Designing for beam propagation in periodic and nonperiodic photonic nanostructures: extended Hamiltonian method.
ABSTRACT We use Hamiltonian optics to design and analyze beam propagation in two-dimensional (2D) periodic structures with slowly varying nonuniformities. We extend a conventional Hamiltonian method, adding equations for calculating the width of a beam propagating in such structures, and quantify the range of validity of the extended Hamiltonian equations. For calculating the beam width, the equations are more than 10(3) times faster than finite difference time domain (FDTD) simulations. We perform FDTD simulations of beam propagation in large 2D periodic structures with slowly varying nonuniformities to validate our method. Beam path and beam width calculated using the extended Hamiltonian method show good agreement with FDTD simulations. By contrasting the method with ray tracing of the bundle of rays, we highlight and explain the limitations of the extended Hamiltonian method. Finally, we use a frequency demultiplexing device optimization example to demonstrate the potential applications of the method.
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ABSTRACT: The dispersive properties of planar photonic crystals (PhCs) have been envisaged for years. In particular, the superprism effect has been considered to obtain a strong influence of input beam conditions (e.g. wavelength or input angle) on the light group velocity direction, enabling the design and fabrication of on-chip infra-red spectrometers and integrated optical demultiplexers. We extend here the properties of PhCs to the study of graded photonic crystals (GPhCs) made of a two-dimensional chirp of lattice parameters and show that GPhCs enable solving several drawbacks of dispersive PhCs like the beam divergence issues or the need of long preconditioning regions to precompensate beam diffraction effects. The proposed approach is applied to a square lattice air-hole PhC with a gradual filling factor that was fabricated using ebeam lithography and ICP etching techniques. A nearly-constant 0.25μm/nm spatial dispersion is demonstrated for a 60μm square GPhC structure in the 1470-1600nm spectral range without noticeable spatial or spectral spreading. Moreover, contrary to PhC superprism structures, a linear dispersion is obtained in the considered wavelength range.Proceedings of SPIE - The International Society for Optical Engineering 11/2012; · 0.20 Impact Factor
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ABSTRACT: We show that the effective gauge field for photons provides a versatile platform for controlling the flow of light. As an example we consider a photonic resonator lattice where the coupling strength between nearest neighbor resonators are harmonically modulated. By choosing different spatial distributions of the modulation phases, and hence imposing different inhomogeneous effective magnetic field configurations, we numerically demonstrate a wide variety of propagation effects including negative refraction, one-way mirror, and on- and off-axis focusing. Since the effective gauge field is imposed dynamically after a structure is constructed, our work points to the importance of the temporal degree of freedom for controlling the spatial flow of light.Physical Review Letters 11/2013; 111(20):203901. · 7.73 Impact Factor
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ABSTRACT: We design an all-dielectric L\"uneburg lens as an adiabatic space-variant lattice explicitly accounting for finite film thickness. We describe an all-analytical approach to compensate for the finite height of subwavelength dielectric structures in the pass-band regime. This method calculates the effective refractive index of the infinite-height lattice from effective medium theory, then embeds a medium of the same effective index into a slab waveguide of finite height and uses the waveguide dispersion diagram to calculate a new effective index. The results are compared with the conventional numerical treatment - a direct band diagram calculation, using a modified three-dimensional lattice with the superstrate and substrate included in the cell geometry. We show that the analytical results are in good agreement with the numerical ones, and the performance of the thin-film L\"uneburg lens is quite different than the estimates obtained assuming infinite height.11/2011;
Designing for beam propagation in periodic and nonperiodic photonic nanostructures:
Extended Hamiltonian method
Yang Jiao, Shanhui Fan, and David A. B. Miller
Ginzton Laboratory, Stanford University, Stanford, California 94305-4088, USA
(Received 6 February 2004; published 23 September 2004)
We use Hamiltonian optics to design and analyze beam propagation in two-dimensional (2D) periodic
structures with slowly varying nonuniformities. We extend a conventional Hamiltonian method, adding equa-
tions for calculating the width of a beam propagating in such structures, and quantify the range of validity of
the extended Hamiltonian equations. For calculating the beam width, the equations are more than 103times
faster than finite difference time domain (FDTD) simulations. We perform FDTD simulations of beam propa-
gation in large 2D periodic structures with slowly varying nonuniformities to validate our method. Beam path
and beam width calculated using the extended Hamiltonian method show good agreement with FDTD simu-
lations. By contrasting the method with ray tracing of the bundle of rays, we highlight and explain the
limitations of the extended Hamiltonian method. Finally, we use a frequency demultiplexing device optimiza-
tion example to demonstrate the potential applications of the method.
