Angular momentum of an incoherent Gaussian beam I. Mokhun,* A. D. Arkhelyuk, Yu. Galushko, Ye. Kharitonova, and Yu. Viktorovskaya Chernivtsi University, Kotsybinsky Str. 2, Chernivtsi 12, 58012, Ukraine *Corresponding author: [email protected] Received 14 November 2013; revised 17 December 2013; accepted 17 December 2013; posted 20 December 2013 (Doc. ID 201336); published 3 February 2014
The relations for the components of the Poynting vector of a quasi-monochromatic wave are obtained. It is shown that in this case the behavior of the transversal Poynting component may be defined similarly to that in the coherent case. The total angular momentum of the quasi-monochromatic wave may be divided into the orbital and spin parts. The example of a Gaussian beam shows that the value of the spin angular momentum is connected to the coherence characteristics of the beam. Experimental results are presented. © 2014 Optical Society of America OCIS codes: (260.6042) Singular optics; (260.5430) Polarization. http://dx.doi.org/10.1364/AO.53.000B38
1. Introduction
The interest in singular optics continues to increase, due to a great extent to an interesting and prospective optical application—optical tweezers (see, for example, Ref. [1]). In its turn, this kind of application caused the advanced study of the fine structure of optical fields, the characteristics of which usually may be considered as some special distributions. It has been shown that in the vicinity of optical singularities the field acquires a specific value of the averaged angular momentum (maximal or minimal in the area) [2–4]. It has been noted that the overwhelming majority of the performed investigations are devoted to the monochromatic case [2–5]. In other words, the investigations of the angular momentum—energy currents that create it in the polychromatic waves— are the first stage [6–8]. At the same time, the study has quite good prospects fundamentally. In our opinion, one of them is related to the statement that the value of angular momentum must be connected to the coherence characteristics of a polychromatic wave. The existence of such a relationship is obvious. As it is known, the angular momentum may be separated into orbital and spin parts [2,3,5]. At the least, 1559-128X/14/100B38-05$15.00/0 © 2014 Optical Society of America B38
APPLIED OPTICS / Vol. 53, No. 10 / 1 April 2014
this statement follows from the fact that the spin angular momentum is defined by the determinate circulation of the field vector (see, for example, Ref. [2]). Naturally, the “level of such determinacy” must be connected to coherency. It is well known that an instantaneous Poynting vector is defined by the relation [9] ⃗ ⃗p E⃗ × H;
(1)
⃗ H ⃗ are the instantaneous strengths of elecwhere E, tric and magnetic fields, respectively. In optics, the value of this vector averaged by time is relevant, as a rule. The general expression for this value may be (at least, for the instantaneous Poynting vector) rather simply obtained if we take into account the fact that Ek t is a real perturbation, and each Cartesian component of the electric or magnetic field may be presented as a Fourier integral [10,11] Z Ei t
∞ 0
aν cosφν − 2πνtdν;
(2)
where i x; y; z. However, this relation is a cumbersome expression that does not allow us to carry out a simple and detailed analysis.
The situation is essentially simplified if the case of a quasi-monochromatic wave is considered. As it is known, the wave may be defined as the one with a relatively narrow spectrum, when [9,10] Δν ≪ 1; ν¯
(3)
⃗ where Δν = frequency range; while Et is essentially different from zero; and ν¯ = mean or main frequency. In the first part of this research, we will show that under the assumption that when the paraxial approximation is satisfied, the components of the averaged Poynting vector may be written in the form similar to one of a completely coherent wave. After that, in the example of the quasimonochromatic Gaussian wave, we will demonstrate the relationship between the value of the field angular momentum and the spectral range of the wave. 2. Poynting Vector Components of a Quasi-Monochromatic Field
Let us consider the quasi-monochromatic wave, which also obeys the paraxial approximation. In this case, Ei t, each Cartesian component of an electric or magnetic field may be represented as [9,10] Ei t Ai t cosΦi t − 2π ν¯ t;
(4)
where Ai t and Φi t = slowly changing functions [in comparison with cos2π ν¯ t]. Correspondingly, under this assumption, the instantaneous Poynting vector may be derived similarly to one of a strongly coherent case [4] 8 c P ≈ − 4πk fEx T 2 − Ey T 1 g > < x c Py ≈ − 4πk fEy T 2 Ex T 1 g ; > : c fE2x E2y g Pz ≈ 4π
5
where (
y
Ax
T 1 Ex Φyx − Ey Φxy AAxx Ex;2π − Ayy Ey;π2 x
Ay
T 2 Ex Φxx Ey Φyy AAxx Ex;2π Ayy Ey;2π
;
6
Ei Ai t cosΦi t − 2π ν¯ t Ei;π2 Ai t sinΦi t − 2π ν¯ t
.
