Kitabı oku: «Optical Engineering Science», sayfa 13

6.6.2 The Maréchal Criterion

The Strehl ratio is of great practical significance. It affords a useful, practical, but somewhat arbitrary definition of ‘diffraction limited’ imaging. Where the Strehl ratio is 0.8 or greater, the image is said to be diffraction limited. This is the so-called Maréchal criterion. It can be expressed as follows:

(6.23)

This condition is widely accepted as the basis for the definition of diffraction limited imaging. As a measure of system wavefront error, the peak to valley wavefront error, Φ_pv ≪ λ/4 is often preferred, in practice. This is very much a traditional description of system wavefront error, preserved for historical reasons, primarily on account of the ease of reckoning peak to valley fringe displacements on visual images of interferograms. This consideration has, of course, been displaced by the ubiquitous presence of computer-based interferometry which has rendered the calculation of rms wavefront errors a trivial process. Nonetheless, the peak to valley description remains prevalent. Although dependent upon the precise distribution of wavefront error, as a rule of thumb, the peak to valley wavefront error is about 3.5 times the standard deviation or rms value. Therefore, we can set out another condition for diffraction limited imaging.

(6.24)

Worked Example 6.2 A simple ×10 microscope objective is to be made from a BK7 plano-convex lens and is to operate at the single wavelength of 589.3 nm. The refractive index of BK7 at 589.3 nm is 1.518 and the assumed microscope tube length is 160 mm. We also assume that only the microscope objective contributes to system aberration. What is the maximum objective numerical aperture consistent with diffraction limited performance, assuming on-axis spherical aberration as the dominant aberration?

Firstly, for a wavelength of 589.3 nm, from Eq. (6.22):

For a ×10 objective, the focal length should be 160/10 = 16 mm for a 160 mm tube length. The tube length is much longer than the objective focal length, so, for all intents and purposes, the image is at the infinite conjugate, as illustrated.

Note that the curved surface faces the infinite conjugate in order to minimise spherical aberration. Thus, in this instance, the conjugate parameter is 1 and the lens shape factor is −1. From Eq. (4.30a)

Substituting the values of s and t:

Substituting the values of n (1.518) and f (16 mm) into the equation, we get:

Further assuming that we are using defocus to ‘balance’ and minimise aberrations, we can relate the rms wavefront error to the maximum OPD:

For the ‘diffraction limited’ condition to be fulfilled the rms wavefront error must be less than 41.94 nm.

The minimum numerical aperture is thus 0.107. Substituting this value into Eq. (5.17), then it is clear that the resolution of the objective is 3.37 μm. The design of complex objectives does, of course, generally proceed by virtue of ray tracing. However, this illustration does provide some insight into the limitations of simple optics and the value added by more complex designs.

6.6.3 The Huygens Point Spread Function

The Huygens point spread function is the diffraction pattern produced in the image plane by a point source located at the object plane. Determination of the point spread function proceeds as per Eq. (6.18) and so is fundamentally influenced by the system wavefront error across the pupil. Of course, as previously argued, any wavefront error reduces the flux at the axial location in proportion to the Strehl ratio. In concert with this, the width of the diffraction pattern increases with increasing wavefront error.

As will be discussed in more detail later, where the wavefront error is small, purely geometrical ray tracing produces a very different spatial distribution when compared to the point spread function. Naturally, in the limit where the wavefront error becomes large, the Huygens point spread function tends to the geometrical spot distribution. This is quite an important consideration, as it impacts how optical systems are optimised with regard to optical performance. If the intention is to design a diffraction limited system, the most efficient optimisation proceeds by minimising the wavefront error. However, where the intended wavefront error is large when compared to the diffraction limit, it is best to optimise the system by minimising the geometrical spot size.

6.7 Laser Beam Propagation

6.7.1 Far Field Diffraction of a Gaussian Laser Beam

The far field divergence of a laser beam may be accounted for by the Fraunhofer approximation according to Eq. (6.13). In the treatment of a laser beam, we consider the beam to have a ‘near field’ location, or beam waist, where the phase is uniform across a plane perpendicular to the propagation direction. That is to say, the wavefronts are planar at this beam waist location. In addition, it must be assumed that the wavefront is ‘coherent’, i.e. that an unambiguous phase relation exists across the wavefront at all times. We now wish to know the disturbance produced by this near field distribution in the far field. To make the problem tractable, we assume that the near field distribution of the laser beam waist may be described by a Gaussian function. This is a useful approximation, however, it must be emphasised that this is an approximation. In practice, the profile of a laser beam is not quite Gaussian, with more flux being in the wings, far from the centre, than would be expected from a Gaussian distribution.

