A new nonspreading beam is proposed for the case in which diffraction occurs only in one transverse coordinate. The beam has the shape of a pulse in one dimension and is constant in the other (slitlike shape). The intensity of the pulse’s peak remains almost constant along a finite interval on the propagation axis. The proposed beam is analyzed and demonstrated experimentally. The analogy between this beam and the temporal pulse in a dispersive medium is discussed

Liu, Hua-Kuang

Rosen, Joseph

Salik, Boaz

Yariv, Amnon

English

Caltech Authors - Main

March 1, 1995 / Vol. 20, No. 5 / OPTICS LETTERS 423Pseudonondiffracting slitlike beam and its analogy
to the pseudonondispersing pulse
Joseph Rosen, Boaz Salik, and Amnon Yariv
California Institute of Technology, MyS 128-95, Pasadena, California 91125
Hua-Kuang Liu
Jet Propulsion Laboratory, MyS 303-310, California Institute of Technology, Pasadena, California 91109
Received September 27, 1994
A new nonspreading beam is proposed for the case in which diffraction occurs only in one transverse coordinate.
The beam has the shape of a pulse in one dimension and is constant in the other (slitlike shape). The intensity
of the pulse’s peak remains almost constant along a finite interval on the propagation axis. The proposed beam
is analyzed and demonstrated experimentally. The analogy between this beam and the temporal pulse in a
dispersive medium is discussed.The analogy between temporal pulse propagation in
a dispersive medium and spatial Fresnel diffraction
has recently yielded new methods to process short
pulses.1–4 One diffraction phenomenon that has not
been imitated in the time domain is the so-called
nondiffracting beam.5 The original nondiffracting
beam, also called the Bessel beam, is a solution, of
the form Esr, zd ­ expsjbzdJ0sard, to the free-space
scalar wave equation in which a2 1 b2 ­ k2, k is the
wave number, J0 is the zero-order Bessel function,
and (r, u, zd are the cylindrical coordinates. This so-
lution is attractive because the intensity distribution
in the entire space is independent of the z coordi-
nate and the transverse intensity profile has the
shape of a pulse (beamlike shape). This solution,
however, exists in three-dimensional space, whereas
the temporal pulse lives in the two-dimensional (2-
D) time propagation axis domain (t, zd. Therefore,
the analogy of the temporal dispersive domain holds
completely only for the 2-D spatial diffractive do-
main (x, z). As we see, reducing the dimensions
of the Bessel beam strictly from (r, u, z) to (x, z)
does not lead to satisfactory results. The aim of
this Letter is to point out a new kind of pseudonon-
diffracting (PND) beam, characterized by a constant
intensity along the propagation axis z and a beam-
like shape in one of its transverse dimensions, say,
x. Based on the space–time analogy, and having a
PND slitlike beam, we can announce the discovery of
a novel pseudonondispersing temporal pulse, whose
peak value remains constant along some predeter-
mined propagation distance. The following treat-
ment refers to the spatial case, and the temporal
pulse will be considered after that.
Using Fresnel’s approximation, we calculate the
2-D complex amplitude distribution around focal
plane Pf (Fig. 1) to be6