8 ¯ = effective local phase difference and Φ ¯iΦ ¯l = where Δ averaged by time phase functions of components. After averaging Eqs. (4)–(7), transform for the following: n o 8 ¯ x ≈ − 1 s0 s1 Φ ¯ xx s0 − s1 Φ ¯ xy − ∂s3 P > ∂y 16π ω ¯ > < n o ¯ y ≈ − 1 s0 s1 Φ ¯ yx s0 − s1 Φ ¯ yy ∂s3 ; 9 P ∂x 16π ω¯ > > : c P¯ z ≈ 8π s0 where si = Stokes parameters and ω¯ 2π ν¯ . Thus, it can be stated that the notation of averaged Poynting components of a quasi-chromatic wave is the same for a strongly monochromatic wave [12] with corresponding determinate parameters. The terms in the square brackets of the first and second equations can be called the structure or orbital transversal part of the field energy density [5]. Just these terms, in the coherent case, are responsible for the appearance of orbital momentum in the area of vortex (scalar field) or C-point, point of circular polarization, and an inhomogeneously polarized field [2,3,13,14]. The last terms in the expressions of transversal components cause spin energy currents, which define the spin angular momentum of the field. Correspondingly, if one takes into account that angular momentum density is defined as follows (see, for example, Ref. [2]): ¯jz xP¯ y − yP¯ x ;
and
We took into account the fact, that in this case, when the time of averaging is significantly more than the coherence time, the spectral components may be considered as statistically independent. As a result, it can be rather easily shown that under our assumptions [Eqs. (3) and (4)], the following expressions as the “base” of averaging procedure take place: 8 hcos2 Φi −2π ν¯ ti 12 > > > > < hsinΦ −2π ν¯ tcosΦ −2π ν¯ ti0 i i ; ¯ 1 cosΦ ¯ i −Φ ¯ l > hcosΦi −2π ν¯ tcosΦl −2π ν¯ ti 12 cos Δ > 2 > > : ¯ 1 sinΦ ¯ i −Φ ¯ l hsinΦi −2π ν¯ tcosΦl −2π ν¯ ti 12 sin Δ 2
7
Ai and Φi are interpreted according to Eq. (4), and Ali and Φli are partial derivatives of Ai t and Φi t, i; l x; y. It has been noted that only x- and y- electric components are identified in Eqs. (5)–(7). Let us carry out the averaging by time of Eqs. (5)–(7).
(10)
one can state that for the polychromatic beam (similar to the coherent case), at least, for the wave with narrow spectrum and when paraxial approximation is satisfied, the total angular momentum may be divided in the orbital and spin parts. 3. Angular Momentum of a Quasi-Chromatic Gaussian Beam
Let us repeat the statement—the obviously angular momentum of the beam or the angular momentum of the field area must be related to the coherent characteristics of the wave. For example, in the limit case 1 April 2014 / Vol. 53, No. 10 / APPLIED OPTICS
B39
of an absolutely depolarized wave, the spin momentum is zero. As it follows from expressions for Poynting vector components in Eq. (9) at least, spin energy currents, and spin angular momentum defined by them are connected by the coherence characteristics of a wave. It follows from the fact that different spectral components insert different contributions to the total angular momentum of a field. This situation necessarily occurs if the analyzed field passes through setup: polarizer, quarter-wave plate, oriented by 450 to the polarizer axis. Let us assume that the thickness of the plate is chosen in such a manner that the spectral component with frequency ν¯ becomes circularly polarized after the plate. Correspondingly, other spectral components are elliptically polarized. Now we will demonstrate the efficiency of such a procedure for the Gaussian beam. Let the frequency spectrum of this beam be distributed by normal distribution. It has been noted that the presence of “negative” frequencies in such a distribution does not lead to an essential inaccuracy because the energy rapidly decreases when the frequency changes from the central one. In this case, the Cartesian components of a wave normalized to a unit may be represented in the following form: 8 2 2 1 expf− x y2 g Ex p > 3∕2 σ 2 σ 2σ > 2π ν > o n > R∞ > ν2 > < × expf jΦ0ν − 2πνtgdν × −∞ exp − ν−¯ 2 n 2σ2 ν 2 o ; x y 1 > Ey p exp − > 2 > 2σ 2π 3∕2 σ 2 σ ν > > o n > R∞ : ν−¯ν2 × −∞ exp − 2σ 2 × expf jΦ0ν − 2πνt Δν gdν ν
11 where σ = «space» width of the beam; σ ν = «spectral» width of the beam, defining the effective frequency band Δν; Φ0ν = constant phase, associated with each spectral component; and Δν = phase shift between orthogonal components, which appears when the spectral component passes through a quarter-wave plate. This shift may be identified by this relation Δv
