Fiber_Optics_Physics_Technology

Example

Consider an amplifier that boosts some input signal from 1 to 20 mV; source and load impedance are equal. When specifying the gain factor, one needs to make the distinction between voltage gain and power gain: Gamplitude = 20, Gpower = 400. Using decibels the gain factor is simply

20mV

δ= 20 log10 1 mV = 26 dB

or

Two di erent numerical values are replaced with a single value in dB. On first encounter, students tend to find this irritating, but in practical usage it is a real simplification. Of course we had to assume equal impedances at input and output, but in radio engineering that is very frequently true.

Electrical and Optical dB Units

The second source of confusion occurs when light is converted to an electric signal. As described in Sect. 8.10, common photodetectors such as, e.g., photo diodes, convert the impinging light power to a proportional electric current. The electric power delivered by the detector is thus proportional to the square of the received optical power. One therefore needs to specify whether one speaks of optical or electric dB. Consider a statement about the dynamic range of some detector, i.e., the ratio of maximum received power before severe distortion sets on and the smallest detectable power that is not masked by noise. One “dBopt” corresponds to two “dBelect.”

A.4 Beer’s Attenuation and dB Units

When light is impinging on a more or less transparent material, the reduction of power P (L) with increasing penetration depth L is described by the well-known Beer’s law:

P (L) = P0 exp (−αL) .

Here, α is called Beer’s absorption coe cient; its reciprocal value is that particular penetration depth where the initial power P0 has decayed to the fraction 1/e ≈ 37%.

By taking the logarithm, one immediately sees that

On the other hand, using the definition of the dB we can write

P αdBL = 10 log10 P0 .

Comparing terms yields the conversion formula

αdB = −α10 log10 e ≈ −4.34α.

Appendix B

Skin E ect

When alternating current flows through a conductor, the current density is not necessarily constant across its entire cross-section. When J (0) denotes the current density at the surface, the current density at some depth x below the

surface is given by

J (x) = J (0)e−x/δ e−ix/δ .

Here, δ is a characteristic penetration depth, i.e., that depth where the current density is reduced by a factor of 1/e ≈ 37 % in comparison to the surface. At the same time, at this depth there is a phase shift of 1 rad. This depth is given by

Here, in turn, μ0 = 4π 10−7 Vs/Am is the vacuum permeability, μr the relative permeability of the material, ρ the specific resistance of the material, and ω the current’s angular frequency.

It is straightforward to realize that at a depth of πδ, there is a phase shift of 180◦. This implies that at this depth the current flows in opposite phase to that at the surface. This is, of course, a hindrance for the current flow through the conductor and is felt as an e ective increase of resistance. Somewhat paradoxically, a massive conductor, such as a solid wire, conducts current less well than a hollow conductor (a tube) of the same outside diameter!

The e ective resistance is given by

with l the length and s the circumference of the conductor. This should be compared with the usual expression for direct current resistance,

R = l Aρ ,

where A is the conductor’s cross-sectional area. In e ect, only a surface layer of thickness ρ contributes to the current flow.

Inserting Eq. (B.1) into Eq. (B.2) yields

For our consideration the relevant feature is the relation

√

R(ω) ω.

The e ective resistance grows with increasing frequency (Fig. B.1). This limits the usefulness of electrical conductors at very high frequencies. Optical fibers do not su er from a comparable limitation.

Figure B.1: Skin e ect causes the e ective resistance of a cable to rise with increasing frequency. Optical fiber is superior, in particular at high frequencies.

C.3 Recursion Formulae

With Zm denoting some cylinder function of type Jm, Nm, Hm, Im, or Km and with Zm(x) the derivative of Zm(x) with respect to the argument x,

C.4 Properties of Jm and Km

The functions Jm describe standing waves. J0 “looks like” cosine, J1 like sine. Asymptotically (for very large x) the following approximations hold:

Functions with negative index are defined as

J−n(x) = (−1)n Jn(x).

There is a relation with angular functions of the form

which is relevant in context with frequency modulation (see Sect. 11.1.2). Functions Km describe decaying cylinder waves and “look like” decaying

exponential functions. Asymptotically (for very large x), for all m the following approximations hold:

As long as the argument is larger than 2, the error is less than 5%.

