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Ординатура / Офтальмология / Английские материалы / Optics Learning by Computing with Examples using MATLAB_Dieter Moller_2007

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7.6. CONFOCAL CAVITY, GAUSSIAN BEAM, AND MODES

299

where zR is a constant. We express the function q(z) as a combination of the radius of curvature of the wavefront R(z) and the beam waist w(z)2 as

1/q(z) 1/(izR + z) 1/R(z) − iλ/(πw(z)2)

(7.73)

and determine R(z) and w(z)2 from Eq. (7.72) and (7.73), after separation of real and imaginary parts, we have

w(z)2 (λ/π){zR + z2/zR }

(7.74)

and

 

R(z) z + zR2 /z.

(7.75)

The function w(z)2 is the beam waist, which is the width of the beam depending on z (Figure 7.16). R(z) is the curvature of the wavefront of the beam depending on z (Figure 7.17).

7.6.2 Fundamental Mode in Confocal Cavity

The confocal cavity was discussed in the chapter on geometrical optics. It is a stable cavity with radii of curvature of the mirrors equal to the length d of the cavity. The fundamental mode of the solution of the paraxial wave equation (see Eq. (7.68)), is the same for rectangular mirrors of a cavity for which Cartesian coordinates are used and for circular mirrors for which cylindrical coordinates are used.

We show that the radius of curvature of the wavefront of the Gaussian beam in the confocal cavity matches the curvature of the mirrors at distance z d/2 and −d/2, counting z from the middle at 0.

7.6.2.1 Beam Waist

The beam waist is indicated in Figure 7.16. Inserting Eq. (7.73), into Eq (7.69) and taking the real part one has (exp −kr2λ/2πw(z)2). With q kλ/2π and setting q 1 for simplicity we get (exp −r2/w(z)2). This factor decreases in the transversal direction and is 1 for r 0 and 1/e for r w(z). The beam is attenuated from its value at r 0 at the axis to 1/e at distance r from the axis.

7.6.2.2 Wavefront of Beam at Center and at Mirror

Wavefront at Center

The wavefront is plain when R(z) ∞. If we choose this value at z 0, one gets from Eq. (7.73)

1/q(0) −iλ/(πw(0)2)

(7.76)

From Eq (7.74) we have for the total waist in the middle w02 w(0)2 (λ/π)zR .

300

 

7. BLACKBODY RADIATION, ATOMIC EMISSION, AND LASERS

 

 

Wavefront at Mirrors

 

 

 

For z zR we have

 

 

 

R(z zR ) 2zR and w(z zR )2 (λ/π)2zR 2w02.

(7.77)

 

We choose 2zR d equal to the distance d between the two mirrors, which is

 

 

also the radius of curvature of the spherical mirrors in the confocal cavity. The

 

 

radius of curvature of the wavefront at z d/2 matches the radius of curvature

 

 

of the mirrors.

 

 

 

 

Confocal Cavity

 

 

 

 

For the confocal cavity the radius of curvature of the wavefront, when approach-

 

 

ing the mirror, is equal to the radius of curvature of the mirror (see Figure 7.17).

 

The beam waist is 2w02 at the point z d/2, which is twice as large as at its

 

 

minimum at z 0. In FileFig 7.10 we have plotted w(z) and R(z) over the range

 

 

z −100 to 100, which is the distance between the mirrors equal to the length

 

 

d 2zR 200. One observes the beam waist at z 0 and one obtains for the

 

 

radius of the wavefront at z 100, that is, R(100), the value 200, corresponding

 

 

to the radius of curvature of the mirror Rm d.

 

 

 

The half angle of the opening of the beam in the far field approximation is the

 

 

angle θap of the asymptote of w(z), which passes through z 0. It is obtained

 

 

as

 

 

 

 

zlim w(z)/z w0/zR λ/πw0.

(7.78)

 

 

→∞

 

 

 

The asymptote y zw0/zR is also indicated in FileFig 7.10.

