We propose several all-pass spectrally-periodic optical structures composed of simple optical cavities for the implementation of repetition rate multipliers of periodic pulse train with uniform output train envelope by phase-only filtering, and analyze them in terms of robustness and accuracy

Muriel, Miguel A.

Preciado, Miguel

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6ª Reunión Española de Optoelectrónica, OPTOEL’09   M. A. Preciado et al. Repetition Rate Multiplication by Phase-only Filtering with Multi-cavity Optical Structures  Miguel A. Preciado and Miguel A. Muriel ETSI Telecomunicacion, Universidad Politecnica de Madrid (UPM), 28040 Madrid, Spain. Contact name: Miguel A. Preciado ma.preciado@upm.es ABSTRACT:  We propose several all-pass spectrally-periodic optical structures composed of simple optical cavities for the implementation of repetition rate multipliers of periodic pulse train with uniform output train envelope by phase-only filtering, and analyze them in terms of robustness and accuracy.  Key words: Talbot and self-imaging effects, Optical resonators, Pulsed laser, All-optical devices, Ultrafast devices  1.- Introduction The generation of periodic pulse trains at repetition rates beyond those achievable by mode locking or direct modulation is highly sought after for future ultrahigh-speed optical communication systems. Several strategies for generating periodic pulse trains at repeti-tion rates beyond those achievable by mode locking or direct modulation have been ex-plored. One alternative is pulse repetition rate multiplication (PRRM) of a lower rate source by applying phase-only spectral filter-ing, usually based on the temporal Talbot effect [1-4].   Fig. 1. Architecture of the system. The peri-odic pulse train is processed by the all-pass optical structure.  We propose and analyze several all-pass spectrally-periodic optical structures of sim-ple optical cavities, concretely ring resona-tors (RRs), for uniform envelope PRRM. 2.- Theory The complex envelope of a periodic input pulse train can be expressed as 1 0( ) ( ) ( )na t a t t nTδ∞=−∞= ⊗ −∑ , where  0 ( )a t  represents the complex envelope of an indi-vidual pulse, ⊗ is the convolution operator, and T is the temporal period of the signal.  Using the Fourier transform, we can express in spectral domain ( )1 0( ) ( )T TmA A mω ω ω ω ω∞=−∞= −∑ , where ωT=2pi/T, and ω is the base-band angular pulsation, i.e., ω=ωopt-ω0, ωopt is the optical angular pulsation, ω0 is the central angular pulsation of the input pulse train.  When a spectrally-periodic optical structure of RRs, with an spectral response H(ω)=H(ω+2piFSR), is applied to the input pulse train 1( )a t, we obtain an output pulse train 2 0( ) ( )nna t C a t nT N∞=−∞= −∑ , with {Cn}=IDFTn{H(2pim/T)}, where FSR is the free spectral range, with FSR≈N/T, N is the desired multiplication factor, IDFTn denotes the n-th inverse discrete Fourier transform [5], {} denotes a sequence of N elements, Cn are complex coefficients, with Cn=Cn+N, and m=1, 2,…, N. We have to impose that all the terms of the sequence {|Cn|} have the maxi-mum uniformity for uniform envelope PRRM. Thus, we can define a figure of merit (FM) for PRRM with uniform envelope as { }( ) { }( )var mean= n nFM C C , where var({Cn}) and mean ({Cn}) denote the va-6ª Reunión Española de Optoelectrónica, OPTOEL’09   M. A. Preciado et al. riance and mean of the sequence, respective-ly, and the function the optimum is FM=0. The variability of the solution can be esti-mated with the gradient magnitude |∇FM|. Both functions, FM and |∇FM| must be taken into account in the optimization, indicating accuracy and robustness respectively. 3.