peaks and the right prevails, which means the existence of blue shift; while the observed point moves to (r' = 0.0004mm, θ' = π) and to (r' = 0.0002mm, θ' = π), the left peak tends to level up gradually. Finally it reaches the same level with the right peak at the point (r' = 0mm) and spectral switch appears. With continuous move of the observed point to (r' = 0.0002mm, θ' = 0) and to (r' =
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20202
0.0004mm, θ' = 0), the right peak levels down gradually, which indicates the existence of red shift. It should be addressed that from Figs. 7(a)-7(f), the move distance of the observation point is very small, but the consequent change on spectral property is sharp. Unlike the spectral property along “main line”, the spectra in the direction θ' = 3π/2 (π/2) show absolutely different characteristic, as shown in Fig. 8
Fig. 8. Normalized spectral distribution of different observation points in the direction θ' = 3π/2(π/2). (a).r' = 0.008mm, θ' = 3π/2; (b) r' = 0.005mm, θ' = 3π/2; (c) r' = 0.002mm, θ' = 3π/2; (d) r' = 0mm; (e) r' = 0.002mm, θ' = π/2; (f) r' = 0.005mm, θ' = π/2. The red curves indicate the original spectra S0(ω) and the blue curves indicate the spectra S(ω).
At the point (r' = 0.008mm, θ' = 3π/2), the observed spectrum has only one peak; while at the point (r' = 0.005mm, θ' = 3π/2), the peak tends to split, slightly. With continuous move of the observation point to (r' = 0.002mm, θ' = 3π/2), the spectral intensity at the center of the curve levels down and two discrete peaks appear. At the switch point, the center of the curve approaches zero. If the observation point moves to (r' = 0.002mm, θ' = π/2) and to (r' = 0.005mm, θ' = π/2), the center of the curve levels up and the two peaks tend to merge gradually until finally a center peak is formed. It should be pointed out that the two peaks in Figs. 8(c) and 8(e) are not at the same level, and the right prevails slightly, which means that there is no corresponding spectral switch. Except for the different characteristic of spectral property between Fig. 7 and Fig. 8 described above, another difference is also straightforward: from Figs. 7(a)-7(f), movement of the observed point is 0.0009mm, which is big enough to show the outstanding spectral variation; however the observed point moves 0.013mm in Figs. 8(a)-8(f) to show the corresponding spectral variation. It reveals that in the vicinity of the spectral switch point, the main line is most sensitive where spectral property varies sharply. This conclusion agrees with the discussion on the contour of spectral shift in Fig. 6. As for other direction in the vicinity of spectral switch point, the spectral property can be by the large estimated. 7. Conclusion
In conclusion, the properties of spectral anomaly and spectral switch have been studied, for a composite field, superposed by a polychromatic Gaussian beam and a polychromatic Gaussian beam carrying a mth-order vortex. It is found that, in the overall spectral shift distribution, there
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20203
are m small areas where there exist maximum shift width (including both red shift and blue shift), and the number of the small areas is always equal to the topological charge of the vortex of one composite beam. The spectral property of the small areas is similar to each other, and their locations are dependent on the ratio of the respective amplitudes of the two composite beams and the corresponding relative phase. During the detailed investigation of the property in one small area for an instance, the sharp variance of spectral distribution is found along the so called “main line”, which means that all m spectral switches can be explored in the same way. The research introduces a new way to generate spectral switches, of which the number and the location can be controlled by changing the parameters of the combined beam. Acknowledgments
This research was supported by the NSC of ROC under contract no. NSC 101-2221-E-005-062-MY3; the National Natural Science Foundations of China under Grant Nos. 61307001 and 61178015; the National Natural Science Foundation of Fujian Province under Grant No.2013J05094; and the National Chung Hsing University, Taiwan.
#213037 - $15.00 USD Received 28 May 2014; revised 4 Aug 2014; accepted 4 Aug 2014; published 13 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020193 | OPTICS EXPRESS 20204