Appendix D

Optics with Gaussian Beams

The general public holds the notion of a laser beam as a cylindrical bundle of rays that travels arbitrary distances without any change of its diameter so that it delivers the same power density wherever it hits. Take note, James Bond: There is no such thing as a cylindrical beam. The laws of di raction make sure that any beam with a finite diameter widens as it propagates; the wider the beam starts out, the more gradual is its spreading, but it is always there.

D.1 Why Gaussian Beams?

For the sake of a quantitative description let us consider a Gaussian beam , i.e., a light beam with a transverse distribution of power following a Gaussian bellshaped curve. Such beams are ubiquitous in laser physics; they originate from laser resonators for which the Gaussian profile is the lowest transverse mode. As light is coupled out, one automatically gets a Gaussian beam. The first concise treatment of the situation, still worthwhile reading today, is given in [89]; for a particularly transparent description see [135] or [125].

One might think of a Gaussian beam as created in the following way: Start with a plane wave and send it through an amplitude mask (like a photographic slide). Let the mask have a circularly symmetric gray scale so that a bell-shaped beam is carved out of the plane wave.

Once a beam has a Gaussian profile, it will stay that way. Propagation across free space will change its size, but not the functional form. In other words, the Gaussian is invariant (except for scale factors) under di raction. The reader is reminded that in the far-field limit of di raction, the field distribution acquires a shape given by the Fourier transform of the initial shape. It is well known that the Fourier transform of a Gaussian is again a Gaussian. Therefore, the near field Gaussian profile is preserved in the far field – and not just in the limiting cases, but everywhere in between, too. A Gaussian beam is a di raction-limited beam. It is even better: Among all di raction-limited beams, it is the one with the least change of shape.

In analogy to the quantum mechanic uncertainty relation and thinking of light as a stream of photons, one can multiply the transverse localization error

c Springer-Verlag Berlin Heidelberg 2009

Figure D.1: The contour of a Gaussian beam at r = w(z) takes its narrowest width w0 at the waist (z = 0). In the far field, the beam radius w(z) increases at a fixed angle θ, called the beam divergence.

Obviously,

lim R(z) = ∞ and

z→0

lim R(z) = z.

z→∞

This implies that the wave fronts are indeed plane at the waist; at very large distance they form segments of spheres centered around the waist. At some intermediate distance, the curvature has a minimum: This is the case at z = z0 where R(z) = 2z.

Example

Find the radius of the bright spot on the lunar surface when we aim the beam of a He–Ne laser (λ = 633 nm) at the moon (z = 384, 000 km)!

At w0 = 1 mm, one obtains z0 = 4.96 m and wmoon = 77.4 km; at w0 = 1 m, one gets z0 = 4.96 ×106 m and wmoon = 5.99 m. Lesson learned: In the absence of a beam expansion by way of a telescope, the beam is scattered about as to be undetectable (nearly 80 km), but using a telescope one can illuminate a reasonably small spot of 6 m radius. This allows, e.g., to hit a retro reflector such as placed on the lunar surface by Apollo astronauts and still get a detectable back-reflected signal.

D.3 Gaussian Beams and Optical Fibers

Fibers are waveguides. Indeed, the waves are weakly guided because the index contrast between core and cladding is very small. In this situation, the fundamental mode profile is nearly Gaussian. Consider as a thought experiment that the index di erence shrinks to zero: Then one would expect the same shape as in free space. It is of course simpler to deal with a Gaussian, rather than the complicated composition of Bessel functions described in Chap. 3. Therefore, this approximation is popular and sometimes good enough.

When a beam is coupled from free space into a fiber, one is usually faced with the matching problem between a true Gaussian beam in free space and a less- than-exact almost-Gaussian profile of the fiber’s fundamental mode. (However,

it is usually safe to assume that in a perpendicularly cleaved fiber surface the wave fronts are plane, so that for incoupling as well as outcoupling of light the fiber face can be identified with the position of a beam waist.) The mismatch causes a reduction in coupling e ciency. Even in the presence of ideal lenses without aberrations, this limit cannot be surpassed [120]. Part of the light winds up in the cladding and is eventually scattered out of the fiber, rather than guided in it.

If light propagating inside the fiber is coupled out at the other end and hits a screen at some distance, the pattern on the screen again is similar to a Gaussian. This is because the far-field pattern is the Fourier transform of the near-field pattern as discussed in Sect. 7.4.2 where it was also shown how deviations from the Gaussian pattern can be exploited to gauge the mode profile.