Whatever tool you use, you will need to calculate eigen frequencies for the modes in your lattice as a function of the Bloch wave vector. This is sometimes called "k-space" or "reciprocal space." The group velocity comes from the slope of this data (i.e. dw/dk). Once you have group velocity, you can calculate the group index.

I posted some lectures on this topic here:

http://emlab.utep.edu/ee5390em21.htm

See Lectures 7, 8, and 14.

Missing common complications, photonic crystals reflect the part of passing light inside them many times, both in backward and forward directions, due to Fresnel reflections on the steps of refractive index. Evidently, this must affect the group velocity. If light frequency falls into bandgap, no transmission is expected, but in the vicinity of the bandgap decrease of group velocity is more interesting for observation

Do you think that decreasing group velocity of the light would be the photovoltaic efficiency of solar cells thanks to one stronger light-matter interacction?

I don't really like to do auto promotion but I think my article: http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-15-10-6102, which is open access, could shed some light on the subject.

If periodical structures slow down light then FBGs implemented in the fiber lasers may do the same to induce further time stretching of optical pulses.

If you mean Bragg mirror of a fiber laser, then you must account for that fiber laser is in some sense equivalent to common laser with long cavity. Temporal characteristics of fiber laser are formed by the whole process of generation, that is, by amplification, feedback, fiber modes, the method of modulation, and possible slowing down in the Bragg reflector , as I think, will play small ( though in some cases not negligible) role

FBG stands for fiber bragg grating. Bragg mirrors of the fiber laser may induce time stretching on the short pulses and this effect may be important in the design of the fiber laser gyroscopes. The FBGs may be employed along the SM fibers as delaying elements . However, may you quantify this effect ?

I remember old paper on superluminal behavior of biphotons in a single-pass Bragg reflector, but I am sure that in a laser cavity the effect must be incorporated into. In any case, I place this reference - VOLUME 71, NUMBER 5 PHYSICAL REVIEW LETTERS

Measurement of the Single-Photon Tunneling Time 2 AUGUST 1993 A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao , p.708

1) How do you modify the concept of enrgy density,energy velocity and dispersion for lossy media?

2)A lucid example is chirped pulse amplification. How do you extend those cocepts for CPA?

the Energy density is the amount of energy stored in a given system or region of space per unit volume or mass which is an integral over the fields divided by a volume, so the nature of the media shouldn't change this definition.

For the energy velocity it will depend on how you defined because you can find different definition for it depending if you want consider the average of the speed of the energy or you consider an integral of this energy over the energy density. In certain particular cases (non-linear/non-isotropic media) it can change the results.

As for the dispersion it will depend on which dispersion you are referring to.

For point 2) it will depend on the definition you take for the energy velocity and which dispersion you are interested to.

Fortunately, there are adequate formalisms and software to model various photonic crystals. For instance, you may check following article:

B. Eftekharinia, P. Parvin, H. Habibiyan, H. Hoseinzadeh, " Analysis of amplification properties of a photonic crystal fiber"

Optik 01/2014; 125(4):1565– 1571.

Waves slow down in photonic crystals because they are "bouncing around" instead of propagating straight. A more complicated answer is that waves slow down in the presence of any kind of dispersion. There are things other than photonic crystals that can slow a wave. I put together an introductory lecture on this exact topic. You can get the notes at the link below. See Lecture 22. I will be recording this lecture maybe some time this week.

http://emlab.utep.edu/ee5390em21.htm

Thank you Raymond.

which software can calculate the group index of photonic crystal structure?

I am using COMSOL, but i can not get the desired results...

Whatever tool you use, you will need to calculate eigen frequencies for the modes in your lattice as a function of the Bloch wave vector. This is sometimes called "k-space" or "reciprocal space." The group velocity comes from the slope of this data (i.e. dw/dk). Once you have group velocity, you can calculate the group index.

I posted some lectures on this topic here:

http://emlab.utep.edu/ee5390em21.htm

See Lectures 7, 8, and 14.

Hello Raymond sir

i got w-k diagram for a photonic crystal line defect waveguide  using optifdtd,now how to calculate slope of w-k diagram to find group velocity at desired wavelength.

I have never used optifdtd, but the group velocity is the derivative of w in k-space.  That is...

vg = dw/dk

Check out Lecture 22 here:

http://emlab.utep.edu/ee5390em21.htm

Also, I just wrote a book chapter that covers this sort of thing in pretty good detail.  I am not sure it is available yet, but you can get it here:

http://dx.doi.org/10.1016/bs.ssp.2015.02.002

Thank you raymond sir 

First link is interesting and useful for slow light waveguides.

sir i don't have access for second link so please upload pdf.

I tried.  I don't access either.

Ok Raymond sir

while we are calculating group velocity,we need to calculate dw/dk. but in our w-k diagram w is normalized by a/2*pi*c.so the slope of this normalized diagram will give equal result or first we have to change it in w again.

The slope will only be off by a constant.  You can denormalize the frequency axis before calculating slope if that makes more sense to you.

thank you Raymond sir, all the best

hello Raymond sir

while i was calculating group velocity for desired wavelength (1550 nm) using w-k diagram,i came across an interesting problem.

in w-k diagram there are more than one phase constant corresponding to single frequency.

so by calculating slope of w-k diagram we got more than one group velocity for a single frequency.

sir is it practically possible????

if yes then which group velocity i should consider for first order dispersion analysis.

here i am attaching w-k diagram find using optifdtd tool.

I think the problem you are running into is called branching, which means there are actually multiple correct answers as far as the parameter retrieval is concerned.  From a band diagram, you have to choose your k-point after unfolding the bands.  Parameter retrieval for photonic crystals is not easy and many times not meaningful.  Here is a good paper on the subject if you want more information.

Brian T. Schwartz, R. Piestun, "Dynamic properties of photonic crystals and their effective refractive index," J. Opt. Soc. Am. B, Vol. 22, No. 9, pp. 2018-2026, 2005.

Thank you, dear Raymond.  Your reference is found free at http://libgen.org/scimag/get.php?doi=10.1364/josab.22.002018

Hello Raymond sir 

 How light is slow down in  photonic crystal wave guides.

and why it deepens on period of air holes we are using. 

The light slows down in waveguides for exact same reasons.  It is bouncing around and interfering with itself.  It is not so much that that deepens with the period of the air holes. The period is simply tuning the wavelength at which it slows.  I would expect the fill factor to have more of a role in maximizing the effect.

thank you sir

what is fill factor??

Fill factor is the fraction of area occupied by dielectric in the lattice.  High fill factor means small holes and thus more dielectric.