πν : 2 ν¯
(12)
In this case, the components of the Poynting vector may be represented in the following form: 8 1 ∂s3 ¯ > > < Px ≈ 16πω¯ ∂y 1 ∂s3 ; P¯ y ≈ − 16π ¯ ∂x ω > > : P¯ ≈ c s z
13
8π 0
due to the fact that the phase of all spectral components is practically constant. B40
APPLIED OPTICS / Vol. 53, No. 10 / 1 April 2014
In other words, the transversal energy currents and the angular moment density depend only on the space changes of the 4th Stokes parameter. Let us determine the 4th Stokes parameter in our case. As shown by Born and Wolf [9], the resulting elements of the coherence matrix are the sum of elementary elements corresponding to each frequency ν. Due to that, the 4th Stokes parameter may be presented in the form s3 jJ yx − J xy 2 2 2 1 x y2 π σν exp − : − 2 exp − 2 σ 16¯ν2 πσ
(14)
Transversal components may be written as the relations n 2 2o 2 2 8 π σν < P¯ x ≈ − 16π12 ωσ exp − x σy y exp − 16¯ 2 ¯ 4 ν2 n 2 2o : 2 2 : P¯ ≈ 1 exp − π σ ν exp − x y x y ¯ 4 16¯ν2 σ2 16π 2 ωσ
15
In other words, the components of the Poynting vector accurate within the factor
π 2 σ 2ν α exp − 16¯ν2
≤1
(16)
are very similar to those of a monochromatic Gaussian beam [3]. As a consequence, the longitudinal components of the angular momentum density and angular momentum differ on the same factor. The magnitude of this factor defined by the width of spectral range and frequency of radiation shows the decrease of the value of angular momentum in relation to the coherent case. Thus, the magnitude of the angular momentum of a quasi-monochromatic Gaussian beam is strongly connected with the coherence of a wave. 4. Discussion
Let us estimate the decreasing of value of angular momentum. As it is known [15], the indeterminacy principle for wave processes has the form Δντ0 ≈ 1;
(17)
where τ0 = coherence time. Under quasi-monochromatic approximation, this parameter is significantly less than the period of wave vibration T 0 , where λ¯ T0 ; c
(18)
where λ¯ is the central wavelength of the spectral range.
In other words, τ0 ∼ aT 0 ;
(19)
where coefficient a is large enough. ¯ ∼ 1∕a or in much It can be easily shown that Δλ∕λ the same way in terms of ν Δν ∼ 1∕a: ν¯
Fig. 2. Optical trap configuration and stages of micro-object (red blood cell) capturing.
(20)
Let us assume that a is close to 10. Δλ-spectral range of radiation is close to 60–70 ηm for red light in this case. Such a spectral range is inherent to the radiation passing though the interference filter, or it is close to the spectral range of a conventional lightemitting diode. At the same time, as it follows from Eq. (16), the exponent in this expression is very small (about 0.001). Correspondingly, the value of angular momentum is practically the same as in the coherent case. This situation is similar even for a 2. It has been noted that this value of a is on the verge of physical sense. Nevertheless, the formally calculated “decreasing” coefficient is equal to 0.975. In other words, the value of angular momentum for quasi-monochromatic Gaussian wave is practically independent on spectral range. 5. Experimental Results
The experimental confirmation was realized in the arrangement presented in Fig. 1. The Gaussian beam, produced by a He–Ne laser directed at an interferometer with a delay line, formed in one of its legs (mirrors 3 and 4). The beams converged after beam splitter 5. After that, the resulting beam was put in an intensity “regulator”, which contains two polarizers 7, 8. Such a system allows us the smooth changes of intensity of the resulting beam. Quarter-wave plate 8 formed the elliptically polarized beam with different ellipticity due to the different orientation of plate relatively the axis of polarizer 8. The polarization changed from the left circular to a right circular one. The intensity of the beam is controlled by systems 9 and 10. Further, the beam focused by micro-objective 14 into the operation plane where the sample with micro-objects was placed. We used the micro-objects as the solution of red blood cells in water. The result of object
Fig. 1. 1, He–Ne laser; 2,5,9,11, beam splitters; 3,4, mirrors; 6,7, intensity “regulators” (two polarizers); 8, quarter-wave plate; 10, reference photo detector; 12,13, system of incoherent illumination; 14, focusing micro-objective; 15, sample with micro-objects; 16, objective of microscope; 17, blue filter; 18, CCD camera.