DOI: 10.1103/PhysRevE.70.036612PACS number(s): 42.70.Qs, 42.15.Dp, 42.25.Dd, 61.50.Ah
Radical dispersion properties of periodic photonic nano-
structures [1,2] has recently generated much research interest
. For applications such as group velocity dispersion com-
pensation and frequency demultiplexing, however, periodic
structures arguably do not offer enough design freedom to
achieve the desired input/output dispersion relation over
large frequency ranges. In such cases, nonperiodic structures
could perform much better than periodic structures [4,5]. In-
troducing nonuniformities in two-dimensional (2D) and
three-dimensional (3D) periodic structures, such as high-
index cylinder/sphere arrays, will also allow more freedom
in the design of the device input/output characteristics. There
has, however, been much less study of 2D and 3D nonuni-
form structures, partly due to the difficulty in analyzing
them. Finite difference time domain (FDTD) simulations can
be used to analyze such structures, but it is very time con-
suming . In this work, we use a Hamiltonian optics
method to design and analyze the beam path in for Gaussian
beams propagating in 2D periodic structures with slowly
varying nonuniformities. Furthermore, we extend the con-
ventional Hamiltonian method with equations for calculating
the beam width in periodic structures. For a beam propagat-
ing in a nonuniform structure, the extended Hamiltonian
method can calculate the beam path and the beam width at
least 103times faster than FDTD simulations. Where slow-
ness of the FDTD method restricts it to a tool for analyzing
given structures, the speedup allows the extended Hamil-
tonian method to be used as a design tool for the intentional
introduction of nonuniformities into periodic photonic struc-
tures for exciting device functions.
The efficiency of the extended Hamiltonian method relies
on the local periodicity of the structure. If the nonuniformity
occurs over many periods, locally the structure is approxi-
mately periodic. Therefore, we can define and calculate the
Bloch wave dispersion relationship, locally, for each position
inside the structure. With the dispersion relationship known,
the classical optical Hamiltonian equations can be used to
track the beam path for a ray propagating in the nonuniform
structure. With the beam path known, we can then use the
extended Hamiltonian equations that we developed to track
the beam width for a Gaussian beam propagating in a non-
uniform structure. The computational cost for calculating the
dispersion relationship is amortized over all designs based on
the same type of nonuniformities, e.g., all designs based on
nonuniform distribution of the high index cylinder radius.
After the dispersion relationships are calculated for each pos-
sible type of local structure, the speed advantage of the ex-
tended Hamiltonian method over FDTD is dramatic.
Previous work on applying Hamiltonian optics to periodic
media solely dealt with the calculation of the ray path .
The classical Hamiltonian method is also useful for calculat-
ing the time of flight of a ray in nonuniform periodic media
and the frequency dependency of the ray path . The
method has also been used to show that chaotic ray paths can
be easily achieved in nonuniform periodic media . How-
ever, without an understanding of the evolution of the beam
width, practical design with such periodic media with non-
uniformities would be very limited. For example, the beam
width is critical for any beam steering device. Therefore, the
beam width equation we introduce here is a critical refine-
ment to the Hamiltonian method for analyzing propagation
in periodic media with nonuniformities.
To the best of our knowledge, there are no previous pub-
lished large-scale FDTD simulations of beam propagation in
periodic media with gradual nonuniformity. In this work, we
perform FDTD simulations of 2D periodic structures with
nonuniformities, with dimensions on the order of 100?100
lattice constants. These simulations validate the beam path
and the beam width calculated with the extended Hamil-
tonian equations. FDTD simulations of structures at this
scale are very time consuming, whereas the computational
complexity of the extended Hamiltonian equations is essen-
tially independent of the device size.