usx, zd ­
expfjksz 1 2f dgp
jlf
3
Z `
2`
gsxidexp
∑
2j
k
f
µ
zxi2
2f
1 xxi
∂∏
dxi ,
(1)0146-9592/95/050423-03$6.00/0where k ­ 2pyl, l is the wavelength, f is the focal
length of the cylindrical lens, xi is the transverse
coordinate of plane Pi, and (x, z) are the coordinates
behind the lens, with the focus as their origin. The
longitudinal profile is obtained by substituting x ­ 0
into Eq. (1):
us0, zd ­
expfjksz 1 2fdgp
jlf
3
Z `
2`
gsxidexp
√
2j
kzxi2
2f2
!
dxi. (2)
We look for a field distribution usx, zd with the
shape of a pulse, i.e., jus0, zdj2 .. jusx, zdj2, for all
x . Wp, where Wp is the width of the pulse and the
axial distribution of the intensity is constant along
a finite interval Dz, i.e., jusx ­ 0, zdj2 ­ const. for
z [ Dz. A function with useful properties for our
purposes is of the form
gsxid ­ exp
(
2j2p
"É
xi
b
Ép
2
√
xi
a
! 2#)
, p . 2 ,
(3)
where a, b, and p are real positive parameters of the
beam. When Eq. (3) is substituted into Eq. (2), the
method of stationary phase approximation7 yields
Fig. 1. Schematic system used to obtain the PND beam. 1995 Optical Society of America
424 OPTICS LETTERS / Vol. 20, No. 5 / March 1, 1995us0, zd ø
8>>>>>>>><>>>>>>>>:
B expsjkzdp
2sz 2 1ya2d
expsjkzd
∑
C 1 D
µ
z 2
1
a2
∂∏
B expsjkzdp
2sz 2 1ya2d
√
1
p
2
exp
Ω
2j2p
∑µ
z 2
1
a2
∂ µ
2b2
p
∂∏p/sp22d
s1 1 pd
æ
1 1
!
z , 2aL 1
1
a2
2 aL 1
1
a2
, z , L 1
1
a2
,
L 1
1
a2
, z
°
4
¢
where z ­ zy2lf2; L ­ 0.5b22pf4s1 1 pdgs22pd/p;
B, C, D are complex constants whose precise value
is not important here; and 0 , a , 1. Note that
the functional behavior of us0, zd is different in each
of the three regions. At the central region us0, zd
is approximately linear. The length of this region
is Dz ­ 2s1 1 adLlf 2, and its center is located at
lf 2fLs1 2 ad 2 2a22g from the focal point. In the
same manner it can be shown that substituting the
conjugate function of Eq. (3) into Eq. (2) yields the op-
posite profile, u*s0, 2zd, of that given in relation (4).
Therefore, by taking a mask of the combination
gsxid 1 g*sxid, where a ­
p
2fs1 2 adLg21/2, we can
combine the two linear curves with opposite slope
signs to create a constant distribution. The length
of the constant-intensity region is equal to that of
the linear region.
The transverse shape of the beam near the focus
is our next consideration. For mathematical conve-
nience we investigate the beam at the point z0 ­ 0.
Substituting z0 and gsxid 1 g*sxid [from Eq. (3)] into
Eq. (1) yields
usx, z0d ­
2 expfjksz0 1 2fdgp
jlf
3
Z `
2`
cos
(
2p
"√
xi
b
! p
2
√
xi
a
! 2#)
3 exp
√
2j
k
f
xxi
!
dxi . (5)
Assuming an infinite aperture, this integral can be
approximated7 at the region far from the origin
(x .. 0) as
usx, z0d ~ x22p/2sp21d cos
"
2ps1 2 pd
√
bx
plf
! p/sp21d#
.
(6)
For an infinite aperture the envelope of the am-
plitude of the pulse decreases as xs22pd/2sp21d. A
cumbersome analysis shows that this asymptotic
behavior (for x .. 0) stays valid for any z [ Dz,
although the field distribution changes substantially.
When the finite aperture D is taken into account, one
can estimate the overall width of the beam at z0 by
calculating the maximum spatial frequency of gsxid at
the point xi ­ Dy2, and that is W0 ­ lfpDp21y2p21bp.
On the other hand, the width of the pulse (the sharp