capturing was observed by microscope 16 and charge-coupled device (CCD) camera 18. The operation plane was illuminated by incoherent systems 12 and 13. Blue filter 17 practically cut the red radiation of the optical trap completely. In Fig. 2, the intensity distribution in the trap and the stages of capturing the red blood cell are presented. At the first stage, the access of the beam to the delay line (elements 3 and 4) was closed, and “traditional” rotation [16] of the red blood cell due to the spin angular momentum was observed. After that, the access to the delay line was open, and influence of the resulting beam was observed. The length of the delay line was significantly more than the coherence length of the radiation produced by the He–Ne laser. In addition, some preparatory interferometric experiments were presented. The resulting field after the interferometer was analyzed interferometrically. Some small angle between the beams was introduced, and the intensity distribution was tested. As a result, it can be stated that delay length of the interferometer is enough for coherence destroying because the intensity distribution of the resulting field does not have any interference fringes, which may be fixed experimentally. Thus, due to that, the coherence of the resulting wave was destroyed. The intensities of beams forming the output (resulting) wave were practically the same. In addition, the intensity of the output wave decreases by systems 6 and 7 to one that is observed in the case when the delay line was undamped. The polarization of the resulting beam, which is focused by objective 14, changed from the left circular to the right circular one. The rotation of the captured object is demonstrated by Figs. 3 and 4. It must be emphasized that in both cases the optical trap influenced the same micro-object. The main result of our experimental investigation is the following: we did not see the significant
Fig. 3. Rotation of a red blood cell due to the spin angular momentum in the clockwise direction. 1 April 2014 / Vol. 53, No. 10 / APPLIED OPTICS
B41
Fig. 4. Rotation of a red blood cell due to the spin angular momentum in the anticlockwise direction.
changes in rotation speed of the micro-object. Correspondingly, we can state that the time coherence, at least for beams with a relatively simple structure, does not influence the value of spin angular momentum. 6. Conclusions
1. In the quasi-monochromatic case, the behavior of a transversal Poynting component may be defined similarly to the coherent case. Correspondingly, the total angular momentum of the polychromatic wave may be divided into the orbital and spin ones. This statement is true, at least, for the wave that satisfies the paraxial approximation. 2. The value of the spin angular momentum of a quasi-monochromatic wave may be connected with the coherence characteristics of the beam, and it is defined only by the space changes of the 4th Stokes parameter. 3. The value of the angular momentum of the quasi-monochromatic Gaussian beam is practically independent of spectral range and time coherence. References 1. M. J. Lang and S. M. Block, “Resource letter: LBOT-1: laserbased optical tweezers,” Am. J. Phys. 71, 201–215 (2003).
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2. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1999), Vol. XXXIX, pp. 291–372. 3. A. Bekshaev, M. Soskin, and M. Vasnetsov, Paraxial Light Beams with Angular Momentum. (Nova Science, 2008). 4. I. I. Mokhun, “Introduction to linear singular optics,” in Optical Correlation Techniques and Applications, O. V. Angelsky, ed. (SPIE, 2007), pp. 1–132. 5. A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation light beams,” J. Opt. 13, 053001 (2011). 6. J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26, 405–407 (2001). 7. O. V. Angelsky, M. P. Gorsky, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and C. Yu. Zenkova, “Investigation of optical currents in coherent and partially coherent vector fields,” Opt. Express 19, 660–672 (2011). 8. O. V. Angelsky, S. G. Hanson, C. Yu. Zenkova, M. P. Gorsky, and N. V. Gorodyns’ka, “On polarization metrology (estimation) of the degree of coherence of optical waves,” Opt. Express 17, 15623–15634 (2009). 9. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980). 10. J. Perina, Coherence of Light, 2nd ed. (D. Reidel, 1985). 11. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, (Wiley, 1981). 12. R. Khrobatin, I. Mokhun, and Ju. Viktorovskaya, “Potentiality of experimental analysis for characteristics of the Poynting vector components,” Ukr. J. Phys. Opt. 9, 182–186 (2008). 13. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics (Elsevier, 2009), Vol. 53, pp. 293–363. 14. I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A 10, 064015 (2008). 15. F. S. Crawford, Waves, Berkeley Physics Course (McGraw-Hill, 1968), Vol. 3, pp. 337–341. 16. O. V. Angelsky, A. Ya. Bekshaev, P. P. Maksimyak, A. P. Maksimyak, I. I. Mokhun, S. G. Hanson, C. Yu. Zenkova, and A. V. Tyurin, “Circular motion of particles suspended in a Gaussian beam with circular polarization validates the spin part of the internal energy flow,” Opt. Express 20, 11351–11356 (2012).
