Abstract

A new approach towards the design of optimized distributed Bragg reflector (DBR) structures is proposed by taking advantage of recent developments related to the concept of parity-time (PT) in optics. This approach is based on using unidirectional gratings that provide coupling between co-propagating modes. Such couplers with PT symmetric gratings can provide co-directional mode coupling occurring only in one direction. This specific coupling property is achieved through a combined contribution of superimposed index and gain/loss modulations with same grating periodicity, but shifted with respect to one another by a quarter periods. Based on the transfer matrix approach, the transmission and reflection properties of the structure are modeled. One of the unique characteristics of the structure is very low lasing threshold. Such low threshold can be achieved by 100% reflectivity of the both Bragg grating mirrors, and by releasing the amplified signal in one single direction through a PT symmetric grating assisted co-directional coupler. Besides the lasing applications, the proposed structure can be implemented as an optical memory unit of replicating any input optical waveform.

© 2013 Optical Society of America

1. Introduction

Light absorption intrinsically present in optical materials is generally seen as a disadvantage that degrades the device’s performance. However, the balanced spatial alteration of gain and loss regions produces a new type of metamaterials with unique optical characteristics not found in nature [1, 2]. The concept of parity time (PT) symmetry borrowed from quantum field theory [3] has been applied to design this type of optical structures. In optics, PT symmetry requires that the complex refractive index n(r)=nRe(r)+jnIm(r) obeys the condition, n(r)=n(r)i.e. the real index profile must be an even function of position while the gain/loss must be odd. PT-symmetric materials can produce interesting and unusual optical functionalities. These include double refraction and “nonreciprocal” diffraction [2], or unidirectional Bloch oscillations [4]. In particular, Paladian [5] predicted unidirectional diffraction of waveguide reflective Bragg gratings when they combine balanced periodic modulations of refractive index and loss/gain which was later theoretically analyzed in details [6] showing the complete reflectiveness of such Bragg gratings when probed from one side. This zero reflectivity is combined with an ideal signal transmission through the grating with no changes to its phase and amplitude, exhibiting therefore practically a unidirectional invisible medium. On the other side, probing the same grating from the opposite side produces very strong back diffraction combined again with perfect signal integrity transmission. This concept of unidirectionality has been experimentally demonstrated [7]

The same concept of PT-symmetry applied to a mode co-directional interaction [9] produces a unique phenomenon where PT symmetry enables only a unidirectional excitation of mode k by mode l, while blocking the inverse excitation of mode l by mode k. This concept is the basis of the design of unidirectional grating assisted co-directional couplers (GACC) [10].

Along with a great number of publications on different aspects of PT-symmetry and their optical properties, there has been relatively few works [11, 12] on new devices with unique functionalities based on the PT-symmetry properties as well as on innovative ideas how these properties can qualitatively improve characteristics of the existing photonic devices.

In this article we will focus on improvement to distributed Bragg reflector lasers through implementing PT-symmetry.

2. Grating assisted codirectional coupler with PT-symmetry

The GACC is shown schematically in Fig. 1. It consists of two asynchronous waveguides (i.e. the propagation constants of both waveguides are not the same) which can interact over a distance LG via a complex grating coupler having index and gain/loss modulations. The waveguides are single mode ones. The propagation constants of the fundamental propagating modes are respectively β1 and β2, where it will be assumed that β1 is greater than β2.

 

Fig. 1 Grating assisted codirectional coupler (GACC). The propagation constants of each waveguides are given by βi . The coupler is asynchronous 1 ≠ β2). The grating length is given by LG.

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The complex grating introduces a perturbation to the complex refractive index, as follows:

Δn=Δn1cos(kz)jΔα1sin(kz),
where Δn1 and Δα1 are respectively modulation amplitudes of the refractive index and of the loss/gain, k = 2π/ΛG – is the wave number, and ΛG is the grating period, j = (−1)1/2. It is important to point out that Eq. (1) does not include gain or losses saturation. The perturbation given by Eq. (1) will enable the coupling between the optical propagating modes in both channel waveguides by matching the propagating constants: β1β2=2π/ΛG.