central lobe of the beam) is approximated under the
assumption that its shape around the origin is de-
termined mainly by the Fourier transform of the
central lobe of gsxid. Therefore, the width of thepulse Wp (at the focus) is Wp ­ 2lfyH , where H is
the width of the central lobe of gsxid obtained as the
solution of the equation sHy2bdp 2 sHy2ad2 ­ 1y4.
Note that only the overall width of the beam W0 is
related to the aperture width D. The shape of the
pulse itself (Wp, Dz) is independent of the aperture
width if D is above some value that empirically is
found to be twice the width of the first cycle of gsxid.
A complete simulation of the PND beam was con-
ducted for p ­ 4, b ­ 128 pixels, a ­ 250 pixels, and
s2lf 2d ­ 1283 pixel3. The behavior of the wave along
the propagation axis can be studied from the solid
curve of Fig. 2. For comparison, the longitudinal
intensity of an ordinary focused beam resulting from
the central lobe of gsxid only [i.e., rect(xiyH dgsxidg
is shown by the dashed curve. The latter beam is
chosen for comparison with the PND beam because
the two have similar transverse shapes around the
focal point. The insets of Fig. 2 show the transverse
profiles of the PND beam (solid curves) and the ordi-
narily focused beam (dashed curves). Although their
width Wp is equal, the PND beam keeps its beam-
like shape for a longer distance than the ordinary
focused beam.
Next, we demonstrate the PND beam experimen-
tally. Looking at Fig. 1, we note that if gsxid is il-
luminated by a plane wave, the field distribution at
plane PL just behind the lens is
tsxLd ­ fg*Ls f dgLs2f d ,
Ls f d ­ exp
µ
j
kx2
f
∂
. (7)
Therefore the combination of the mask gsxid plus the
Fig. 2. Computer-simulated axial intensity distribution
of the PND beam (solid curve) and the ordinary focused
beam (dashed curve). The insets show the transverse
cross section of the PND beam (solid curves) and the
ordinary focused beam (dashed curves) at some points
along the axis.
March 1, 1995 / Vol. 20, No. 5 / OPTICS LETTERS 4250
z
0
z
(a) (b)
(c)
(d)
x
y 1
x
-21 21 [cm]
24 [cm]
x1 x1
-24
y 1
Fig. 3. Holograms that produce (a) the PND beam and
(b) the ordinary focused beam. The 2-D intensity distri-
bution of (c) the PND and (d) the ordinary focused beam.
lens can be replaced by a single mask with the dis-
tribution tsxLd. For an amplitude-only transparency
the mask that we use is given by
ssxLd ­ 1 1
1
2
tsxLd 1
1
2
t*sxLd . (8)
Assuming that tsxLd is normalized, ssxLd is a real
positive distribution that is easily implemented. We
see from Eq. (8), however, that there are two use-
less terms, t*sxLd and a constant term. Fortunately,
t*sxLd is responsible for the appearance of a virtual
image on plane Pi, so it contributes only a weak scat-
tering light around plane Pf. A similar but more uni-
form contribution is due to the constant term.
The hologram used in the experiment is shown
in Fig. 3(a). We obtained the distribution of the
hologram by substituting gsxid ­ cosh2pfsxiybd4 2
sxiyad2gj sxi , Dy2d into Eqs. (7) and (8), where l ­
0.63 mm, b ­ 5 mm, a ­ 9.77 mm, D ­ 16 mm, and
f ­ 1.2 m. The behavior of the PND beam passing
the hologram around the distance 1.2 m is demon-
strated in Fig. 3(c). As was done in the simulation,
