Dispersion of the propagation constants of waveguide modes of a thin left-handed film is considered. It is shown that by certain frequency the group velocity of mode can be vanished. An analysis of spatio-temporal transformation of intramode packet with spectrum near this frequency is performed. It is shown that a propagation velocity of the packet is proportional to a square root of a width of the packet spectrum and for packets with spectrum about 10 MHz can be by four orders less than the group velocity of the light in a volume left-handed material. Modern technology of metamaterials allows creating thin left-handed waveguides on the volume substrate. These waveguides have both negative values of permittivity and permeability in optical range of wavelengths [1]. The waveguides have unique properties which considerably differ from usual waveguides. The features of dispersive characteristics of such waveguides and the features of the propagation of optical pulses in them are considered below. It is shown in [2, 3] that the propagation constants β of waveguide modes of TE -type of thin films with thickness h including on the basis of left-handed metamaterials are the real roots of the dispersion relation which can be represented in the form χs µf χ−1 −1 −1 −1 f µs + χc µf χf µc tanh χf h + = f (β, ω) = 0, (1) 1 + χs χc µ2f χ−2 −1 −1 f µs µc ³ ´1/2 where χ∗ = β 2 − ε∗ µ∗ (ω/c)2 ; µ∗ and ε∗ are the relative permeability and the relative permit- tivity, respectively; c is the velocity of light in a vacuum; sign “∗” at subscript is used instead of letters “c” (coating medium), “f ” (film) and “s” (substrate). We assume that the dispersion dependences of permittivity εf and permeability µf of a volume medium of left-handed metamaterial on the frequency of electromagnetic field ω are described by the well-known relations [4] ± ¡ ¢ 2 −1 εf = 1 − ωp2 ω 2 , µf = 1 − F ω 2 ω 2 − ωm , (2) where ωp is a plasma frequency, ωm is a frequency of magnetic resonance, F is a filling factor of metamaterial. The dispersion of material parameters of the usual right-handed material is negligible in comparison with the dispersion of the left-handed material and isn’t taken into consideration below. The dispersion dependences of propagation constants β(ω) of waveguide modes of TE -type for the left-handed film with thickness h = 330 nm in the case of air coating media (εc = 1 and µc = 1) and non-magnetic substrate (µs = 1) with relative permittivity equaled εs = 2 are represented in Fig. 1. The dependences have been calculated for parameters ωp = 3.46·1015 rad/s, ωm = 1.63·1015 rad/s and F = 0, 5 which are in accord with data of papers [1, 5]. In this case the frequency range in which both conditions εf ≤ 0 and µf ≤ 0 are satisfied is within the limits from 1.63·1015 rad/s (the wavelength equals λ = 1.16 µm) to 2.31·1015 rad/s (λ = 0.82 µm). The dispersion dependences β(ω) for modes TE 0 and TE 1 are not only in the range of fast modes (ks < β < kf ) that typically for usual films of right-handed materials but also in the range of slow modes (β > max (kf ; ks )) which is absent in the usual films [6]. At the same time the dependence β(ω) for mode TE 1 is divided into two branches by point with infinite derivative dβ/dω = ∞ in which the group velocity of mode is vanished υg = dω/dβ = 0. The phase and group velocities of mode which correspond to a lower branch of the dispersion dependence have the same directions, and which correspond to a upper branch have opposite directions. The dispersion dependences β(ω) of remaining modes are in the fast range. However, in contrast to usual film waveguides of right- handed materials the dispersion dependences for modes of them are monotone the dependences 1826 PIERS Proceedings, Prague, Czech Republic, July 6–9, 2015 40