Assuming that the input plane defined by z = 0 is located at the beginning of the grating in ports A and B, the matrix describing the transmission through the grating of the length LG is described by the following transfer matrix:

[DC]=[Т11Т12Т21Т22][AB],
where the transmission matrix T is linking the fields between input ports A and B and output ports C and D and has following elements [11]:
T11=[cos(γGLG)+jΔβγGsin(γGLG)]exp[j(β1Δβ)LG],T12=jκ12γGsin(γGLG)exp[j(β1Δβ)LG],T21=jκ21γGsin(γGLG)exp[j(β2+Δβ)LG],T22=[cos(γGLG)jΔβγGsin(γGLG)]exp[j(β+2Δβ)LG],
where Δβ=(β1β2)/2π/ΛGand γG=((Δβ)2+κ12κ21)1/2. The coupling coefficients κ12and κ21 are proportional to the overlapping between the spatial mode distributions of each waveguide and the index modulation component of Eq. (1): κ12=κnκαandκ21=κn+κα, where κnand καare contributions to the coupling coefficients κ12and κ21from the index and gain/loss gratings, respectively.

If a signal is injected in Port A, then the resulting signal power in Port C is given by |T21|2 and in Port D by |T11|2. On the other hand, if the signal is injected in Port B, then the signal power in Port C is given by |T22|2 and in Port D by |T12|2. In order to fully describe the uni-directionality of the GACC, the transmission matrix T is inverted in order to compute the propagation from Ports C and D toward Ports A and B. Therefore, the transmission matrix for the case where the signal is launched in Port C or D is given by the inverse expression of Eq. (3):

T111=[cos(γGLG)jΔβγGsin(γGLG)]exp[j(β1Δβ)LG],T121=jκ12γGsin(γGLG)exp[j(β2+Δβ)LG],T211=jκ21γGsin(γGLG)exp[j(β1Δβ)LG],T221=[cos(γGLG)+jΔβγGsin(γGLG)]exp[j(β2+Δβ)LG].
In this case, if a signal is injected in Port C, then the signal power in Port A is given by |T12−1|2 and in Port B by |T22−1|2. And finally, if the signal is launched in Port D, the signal power in Port A is given by |T11−1|2 and in Port B by |T21−1|2. The correspondent bar and cross transmission spectra are shown in Fig. 2(a).

 

Fig. 2 Transmission spectra of a traditional grating assisted coupler(a) where κα = 0 and (b) uni-directional grating assisted coupler where κn = κα GACCs for the bar-state (solid, red) and the cross-state (dash, blue) for κ12LG = π/2 and LG = 30 mm and the central wavelength of 1550 nm. The signal is launched into Port A.

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In the case of PT symmetry breaking point, when the modulation of the real grating is equal to the modulation of the imaginary grating Δn1=Δα1, and the coupling coefficient κ12 is cancelled: κnκα = 0, the transfer matrix takes the following form:

T=[exp[jβ1LG]0jκ21Δβsin(ΔβL)Gexp[j(β1+β22πΛG)LG]exp[jβ2LG]].
The most noticeable distinction between these two spectral behaviors is that the uni-directional design allows the coupling of signal into the neighbor waveguide without depletion. Furthermore, it is worth noting that the power transferred from Port A and Port C is wavelength dependent. At the same time, the transmission in the excited channel between Port A and Port D remains unaffected by the grating. Indeed, as it can be seen from the transfer matrix Eq. (5), light propagates between Port A and Port D as if there is no modulation in the GACC. It is interesting to note that the power conservation law is not violated since the complex grating is an active structure and power is supplied to maintain optical gain.

If the signal is injected into Port C or D, the transfer matrix becomes:

T1=[exp[jβ1LG]0jκ21Δβsin(ΔβLG)exp[j(β1+β22+πΛG)LG]exp[jβ2LG]].
The signal is propagating unaffected through the upper guide of Fig. 1 between Port B and C, as if the grating would not be there. This can be seen by looking at the matrix elements of Eqs. (5) and (6). In the case where the signal enters Port B, since T12 = 0, therefore the power at Port D will be zero showing no coupling between the two waveguides. In the case where the signal enters Port C, since T12−1 = 0, the power at Port A will be zero showing again no coupling between the two waveguides. The correspondent bar and cross transmission spectra are shown in Fig. 2(b).

Such unique transmission property of PT-symmetric GACC has the potential to create numerous opportunities in designing either new devices with very unique functionalities, (as our proposal of optical memory cell [11]), or in a substantial improvement of existing Planar Lightwave Circuits (PLC). In this paper we propose a design that can significantly reduce the threshold power of DBR lasers.