The relations for the components of the Poynting vector of a quasi-monochromatic wave are obtained. It is shown that in this case the behavior of the transversal Poynting component may be defined similarly to that in the coherent case. The total angular momentum of the quasi-monochromatic wave may be divided into the orbital and spin parts. The example of a Gaussian beam shows that the value of the spin angular momentum is connected to the coherence characteristics of the beam. Experimental results are presented. © 2014 Optical Society of America OCIS codes: (260.6042) Singular optics; (260.5430) Polarization. http://dx.doi.org/10.1364/AO.53.000B38
1. Introduction
The interest in singular optics continues to increase, due to a great extent to an interesting and prospective optical application—optical tweezers (see, for example, Ref. [1]). In its turn, this kind of application caused the advanced study of the fine structure of optical fields, the characteristics of which usually may be considered as some special distributions. It has been shown that in the vicinity of optical singularities the field acquires a specific value of the averaged angular momentum (maximal or minimal in the area) [2–4]. It has been noted that the overwhelming majority of the performed investigations are devoted to the monochromatic case [2–5]. In other words, the investigations of the angular momentum—energy currents that create it in the polychromatic waves— are the first stage [6–8]. At the same time, the study has quite good prospects fundamentally. In our opinion, one of them is related to the statement that the value of angular momentum must be connected to the coherence characteristics of a polychromatic wave. The existence of such a relationship is obvious. As it is known, the angular momentum may be separated into orbital and spin parts [2,3,5]. At the least, 1559-128X/14/100B38-05$15.00/0 © 2014 Optical Society of America B38
APPLIED OPTICS / Vol. 53, No. 10 / 1 April 2014
this statement follows from the fact that the spin angular momentum is defined by the determinate circulation of the field vector (see, for example, Ref. [2]). Naturally, the “level of such determinacy” must be connected to coherency. It is well known that an instantaneous Poynting vector is defined by the relation [9] ⃗ ⃗p E⃗ × H;
(1)
⃗ H ⃗ are the instantaneous strengths of elecwhere E, tric and magnetic fields, respectively. In optics, the value of this vector averaged by time is relevant, as a rule. The general expression for this value may be (at least, for the instantaneous Poynting vector) rather simply obtained if we take into account the fact that Ek t is a real perturbation, and each Cartesian component of the electric or magnetic field may be presented as a Fourier integral [10,11] Z Ei t
∞ 0
aν cosφν − 2πνtdν;
(2)
where i x; y; z. However, this relation is a cumbersome expression that does not allow us to carry out a simple and detailed analysis.
The situation is essentially simplified if the case of a quasi-monochromatic wave is considered. As it is known, the wave may be defined as the one with a relatively narrow spectrum, when [9,10] Δν ≪ 1; ν¯
(3)
⃗ where Δν = frequency range; while Et is essentially different from zero; and ν¯ = mean or main frequency. In the first part of this research, we will show that under the assumption that when the paraxial approximation is satisfied, the components of the averaged Poynting vector may be written in the form similar to one of a completely coherent wave. After that, in the example of the quasimonochromatic Gaussian wave, we will demonstrate the relationship between the value of the field angular momentum and the spectral range of the wave. 2. Poynting Vector Components of a Quasi-Monochromatic Field
Let us consider the quasi-monochromatic wave, which also obeys the paraxial approximation. In this case, Ei t, each Cartesian component of an electric or magnetic field may be represented as [9,10] Ei t Ai t cosΦi t − 2π ν¯ t;
(4)
where Ai t and Φi t = slowly changing functions [in comparison with cos2π ν¯ t]. Correspondingly, under this assumption, the instantaneous Poynting vector may be derived similarly to one of a strongly coherent case [4] 8 c P ≈ − 4πk fEx T 2 − Ey T 1 g > < x c Py ≈ − 4πk fEy T 2 Ex T 1 g ; > : c fE2x E2y g Pz ≈ 4π
5
where (
y
Ax
T 1 Ex Φyx − Ey Φxy AAxx Ex;2π − Ayy Ey;π2 x
Ay
T 2 Ex Φxx Ey Φyy AAxx Ex;2π Ayy Ey;2π
;
6
Ei Ai t cosΦi t − 2π ν¯ t Ei;π2 Ai t sinΦi t − 2π ν¯ t
.
