## 4 Conclusions  
In summary, light propagation in quasi-1D PCs has been investigated quantitatively. The transmittances for both the TE and TM modes through a periodic multilayer heterostructure in a rectangular waveguide are calculated using the transfer matrix method. The corresponding band structures are obtained by solving the eigenvalue equations with proper periodic boundary conditions following the Bloch theorem. The formulas for determining the PDOS have been obtained to facilitate identifying the photonic band gaps for all the modes residing in the system. With our approach, the quantitative determination of PDOS can be performed very accurately and efficiently. A complete PBG can exist in these quasi-1D PCs, but the determination must be carefully conducted and verified. It is demonstrated that three types of \"complete PBG\" can be found and categorized as follows. The first type is the frequency range within which the TE and TM modes are both cutoff, the second type is for which the fundamental TE mode has a PBG while other higher order modes are cutoff, and the last type is an overlap of the PBGs of either TE modes, TM modes, or both. The model might be easier for an experimental validation in a millimeter wave frequency range while the optical counterpart might  
Figure 4: Band structures of (**a**) TE\({}_{10}\) and (**b**) TM\({}_{11}\) modes for the quasi-1D PC with the dielectric constants \(\varepsilon_{1}=3.8\), \(\varepsilon_{2}=1\) and the thicknesses \(t_{1}=1.00\) mm, \(t_{2}=3.30\) mm, corresponding to a filling ratio of 23.26%. The PBGs of TE\({}_{10}\) mode show no overlap with those of TM\({}_{11}\) mode.  
Figure 5: PDOS of TE\({}_{10}\), TE\({}_{01}\), TE\({}_{11}\), and TM\({}_{11}\) modes for the quasi-1D PC with the dielectric constants \(\varepsilon_{1}=3.8\), \(\varepsilon_{2}=1\) and the thicknesses \(t_{1}=1.00\) mm, \(t_{2}=3.30\) mm, corresponding to a filling ratio of 23.26%. The combined PDOS of TE\({}_{10}\), TE\({}_{01}\), TE\({}_{11}\), and TM\({}_{11}\) modes shows no photon states in some frequency ranges.  
be possibly pursued as well. We believe these results are of general importance and relevance to the dipole radiation or spontaneous emission by an atom in quasi-1D periodic structures and may have applications in future photonic quantum technologies.  
Conceptualization, M.-C.L.; methodology, R.-F.J. and M.-C.L.; software, R.-F.J. and M.-C.L.; validation, R.-F.J. and M.-C.L.; formal analysis, R.-F.J.; investigation, R.-F.J. and M.-C.L.; resources, R.-F.J. and M.-C.L.; data curation, R.-F.J.; writing--original draft preparation, R.-F.J. and M.-C.L.; writing--review and editing, R.-F.J. and M.-C.L.; visualization, R.-F.J.; supervision, M.-C.L.; project administration, M.-C.L.; funding acquisition, R.-F.J. and M.-C.L.  
This work was partially supported by Guangdong Industry Polytechnic, P. R. China, under Grant Nos. KYRC2018-001 and RC2016-005, the research fund of Hanyang University (HY- 20140000002393), National Research Foundation of South Korea (2015R1D1A1A01061017), and the Alexander von Humboldt Foundation of Germany.  
The authors would like to thank the late [PERSON] at the Institute of Physics, CAS and [PERSON] at the Department of Physics, HKUST for the helpful comments, discussions, and encouragement.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.  
The following abbreviations are used in this manuscript:  
1D one-dimensional  
2D two-dimensional  
3D three-dimensional  
BS band structure  
EM electromagnetic  
PBG photonic band gap  
PC photonic crystal  
PDOS photon density of states  
TE transverse electric  
TM transverse magnetic