3. Threshold condition in DBR lasers

In DBR lasers two Bragg gratings are used to form a cavity in a fiber or waveguide section with active amplifying medium. The grating sections act as high-reflectance “mirrors” for waveguide modes at frequencies within the grating spectral reflection bandwidth. To get better physical insight, it is illustrative to consider the two grating reflectors with complex reflectances r1,2(ω)=|r1,2(ω)|exp(jφ1,2(ω)) and look for the threshold condition by setting the round trip gain equal to zero:

|r1(ω)||r2(ω)|exp(j(φ1(ω)+φ2(ω))exp(2d(g0(ω)α0(ω)))exp(2jβ(ω)d)=1
Whereas the imaginary part of this equation is responsible for the lasing frequency (ωi) selection, the real regulates the power balance between the gain and loss in the DBR structure:
g0(ωі)α0(ωі)=12dln(1|r1(ωі)||r2(ωі)|)
where g0 is the total generated gain and α0 is the compound losses in the DBR (absorption, scattering losses etc.). As we can see from Eq. (8), the lowest lasing threshold will occurs when the gratings provides 100% reflection, i.e.|r1(ω)|=|r2(ω)|=1. Obviously such arrangement is not possible in the case of regular index modulated Bragg gratings because no light will be released from such resonator.

4. DBR lasers with PT-symmetrical GACC output

Here a traditional GACC architecture is used in combination with a DBR laser for tuning purposes [12, 13], as shown in Fig. 3(a). Although this traditional GACC allows the use of Bragg gratings with |r1(ω)|=|r2(ω)|=1 and output of lasing power through the GACC, it does not really help to reduce the lasing power condition as it removes a portion of the amplifying signal from the resonator every time the wave propagates from left to right and from right to left. Besides, lasing will be split between the left and right GACC ports.

 

Fig. 3 Schematic structure of a DBR laser with (a) traditional index grating GACC and (b) PT-symmetric GACC as well as mode interaction pattern.

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The situation is completely different when the traditional GACC is replaced by the PT-symmetric GACC, as shown in Fig. 3(b). The power is here coupled out from the cavity into the left port only when the wave is traveling from right to left. There is no coupling into the GACC from the cavity wave travelling in opposite direction. As follows from the PT symmetrical GACC characteristics, the coupling is taking place without depleting the power of the donor wave, and at the same time the output signal can also be amplified if κ21LG> 1. Such configuration allows 100% reflection of the both Bragg gratings |r1(ω)|=|r2(ω)|=1and provides the lowest possible lasing threshold defined only by compound losses in the cavity. Such improvement is achieved by adding the active structure – PT symmetric GACC, however, its coupling characteristics can be controlled independently from the gain inside the cavity. One could argue that the same effect might be achieved by adding extra gain stages into the arms of a conventional non-PT coupler. Indeed, using this approach the same power output might be achieved above the threshold, but it would not trigger any lasing at lower pumping powers within the cavity. As can be seen from Eq. (7), the threshold is solely defined by the active cavity, any external amplification will not affect the threshold level. The PT-symmetric grating reduces the threshold because this grating is integrated into the cavity.

Another advantage of the proposed configuration is that it has an additional degree of freedom in its design. The second waveguide where the lasing power is coupled out can be designed to match the mode geometry of the output device (optical fiber, microchip etc.). As we can see, the proposed design provides significant operational improvement but will require a more in-depth mathematical analysis.

5. Transfer matrix description of the DBR structure with PT-symmetrical GACC output

Coupling a PT-symmetric GACC with a DBR or Fabry-Perot cavity can be presented in a broader sense as shown in Fig. 4, where it can be employed also below, or very close, to the lasing condition. An input signal can then be injected and retrieved through the GACC port.

 

Fig. 4 Schematic structure of a Fabry-Perot cavity based signal repeater with PT-symmetric co-directional coupler as well as mode interaction patterns.

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With the origin of the coordinate system located at the left side of the left Bragg grating (see Fig. 4) the transfer matrix equation of the left Bragg grating can be presented in the following form:

B(+)(L)B()(L)=r(+)=m12(1)m22(1),
where mij(1) is the transfer matrix elements of the left Bragg grating, and B(+) and B() are the cavity waves propagating from left to right and from right to left, respectively. A similar equation for the right Bragg grating can be written as follow:
B()(L+d)B(+)(L+d)=r()=m21(2)m22(2)
The GACC relates theB(+) and B() waves to the waves A(+) and A() propagating in the lower guide from left to right and from right to left respectively. For the waves propagating from left to right with an input signalA(+)(0)=1 or A(+)(L+d1)=exp(jβ1(L+d1)) the transfer matrix equation becomes:
[A(+)(L+d1+LG)B(+)(L+d1+LG)]=[Т11Т12Т21Т22][exp(jβ1(L+d1))B(+)(L+d1)],
and for the waves propagation from right to left it takes on the following form
[A()(L+d1)B()(L+d1)]=[Т111Т121Т211Т221][0B()(L+d1+LG)].
where
B(+)(L+d1)=B(+)(L)exp(jβ2d1),B(+)(L+d)=B(+)(L+d1+LG)exp(jβ2(d1LG)),B()(L+LG+d1)=B(+)(L+d)exp(jβ2(d2LG)),
Equations (9)-(13) can be solved in respect to the output signal
A()(L+d1)=T211T21m21(2)m22(2)exp(jβ2(d2LG+d2))1T111T22m21(2)m12(1)m22(2)m22(1)exp(2jβ2(dLG)).
Unlike in the traditional (index) GACC, in the case of PT-symmetric GACC the bar transmission occurs without any loss, therefore leaving the input signal unchanged:
Т111=exp(jβ2LG);T22=exp(jβ2LG),
and Eq. (14) is simplified to the following form:
A()(L+d1)=κ122Δβ2r()sin2(ΔβLG)exp(j(β1β2)LG)exp(2jβ2d)1r()r(+)exp(2jβ2d),
Using well-known expansion formula:
n=0xn=11x
It can also be presented in the following series form
A()(L+d1)=κ122Δβ2r()sin2(ΔβLG)exp(j(2β2d+(β1β2)LG))m=0(r()r(+))mexp(2jmβ2d),
The spectral response in this form, unlike the one presented by Eq. (16), remains valid even for the case of operation above the threshold, i.e
|r()r(+)exp(2jmβ2d)|>1,
In the case of identical Bragg gratings forming the Fabry-Perot cavity
r(±)=j(κ/γ)sinh(γL)cosh(γL)j(σ/γ)sinh(γL)
whereγ=(κ2σ2)1/2, mismatch factor σ=β2π/Λ, κ and Λ are the coupling coefficients and the period of the Bragg gratings, respectively. In the proposed configuration both Bragg gratings can have 100% reflection, and according to Eq. (19) the lasing starts as soon as total gain in the resonator exceed the total propagation losses.

The transmission (A(-)(L + d1)) of the DBR structure with PT-symmetric coupling shown in Fig. 4 is presented in Fig. 5(a). The resulting spectrum is typical of a Fabry-Perot cavity formed by two identical Bragg gratings. The reflection spectrum of these identical Bragg gratings is shown in Fig. 5(b) by the red (solid) curve. They provide 100% reflection at the resonance wavelength (1550 nm), with the Bragg grating length L = 1 mm, the grating strength κL = π and the period 0.5 µm. The cavity length (d) is 20 mm. In our simulation we neglected the material dispersion and consider the propagation constant in the cavity guide as β2 = 2πn2/λ, where the effective refractive index was chosen to be n2 = 1.55. In the portion of the guide between the Bragg gratings there is uniform gain α = 2.206 mm−1. The PT-symmetric co-directional coupling is taking place within the guide with β1 = 2πn1/λ with n1 = 1.4875. Its cross transmission spectrum is shown by the blue (dashed) curve in Fig. 5(b). This PT-symmetric coupling structure is built by a hybrid grating composed of both phase and amplitude modulations with period ΛG = 24.8 µm, length LG = 5 mm and grating strength κGLG = 0.446π = 1.402. As one can see from Fig. 5(b), such grating produces nearly a 100% signal amplification at the resonance wavelength.

 

Fig. 5 (a) Output transmission spectrum of the proposed Fabry-Perot structure with PT-symmetric coupling. (b) The reflection spectrum of two identical Bragg grating-reflectors (red, solid), cross-transmission spectrum of the PT-symmetric codirectional coupler (blue, dash) and the spectrum of the input Gaussian pulse 10 ps FWHM.

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The temporal response of the proposed Fabry-Perot structure with PT-symmetric coupling to an optical signal launched into the lower left branch of the coupler can be calculated by taking the inverse Fourier transform (F−1{…}) of the product of the input pulse spectrum Hin(ω) by the spectral transmission response (Eq. (18)):