the performance of the PND beam is compared with
the ordinary focused beam shown in Fig. 3(d). We
created this beam by illuminating the mask shown
in Fig. 3(b) by a plane wave. We obtained the mask
of Fig. 3(b) by the same projection process as for the
mask of Fig. 3(a), but this time the initial distribution
was the central lobe only of gsxid.
The analogy between temporal pulse propagation
in a dispersive medium and Fresnel diffraction is
our next consideration. Based on Ref. 3, the pulse
is represented as a sum of plane waves, each propa-
gating with a different phase delay bsvd, and bsvd
is approximated as a polonium series, bsvd ­ b0 1
b1v 1 b2v
2y2, where v is the angular frequency.
With these assumptions the field of the pulse is
ust, zd ­ expsjb0zd
Z `
2`
U svdexp
µ
2j
b2z
2
v2
∂
3 expfjvst 2 b1zdg dv. (9)The pulse distribution is represented as a Fourier
transform of its temporal spectrum (at z ­ 0) mul-
tiplied by a quadratic phase factor. According to the
convolution theorem, it can be represented by a con-
volution integral,
ust-, zd ­
expsjb0zdp
2jpb2z
Z `
2`
ust0dexp
∑
j
st- 2 t0d2
2b2z
∏
dt0, (10)
where t- ­ t 2 b1z, i.e., the time variable minus the
delay that the pulse takes to reach to z. This ex-
pression has the same form of the Fresnel integral
describing the propagation of an electromagnetic field
in free space, where t- is equivalent to x. Based on
this analogy, there must be an equivalent expression
to Eq. (1) in the (t, z) domain. To obtain it, assume
that the initial pulse distribution exists at a distance
f from a temporal lens of the form exps2jt2y2b2fd.
After the analyses of Ref. 6, the pulse distribution af-
ter it propagates the distance f and is multiplied by
the temporal lens (at time t ­ b1f d is
ust-, zd ­
expfjb0sz 1 2fdgp
2jpb2f
3
Z `
2`
ust0d exp
"
2
j
b2f
√
t0z
2f
2 t-t0
!#
dt0,
(11)
where the temporal lens is at z ­ 2f . Equation (11)
has indeed the same form as Eq. (1). Therefore, in
general we can reshape a profile of the pulse along
the propagation axis, inside a dispersive medium, in
the same manner as has been done to the beam in
the spatial domain.6,8,9 Specifically, we can create
a pseudonondispersive temporal pulse by using the
theory of the pseudonondiffracting beam in the 2-D
space analyzed at the beginning of this Letter. Anal-
ogously to the spatial pulse, we expect the temporal
pulse to remain constant for a distance Dz ­ 4pLs1 1
adb2f 2 if a pulse of the form cosh2pfstybdp 2 styad2gj is
processed by the temporal analogy of Eqs. (7) and (8).
This research was sponsored in part by the NASA
Office of Advanced Concepts and Technology.
References
1. E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454
(1969).
2. B. H. Kolner and M. Nazarathy, Opt. Lett. 14, 630
(1989).
3. A. W. Lohmann and D. Mendlovic, Appl. Opt. 31, 6212
(1992).
4. M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M.
Bloom, Appl. Phys. Lett. 64, 270 (1994).
5. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev.
Lett. 58, 1499 (1987).
6. J. Rosen and A. Yariv, Opt. Lett. 19, 843 (1994).
7. A. Papoulis, Systems and Transforms with Applica-
tions in Optics, 1st ed. (McGraw-Hill, New York, 1968),
Chap. 7, p. 222.
8. J. Rosen, Opt. Lett. 19, 369 (1994).
9. R. Piestun and J. Shamir, Opt. Lett. 19, 771 (1994).


