Dear Steven,
Thanks for your suggestion before. As you mentioned, I really need to fix
a frequency first, but the wavelength equals to 0.5um, and the lattice period
is 11um( I am now repeating a result from PRL paper), which makes the frequency
22 in MPB, too big to find the root? On the other hand, I am not sure about,
since I define kdir in 'find-k' as (kx,ky,0) to find k-magnitude(since it's 2D,
kz should be 0 here, am I right?) does the k-magnitude equal to kz(q)?
I have attached my code below, I have also attached the band structure
picture they got in attachment. Their values of q completely have no comparison
with the results got from MPB, which makes me more confused now.
So could you kindly give me some further suggestion to solve this problem?
I really appreciate your help.
Best wishes
Yiling
========================================================================
; simulate the band structure in paper PRL-91-213906-2003-Two-dimensional
Optical lattice soliton
; Sinusoidal lattice
(define pi (deg->rad 180.0))
; Define a function of position p (in the lattice basis) which returns
; the material refractive index at that position.
; This is periodic, and also has inverse symmetry.
; RI modulation is directly related to the voltage applied to the
photorefractive crytal,
; which is called as index potential in paper-----V(x,y)
; For sinusoidal, V(x,y)=-(V0/2)[sin(pi*x)^2+sin(pi*y)^2]
; and the RI modulation is related to the normalized potential depth via
V0=(2*pi*n0*a/lamda0)^2*delta.
; Then RI modulation: Delta=-(lamda0/2/pi/n0/a)^2*V0/2*[sin(pi*x)^2+sin(pi*y)^2]
(define n0 2.3)
(define a 11)
(define lamda0 0.5)
(define-param V0 21.6)
(define rad 3.5)
(define r (/ rad a))
(define (sinusoidal-func p)
(make dielectric
(index (+ n0 (* -0.5 V0 (sqr (/ lamda0 2 pi n0 a)) (+ (sqr (sin (* pi
(vector3-x p)))) (sqr (sin (* pi (vector3-y p))))))))))
; The periodicity is 11um, and the radius is 3.5um for the dielectric-function.
(set! geometry (list
(make cylinder
(center 0 0 0)
(radius r)
(height infinity)
(material (make material-function (material-func
sinusoidal-func))))))
(set! default-material (make dielectric (index n0)))
(set! geometry-lattice (make lattice(size 1 1 no-size)))
(set! resolution 64)
; the wavelength is 0.5um, according to the implicit periodicity, omega shoule
be a/lamda=22(???? is it too big)
(define-param omega 22) ;????it's too big to find the root
(define-param band-min 1)
(define-param band-max 5)
(define-param kdir (vector3 0.5 0 0))
(define-param tol 1e-4)
(define-param kmag-guess 0.5)
(define-param kmag-min -1.0)
(define-param kmag-max 1.0)
;(find-k p omega band-min band-max kdir tol kmag-guess kmag-min kmag-max)
(find-k NO-PARITY omega band-min band-max kdir tol kmag-guess kmag-min kmag-max)
****************************************************************************8
MPB can certainly solve any problem of that form. However, you still
need a frequency in order to define your mode; given K (the in-plane
wavevector with components kx and ky) and q (the z wavevector,
although have switched the sign convention so that q = -kz) there are
infinitely many eigenmodes with different frequencies. Without a
frequency your problem seems to be ill-posed.
MPB solves the problem where, given (kx,ky,kz) you find the
frequencies. Alternatively, with find-k you can give (kx,ky) and the
frequency and then solve for kz (q).
Steven
------------------------------
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<<attachment: bandstructure.JPG>>
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