a()(L+d1,t)=F1{Hin(ω)A()(L+d1,ω)}
We assume that the input optical signal (centered at the resonance wavelength of the gratings λ0 = 1.55µm) has a Gaussian envelope |hin(t)|2 with a full width half-maximum (FWHM) time width τ0 and without initial chirp:
hin(t)=exp[t2τ022ln(2)]exp(jω0t)
The Fourier transform Hin(ω) of this pulse is given by:
Hin(ω)=(πτ022ln(2))1/2exp[τ028ln(2)(ωω0)2]
The temporal response on a single Gaussian pulse with duration τ0 = 10 ps is shown in Fig. 6. As one can see, the PT-symmetric coupler produces two signal replicas, one will exit through the bottom right branch A( + ), and the second one will be amplified and coupled into the Fabry-Perot cavity. In the cavity it will reach the right Bragg grating which reflects the signal back towards the left Bragg grating. Reaching the PT-symmetric coupler on its way back, it will be amplified again and coupled back into the lower left branch of the coupler producing a first replica of the input signal (numbered #“1” in Fig. 6(a)). At the same time, due to the PT-symmetry, another replica of the pulse will pass through the coupler grating and reach the left Bragg mirror which reflects the signal back into the opposite direction. This time it will pass the coupler without any changes, since there is no coupling occurring from this direction of incidence. Then again, the pulse will reach the right Bragg mirror and will be reflected again, and again it will be coupled into the output producing the replica #“2” in Fig. 6(a). This process will be repeated over and over, producing replicas the first ten of which are shown in Fig. 6(a) and 508th through-518th in Fig. 6(b). Each replica is separated by the time it takes to cover the distance between the Bragg gratings in the right and left directions: 2dn2/c = 206.8 ps, where c is the light velocity in vacuum. In reality they are separated by ~210 ps, because the pulse penetrates certain distance into the Bragg grating before it is fully reflected.

 

Fig. 6 Temporal response of the proposed Fabry-Perot structure with PT-symmetric coupling to an input Gaussian pulse 10 ps FWHM. (a) First ten replicas and (b) 508th – 518th replicas.

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As we can see, these echo-signals are not exact replicas of the input signal. Whereas the pulse amplitudes are practically the same due to amplification in the waveguide between the Bragg mirrors, their shape gradually evolves, producing a spread which can be clearly seen in Fig. 6(b). The main distinction between the pulses is the apparition of four small pulses trailing the main pulse. The apparition of these pulses can be explained by out of resonance reflection from the Bragg grating ends. Small portions of the signal mainly outside of the resonance wavelength, λ0 = 1.55µm, are not fully reflected by the Bragg grating propagating inside this grating and then being reflected from its external ends. These small reflected pulses take longer to propagate larger distance thus producing the main pulse trailing signals. They can be easily suppressed, if the Bragg gratings are apodised to avoid the reflection from their ends. Another distinction between the first 10 pulses and the five hundredth one is the temporal width. The longer the signal stays in the cavity, the broader it becomes due to the waveguide and material (neglected here) dispersions as well as the filtering effect of the Bragg mirrors. This problem also can be resolved by inserting a dispersion compensation element into the cavity.

This same structure could be used also to produce an echo a data sequence. If instead of a single pulse, a pulse train with an encoded byte of information is injected, it can be used as an optical memory buffer that periodically echoes the stored information. The time duration over which the information is stored is limited by the pulse broadening discussed previously and the ability to compensate the dispersion preventing each individual pulse from broadening.

6. Conclusions

We analyzed a novel structure consisting of a DBR resonator connected uni-directionally to a parallel guide through a PT symmetrical grating. Such grating assisted coupler provides co-directional mode coupling in only one direction. Such uni-directional coupling is achieved through a combined contribution of superimposed index and gain/loss modulations with the same spatial period shifted in respect to one another by a quarter period. A transfer matrix approach has been used to analyze its spectral property. It has been shown that one of the unique characteristics of the structure is its lowest possible lasing threshold, due to 100% reflectivity of both Bragg grating mirrors and to releasing the lasing power uni-directionally through the PT symmetric grating assisted co-directional coupler. Besides the lasing applications, the proposed structure could perform as an optical memory unit or a data buffer unit that is capable to replicating an input waveform.

Another unique characteristic when comparing it to ‘traditional’ DBR structures is that any incident signal enters uni-directionally into the cavity, suggesting the development of potential interesting applications such as optical memory or other signal processing functionalities. We found that the limitation of the duration over which the information could be stored inside such novel resonator structure is given by the unique spectral filtering characteristics of the gratings used to form the cavity. There are numerous way to design specific filtering characteristics from various type of Bragg reflector gratings. We believe that by integrating a suitable non-linearity in the system, the temporal broadening of the signal replicas could be overcome.

Although we discuss only some of the unique characteristics of these novel structures, we believe there may be also other interesting properties that could be of interest to other fields of applications.