In Sec. II we develop the extension to the Hamiltonian
equations in the context of bulk media with slowly varying
PHYSICAL REVIEW E 70, 036612 (2004)
1539-3755/2004/70(3)/036612(9)/$22.50 ©2004 The American Physical Society
nonuniformities. We explain the applicability of the extended
Hamiltonian equations to periodic media with nonuniformi-
ties in Sec. III. In Sec. IV, we provide a measure for the
range of validity of the extended Hamiltonian method, vali-
date the method with FDTD simulations, and explain limita-
tions of the method with a comparison to ray bundle tracing.
Finally, we use a numerical optimization example in Sec. V
to demonstrate the potential applications of the method.
II. THE HAMILTONIAN EQUATIONS
A. Hamiltonian optics
Hamiltonian optics has traditionally been used to describe
the optical ray path in bulk dielectric media with slowly
varying properties . We will define the characteristic inho-
mogeneity length L as the shortest distance over which the
nonuniformity changes the medium significantly. We assume
the field behaves similar to a plane wave locally, and has the
form E?x?=a?x?exp?is?x??, where x=?x1,x2,x3? is a dimen-
sionless coordinate system, scaled from real spatial coordi-
nates ?X1,X2,X3? as in ?x1,x2,x3?=?X1/L,X2/L,X3/L?. By
substituting the assumed form of E?x? into Maxwell’s equa-
tions, and using the slowly varying approximation [e.g., the
wave amplitude a?x? varies much more slowly than the
phase, and therefore the spatial derivative of a?x? is ignored],
we will get a equation for a?x?:
??k?2I − kkT? − ??x?? a = 0,
where ? is the frequency, ??x? is the dielectric tensor, I is the
identity matrix, we defined k=?s as the wave vector, and kT
is the transpose of k (so that kkTforms a matrix or tensor).
We assumed the nonuniformity comes from the spatial de-
pendence of the dielectric tensor. Now we define the math-
ematical entity that will become the effective Hamiltonian
for this problem, i.e., we choose
H?x,k,?? ? det??c
??k?2I − kkT? − ??x??.
In order for a nontrivial solution to exist for a, we must have
??k?2I − kkT? − ??x??? H?x,k,?? = 0.
For fixed ?, we look for the parameterized curve ?r???,q????
in the 6D space ?x,k? such that Eq. (3) is satisfied. Here ? is
the parameterization variable. If Eq. (3) is satisfied, then r???
describes the path followed by the wave, and q??? describes
the wave vector along the path. We can now find that Eq. (3)
is satisfied on the curve ?r???,q???? if the curve obeys the
d?= −? H?x,k,??
It can be verified that Eq. (4) is a solution of Eq. (3) by
taking the full derivative of the left-hand side of Eq. (3) with
respect to ?, and substituting Eq. (4) into the expression,
therefore justifying our choice for the effective Hamiltonian
of the problem.
The widely used dispersion relationship ??k,x? is ob-
tained by solving Eq. (3) for fixed x. We can write Eq. (4) in
terms of the ??x,k? rather than H?x,k,??. Using the fact
that, for a solution we must have H(x,k,??x,k?)=0, and
taking partial derivatives of H(x,k,??x,k?) with respect to x
and k, we get
? x= C?x,k?? H
? k= C?x,k?? H
where the scaling factor C?x,k?=−??H/???−1is the same for
all six partial derivatives. Therefore, ??H/?x,?H/?k? and
???/?x,??/?k? always point in the same direction. Thus Eq.
(4) can also be written as
dt= −? ??x,k?
The curve ?r?t?,q?t?? describes the same curve as
?r???,q????, but parametrized by the parameter t instead of ?.
Using the definition of group velocity Vg=??/?k, dX/dt can
now be identified with the group velocity, t can be identified
simply as time, and r?t?, q?t? can be identified with the
optical ray path and the wave vector along the ray path,
Because the Hamiltonian equation, (4) is satisfied for ei-
ther the H?x,k? obtained in Eq. (3) or the dispersion relation
??x,k?, we will refer to both quantities as the Hamiltonian.
We will consistently use H?x,k? in the rest of the paper to
denote the Hamiltonian. This notation signifies the equiva-
lence of Eq. (4) to classical Hamiltonian equations in me-
If we know H?x,k? or the dispersion relation at each lo-
cation in the bulk dielectric medium, then Eq. (4) can be
integrated to give the ray path for all later times given the
initial ray location, and the initial wave vector. For Eq. (4) to
be valid, L needs to be large compared to the wavelength ?.