Pseudonondiffracting slitlike beam and its analogy to the pseudonondispersing pulse

March 1, 1995 / Vol. 20, No. 5 / OPTICS LETTERS 423Pseudonondiffracting slitlike beam and its analogy
to the pseudonondispersing pulse
Joseph Rosen, Boaz Salik, and Amnon Yariv
California Institute of Technology, MyS 128-95, Pasadena, California 91125
Hua-Kuang Liu
Jet Propulsion Laboratory, MyS 303-310, California Institute of Technology, Pasadena, California 91109
Received September 27, 1994
A new nonspreading beam is proposed for the case in which diffraction occurs only in one transverse coordinate.
The beam has the shape of a pulse in one dimension and is constant in the other (slitlike shape). The intensity
of the pulse’s peak remains almost constant along a finite interval on the propagation axis. The proposed beam
is analyzed and demonstrated experimentally. The analogy between this beam and the temporal pulse in a
dispersive medium is discussed.The analogy between temporal pulse propagation in
a dispersive medium and spatial Fresnel diffraction
has recently yielded new methods to process short
pulses.1–4 One diffraction phenomenon that has not
been imitated in the time domain is the so-called
nondiffracting beam.5 The original nondiffracting
beam, also called the Bessel beam, is a solution, of
the form Esr, zd ­ expsjbzdJ0sard, to the free-space
scalar wave equation in which a2 1 b2 ­ k2, k is the
wave number, J0 is the zero-order Bessel function,
and (r, u, zd are the cylindrical coordinates. This so-
lution is attractive because the intensity distribution
in the entire space is independent of the z coordi-
nate and the transverse intensity profile has the
shape of a pulse (beamlike shape). This solution,
however, exists in three-dimensional space, whereas
the temporal pulse lives in the two-dimensional (2-
D) time propagation axis domain (t, zd. Therefore,
the analogy of the temporal dispersive domain holds
completely only for the 2-D spatial diffractive do-
main (x, z). As we see, reducing the dimensions
of the Bessel beam strictly from (r, u, z) to (x, z)
does not lead to satisfactory results. The aim of
this Letter is to point out a new kind of pseudonon-
diffracting (PND) beam, characterized by a constant
intensity along the propagation axis z and a beam-
like shape in one of its transverse dimensions, say,
x. Based on the space–time analogy, and having a
PND slitlike beam, we can announce the discovery of
a novel pseudonondispersing temporal pulse, whose
peak value remains constant along some predeter-
mined propagation distance. The following treat-
ment refers to the spatial case, and the temporal
pulse will be considered after that.
Using Fresnel’s approximation, we calculate the
2-D complex amplitude distribution around focal
plane Pf (Fig. 1) to be6
usx, zd ­
expfjksz 1 2f dgp
jlf
3
Z ‘
2‘
gsxidexp
•
2j
k
f
µ
zxi2
2f
1 xxi
¶‚
dxi ,
(1)0146-9592/95/050423-03$6.00/0where k ­ 2pyl, l is the wavelength, f is the focal
length of the cylindrical lens, xi is the transverse
coordinate of plane Pi, and (x, z) are the coordinates
behind the lens, with the focus as their origin. The
longitudinal profile is obtained by substituting x ­ 0
into Eq. (1):
us0, zd ­
expfjksz 1 2fdgp
jlf
3
Z ‘
2‘
gsxidexp
ˆ
2j
kzxi2
2f2
!
dxi. (2)
We look for a field distribution usx, zd with the
shape of a pulse, i.e., jus0, zdj2 .. jusx, zdj2, for all
x . Wp, where Wp is the width of the pulse and the
axial distribution of the intensity is constant along
a finite interval Dz, i.e., jusx ­ 0, zdj2 ­ const. for
z [ Dz. A function with useful properties for our
purposes is of the form
gsxid ­ exp
(
2j2p
"
xi
b
p
2
ˆ
xi
a
! 2#)
, p . 2 ,
(3)
where a, b, and p are real positive parameters of the
beam. When Eq. (3) is substituted into Eq. (2), the
method of stationary phase approximation7 yields
Fig. 1. Schematic system used to obtain the PND beam. 1995 Optical Society of America
424 OPTICS LETTERS / Vol. 20, No. 5 / March 1, 1995us0, zd ø
8>>>>>>>><>>>>>>>>:
B expsjkzdp
2sz 2 1ya2d
expsjkzd
•
C 1 D
µ
z 2
1
a2
¶‚
B expsjkzdp
2sz 2 1ya2d
ˆ
1p
2
exp
‰
2j2p
•µ
z 2
1
a2
¶ µ
2b2
p
¶‚p/sp22d
s1 1 pd
¾
1 1
!
z , 2aL 1
1
a2
2 aL 1
1
a2
, z , L 1
1
a2
,
L 1
1
a2
, z
¡
4
¢where z ­ zy2lf2; L ­ 0.5b22pf4s1 1 pdgs22pd/p;
B, C, D are complex constants whose precise value