In the Gaussian approximation, we may, for example, describe the laser beam emerging from the end of a single mode fibre or from the output of a semiconductor laser facet or from a helium neon laser. The size of the beam waist is described by the parameter, w₀, namely the radial distance at which the amplitude, A(r), falls to 1/e, or 37% of the peak value. When expressed in terms of flux, the size of the beam waist, r₀ defines the radius at which the flux, I(r) falls to (1/e)² or 13% of the peak value. For example, in the case of a single mode telecoms fibre, a laser beam emerging from the end of the fibre might have a beam waist, w₀ of 5.5 μm. Equation (6.25) expresses the form of the laser beam profile:

(6.25)

Having characterised the beam waist in this way, it is useful to relative it to both the FWHM, dFWHM, and the rms radius, rrms.

(6.26)

In the Fraunhofer approximation, the far field may be derived from the Fourier transform of the near field. At this point, the significance of the Gaussian representation becomes clear, as the Fourier transform of a Gaussian is another Gaussian. The far field representation is thus given by another Gaussian, with the divergence expressed by a characteristic numerical aperture, NA₀.

(6.27)

From the Fourier transform, it is possible to derive a clear relationship between the near field beam waist, w₀, and the far field divergence, NA₀. This is given by Eq. (6.28).

(6.28)

As one might expect, the far field divergence is inversely proportional to the size of the beam waist, a smaller beam waist effecting a larger divergence.

Worked Example 6.3 – Beam Divergence of a Fibre Laser

A laser beam with a wavelength of 1.55 μm emerges from a single mode fibre. The laser beam, at this point can be characterised as a beam with a waist size of 5.25 μm. What is the characteristic numerical aperture, NA₀, associated with the far field divergence?

Substituting the relevant values into Eq. (6.28), we get:

The numerical aperture is thus 0.094 and this corresponds to a divergence angle of 5.39°.

Expressed as a FWHM, the near field beam width is 6.18 μm, and the rms radius is 4.37 μm. Similarly in the far field the FWHM divergence angle is 6.35° and the rms divergence angle is 3.81°.

6.7.2 Gaussian Beam Propagation

Thus far, we have analysed the relationship between the near field and far field dispositions of a Gaussian beam. We now turn to the more general case of the propagation of a Gaussian beam. As in the previous analysis, the laser beam is defined by a characteristic Gaussian radius. In this case, the characteristic radius, w(z), is a function of the axial propagation distance, z. To describe the laser beam, we introduce an envelope function, A₀(x, y, z) that describes the radial profile of a propagating laser beam.

(6.29)

In practice, in most cases, the size, w(z), of the laser beam is much larger than the wavelength, and we can use the slowly varying envelope approximation. This is, in effect, a paraxial approximation, where far field divergence angles are necessarily small and can be formally expressed as:

(6.30)

Applying Eq. (6.29) to the scalar version of Maxwell's equation and taking into account the above approximation, we obtain the so-called paraxial Helmholtz equation:

(6.31)

As suggested, in the case of a Gaussian beam, the amplitude may be expressed a characteristic width, w(z) which varies slowly with respect to z. Most significantly, w(z) can be complex, giving rise to wavefront curvature. This may be easily seen if the complex part of the envelope is subsumed within the sinusoidal propagation term, leading to a quadratic phase variation across the beam. In the Fresnel and paraxial approximation, these quadratic wavefronts may be viewed as approximately spherical. This is illustrated in Figure 6.12.

Mathematically, the Gaussian beam envelope is expressed as follows:

(6.32)

The component A₀(z) represents the peak on axis amplitude and this would be expected to diminish as the beam expands. To make the analysis more tractable, we subsume this axial variation into the exponential function as a complex function of z, β(z). Furthermore, both real and imaginary parts of the Gaussian are combined into a single complex function, α(z). Hence, to solve the paraxial Helmholtz equation in this instance we re-cast Eq. (6.32) in the following form:

Figure 6.12 Gaussian beam.

And substituting this into the paraxial Helmholtz equation:

This equation must hold for all values of r. Both the quadratic terms and those with no dependence in r must sum to zero. Taking the quadratic element only, we may derive a very simple relationship for α(z).