It should be noted that the Hamiltonian equations in op-
tics are analogous to the WKB method of quantum mechan-
ics. Both methods start by assuming a field that has an am-
plitude that varies much slower than the phase, and that
scattering can be neglected. Useful approximations of the
field pattern are then found by substituting into either Max-
well’s equations or the Schrödinger equation as appropriate.
We will show in Sec. III that, if a packet of Bloch waves
has a Gaussian distribution of the transverse k vector, the
envelope of the field will behave like a Gaussian beam field.
Therefore, by using the photonic crystal dispersion surface
instead of the bulk medium dispersion surface, and Bloch
waves instead of the plane wave components in a Gaussian
beam, we can apply Eq. (4) to ray propagation in nonuniform
photonic crystals. The same argument applies to the extended
Hamiltonian equations, which we will develop next, in part
B of this section. For clarity, we will introduce the extended
Hamiltonian equations for Gaussian beam propagation in
JIAO, FAN, AND MILLERPHYSICAL REVIEW E 70, 036612 (2004)
bulk nonuniform dielectric media, keeping in mind that ap-
plication to photonic crystals is straightforward.
B. Extension to beam width calculations
First we introduce some notations. We define a general-
ized coordinate system ?? ?x?,w1?x? w2?x?? that is attached
to the ray path R. The ray path R is given by the coordinate
curve (?, w1=0, w2=0). For the derivation of the extended
Hamiltonian equations, we do not need an explicit expres-
sion for the mapping from the Cartesian system ?x1,x2,x3? to
??,w1,w2?. But we assume that, in the proximity of the ray
path, the mapping from ?x1,x2,x3? to ??,w1,w2? is unitary.
To simplify notation, we will use the subscripts to denote
first and second partial derivatives with respect to x?, e.g.,
When we treat g as dependent on x, we will use the symbol
??to denote the partial derivative with respect to x?, e.g.,
In Eq. (8) and all subsequent equations, Einstein’s summa-
tion convention is adopted. That is, if an index occurs more
than once in a product of terms, such as the index ? above,
summation over all values of the index is implied.
When we change coordinates to ??,w1,w2?, we will use
the symbol ?ito denote the partial derivative with respect to
?ijwill similarly be the second derivative with respect to wi
We now extend the Hamiltonian optics method with equa-
tions for obtaining the beam width and the radius of curva-
ture in periodic structures with slowly varying nonuniformi-
ties. To the best of our knowledge, beam width equations
have not previously been applied to the analysis of periodic
nanostructures with nonuniformities. We follow the exten-
sion of Hamiltonian optics developed by Poli et al. in Ref.
 for plasmas. Assume again the nonuniformity is caused
by a slowly varying dielectric tensor ??x?. Under the as-
sumption that L is much greater than the wavelength ?, a
Gaussian beam traveling in nonuniform media should re-
semble a Gaussian beam whose center is following some
path. Therefore, we assume the field has the form
E?x? = a?x?exp?− ???x? + is??x?? = a?x?exp?− ig??x??,
where a?x? is the amplitude of the beam, s??x? is the phase of
the beam, exp?−???x?? describes a Gaussian profile perpen-
dicular to a central ray path, and g?=s?+i?? defines a com-
plex phase for this problem. Since we assume a Gaussian
transverse beam profile, we have, in the coordinate system
exp?− ???x?? = exp?−1
where ?ij? specifies the beam width.
It is important to estimate and then keep track the relative
rates of variation for the terms in Eq. (10). The medium
property changes by a significant percentage over a distance
of 1 unit (remember the coordinate system is scaled by L).
Therefore we expect a?x? should also change by O?1? (i.e.,
of order unity) over unit length. We can express this assump-
The optical phase s?, on the other hand, should change by
O?2?L/?? over unit length. Furthermore, we expect the
beam picture to be valid when, perpendicular to the ray path,
?? causes the field to decay on a distance much shorter than
L, but longer than ?. In the interest of keeping track of the
relative sizes of the terms for later approximations in the
derivation, we scale the terms in Eqs. (10) and (11):
g = s + i? =1
After the scaling, we will assume that s?x?, ?ij?x? are of order
the same size as a?x?, and s?x?, ??x?, ?ij?x? vary at of order
the same rate as a?x?. The ansatz for the field becomes
E?x? = a?x?exp?− ?? + i?s? = a?x?exp?− i?g?.
