Controllable rotating Gaussian Schell-model beams

Abstract

Optical twists are the rotation of light structures along the beam axis, which can be caused by the quadratic twist phase of a partially coherent field. Here, we introduce a new class of partially coherent beams whose spectral density and degree of coherence tend to rotate during propagation. Unlike the previously reported twisted Gaussian Schell-model beams, this family of rotating beams is constructed without the framework of rotationally invariant cross-spectral density functions. Thus, these beams have different underlying physics and exhibit distinctive twist effects. It is shown that such beams can undergo a twist of more than 90 deg, providing larger degrees of freedom for flexibly tailoring the beam twist. Our results may pave the way toward synthesizing rotating beams for applications in optics and, in particular, inspire further studies in the field of twist phase proposed 25 years ago.

Full text

Controllable rotating Gaussian Schell-model beams
LIPENG WAN AND DAOMU ZHAO*
Department of Physics, Zhejiang University, Hangzhou 310027, China
*Corresponding author: zhaodaomu@yahoo.com
Received 14 November 2018; revised 17 December 2018; accepted 21 December 2018; posted 3 January 2019 (Doc. ID 351892);
published 4 February 2019
Optical twists are the rotation of light structures along the
beam axis, which can be caused by the quadratic twist phase
of a partially coherent field. Here, we introduce a new class
of partially coherent beams whose spectral density and de-
gree of coherence tend to rotate during propagation. Unlike
the previously reported twisted Gaussian Schell-model
beams, this family of rotating beams is constructed without
the framework of rotationally invariant cross-spectral den-
sity functions. Thus, these beams have different underlying
physics and exhibit distinctive twist effects. It is shown that
such beams can undergo a twist of more than 90 deg, pro-
viding larger degrees of freedom for flexibly tailoring the
beam twist. Our results may pave the way toward synthesiz-
ing rotating beams for applications in optics and, in par-
ticular, inspire further studies in the field of twist phase
proposed 25 years ago. © 2019 Optical Society of America
https://doi.org/10.1364/OL.44.000735
In 1993, Simon and Mukunda introduced a position-dependent
quadratic phase by studying the most general cross-spectral den-
sity (CSD) function that is rotationally invariant about the
propagation axis [1]. Perhaps one of the most remarkable fea-
tures of this phase is its ability to twist the beam; thus, they called
this phase the twist phase. Impressing such a phase term on
Gaussian Schell-model (GSM) sources leads to the first type of
twisted sources termed twisted GSMs (TGSMs).
Recent years have witnessed significant conceptual progress
in twisted partially coherent beams. For instance, Gori and
Santarsiero demonstrated a modeling procedure to devise bona
fide twisted sources [2]. Borghi et al. derived the necessary and
sufficient condition of the twisted CSDs [3,4]. Mei and
Korotkova created the genuine twisted sources through a super-
position integral [5]. Nontrivial twisted beams such as twisted
flat-top beams [5] and twisted GSM array beams [6,7] were also
proposed.
On the other hand, research on the phase of the source
degree of coherence (DOC) differing from the twist type
has increasingly attracted attention and advanced in recent
years. These studies include partially coherent vortex beams
[8,9], self-steering beams [10], optical coherence lattices
[11–13], azimuthal beams [14], and crescent beams [15].
Further, the construction of partially coherent beams whose
complex DOC possesses a nontrivial phase structure was also
reported in a recent study [16]. It is expected that the studies
involving twist phase and other phases will advance the field of
partially coherent beams considerably.
The aim of this Letter is to introduce a class of partially
coherent beams whose spectral density and degree of coherence
rotate during propagation. The CSD function of this family of
beams differs from that of the previously reported TGSMs
in the quadratic phase structure, which leads to striking
differences in terms of physical properties of the beam.
We begin our analysis with a statistically stationary scalar
source located in the plane z0. The statistical properties
of the source, up to second order, can be described by the
CSD function W0r1,r2. It was demonstrated by Gori and
Santarsiero that, in order for the CSD function to be physically
genuine, the following integral representation must hold for an
arbitrary kernel function H0and a nonnegative weight func-
tion p[17]:
W0r1,r2ZpvH
0r1,vH0r2,vd2v:(1)
To introduce the sources generating partially coherent beams
endowed with the twist property, one may employ the
following form of H0:
H0r,vτrexp−2πiv·r
×expiuxcos θ−ysin θxsin θycos θ,
(2)
where τrdepicts the source intensity profile, the real constant
ucharacterizes the beam twist, and an arbitrary angle θchar-
acterizes the rotation of phase structure around the beam axis
that is performed by the rotation matrix Rz:
Rzθcos θ−sin θ
sin θcos θ:(3)
Without loss of generality, the source intensity takes the form of
an anisotropic Gaussian profile:
τrexp−
x2
4σ2
xexp−
y2
4σ2
y,(4)
where σxand σyare the spectral density widths along xand y
directions in the source plane. On substituting Eq. (4) first into
Eq. (2) and then into Eq. (1), we obtain for the CSD function
the formula
Letter Vol. 44, No. 4 / 15 February 2019 / Optics Letters 735
0146-9592/19/040735-04 Journal © 2019 Optical Society of America