- Proposed Optical Structures The proposed optical structures are com-posed of one or two proposed basic building blocks, which are showed in Fig. 2, and re-spectively consist of a single RR, or two identical RRs in cascade or coupled configu-ration. From the spectral response of a single RR, the spectral responses of cascade and coupled configurations can be obtained, by directly multiplying them in case of cascade configuration, or by using the transfer matrix model method for coupled configuration [6,7]. Each building block can be paramete-rized with k, a, and φ0, where k is the power coupling factor, a=exp(-αLc/2) is the round-trip amplitude transmission factor, α is the power loss coefficient, Lc is the length of the round-trip length, and ( ) ( )0 0φ ω ω ω ω ω φ= = + = +opt FSR FSR FSR  is the round trip phase, where φ0=ω0/FSR is the round-trip phase at ωopt=ω0. Applying the theory previously exposed, it can be demon-strated that a structure composed of one of the previous building blocks can perform 2x, 3x, and 4x uniform envelope PRRM [2,3]. Table 1 shows the optimum structure para-meters set for loss-less filtering (a=1).   Fig. 2. Basic building blocks of the proposed optical structures for uniform PRRM. Moreover, by combining the previous opti-mum building blocks of 2x, 3x, and 4x, we obtain the optical structures showed in Fig. 3, which can perform 6x (2x with 3x building blocks) and 12x (3x with 4x building blocks) uniform envelope PRRM 0. Note that the structures RRs have different sizes, since the building blocks combined have different FSRs. It is worth noting that that required parameters may be a fabrication challenge for current technology, depending on the RRs size required in the concrete application. Thus, RR losses affect significantly to the energy efficiency [8], and it would be desira-ble to obtain RRs with a>0.95.  Table 1. Summary of the Optimum Build-ing Blocks Parameters Corresponding to 2×, 3×, and 4× Uniform Envelope PRRM. N Configuration k φ0 [rad] 2 Single 0.8284 1.571 3 Coupled 0.7393 0.8435 3 Cascade 0.8571 0.3335 4 Coupled 0.8284 0 4 Cascade 0.8284 0   Fig. 3. All-pass optical structures for uniform envelope PRRM for (a) 6× multiplication and  (b) 12× multiplication. The basic building blocks are marked with a dashed box.  4.- Conclusion We have presented a simple method for a uniform envelope PRRM with a high ener-getic efficiency (ideally 100%, with lossless cavities). We have found that the proposed optical structures can achieve several factors of repetition-rate multiplication (2, 3, 4, 6, and 12). Acknowledgements: This work was supported by the Ministerio de Ciencia e Innovacion of Spain under Project “Plan Nacional de I+D+I TEC2007-68065-C03-02”. References [1] J. Azaña and M. A. Muriel, “Temporal Talbot effect in fiber gratings and its applications,” Appl. Opt., vol. 38, pp. 6700-6704,  1999.  [2] M. A. Preciado and M. A. Muriel, “Repetition rate multiplication using a single all-pass optical cavity,” Opt. Lett., vol 33, pp. 962-964, 2008. 6ª Reunión Española de Optoelectrónica, OPTOEL’09   M. A. Preciado et al. [3] M. A. Preciado and M. A. Muriel, "All-pass optical structures for repetition rate multiplication," Opt. Express, vol. 16, pp. 11162-11168, 2008.  [4] M. A. Preciado and M. A. Muriel, "Repetition Rate Multiplication Using All-Pass Optical Structures," Optics & Photonics News, vol. 19, pp.  37, 2008.    [5] A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing (Prentice-Hall ,1999). [6] J. Capmany and M. A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol., vol. 8, pp. 1904–1919, 1990. [7] J. Capmany, P. Muñoz, J. D. Domenech, and M. A. Muriel, “Apodized coupled resonator waveguides,” Opt. Express, vol. 15, pp. 10196-10206, 2007. [8] S. Darmawan and M. K. Chin, “Critical coupling, oscillation, reflection, and transmission in optical waveguide-ring resonator systems,” J. Opt. Soc. Am. B, vol. 23, pp. 834-841, 2006. 

Repetition rate multiplication by phase-only filtering with multi-cavity optical structures

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Repetition rate multiplication by phase-only filtering with multi-cavity optical structures

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