G. Jacobsen

Theory for optical heterodyne narrow-deviation FSK receivers with delay demodulation

A theoretical model is presented that includes the effects of laser phase noise, receiver noise, imperfect modulation, IF bandwidth, and postdetection filtering. Detailed numerical results for 140-Mb/s and 400-Mb/s systems are presented, showing excellent agreement with independent published experimental results and strongly supporting the theoretical analysis. It is found that an IF linewidth of less than 0.25% of bit rate is required to avoid degrading the receiver sensitivity by more than 1 dB in a system with a strong local oscillator and modulation index of 0.7. A larger modulation index allows a larger linewidth to be accommodated. If the demodulation is not optimal, a narrower linewidth is necessary. >

Performance of DPSK and CPFSK systems with significant post-detection filtering

A rigorous model for preamplifier implemented DPSK and CPFSK systems with delay-detection and postdetection filtering is presented. The system performance is impaired by amplifier spontaneous emission noise as well as laser phase noise. A best sensitivity for the system is 20 (DPSK) and 36 (CPFSK) received signal photons for a 10/sup -9/ bit error probability. It is found that, using the CPFSK system, significant amounts of phase noise can be tolerated provided that sufficient bandpass bandwidth is allocated and a large modulation index value is used. A linewidth as large as five times the bit rate causes only a 4.2-dB penalty for a receiver employing polarization control and a 5.3-dB penalty for a polarization diversity receiver. >

Propagation constants and group delays of guided modes in graded-index fibers: a comparison of three theories

For a general class of graded-index fiber profiles analytical expressions for propagation constants of guided modes have been evaluated by the evanescent field theory to the order 0(k(-9)) in asymptotic form. The results are compared with the analytical result of the third order perturbation theory. Analytical asymptotic expressions for group delays are also derived. The asymptotic expressions have been applied for a numerical comparison with the WKB theory and the perturbation theory to the third order. Herefrom, we have estimated the accuracy of the WKB calculations of group delays to be approximately 10 psec/km for near parabolic profiles. We show that the evanescent field theory gives a very accurate determination of the propagation constants and group delays of the lower order modes in a near parabolic profile. For profiles which are far from a parabolic shape, we find that the perturbation theory gives wrong results and that the asymptotic error in the evanescent field theory increases. The accuracy of the WKB theory is found to be within the asymptotic error for higher order modes of near-parabolic profiles and all modes of profiles which are far from a parabolic shape.

Modal propagation constants, group delays, and eigenfields for practical multimode graded-index fibers

Using a Rayleigh–Ritz formulation, modal propagation constants, group delays, and eigenfields are evaluated for practical multimode graded-index profiles by a solution of the scalar-wave equation. The fiber-index core may consist of a Taylor port, a central Gaussian dip, and damped ripples and is surrounded by a uniform cladding. Laguerre Gaussian expansion functions are applied, permitting analytical simplification of the general formulation. For a graded-index fiber with dip and V value of 20.8, modal group delays are determined with an accuracy of the order of 0.5 psec/km. For a larger fiber with V = 69, the method permits accurate and fast interpretation of differential-mode delay measurements.

Evanescent-wave and nonlinear transformation analysis of graded-index fibers: errata

For a general class of graded-index fiber profiles, modal propagation constants and group delays were previously evaluated asymptotically to O(k−9) by using the systematic calculation scheme of evanescent wave theory. Here, the numerical results obtained directly are improved by using a nonlinear transformation technique developed by Shanks. This technique extracts information from the diverging as well as converging part of an asymptotic series, whereas the direct method utilizes the converging part alone. We obtain accurate results for a profile far from parabolic shape as well as for one of near-parabolic shape. For the profile far from parabolic shape the results quantify the error in group delay for the WKB technique for the lowest-order mode to approximately 30 ps/km. For high-order modes the WKB results agree with the transformed results.

Transfer function of a multimode step-index fibre: a geometrical optical investigation

The transfer function of a multimode step-index fibre has been calculated by means of geometrical optics taking into account mode coupling and leaky rays. The light source may have axial symmetry (light emitting diode) or be line shaped (semiconductor laser). Radial displacement of the line source relative to the fibre axis is allowed. Calculations performed for a large numerical aperture fibre (N.A.=0.475) with a coupling length of 2 km show that leaky rays and radial displacement of the line source are significant for fibre lengths shorter than 200m. The influence of the width of the entrance beam is significant for lengths up to 5 km.Varying coupling length and steady state angle for a fibre length of 150 m calculations show that the bandwidth varies between 15 and 95 MHz for penalty losses between 8 and 18 dB.Calculations have been compared with direct baseband frequency measurements for a high-loss fibre with a large numerical aperture and excellent agreement has been found.

Multimode graded-index optical fibers: comparison of two Wentzel–Kramers–Brillouin formulations

Light propagation in graded-index optical fibers is described by two WKB methods, a classical formulation and the Langer–McKelvey approach. Explicit errors in these methods are evaluated for a fiber profile of near-parabolic shape and for one far from parabolic shape by comparing the methods with the results of evanescent wave theory that involve asymptotic error estimates. The most accurate formulation in general is the classical method, but in special cases the Langer–McKelvey approach is slightly more accurate. One example shows an inaccuracy of −672.5 to −672.1 psec/km in modal group delay for the latter method when the maximum intermodal group delay difference is 2.84 nsec/km.

Evanescent-wave analysis of general clad graded-index fibers

The evanescent wave theory (ewt) is extended to treat modal propagation along a general graded-index optical-fiber structure surrounded by a uniform cladding. A step or a valley may be present at the core–cladding interface. The fiber core may be specified in a more general form than was the case for untruncated profiles. The reformulation of the original ewt method uses the uniform corrective method, developed by Arnold for WKB calculations, iteratively to specify nonintegral radial mode numbers for the clad index profile. The integral radial mode numbers of the unclad profile are used as first approximations. Our method gives corrected values for propagation constants and group delays of guided modes and complex propagation constants of leaky modes. Numerical results show good agreement with a numerical solution for a parabolic-core fiber. For other profiles they specify accurate asymptotic upper and lower limits for the cladding influence. For leaky modes of a truncated near-parabolic profile we show agreement with results of a simple WKB formulation. A valley, present at the core–cladding interface, causes a reduction in the maximum intermodal group-delay difference from 3.1 to 1.9 nsec/km because of a strong attenuation of high-order (leaky) modes.

Leaky modes in non-circular graded index optical fibers

Abstract A new theoretical method using WKB technique is used for calculation of attenuation constants of leaky modes in non-circular graded index fibers. Results for a parabolic profile show no severe change in attenuation constants for the non-circular case compared to the circular case and for a main axis ratio less than 1.06. The biggest relative change is an increase from 7.22 × 10 -5 dB / km to 3.38 × 10 -3 dB / km for the least leaky mode. The change in the real part of the propagation constant is shown to be negligible. Near field measurements for a graded index fiber with axis ratio 1.10 show that leaky modes exist for short fiber lengths in qualitative agreement with the calculations.

The influence of the input beam width on the bandwidth of multimode step index fibers

Abstract A detailed theoretical investigation of the influence of the input on the bandwidth of multimode step index fibers is given. The calculations take into account mode coupling, absorption and leaky modes. For 100 m of a typical large numerical aperture fiber the bandwidth can vary with excitation from 1.8 GHz to 15 GHz. The numerical calculations agree with published experimental results.