Engineering & Scientific
Hyperbolic Secant Envelope & Complex Hyperbolic Reciprocals (SECH, IMSECH & IMCSCH)
Model optical soliton pulse envelopes sech^2(t/tau) with SECH(x) = 2 / (EXP(x) + EXP(-x)) and compute complex hyperbolic secant and cosecant components (IMSECH & IMCSCH).
Real sech(x) pulse profiles and complex waveguide reciprocals
The real hyperbolic secant sech(x) = 1 / cosh(x) = 2 / (e^x + e^-x) peaks at 1 when x = 0, decays symmetrically toward 0 without negative sidelobes, and defines the intensity profile I(t) = I_0 * sech^2(t / tau) of mode-locked laser solitons and shallow-water solitary waves.
Extending to complex propagation constant z = a + b*i, cosh(z) = cosh(a)*cos(b) + i*sinh(a)*sin(b) and sinh(z) = sinh(a)*cos(b) + i*cosh(a)*sin(b), whose reciprocals IMSECH(z) and IMCSCH(z) rationalize cleanly into real and imaginary cells.
| Quantity | Expression | At a = ln(2), b = 0 | At a = ln(2), b = pi/2 |
|---|---|---|---|
| Real SECH(a) | 2 / (EXP(a) + EXP(-a)) | 0.8000 | 0.8000 |
| Soliton Intensity SECH^2(a) | SECH(a)^2 | 0.6400 | 0.6400 |
| IMSECH(a + b*i) | 1 / cosh(a + b*i) | 0.8000 + 0i | 0 - 1.3333i |
| IMCSCH(a + b*i) | 1 / sinh(a + b*i) | 1.3333 + 0i | 0 - 0.8000i |
Real Hyperbolic Secant & Soliton Intensity (SECH)
Evaluate 2 / (EXP(x) + EXP(-x)) directly to prevent overflow for moderate arguments and compute normalized intensity SECH(x)^2.
=ROUND(2 / (EXP(B2) + EXP(-B2)), 4)Complex Hyperbolic Secant & Cosecant (IMSECH & IMCSCH)
Rationalize 1 / cosh(a + b*i) and 1 / sinh(a + b*i) into exact real and imaginary parts.
=LET(re_a, B2, im_b, B3, ch_a, (EXP(re_a) + EXP(-re_a)) / 2, sh_a, (EXP(re_a) - EXP(-re_a)) / 2, ur, ch_a * COS(im_b), ui, sh_a * SIN(im_b), ROUND(-ui / (ur ^ 2 + ui ^ 2), 4))Try the example
Use the original values or change them to test your understanding.