W0r1,r2exp−
x2
1x2
2
4σ2
xexp−
y2
1y2
2
4σ2
yFr1−r2
×expf−iux1cos θ−y1sin θx1sin θy1cos θ
−x2cos θ−y2sin θx2sin θy2cos θg,(5)
where Fr1−r2corresponds to the Fourier transform of the
weight function pv:
Fr1−r2Zpvexp2πiv·r1−r2d2v:(6)
Let us consider an anisotropic Gaussian correlated field,
pv2πδxδyexp−2π2δ2
xv2
xexp−2π2δ2
yv2
y,(7)
with δxand δybeing positive real constants whose values are
related to the spatial coherence widths along xand ydirections.
On substituting Eq. (7) first into Eq. (6) and then into
Eq. (5), we obtain the CSD function of the form
W0r1,r2exp−
x2
1x2
2
4σ2
xexp−
y2
1y2
2
4σ2
yμ0r1,r2,(8)
where a complex function μ0gives the DOC of the field, viz.,
μ0r1,r2
exp−x1−x22
2δ2
xexp−y1−y22
2δ2
y
×expf−iux1cos θ−y1sin θx1sin θy1cos θ
−x2cos θ−y2sin θx2sin θy2cos θg:(9)
The CSD function of this family of sources differs from that of
the previously reported TGSMs in the quadratic phase struc-
ture, which does not seem to have been encountered in pre-
vious studies. However, as we shall see, beams radiated by
such sources exhibit an optical twist as well. In order to avoid
confusion with the TGSMs henceforth, we use the term
“rotating anisotropic Gaussian Schell-model (RAGSM)”for
the source described by Eqs. (8) and (9).
From Eqs. (8) and (9), we can unravel the underlying phys-
ics of the RAGSM beam. The simple fact is that this family of
model beams has two interesting special cases: the states of θ
and θ0θNπ(N0,1,2…) describe the same
beams. It is therefore convenient to introduce the main value
of θin the domain 0≤θ≤π. Two states θand θ0θπ∕2
differ only by the sign of u. This means that for any beam car-
rying a positive twist factor uwithin the domain 0≤θ≤π∕2,
one can find a corresponding beam carrying a negative twist
factor within π∕2≤θ≤π. Importantly, the RAGSM beam
is essentially different from the TGSM one. The CSD function
of the RAGSM beam is rotationally variant about the beam
axis, and the twist strength of the beam is, in principle,
unbounded by the transverse coherence width in the source
field. It is thus clear that although both RAGSMs and
TGSMs can exhibit beam twists, their twist effects may be
quite different from each other. To obtain some insight into
the twisted beams, we should investigate the OAM properties
of the beams. In the paraxial approximation, the OAM density
along the optical axis may be shown to be [18,19]
Morbitr−
ε0
kImfy1∂x2Wr1,r2−x1∂y2Wr1,r2gr1r2,
(10)
where ∂jrepresents the partial derivation with respect to j,ε0
is the dielectric constant in vacuum, and Im denotes the
imaginary part.
Substituting Eqs. (8) and (9) into Eq. (10) yields
Morbitr−uε0
kexp−
x2
2σ2
xexp−
y2
2σ2
y
×xsin θycos θ2−xcos θ−ysin θ2:
(11)
A more intuitive result, i.e., the OAM density per photon
morbit, can be readily found by normalizing Morbit:
morbitruxcos θ−ysin θ2−xsin θycos θ2ℏ:
(12)
Equation (12) suggests that the OAM density of a RAGSM
beam takes both positive and negative values across the source.
This is quite different from TGSMs (with twist strength n) and
partially coherent vortex: the former has an OAM density of
nr2ℏper photon varying quadratically as a function of r,
and the latter behaves like a Rankin vortex [20].
The total average OAM Lorbit per photon, found by integrat-
ing Eq. (11)withrespecttoxand yand a normalization, is
Lorbit uℏσ2
x−σ2
ycos 2θ:(13)
From Eq. (13), the total OAM carried by the beam is seen to be
independent of source coherence properties. It is important to
appreciate that even when u≠0, the beam can carry zero
OAM as long as θ45,135°.
Let us now consider the field in the half-space z>0into
which the RAGSM source radiates. To do that, we invoke the
Fresnel diffraction formula [20]:
Wρ1,ρ2,zZZ Wr1,r2G
zρ1,r1,zGzρ2,r2,zd2r1d2r2,
(14)
where ρ≡x,yis an arbitrary position vector in the plane z,
and Gzis a free-space propagator that takes the following form
in the paraxial approximation:
Gzρ,r,z−ik∕2πzexpikzexpikρ−r2∕2z,(15)
where k2π∕λdenotes the wavevector with λbeing the
optical wavelength fixed at 632.8 nm. Substituting Eqs. (8),
(9) and (15) into Eq. (14), we obtain
Wρ1,ρ2,z kπσx2
ffiffiffiffiffiffiffiffiffiffi
εΔT
pz2exp−
ik
2zρ2
1−ρ2
2
×exp−
k2σ2
x
2z2x0
1−x0
22
×exp1
4ΔiuGx
2Tγik
zy0
1−y0
22
×exp1
4εik
2zy0
1y0
2−MΓ
4Δ2,
(16a)
where
736 Vol. 44, No. 4 / 15 February 2019 / Optics Letters Letter