References and links

1. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007). [CrossRef]   [PubMed]  

2. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008). [CrossRef]   [PubMed]  

3. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonian having PT symmetry,” Phys. Rev. Lett. 80(24), 5243 (1998). [CrossRef]  

4. E. M. Graefe and H. F. Jones, “PT-symmetric sinusoidal optical lattices at the symmetry breaking threshold,” Phys. Rev. A 84(1), 013818 (2011). [CrossRef]  

5. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996). [CrossRef]   [PubMed]  

6. M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13(8), 3068–3078 (2005). [CrossRef]   [PubMed]  

7. L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012). [CrossRef]   [PubMed]  

8. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012). [CrossRef]   [PubMed]  

9. M. Greenberg and M. Orenstein, “Irreversible coupling by use of dissipative optics,” Opt. Lett. 29(5), 451–453 (2004). [CrossRef]   [PubMed]  

10. M. Greenberg and M. Orenstein, “Unidirectional complex grating assisted couplers,” Opt. Express 12(17), 4013–4018 (2004). [CrossRef]   [PubMed]  

11. M. Kulishov, J. M. Laniel, N. Bélanger, and D. V. Plant, “Trapping light in a ring resonator using a grating-assisted coupler with asymmetric transmission,” Opt. Express 13(9), 3567–3578 (2005). [CrossRef]   [PubMed]  

12. M. A. Miri, P. LiKamWa, and D. N. Christodoulides, “Large area single-mode parity-time-symmtric laser amplifiers,” Opt. Lett. 37, 764–766 (2012).

13. P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett. 7(7), 697–699 (1995). [CrossRef]  

14. L. A. Coldren, G. A. Fish, Y. Akulova, J. S. Barton, L. Johansson, and C. W. Coldren, “Tunable semiconductor lasers: tutorial,” J. Lightwave Technol. 22(1), 193–202 (2004). [CrossRef]  

References

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  1. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
    [CrossRef] [PubMed]
  2. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
    [CrossRef] [PubMed]
  3. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonian having PT symmetry,” Phys. Rev. Lett. 80(24), 5243 (1998).
    [CrossRef]
  4. E. M. Graefe and H. F. Jones, “PT-symmetric sinusoidal optical lattices at the symmetry breaking threshold,” Phys. Rev. A 84(1), 013818 (2011).
    [CrossRef]
  5. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
    [CrossRef] [PubMed]
  6. M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13(8), 3068–3078 (2005).
    [CrossRef] [PubMed]
  7. L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
    [CrossRef] [PubMed]
  8. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
    [CrossRef] [PubMed]
  9. M. Greenberg and M. Orenstein, “Irreversible coupling by use of dissipative optics,” Opt. Lett. 29(5), 451–453 (2004).
    [CrossRef] [PubMed]
  10. M. Greenberg and M. Orenstein, “Unidirectional complex grating assisted couplers,” Opt. Express 12(17), 4013–4018 (2004).
    [CrossRef] [PubMed]
  11. M. Kulishov, J. M. Laniel, N. Bélanger, and D. V. Plant, “Trapping light in a ring resonator using a grating-assisted coupler with asymmetric transmission,” Opt. Express 13(9), 3567–3578 (2005).
    [CrossRef] [PubMed]
  12. M. A. Miri, P. LiKamWa, and D. N. Christodoulides, “Large area single-mode parity-time-symmtric laser amplifiers,” Opt. Lett. 37, 764–766 (2012).
  13. P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett. 7(7), 697–699 (1995).
    [CrossRef]
  14. L. A. Coldren, G. A. Fish, Y. Akulova, J. S. Barton, L. Johansson, and C. W. Coldren, “Tunable semiconductor lasers: tutorial,” J. Lightwave Technol. 22(1), 193–202 (2004).
    [CrossRef]

2012 (3)

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[CrossRef] [PubMed]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[CrossRef] [PubMed]

M. A. Miri, P. LiKamWa, and D. N. Christodoulides, “Large area single-mode parity-time-symmtric laser amplifiers,” Opt. Lett. 37, 764–766 (2012).

2011 (1)

E. M. Graefe and H. F. Jones, “PT-symmetric sinusoidal optical lattices at the symmetry breaking threshold,” Phys. Rev. A 84(1), 013818 (2011).
[CrossRef]

2008 (1)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[CrossRef] [PubMed]

2007 (1)

2005 (2)

2004 (3)

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonian having PT symmetry,” Phys. Rev. Lett. 80(24), 5243 (1998).
[CrossRef]

1996 (1)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
[CrossRef] [PubMed]

1995 (1)

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett. 7(7), 697–699 (1995).
[CrossRef]

Akulova, Y.

Almeida, V. R.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[CrossRef] [PubMed]

Azaña, J.

Backbom, L.

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett. 7(7), 697–699 (1995).
[CrossRef]

Barton, J. S.

Bélanger, N.

Bender, C. M.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonian having PT symmetry,” Phys. Rev. Lett. 80(24), 5243 (1998).
[CrossRef]

Berglind, E.