By substituting the assumed form of E?x? into Maxwell’s
equation, and keeping only the terms on the order of ?2, a
matrix equation for a?x? can be obtained:
???g?2I − ? g ? gT? − ??x??a ? Pa = 0,
which gives our definition of the 3?3 matrix P??g,x?. In
order for a nontrivial solution to exist, we must have
H˜??g,x? ? det?P??g,x?? = 0,
where we have introduced a “complex Hamiltonian” H˜. Note
det?P?=0 is just Eq. (3) with k replaced by ?g. Along R,
?=0, and H˜reduces to the real Hamiltonian H given in Eq.
(2). Therefore, Eq. (16) is satisfied along R. We seek a solu-
tion to Eq. (16) in the neighborhood of R, when ??0. By
expanding Eq. (15) in w1, w2around R, and keeping track of
the size of the terms, it can be shown that Eq. (16) is satisfied
to sufficient accuracy in the neighborhood of R if the follow-
ing expressions hold :
DESIGNING FOR BEAM PROPAGATION IN PERIODIC…
PHYSICAL REVIEW E 70, 036612 (2004)
Because of Eq. (17a), and the fact that ? is measured along
R, all derivatives of H˜with respect to ? are zero. We can now
i?iH˜+ ???? H˜
Similarly, taking one more derivative,
? wi? wj?
Using Eq. (17) in Eqs. (18) and (19) , we have
This statement is trivial if we replace H˜with the real Hamil-
tonian H. But, here we are claiming that the imaginary part
of H˜also has a zero second derivative along R. Expansion of
Eq. (20) using Eq. (8) will give the equations governing the
evolution of the beam width along R.
We can expand Eq. (20c) by applying Eq. (8) twice. Be-
cause H˜is equal to the real Hamiltonian H along R, and ??x?
has zero first derivative along R, we can simplify the expan-
sion to the following expression. The expression involves
only derivatives of the real Hamiltonian H with respect to the
real part of the phase:
????H?R= ???? H˜
? s?? x?g??+
? s?? s?
? x?? x?+
? x?? s?
? s?? s?
? x?+? H˜
? x?? x?+
? x?? s?
? s?? x?g??
In the third step above we used the original Hamiltonian
equation to write
Equation (21) is a system of ordinary differential equations
governing the evolution of g??along the ray path R. We will
write the real and imaginary parts separately, and refer to
these equations as the system of extended Hamiltonian equa-
? x?? x?
? k?? k?
? x?? k?
? x?? k?
? k?? k?
? x?? k?
? x?? k?
? k?? k?
? k?? k?
Note the physical significance of the equations. The second
derivative of ??x? will give the beam width measured along
each spatial coordinate. The second derivative of s?x? gives
the radius of curvature of the phase fronts along each spatial
The beam width is then given by
Here we used the definition of the beam width as the distance
from the center of the beam where the field amplitude has
fallen to 1/e of the maximum.
Equation (23) is exact for a beam propagating in homog-
enous medium, and a good approximation in an inhomoge-
neous medium when L/W?1, where L is the characteristic
inhomogeneity length. For homogenous media, Eq. (23) re-
duces to equations for the beam width and the radius of
curvature of a freely propagating Gaussian beam. As we dis-
cuss in the next section, for a periodic medium with nonuni-
formity, Eq. (23) still describes the beam width, and can be
easily integrated numerically since it is a system of linear
ordinary differential equations.