1
αj1
8σ2
j1
2δ2
j
;βk
2zusin θcos θ;
γcos2θ−sin2θ;ε1
αy
−
u2σ4
x
Tβ2γ2−
u2σ2
x
2γ21
4ΔP2
r;
Γ−2Prk
zy0
1−y0
2−
uGx
2Tγ;
Gxik
2zx0
1x0
22kσ2
x
zβx0
1−x0
2;
T2σ2
xβ21
αx
;MuGxσ2
x
Tβγ −
ukσ2
x
zγx0
1−x0
2;
Δ1
2σ2
yu2
4Tγ2;P
ru2σ2
x
Tβγ2−
k
z2usin θcos θ:
(16b)
From Eqs. (16a) and (16b), one can readily deduce expressions
for the spectral density Sρ,zand DOC μρ1,ρ2,zat any
transverse plane z. In what follows, DOC is evaluated at
two points located symmetrically about the optical axis.
To quantitatively analyze twist effects, we exploit the coor-
dinate transformation to determine angles φsand φc, where φs
denotes the angle formed by the major axis of a spectral density
ellipse with the positive x0-axis, and φcdenotes the angle
formed by the major axis of a coherence ellipse with the positive
x0-axis, viz.,
tan2φs 4kΔT2Φz
4ΔT2Φ2z2−k2u2εγ24ΔTT−ε;
tan2φc 4kTTzΔΦ −2ξPr4ukσ2
xεβγ
4ΔT2Φ24ξ2z2−k2nu2εγ2Ψ4ΔThT1P2
r
Δ2−εΨ−8Tσ2
x4T
Δio,(17a)
where
Ψ16β2σ4
x−1; ξukσ2
x
zγ2σ2
xΦβ;
Φuk
4ΔTzPrγ−
ukσ2
x
Tz βγ:(17b)
We see from Eqs. (16b), (17a), and (17b) that tan2φs→0
and tan2φc→0as u→0or as γ→0(γ→0means
θ→45, 135°). This is an important result, since it suggests
that the rotation dynamics would disappear when such aniso-
tropic beams carry zero OAM. In what follows, we will be pri-
marily interested in quantities ϕsand ϕc,whereϕsdenotes the
rotation angle of the major axis of the intensity ellipse, and ϕc
denotes the rotation angle of the major axis of the coherence
ellipse. These quantities are readily given from Eqs. (17a)and
(17b). Accordingly, the sense of rotation is denoted by the sign
of rate of rotation (angular velocities ωdϕ∕dz) and is clock-
wise for positive values and counterclockwise for negative values.
The propagation characteristics of a RAGSM beam
with σxδx1mm,σyδy0.3mm is depicted in
Fig. 1(a)–1(d). As clearly seen, the spectral density and DOC
rotate all the way around the beam axis upon propagation,
completing a twist of 90 deg in a synchronous manner. Note
that while both RAGSM and TGSM beams exhibit optical
twists, they have different phase structures that lead to a certain
difference in the rotation dynamics. Specifically, the spectral den-
sity and DOC of RAGSM beams rotate in the same direction,
and those of TGSM beams rotate in opposite directions. This
character can serve as an effective way to identify whether a
twisted partially coherent beam is TGSM or RAGSM.
In order to highlight the impressive angle of beam twist, we
note that one can set the beam into a strong twist or a weak
twist by adjusting the parameter θ. Figure 2depicts the propa-
gation of a strongly twisted RAGSM beam with θ120 deg
and σxδx1mm,σyδy0.3mm. Compared to
TGSMs whose beam twists are fundamentally limited to
90 deg [1,6], the spectral density and DOC of such a beam
undergo a clockwise rotation of up to 150 deg during propa-
gation, resulting in a larger freedom for beam twist engineering.
However, for a smaller twist factor, the beam can undergo a
twist of only 90 deg, as seen in Figs. 2(c) and 2(d). This is
expected, since, for an irrotational GSM beam with isotropic
global coherence, both intensity and coherence ellipses stretch
along the minor axis [21].
Since the radiant intensity strongly depends on the spatial-
coherence properties of the source, one can predict that the
twist effect would be greatly different when δx∕σx≠δy∕σy.
An example for σxδy1mm,σy5δx0.5mm,and
θ113 deg is given in Fig. 3. As clearly seen in Fig. 3(a),
the average intensity undergoes a clockwise rotation within
the critical range zs0ϕ0
szs00and a counterclockwise rota-
tion beyond this range. Moreover, the comparison between
Figs. 3(a) and 3(b) shows that the DOC rotates asynchronously
with respect to spectral density.
Having shown the intriguing property of beams, it is in-
structive to study the dependency of the twist effects on θ.
Figure 4(a) depicts the maximal rotating angle ϕM(the angle
of twist when z →∞) of the spectral density as a function of θ,
while for a larger value of u, the maximal rotating angle depends
linearly on θ; as a consequence, the beam can undergo a twist of
angle ranging from −180 deg to 180 deg. For a smaller u, such
beams can undergo a twist of only about 90 deg, regardless
of the value of θ, in agreement with our analysis of the twist
effect shown in Fig. 2(c). Furthermore, similar considerations
apply for the value of source intensity widths, as shown in
Fig. 4(b). Interestingly, it is clear, from the total OAM given
by Eq. (13) as well as results presented in Figs. 4(a) and 4(b),
that the transverse spectral density rotates counter-clockwise
(left-handed beams) for positive OAM values and clockwise
(right-handed beams) for negative OAM values, and the rotat-
ing effect disappears when such anisotropic beams carry zero
OAM. This behavior is analogous to that of the phase front
Letter Vol. 44, No. 4 / 15 February 2019 / Optics Letters 737