(6.33)

By viewing Eq. (6.32) it is straightforward to relate β(z) to the beam size, w(z) and the radius R(z):

(6.34)

To interpret the constant C, we assume that there exists a minimum beam size, the beam waist, where the wavefront curvature is zero. We denote this beam waist by the symbol, w₀. It is clear, then, that the constant C is equal to the square of the beam waist. Eliminating the constant, C, gives:

(6.35)

We note that there some characteristic distance, Rz, over which the beam expands around the beam waist. This distance is known as the Rayleigh distance and the Eq. (6.35) may be finally re-cast to give the following:

(6.36)

The Rayleigh distance is of particular significance. In effect, it sets the demarcation between the near field and the far field. In the case of the far field, z ≫ Z_R the expressions in Eq. (6.36) revert to the Fraunhofer diffraction pattern of a Gaussian beam:

(6.37)

This may be compared with Eq. (6.28) and is very similar in form. Thus Eq. (6.37) represents the far field diffraction pattern of a Gaussian beam. In the near field where z ≪ ZR, the beam is parallel and the size constant at w₀ and, of course, the radius tends to infinity. At the Rayleigh distance, the beam size is increased by a factor corresponding to the square root of two and the wavefront radius is equal to twice the Rayleigh distance. This is illustrated more formally in Figure 6.13.

For values of z that are of a similar magnitude to the Rayleigh distance, then the beam is in an intermediate zone between the near and far fields.

Worked Example 6.4 – Rayleigh Distance of Fibre Laser

In the previous worked example, we introduced a fibre laser with a wavelength of 1.55 μm, with a Gaussian beam size of 5.25 μm. If we assume that the beam waist is located at the exit from the fibre, what is the Rayleigh distance of the laser beam? In addition, what is the beam size, w(z), 50 μm from the exit from the fibre and what is the wavefront radius at that location?

From Eq. (6.36):

The Rayleigh distance is 55.9 μm

Figure 6.13 Form of expanding Gaussian beam and beam waist.

Calculation of the beam size and the wavefront radius also proceeds from Eq. (6.36).

The beam size, w(z), is 7.05 μm

The wavefront radius is 112.4 μm

6.7.3 Manipulation of a Gaussian Beam

The preceding analysis has presented an analysis of the propagation of a Gaussian beam through free space. However, in most practical instances, the beam will be manipulated by optical components, lenses, and mirrors and so on. Therefore it would be useful to be able to understand the impact of an individual optical component or system on the propagation of a Gaussian beam. In the analysis presented here, the component or system is simply represented in its paraxial form by a ray tracing matrix, as presented in Chapter 1. The matrix is populated by four elements, A, B, C, and D and these transform the wavefront radius according to the following equation:

(6.38)

(R1 and R2 are the input and output radii respectively)

For a Gaussian beam, the wavefront curvature can be represented by the complex quantity, q, where q = z + iZ_R. The distance from the beam waist is represented by z and Z_R is the Rayleigh distance. Expression (6.38) may be re-cast in the following form:

(6.39)

where q = z + iZ_R

This is the so-called ABCD law for the manipulation of Gaussian beams.

Worked Example 6.5 – Gaussian Beam Manipulation

A helium neon laser beam at 633 nm has a beam waist of 0.6 mm, located 80 mm from a positive lens of focal length 60 mm. Calculate the size of the beam waist following the lens and its location with respect to the lens.

If we take the origin of the co-ordinate system to be at the original beam waist (z = 0), then the system matrix consists of a translation by 80 mm followed by a 60 mm focal length lens.

A = 1; B = 80; C = −0.0167; D = −0.333

The Rayleigh distance of the original beam is given by:

Using the ABCD law:

The lens lies at −59.9 mm from the beam waist and hence the beam waist is 59.9 mm in advance of the lens. The Rayleigh distance is 2.201 mm. This corresponds to a beam waist size (from Eq. (6.36)) of 0.02 mm or 20 μm.

The new beam waist lies approximately at the focus of the lens. Since the lens lies much closer to the original beam waist than the corresponding Rayleigh distance, then the beam is almost parallel and the new beam waist should lie very close to the focal position.