8 ¯ = effective local phase difference and Φ ¯iΦ ¯l = where Δ averaged by time phase functions of components. After averaging Eqs. (4)–(7), transform for the following: n o 8 ¯ x ≈ − 1 s0 s1 Φ ¯ xx s0 − s1 Φ ¯ xy − ∂s3 P > ∂y 16π ω ¯ > < n o ¯ y ≈ − 1 s0 s1 Φ ¯ yx s0 − s1 Φ ¯ yy ∂s3 ; 9 P ∂x 16π ω¯ > > : c P¯ z ≈ 8π s0 where si = Stokes parameters and ω¯ 2π ν¯ . Thus, it can be stated that the notation of averaged Poynting components of a quasi-chromatic wave is the same for a strongly monochromatic wave [12] with corresponding determinate parameters. The terms in the square brackets of the first and second equations can be called the structure or orbital transversal part of the field energy density [5]. Just these terms, in the coherent case, are responsible for the appearance of orbital momentum in the area of vortex (scalar field) or C-point, point of circular polarization, and an inhomogeneously polarized field [2,3,13,14]. The last terms in the expressions of transversal components cause spin energy currents, which define the spin angular momentum of the field. Correspondingly, if one takes into account that angular momentum density is defined as follows (see, for example, Ref. [2]): ¯jz xP¯ y − yP¯ x ;
and
We took into account the fact, that in this case, when the time of averaging is significantly more than the coherence time, the spectral components may be considered as statistically independent. As a result, it can be rather easily shown that under our assumptions [Eqs. (3) and (4)], the following expressions as the “base” of averaging procedure take place: 8 hcos2 Φi −2π ν¯ ti 12 > > > > < hsinΦ −2π ν¯ tcosΦ −2π ν¯ ti0 i i ; ¯ 1 cosΦ ¯ i −Φ ¯ l > hcosΦi −2π ν¯ tcosΦl −2π ν¯ ti 12 cos Δ > 2 > > : ¯ 1 sinΦ ¯ i −Φ ¯ l hsinΦi −2π ν¯ tcosΦl −2π ν¯ ti 12 sin Δ 2
7
Ai and Φi are interpreted according to Eq. (4), and Ali and Φli are partial derivatives of Ai t and Φi t, i; l x; y. It has been noted that only x- and y- electric components are identified in Eqs. (5)–(7). Let us carry out the averaging by time of Eqs. (5)–(7).
(10)
one can state that for the polychromatic beam (similar to the coherent case), at least, for the wave with narrow spectrum and when paraxial approximation is satisfied, the total angular momentum may be divided in the orbital and spin parts. 3. Angular Momentum of a Quasi-Chromatic Gaussian Beam
Let us repeat the statement—the obviously angular momentum of the beam or the angular momentum of the field area must be related to the coherent characteristics of the wave. For example, in the limit case 1 April 2014 / Vol. 53, No. 10 / APPLIED OPTICS
B39
of an absolutely depolarized wave, the spin momentum is zero. As it follows from expressions for Poynting vector components in Eq. (9) at least, spin energy currents, and spin angular momentum defined by them are connected by the coherence characteristics of a wave. It follows from the fact that different spectral components insert different contributions to the total angular momentum of a field. This situation necessarily occurs if the analyzed field passes through setup: polarizer, quarter-wave plate, oriented by 450 to the polarizer axis. Let us assume that the thickness of the plate is chosen in such a manner that the spectral component with frequency ν¯ becomes circularly polarized after the plate. Correspondingly, other spectral components are elliptically polarized. Now we will demonstrate the efficiency of such a procedure for the Gaussian beam. Let the frequency spectrum of this beam be distributed by normal distribution. It has been noted that the presence of “negative” frequencies in such a distribution does not lead to an essential inaccuracy because the energy rapidly decreases when the frequency changes from the central one. In this case, the Cartesian components of a wave normalized to a unit may be represented in the following form: 8 2 2 1 expf− x y2 g Ex p > 3∕2 σ 2 σ 2σ > 2π ν > o n > R∞ > ν2 > < × expf jΦ0ν − 2πνtgdν × −∞ exp − ν−¯ 2 n 2σ2 ν 2 o ; x y 1 > Ey p exp − > 2 > 2σ 2π 3∕2 σ 2 σ ν > > o n > R∞ : ν−¯ν2 × −∞ exp − 2σ 2 × expf jΦ0ν − 2πνt Δν gdν ν
11 where σ = «space» width of the beam; σ ν = «spectral» width of the beam, defining the effective frequency band Δν; Φ0ν = constant phase, associated with each spectral component; and Δν = phase shift between orthogonal components, which appears when the spectral component passes through a quarter-wave plate. This shift may be identified by this relation Δv