III. GAUSSIAN BEAM IN A LOCALLY PERIODIC
In the discussion above of the extended Hamiltonian
equations, we introduced the Hamiltonian equations and the
extended Hamiltonian equations in the context of Gaussian
beams propagating in slowly varying, anisotropic, bulk me-
dia. The application to slowly varying, locally periodic me-
dia, in which the properties of the unit cell change slowly
from cell to cell, is straightforward by using the Bloch wave
dispersion relation instead of the Hamiltonian for bulk me-
JIAO, FAN, AND MILLER PHYSICAL REVIEW E 70, 036612 (2004)
dia. Eigenstates in a periodic medium for a given wave vec-
tor k can be written in the Bloch form
where ?kis either the E field or the H field, uk?x? is a
periodic function with the same period as the lattice, and x is
the position. The dispersion relation ??k? for the Bloch wave
can be obtained using various methods. For such periodic
media, the dispersion relation corresponds to a band struc-
ture, very similar to the band structures found for the
Schrödinger equation for electrons in a periodic lattice, and
the calculation of such band structures is a core part of the
study of periodic optical media (i.e., photonic crystals). A
key point about the present work is that, if we have calcu-
lated that band structure or dispersion relation for each dif-
ferent possible locally periodic region, we may take all the
results considered above for slowly varying bulk media, and
apply those now to the case of media that are locally peri-
odic. We merely have to substitute the local dispersion rela-
tion of the locally periodic medium for the local dispersion
relation of the material in the bulk case; no other change is
necessary for analyzing the propagation in locally periodic
media, at least within the approximations of the method.
We still need to show that, when a Gaussian beam travel-
ing in free space enters a periodic medium, the transmitted
wave form inside the periodic medium can still be described
by a Gaussian beam equation. A solution of the scalar wave
equation can be written in its plane wave components
A?kx,ky?exp?jk · x?dkxdky, ?27?
where k=?kx,ky,kz? is the wave vector and A?kx,ky? is the
amplitude of the plane wave components. A Gaussian beam
can be described by a field in this form. If we assume the
Gaussian beam travels along the z axis, it is possible to find
a equation for A?kx,ky?:
where W is the beam width. If the W is large, A decays to
negligible values when k?approaches ?k?. In other words,
the plane wave components only span a small angle in k
space. Equation (28) is only valid in the paraxial approxima-
kz? k −kx
In particular, in order for the plane waves described by Eq.
(29) to combine as to a Gaussian beam, we must use the
paraxial approximation to express kzin Eq. (27) as a qua-
dratic function of kxand ky:
2?1/2is the magnitude of the wave vector.
+ j?k −kx
A?kx,ky?exp?jkxx + jkyy
After expressing the incident Gaussian beam in its plane
wave components, we can see how these components trans-
form when they enter the periodic medium.
When a Gaussian beam traveling in a bulk medium im-
pinges on a periodic medium, phase matching requires that
each incident plane wave component only transfers energy to
the Bloch wave with the same transverse wave vector com-
ponent kxand ky(assuming the wave vectors are concen-
trated in the first Brillouin zone):
when T?kx,kx? is the transmission coefficient from a plane
wave to its corresponding Bloch wave and A?kx,ky? is given
by Eq. (28). If the incident beam is wide, then kx, kyspans a
small range, and we can assume that the transmission coef-
ficient T is not a function of kx, ky. We can then express the
field inside the periodic medium as
In the last step above, we assumed uk?r? is constant indepen-
dent of k, again because the Gaussian beam spans a small
angle in k space. In order for the integral term to match the
equation for a Gaussian beam, we need to be able to express
kzas a quadratic function of kxand kyas in Eq. (30). From
the Bloch wave dispersion relation ??k?=?0, we can find kz
in the form kz=f?k,kx,ky?, which is not in general in the
small, we can always approximate f?k,kx,ky? around the av-
erage k vector ?kx0,ky0,kz0? as
2?1/2. But, if the angular spread of k is
f?k,kx,ky? ? kz0+?
? k?? k?
?k?− k?0??k?− k?0?.
Using this approximation in Eq. (32), we can express kzas a
quadratic function of kxand kyas in Eq. (30). Except for the
addition of terms linear in kxand ky, the integral term in Eq.
(32) then corresponds exactly with the equation of a Gauss-
ian beam. Just as in an anisotropic medium, the linear terms
corresponds to changes in beam propagation angle, which
does not alter the Gaussian beam shape. This shows that,
when the incident Gaussian beam is wide, the envelope of
?pcbehaves as a Gaussian beam. Therefore, we can use the
extended Hamiltonian equations developed for the bulk me-
DESIGNING FOR BEAM PROPAGATION IN PERIODIC…
PHYSICAL REVIEW E 70, 036612 (2004)