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett. 7(7), 697–699 (1995).
[CrossRef]

Bersch, C.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[CrossRef] [PubMed]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonian having PT symmetry,” Phys. Rev. Lett. 80(24), 5243 (1998).
[CrossRef]

Chen, Y.-F.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[CrossRef] [PubMed]

Christodoulides, D. N.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[CrossRef] [PubMed]

M. A. Miri, P. LiKamWa, and D. N. Christodoulides, “Large area single-mode parity-time-symmtric laser amplifiers,” Opt. Lett. 37, 764–766 (2012).

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[CrossRef] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[CrossRef] [PubMed]

Coldren, C. W.

Coldren, L. A.

El-Ganainy, R.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[CrossRef] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[CrossRef] [PubMed]

Fegadolli, W. S.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[CrossRef] [PubMed]

Feng, L.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[CrossRef] [PubMed]

Fish, G. A.

Graefe, E. M.

E. M. Graefe and H. F. Jones, “PT-symmetric sinusoidal optical lattices at the symmetry breaking threshold,” Phys. Rev. A 84(1), 013818 (2011).
[CrossRef]

Greenberg, M.

Johansson, L.

Jones, H. F.

E. M. Graefe and H. F. Jones, “PT-symmetric sinusoidal optical lattices at the symmetry breaking threshold,” Phys. Rev. A 84(1), 013818 (2011).
[CrossRef]

Klinga, T.

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett. 7(7), 697–699 (1995).
[CrossRef]

Kulishov, M.

Laniel, J. M.

LiKamWa, P.

Lu, M.-H.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[CrossRef] [PubMed]

Makris, K. G.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[CrossRef] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[CrossRef] [PubMed]

Miri, M. A.

M. A. Miri, P. LiKamWa, and D. N. Christodoulides, “Large area single-mode parity-time-symmtric laser amplifiers,” Opt. Lett. 37, 764–766 (2012).

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[CrossRef] [PubMed]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[CrossRef] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[CrossRef] [PubMed]

Nilsson, S.

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett. 7(7), 697–699 (1995).
[CrossRef]

Oliveira, J. E.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[CrossRef] [PubMed]

Onishchukov, G.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[CrossRef] [PubMed]

Orenstein, M.

Peschel, U.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[CrossRef] [PubMed]

Plant, D. V.

Poladian, L.

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
[CrossRef] [PubMed]

Regensburger, A.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[CrossRef] [PubMed]

Rigole, P.-J.

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett. 7(7), 697–699 (1995).
[CrossRef]

Scherer, A.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[CrossRef] [PubMed]

Stalnacke, B.

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett. 7(7), 697–699 (1995).
[CrossRef]

Stoltz, B.

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett. 7(7), 697–699 (1995).
[CrossRef]

Wallin, J.

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett. 7(7), 697–699 (1995).
[CrossRef]

Xu, Y.-L.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[CrossRef] [PubMed]

IEEE Photon. Technol. Lett. (1)

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett. 7(7), 697–699 (1995).
[CrossRef]

J. Lightwave Technol. (1)

Nat. Mater. (1)

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[CrossRef] [PubMed]

Nature (1)

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. A (1)

E. M. Graefe and H. F. Jones, “PT-symmetric sinusoidal optical lattices at the symmetry breaking threshold,” Phys. Rev. A 84(1), 013818 (2011).
[CrossRef]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[CrossRef] [PubMed]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonian having PT symmetry,” Phys. Rev. Lett. 80(24), 5243 (1998).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Grating assisted codirectional coupler (GACC). The propagation constants of each waveguides are given by βi . The coupler is asynchronous 1 ≠ β2). The grating length is given by LG.

Fig. 2
Fig. 2

Transmission spectra of a traditional grating assisted coupler(a) where κα = 0 and (b) uni-directional grating assisted coupler where κn = κα GACCs for the bar-state (solid, red) and the cross-state (dash, blue) for κ12LG = π/2 and LG = 30 mm and the central wavelength of 1550 nm. The signal is launched into Port A.

Fig. 3
Fig. 3

Schematic structure of a DBR laser with (a) traditional index grating GACC and (b) PT-symmetric GACC as well as mode interaction pattern.

Fig. 4
Fig. 4

Schematic structure of a Fabry-Perot cavity based signal repeater with PT-symmetric co-directional coupler as well as mode interaction patterns.

Fig. 5
Fig. 5

(a) Output transmission spectrum of the proposed Fabry-Perot structure with PT-symmetric coupling. (b) The reflection spectrum of two identical Bragg grating-reflectors (red, solid), cross-transmission spectrum of the PT-symmetric codirectional coupler (blue, dash) and the spectrum of the input Gaussian pulse 10 ps FWHM.