is not important here; and 0 , a , 1. Note that
the functional behavior of us0, zd is different in each
of the three regions. At the central region us0, zd
is approximately linear. The length of this region
is Dz ­ 2s1 1 adLlf 2, and its center is located at
lf 2fLs1 2 ad 2 2a22g from the focal point. In the
same manner it can be shown that substituting the
conjugate function of Eq. (3) into Eq. (2) yields the op-
posite profile, u*s0, 2zd, of that given in relation (4).
Therefore, by taking a mask of the combination
gsxid 1 g*sxid, where a ­
p
2fs1 2 adLg21/2, we can
combine the two linear curves with opposite slope
signs to create a constant distribution. The length
of the constant-intensity region is equal to that of
the linear region.
The transverse shape of the beam near the focus
is our next consideration. For mathematical conve-
nience we investigate the beam at the point z0 ­ 0.
Substituting z0 and gsxid 1 g*sxid [from Eq. (3)] into
Eq. (1) yields
usx, z0d ­
2 expfjksz0 1 2fdgp
jlf
3
Z ‘
2‘
cos
(
2p
"ˆ
xi
b
! p
2
ˆ
xi
a
! 2#)
3 exp
ˆ
2j
k
f
xxi
!
dxi . (5)
Assuming an infinite aperture, this integral can be
approximated7 at the region far from the origin
(x .. 0) as
usx, z0d ~ x22p/2sp21d cos
"
2ps1 2 pd
ˆ
bx
plf
! p/sp21d#
.
(6)
For an infinite aperture the envelope of the am-
plitude of the pulse decreases as xs22pd/2sp21d. A
cumbersome analysis shows that this asymptotic
behavior (for x .. 0) stays valid for any z [ Dz,
although the field distribution changes substantially.
When the finite aperture D is taken into account, one
can estimate the overall width of the beam at z0 by
calculating the maximum spatial frequency of gsxid at
the point xi ­ Dy2, and that is W0 ­ lfpDp21y2p21bp.
On the other hand, the width of the pulse (the sharp
central lobe of the beam) is approximated under the
assumption that its shape around the origin is de-
termined mainly by the Fourier transform of the
central lobe of gsxid. Therefore, the width of thepulse Wp (at the focus) is Wp ­ 2lfyH , where H is
the width of the central lobe of gsxid obtained as the
solution of the equation sHy2bdp 2 sHy2ad2 ­ 1y4.
Note that only the overall width of the beam W0 is
related to the aperture width D. The shape of the
pulse itself (Wp, Dz) is independent of the aperture
width if D is above some value that empirically is
found to be twice the width of the first cycle of gsxid.
A complete simulation of the PND beam was con-
ducted for p ­ 4, b ­ 128 pixels, a ­ 250 pixels, and
s2lf 2d ­ 1283 pixel3. The behavior of the wave along
the propagation axis can be studied from the solid
curve of Fig. 2. For comparison, the longitudinal
intensity of an ordinary focused beam resulting from
the central lobe of gsxid only [i.e., rect(xiyH dgsxidg
is shown by the dashed curve. The latter beam is
chosen for comparison with the PND beam because
the two have similar transverse shapes around the
focal point. The insets of Fig. 2 show the transverse
profiles of the PND beam (solid curves) and the ordi-
narily focused beam (dashed curves). Although their
width Wp is equal, the PND beam keeps its beam-
like shape for a longer distance than the ordinary
focused beam.
Next, we demonstrate the PND beam experimen-
tally. Looking at Fig. 1, we note that if gsxid is il-
luminated by a plane wave, the field distribution at
plane PL just behind the lens is
tsxLd ­ fg*Ls f dgLs2f d ,
Ls f d ­ exp
µ
j
kx2
f
¶
. (7)
Therefore the combination of the mask gsxid plus the
Fig. 2. Computer-simulated axial intensity distribution
of the PND beam (solid curve) and the ordinary focused
beam (dashed curve). The insets show the transverse
cross section of the PND beam (solid curves) and the
ordinary focused beam (dashed curves) at some points
along the axis.
March 1, 1995 / Vol. 20, No. 5 / OPTICS LETTERS 4250
z
0
z
(a) (b)
(c)
(d)
x
y 1
x
-21 21 [cm]
24 [cm]
x1 x1
-24
y 1
Fig. 3. Holograms that produce (a) the PND beam and
(b) the ordinary focused beam. The 2-D intensity distri-
bution of (c) the PND and (d) the ordinary focused beam.
lens can be replaced by a single mask with the dis-
tribution tsxLd. For an amplitude-only transparency
the mask that we use is given by
ssxLd ­ 1 1
1
2
tsxLd 1
1
2
t*sxLd . (8)
Assuming that tsxLd is normalized, ssxLd is a real