πν : 2 ν¯
(12)
In this case, the components of the Poynting vector may be represented in the following form: 8 1 ∂s3 ¯ > > < Px ≈ 16πω¯ ∂y 1 ∂s3 ; P¯ y ≈ − 16π ¯ ∂x ω > > : P¯ ≈ c s z
13
8π 0
due to the fact that the phase of all spectral components is practically constant. B40
APPLIED OPTICS / Vol. 53, No. 10 / 1 April 2014
In other words, the transversal energy currents and the angular moment density depend only on the space changes of the 4th Stokes parameter. Let us determine the 4th Stokes parameter in our case. As shown by Born and Wolf [9], the resulting elements of the coherence matrix are the sum of elementary elements corresponding to each frequency ν. Due to that, the 4th Stokes parameter may be presented in the form s3 jJ yx − J xy 2 2 2 1 x y2 π σν exp − : − 2 exp − 2 σ 16¯ν2 πσ
(14)
Transversal components may be written as the relations n 2 2o 2 2 8 π σν < P¯ x ≈ − 16π12 ωσ exp − x σy y exp − 16¯ 2 ¯ 4 ν2 n 2 2o : 2 2 : P¯ ≈ 1 exp − π σ ν exp − x y x y ¯ 4 16¯ν2 σ2 16π 2 ωσ
15
In other words, the components of the Poynting vector accurate within the factor
π 2 σ 2ν α exp − 16¯ν2
≤1
(16)
are very similar to those of a monochromatic Gaussian beam [3]. As a consequence, the longitudinal components of the angular momentum density and angular momentum differ on the same factor. The magnitude of this factor defined by the width of spectral range and frequency of radiation shows the decrease of the value of angular momentum in relation to the coherent case. Thus, the magnitude of the angular momentum of a quasi-monochromatic Gaussian beam is strongly connected with the coherence of a wave. 4. Discussion
Let us estimate the decreasing of value of angular momentum. As it is known [15], the indeterminacy principle for wave processes has the form Δντ0 ≈ 1;
(17)
where τ0 = coherence time. Under quasi-monochromatic approximation, this parameter is significantly less than the period of wave vibration T 0 , where λ¯ T0 ; c
(18)
where λ¯ is the central wavelength of the spectral range.
In other words, τ0 ∼ aT 0 ;
(19)
where coefficient a is large enough. ¯ ∼ 1∕a or in much It can be easily shown that Δλ∕λ the same way in terms of ν Δν ∼ 1∕a: ν¯
Fig. 2. Optical trap configuration and stages of micro-object (red blood cell) capturing.
(20)
Let us assume that a is close to 10. Δλ-spectral range of radiation is close to 60–70 ηm for red light in this case. Such a spectral range is inherent to the radiation passing though the interference filter, or it is close to the spectral range of a conventional lightemitting diode. At the same time, as it follows from Eq. (16), the exponent in this expression is very small (about 0.001). Correspondingly, the value of angular momentum is practically the same as in the coherent case. This situation is similar even for a 2. It has been noted that this value of a is on the verge of physical sense. Nevertheless, the formally calculated “decreasing” coefficient is equal to 0.975. In other words, the value of angular momentum for quasi-monochromatic Gaussian wave is practically independent on spectral range. 5. Experimental Results
The experimental confirmation was realized in the arrangement presented in Fig. 1. The Gaussian beam, produced by a He–Ne laser directed at an interferometer with a delay line, formed in one of its legs (mirrors 3 and 4). The beams converged after beam splitter 5. After that, the resulting beam was put in an intensity “regulator”, which contains two polarizers 7, 8. Such a system allows us the smooth changes of intensity of the resulting beam. Quarter-wave plate 8 formed the elliptically polarized beam with different ellipticity due to the different orientation of plate relatively the axis of polarizer 8. The polarization changed from the left circular to a right circular one. The intensity of the beam is controlled by systems 9 and 10. Further, the beam focused by micro-objective 14 into the operation plane where the sample with micro-objects was placed. We used the micro-objects as the solution of red blood cells in water. The result of object
Fig. 1. 1, He–Ne laser; 2,5,9,11, beam splitters; 3,4, mirrors; 6,7, intensity “regulators” (two polarizers); 8, quarter-wave plate; 10, reference photo detector; 12,13, system of incoherent illumination; 14, focusing micro-objective; 15, sample with micro-objects; 16, objective of microscope; 17, blue filter; 18, CCD camera.