 

 

 

FileFig 7.10

(L10WRS)

 

 

 

 

 

 

 

Graphs are shown for the radius of the wavefront R(z), beam waist w(z), and the asymptote y(z).

L10WRS

Radius of Curvature and Beam Waist

1.Radius of curvature

Beam waist is normalized to 1; that is, we plot (w(z) w0SQR(1+(z/zR)2) and set w0 .1 in cm, and zR π(ω0)2/λ − .01π/λ, λ in cm. Radius of curvature R(z) z + (zR)2/z

R(z) : z +

zR2

zR : 100

z

z : −100, −99.99 . . . 100.

7.6. CONFOCAL CAVITY, GAUSSIAN BEAM, AND MODES

301

Rm 2zR. At z 1/2 of distance of mirrors, that is, for distance 200 at 100, the radius of curvature must be equal to the distance of the mirrors.

2.Beam waist

Plots of two branches of the beam waist and the asymptote to w(z); that is,

y z/zR.

If z is in cm, we have set for w0 .1, λ 3.14 0.01/zR in cm (about 3 microns for zR 100).

 

 

z

w(z) : .1 ·

1 + (

 

)2

zR

and for the asymptote yy(z) : −y(z),

 

z

2

ww(z) : −.1 · 1 +

.

 

 

zR

302

7. BLACKBODY RADIATION, ATOMIC EMISSION, AND LASERS

Application 7.10.

1.Repeat the calculations for λ 10 microns, zR 100 cm.

2.Repeat the calculations for λ 3 microns, zR 160 cm.

3.Repeat the calculations for λ 10 microns, zR 160 cm.

7.6.3 Diffraction Losses and Fresnel Number

The diffraction losses of a mirror of diameter 2a at distance d are characterized by the Fresnel number F . We assume parallel light incident on an obstacle of diameter 2a and have for the diffraction angle θ λ/2a (see Figure 7.18). The geometrical shadow AG at distance d is 4a2π and the total area AT, enlarged by diffraction, is 4(a + θd)2π. The Fresnel number is defined as F AG/(AT − AG), which is

F 4a2π/{(4a2 + 8ad(λ/2a) + . . .)π − 4a2π}

 

4a2π/(8adλπ/2a) a2/dλ,

(7.79)

where a term in θ2 was neglected in the denominator.

FIGURE 7.18 Diffraction angle for mirror. The geometrical shadow has the area AG. The total area of the light is AT . The Fresnel number is defined as F (AG)/(AT − AG).

7.6. CONFOCAL CAVITY, GAUSSIAN BEAM, AND MODES

303

FIGURE 7.19 Fresnel numbers for circular plain reflectors and confocal reflectors. The confocal cavity is about two orders of magnitude better than a Fabry–Perot cavity.

A confocal resonator has considerably fewer losses for Fresnel numbers larger than 1 compared to Fabry–Perot resonators. For example, at F 1 the confocal resonator does more than 300 times better than the Fabry–Perot (Figure 7.19).

7.6.4 Higher Modes in the Confocal Cavity

We have discussed in Section 7.2 the fundamental mode in a confocal cavity. The fundamental mode is the same for a cavity using rectangular mirrors or spherical mirrors. The higher modes need for their description Cartesian coordinates for rectangular mirrors and cylindical coordinates for round mirrors. In both cases we have TEM modes characterized by three mode numbers, the first two for the transversal modes and the last for the longitudinal modes.

7.6.4.1 Confocal Cavity and Rectangular-Shaped Mirrors (Cartesian Coordinates)

For the higher modes of cavities with rectangular-shaped mirrors, using Cartesian coordinates (x, y, z), the solution of the paraxial wave equation (see Eq. (7.68)), is

√

ψ Hm( 2x/w) · Hn( 2y/w) · exp{−i(P (z) + k(x2 + y2)/2q(z))}, (7.80)

√ √

where Hm( 2x/w) and Hn( 2y/w) are Hermitian polynomials, each a solution of a differential equation written in the variable u as

d2Hm/δu2 − 2u · dHm/du + 2mHm 0.