Fig. 6
Fig. 6

Temporal response of the proposed Fabry-Perot structure with PT-symmetric coupling to an input Gaussian pulse 10 ps FWHM. (a) First ten replicas and (b) 508th – 518th replicas.

Equations (23)

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Δn=Δ n 1 cos(kz)jΔ α 1 sin(kz),
[ D C ]=[ Т 11 Т 12 Т 21 Т 22 ][ A B ],
T 11 =[ cos( γ G L G )+j Δβ γ G sin( γ G L G ) ]exp[ j( β 1 Δβ ) L G ], T 12 =j κ 12 γ G sin( γ G L G )exp[ j( β 1 Δβ ) L G ], T 21 =j κ 21 γ G sin( γ G L G )exp[ j( β 2 +Δβ ) L G ], T 22 =[ cos( γ G L G )j Δβ γ G sin( γ G L G ) ]exp[ j( β + 2 Δβ ) L G ],
T 11 1 =[ cos( γ G L G )j Δβ γ G sin( γ G L G ) ]exp[ j( β 1 Δβ ) L G ], T 12 1 =j κ 12 γ G sin( γ G L G )exp[ j( β 2 +Δβ ) L G ], T 21 1 =j κ 21 γ G sin( γ G L G )exp[ j( β 1 Δβ ) L G ], T 22 1 =[ cos( γ G L G )+j Δβ γ G sin( γ G L G ) ]exp[ j( β 2 +Δβ ) L G ].
T=[ exp[ j β 1 L G ] 0 j κ 21 Δβ sin(ΔβL ) G exp[ j( β 1 + β 2 2 π Λ G ) L G ] exp[ j β 2 L G ] ].
T 1 =[ exp[ j β 1 L G ] 0 j κ 21 Δβ sin(Δβ L G )exp[ j( β 1 + β 2 2 + π Λ G ) L G ] exp[ j β 2 L G ] ].
| r 1 (ω)|| r 2 (ω)|exp(j( φ 1 (ω)+ φ 2 (ω))exp(2d( g 0 (ω) α 0 (ω)))exp(2jβ(ω)d)=1
g 0 ( ω і ) α 0 ( ω і )= 1 2d ln( 1 | r 1 ( ω і )|| r 2 ( ω і )| )
B (+) (L) B () (L) = r (+) = m 12 (1) m 22 (1) ,
B () (L+d) B (+) (L+d) = r () = m 21 (2) m 22 (2)
[ A (+) (L+ d 1 + L G ) B (+) (L+ d 1 + L G ) ]=[ Т 11 Т 12 Т 21 Т 22 ][ exp(j β 1 (L+ d 1 )) B (+) (L+ d 1 ) ],
[ A () (L+ d 1 ) B () (L+ d 1 ) ]=[ Т 11 1 Т 12 1 Т 21 1 Т 22 1 ][ 0 B () (L+ d 1 + L G ) ].
B (+) (L+ d 1 )= B (+) (L)exp(j β 2 d 1 ), B (+) (L+d)= B (+) (L+ d 1 + L G )exp(j β 2 ( d 1 L G )), B () (L+ L G + d 1 )= B (+) (L+d)exp(j β 2 ( d 2 L G )),
A () (L+ d 1 )= T 21 1 T 21 m 21 (2) m 22 (2) exp(j β 2 (d2 L G + d 2 )) 1 T 11 1 T 22 m 21 (2) m 12 (1) m 22 (2) m 22 (1) exp(2j β 2 (d L G )) .
Т 11 1 =exp(j β 2 L G ); T 22 =exp(j β 2 L G ),
A () (L+ d 1 )= κ 12 2 Δ β 2 r () sin 2 (Δβ L G )exp(j( β 1 β 2 ) L G )exp(2j β 2 d) 1 r () r (+) exp(2j β 2 d) ,
n=0 x n = 1 1x
A () (L+ d 1 )= κ 12 2 Δ β 2 r () sin 2 (Δβ L G )exp(j(2 β 2 d+( β 1 β 2 ) L G )) m=0 ( r () r (+) ) m exp(2jm β 2 d) ,
| r () r (+) exp(2jm β 2 d) |>1,
r (±) = j(κ/γ)sinh(γL) cosh(γL)j(σ/γ)sinh(γL)
a () (L+ d 1 ,t)= F 1 { H in (ω) A () (L+ d 1 ,ω)}
h in (t)=exp[ t 2 τ 0 2 2ln(2) ]exp(j ω 0 t)
H in (ω)= ( π τ 0 2 2ln(2) ) 1/2 exp[ τ 0 2 8ln(2) (ω ω 0 ) 2 ]

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