positive distribution that is easily implemented. We
see from Eq. (8), however, that there are two use-
less terms, t*sxLd and a constant term. Fortunately,
t*sxLd is responsible for the appearance of a virtual
image on plane Pi, so it contributes only a weak scat-
tering light around plane Pf. A similar but more uni-
form contribution is due to the constant term.
The hologram used in the experiment is shown
in Fig. 3(a). We obtained the distribution of the
hologram by substituting gsxid ­ cosh2pfsxiybd4 2
sxiyad2gj sxi , Dy2d into Eqs. (7) and (8), where l ­
0.63 mm, b ­ 5 mm, a ­ 9.77 mm, D ­ 16 mm, and
f ­ 1.2 m. The behavior of the PND beam passing
the hologram around the distance 1.2 m is demon-
strated in Fig. 3(c). As was done in the simulation,
the performance of the PND beam is compared with
the ordinary focused beam shown in Fig. 3(d). We
created this beam by illuminating the mask shown
in Fig. 3(b) by a plane wave. We obtained the mask
of Fig. 3(b) by the same projection process as for the
mask of Fig. 3(a), but this time the initial distribution
was the central lobe only of gsxid.
The analogy between temporal pulse propagation
in a dispersive medium and Fresnel diffraction is
our next consideration. Based on Ref. 3, the pulse
is represented as a sum of plane waves, each propa-
gating with a different phase delay bsvd, and bsvd
is approximated as a polonium series, bsvd ­ b0 1
b1v 1 b2v
2y2, where v is the angular frequency.
With these assumptions the field of the pulse is
ust, zd ­ expsjb0zd
Z ‘
2‘
U svdexp
µ
2j
b2z
2
v2
¶
3 expfjvst 2 b1zdg dv. (9)The pulse distribution is represented as a Fourier
transform of its temporal spectrum (at z ­ 0) mul-
tiplied by a quadratic phase factor. According to the
convolution theorem, it can be represented by a con-
volution integral,
ust-, zd ­
expsjb0zdp
2jpb2z
Z ‘
2‘
ust0dexp
•
j
st- 2 t0d2
2b2z
‚
dt0, (10)
where t- ­ t 2 b1z, i.e., the time variable minus the
delay that the pulse takes to reach to z. This ex-
pression has the same form of the Fresnel integral
describing the propagation of an electromagnetic field
in free space, where t- is equivalent to x. Based on
this analogy, there must be an equivalent expression
to Eq. (1) in the (t, z) domain. To obtain it, assume
that the initial pulse distribution exists at a distance
f from a temporal lens of the form exps2jt2y2b2fd.
After the analyses of Ref. 6, the pulse distribution af-
ter it propagates the distance f and is multiplied by
the temporal lens (at time t ­ b1f d is
ust-, zd ­
expfjb0sz 1 2fdgp
2jpb2f
3
Z ‘
2‘
ust0d exp
"
2
j
b2f
ˆ
t0z
2f
2 t-t0
!#
dt0,
(11)
where the temporal lens is at z ­ 2f . Equation (11)
has indeed the same form as Eq. (1). Therefore, in
general we can reshape a profile of the pulse along
the propagation axis, inside a dispersive medium, in
the same manner as has been done to the beam in
the spatial domain.6,8,9 Specifically, we can create
a pseudonondispersive temporal pulse by using the
theory of the pseudonondiffracting beam in the 2-D
space analyzed at the beginning of this Letter. Anal-
ogously to the spatial pulse, we expect the temporal
pulse to remain constant for a distance Dz ­ 4pLs1 1
adb2f 2 if a pulse of the form cosh2pfstybdp 2 styad2gj is
processed by the temporal analogy of Eqs. (7) and (8).
This research was sponsored in part by the NASA
Office of Advanced Concepts and Technology.
References
1. E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454
(1969).
2. B. H. Kolner and M. Nazarathy, Opt. Lett. 14, 630
(1989).
3. A. W. Lohmann and D. Mendlovic, Appl. Opt. 31, 6212
(1992).
4. M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M.
Bloom, Appl. Phys. Lett. 64, 270 (1994).
5. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev.
Lett. 58, 1499 (1987).
6. J. Rosen and A. Yariv, Opt. Lett. 19, 843 (1994).
7. A. Papoulis, Systems and Transforms with Applica-
tions in Optics, 1st ed. (McGraw-Hill, New York, 1968),
Chap. 7, p. 222.
8. J. Rosen, Opt. Lett. 19, 369 (1994).
9. R. Piestun and J. Shamir, Opt. Lett. 19, 771 (1994).


https://authors.library.caltech.edu/2843/1/ROSol95a.pdf

Pseudonondiffracting slitlike beam and its analogy to the pseudonondispersing pulse

Abstract

Similar works

Full text

Available Versions

Caltech Authors - Main

Caltech Authors - Main

Caltech Authors - Main