6.7.4 Diffraction and Beam Quality

All the analysis presented thus far has assumed a Gaussian beam that possesses perfect spatial coherence. Perfect spatial coherence implies that an unambiguous phase relationship exists between all points across the wavefront. A less than perfect wave disturbance is composed of a number of different components whose phase relationship is entirely random. As such, spatial coherence is defined more formally as the correlation between the wave disturbance at two points. The complex amplitude of a wave at a point, A(t), may be expressed in Fourier space in terms of a frequency distribution, S(f). The coherence between two points, x, y is simply given by the cross correlation of the two disturbances:

(6.40)

Perfect coherence is represented by a correlation of one; complete incoherence is represented by a correlation of zero. In many practical problems in Gaussian beam propagation, it may be assumed that the coherence of the laser beam is one. However, this is dependent upon the number of independent ‘modes’ that characterise the laser beam. A single mode is effectively one unique solution to the wave equation and laser devices are often engineered in such a way that only one of these modes is allowed to propagate. The extent to which this is true is a measure of the laser's beam quality. The beam quality of a laser is generally expressed by the parameter M²or M squared and is indicative of the number of modes supported by the beam. If this value is one or close to one, then the beam quality is high and any propagation analysis will proceed as previously described. For a beam with a beam quality defined by the M² parameter, then the spatial coherence is given by:

(6.41)

Returning to the practical question of Gaussian beam propagation, the beam propagation may be expressed entirely in the original form given by Eq. (6.36), except with a revised Rayleigh distance, Z′_R.

(6.42)

It is clear from Eq. (6.42), that, where M² is significantly greater than one, the divergence of the laser beam in the far field is greater than would be expected from a perfect beam. The revised equivalent of 5.68, giving the beam divergence is set out in Eq. (6.43).

(6.43)

In practice, the M² value for a laser beam is measured and then analysed using the relationships set out in Eqs. (6.42) and (6.43). The parameter is generally specified for many commercial laser systems.

6.7.5 Hermite Gaussian Beams

Further exploring the theme of multiple modes as individual solutions to the wave equation, we must recognise that the simple Gaussian beam previously defined is not the only solution to the paraxial Helmholtz equation. In fact, a complete set of orthonormal solutions exist of which the simple Gaussian solution is the first member. Orthonormal, in this context, means that the cross-integral of two different solutions is always zero and that involving the same solutions is always unity. This is an important property, as will be seen later. This set of solutions are defined by the Hermite-Gaussian polynomials, where the original Gaussian amplitude envelope is multiplied by a unique Hermite polynomial in x and y. Each Hermite polynomial solution is defined by its maximum order in x and y, which we will refer to as l and m respectively. Overall, the solution may be represented as:

(6.44)

The expression w(z) is simply the Gaussian beam radius for any specific value of z, as given in Eq. (6.36) and A[w(z)] is a normalising factor. The first few polynomials are set out Table 6.1.

Figure 6.14 shows graphically the form of some low order Hermite polynomials.

The orthogonal nature of the polynomials provides a suggestion as to their utility. As with a Fourier series, any arbitrary beam profile may be represented as a summation of a series of Hermite polynomials. If we represent the full expression contained in Eq. (6.44) as Gl,m(x, y, z), then the series may be represented as:

(6.45)

C_l,m are coefficients describing the amplitude of each term

Table 6.1 Low order Hermite polynomials.

Figure 6.14 A selection of low order Hermite polynomials.

Assuming the beam profile is known as some plane, z₀, then each coefficient may be calculated by exploiting the orthonormal property of the series:

(6.46)

Thus, Gaussian-Hermite polynomials represent a powerful tool for physical optics propagation. Assuming a beam profile is known at some point, the relevant coefficients may be calculated according to Eq. (6.46) and summed according to Eq. (6.45) and then propagated in free space according to Eq. (6.44).

6.7.6 Bessel Beams

An interesting solution to the paraxial Helmholtz equation, Eq. (6.31), is the so-called Bessel beam. Generally, Bessel functions of the first kind form a series of solutions to the equation. Most specifically for an axially symmetric form, the solution is represented by a Bessel function of the first kind and zeroth order. The unique feature of this solution is that the wavefronts are planar and the amplitude envelope of the beam does not change as it propagates. That is to say, the beam appears to be diffractionless and does not diverge. The solution is of the form given by:

(6.47)

J₀ is a Bessel function of the first kind and zeroth order; ; kr is the effective transverse wavevector; β is the propagation wavevector given by:

Another interesting type of beam is the Talbot beam. Rather than retaining a constant profile as it propagates, the Talbot beam replicates itself at specific propagation distances. For further details, the reader is advised to consult the Further Reading section at the end of the chapter.