of Laguerre–Gaussian modes, where the sense of rotation
coincides with the sign of OAM.
We summarize this Letter by saying that a new class of par-
tially coherent beams whose spectral density and DOC rotate
during propagation is introduced. This family of beams is es-
sentially different from TGSMs, since the rotation dynamics is
induced by the phase of DOC differing from the twist type. It is
shown that, for such beams, the control of source parameters
allows beam twist to be tailored arbitrarily, providing larger de-
grees of freedom for beam twist engineering.
It is believed that the RAGSM beam may be particularly
useful in many and diverse applications, such as in optical trap-
ping and conveying, where the beam twist plays a significant
role [22,23]. More importantly, we hope the results presented
here can inspire further studies in the field of twist phase.
Funding. National Natural Science Foundation of China
(NSFC) (11874321, 11474253); Fundamental Research
Funds for the Central Universities (2018FZA3005).
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Fig. 4. (a) Maximal rotating angle of the spectral density versus θ
with σxδx1mmand σyδy0.3mmfor u1.6, 3.8, and
9mm
−2. (b) Maximal rotating angle of the spectral density versus θ
with u3.8mm
−2,δx1mm, and δy0.3mmfor σx0.5,
σx1, and σx1.8mm (δx∕σxδy∕σy). The hollow circles
denote points of discontinuity where rotation dynamics disappear.
Fig. 2. Propagation dynamics of a strongly twisted RAGSM beam
(Visualization 2) with θ120° and u15 mm−2. (a) Spectral
density and (b) DOC. Rotation angles versus zwith u0.3,5,
and 15 mm−2for (c) spectral density and (d) DOC.
Fig. 3. Rotation angles of (a) spectral density and (b) DOC as the
field propagates with u3, 6, and 9mm
−2(Visualization 3).
Fig. 1. Propagation dynamics of a RAGSM beam (Visualization 1)
with θ0and u3mm
−2. (a) Spectral density and (b) DOC.
Rotation angles versus propagation distance zwith u0.03, 0.3,
and 3mm
−2for (c) spectral density and (d) DOC.
738 Vol. 44, No. 4 / 15 February 2019 / Optics Letters Letter