capturing was observed by microscope 16 and charge-coupled device (CCD) camera 18. The operation plane was illuminated by incoherent systems 12 and 13. Blue filter 17 practically cut the red radiation of the optical trap completely. In Fig. 2, the intensity distribution in the trap and the stages of capturing the red blood cell are presented. At the first stage, the access of the beam to the delay line (elements 3 and 4) was closed, and “traditional” rotation [16] of the red blood cell due to the spin angular momentum was observed. After that, the access to the delay line was open, and influence of the resulting beam was observed. The length of the delay line was significantly more than the coherence length of the radiation produced by the He–Ne laser. In addition, some preparatory interferometric experiments were presented. The resulting field after the interferometer was analyzed interferometrically. Some small angle between the beams was introduced, and the intensity distribution was tested. As a result, it can be stated that delay length of the interferometer is enough for coherence destroying because the intensity distribution of the resulting field does not have any interference fringes, which may be fixed experimentally. Thus, due to that, the coherence of the resulting wave was destroyed. The intensities of beams forming the output (resulting) wave were practically the same. In addition, the intensity of the output wave decreases by systems 6 and 7 to one that is observed in the case when the delay line was undamped. The polarization of the resulting beam, which is focused by objective 14, changed from the left circular to the right circular one. The rotation of the captured object is demonstrated by Figs. 3 and 4. It must be emphasized that in both cases the optical trap influenced the same micro-object. The main result of our experimental investigation is the following: we did not see the significant
Fig. 3. Rotation of a red blood cell due to the spin angular momentum in the clockwise direction. 1 April 2014 / Vol. 53, No. 10 / APPLIED OPTICS
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Fig. 4. Rotation of a red blood cell due to the spin angular momentum in the anticlockwise direction.
changes in rotation speed of the micro-object. Correspondingly, we can state that the time coherence, at least for beams with a relatively simple structure, does not influence the value of spin angular momentum. 6. Conclusions
1. In the quasi-monochromatic case, the behavior of a transversal Poynting component may be defined similarly to the coherent case. Correspondingly, the total angular momentum of the polychromatic wave may be divided into the orbital and spin ones. This statement is true, at least, for the wave that satisfies the paraxial approximation. 2. The value of the spin angular momentum of a quasi-monochromatic wave may be connected with the coherence characteristics of the beam, and it is defined only by the space changes of the 4th Stokes parameter. 3. The value of the angular momentum of the quasi-monochromatic Gaussian beam is practically independent of spectral range and time coherence. References 1. M. J. Lang and S. M. Block, “Resource letter: LBOT-1: laserbased optical tweezers,” Am. J. Phys. 71, 201–215 (2003).
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2. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1999), Vol. XXXIX, pp. 291–372. 3. A. Bekshaev, M. Soskin, and M. Vasnetsov, Paraxial Light Beams with Angular Momentum. (Nova Science, 2008). 4. I. I. Mokhun, “Introduction to linear singular optics,” in Optical Correlation Techniques and Applications, O. V. Angelsky, ed. (SPIE, 2007), pp. 1–132. 5. A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation light beams,” J. Opt. 13, 053001 (2011). 6. J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26, 405–407 (2001). 7. O. V. Angelsky, M. P. Gorsky, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and C. Yu. Zenkova, “Investigation of optical currents in coherent and partially coherent vector fields,” Opt. Express 19, 660–672 (2011). 8. O. V. Angelsky, S. G. Hanson, C. Yu. Zenkova, M. P. Gorsky, and N. V. Gorodyns’ka, “On polarization metrology (estimation) of the degree of coherence of optical waves,” Opt. Express 17, 15623–15634 (2009). 9. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980). 10. J. Perina, Coherence of Light, 2nd ed. (D. Reidel, 1985). 11. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, (Wiley, 1981). 12. R. Khrobatin, I. Mokhun, and Ju. Viktorovskaya, “Potentiality of experimental analysis for characteristics of the Poynting vector components,” Ukr. J. Phys. Opt. 9, 182–186 (2008). 13. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics (Elsevier, 2009), Vol. 53, pp. 293–363. 14. I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A 10, 064015 (2008). 15. F. S. Crawford, Waves, Berkeley Physics Course (McGraw-Hill, 1968), Vol. 3, pp. 337–341. 16. O. V. Angelsky, A. Ya. Bekshaev, P. P. Maksimyak, A. P. Maksimyak, I. I. Mokhun, S. G. Hanson, C. Yu. Zenkova, and A. V. Tyurin, “Circular motion of particles suspended in a Gaussian beam with circular polarization validates the spin part of the internal energy flow,” Opt. Express 20, 11351–11356 (2012).