(7.81)

The indices m and n are the transverse mode numbers. The Hermitian polynomials for m equals 0 to 3 are

H0(u) 1

 

H1(u) u

(7.82)

304

7. BLACKBODY RADIATION, ATOMIC EMISSION, AND LASERS

FIGURE 7.20 Schematic of modes for a cavity with rectangular mirrors.

H2(u) 4u2 − 2

H3(u) 8u3 − 12u.

The fundamental mode, discussed above, has the indices m 0 and n 0. The higher modes are labeled by TEMmnq, where n and m refer to the Hermi-

tian polynomials (Eqs. (7.81) and (7.82)). They are transversal mode numbers and correspond to the number of vertical or horizontal zero-intensity lines in the transversal pattern. A schematic representation is given in Figure 7.20. The longitudinal mode is characterized by the large number q. Since the phase has to be the same after one round trip, one obtains the following resonance condition,

2L/λnmq 1/2(m + n + 1) + q.

(7.83)

There are degenerate modes if m + n + 2q m + n + 2q . For the mode separation one has v00 q+1 − ν00q c /2L, where c is the phase velocity in the medium of the cavity and is independent of the Fresnel number.

In FileFig 7.11 we show surface graphs of the higher-order modes in the transversal direction from (0,0) to (2,2).

FileFig 7.11 (L11MOCONFCS)

Modes for confocal cavity and rectangular mirrors. Surface plots of Hermitian polynomials with exponential factor for indices 0, 1, and 2, and scaling factors X and Y .

L11MOCONFCS

Cartesian Coordinates for Rectangular Mirrors in the Confocal Resonator

Field distribution as contour plot. The mode numbers m and n are for hermitian polynomials. The constant in the exponential is simulated by X. Small X

7.6. CONFOCAL CAVITY, GAUSSIAN BEAM, AND MODES

305

corresponds to small waist width.

N : 40

i : 0 . . . N

j : 0 . . . N

 

 

 

 

 

xi : (−20) + 1.00 · i

yj : (−20) + 1.00 · j

 

 

 

 

H 0(x) : 1 H 0(y) : 1 H 1(x) : x ·

 

2

 

H 1(y) : y ·

 

2

 

 

 

 

 

 

 

 

 

 

 

Y

 

 

 

Y

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

2

 

 

 

 

H 2(y) : 4 ·

2

 

 

 

 

 

 

 

 

2

 

 

 

 

 

· y

− 2

H 2(x) :

4 ·

 

· x

− 2

 

 

 

 

 

Y

 

Y

H 00(x, y) : H 0(x) · H 0(y)

H01(x, y) : H 0(x) · H 1(y)

H10(x, y) : H 1(x) · H 0(y)

H20(x, y) : H 2(x) · H 0(y)

H02(x, y) : H 0(x) · H 2(y)

H11(x, y) : H 1(x) · H 1(y)

H21(x, y) : H 2(x) · H 1(y)

H12(x, y) : H 1(x) · H 2(y)

R(x, y) : (x)2 + ((y))2.

Constant X:

X ≡ 100 Y ≡ 100

g(x, y) : e

−R(x,y)

.

X

 

 

 

M00i,j : g(xi , yj ) · H 00(xi , yj ) 2

M10i,j :

g(xi , yj ) · H 10(xi , yj ) 2

306

7. BLACKBODY RADIATION, ATOMIC EMISSION, AND LASERS

M01i,j

: g(xi , yj ) · H 01(xi , yj ) 2

M11i,j : g(xi , yj ) · H 11(xi , yj ) 2

M20i,j : g(xi , yj ) · H 20(xi , yj ) 2

M21i,j : g(xi , yj ) · H 21(xi , yj ) 2

7.6.

CONFOCAL CAVITY, GAUSSIAN BEAM, AND MODES

307

M02i,j : g(xi , yj ) · H 02(xi , yj ) 2

M12i,j : g(xi , yj ) · H 12(xi , yj ) 2

M22i,j : g(xi , yj ) · H 22(xi , yj ) 2

Application 7.11.

1.Compare with Figure 7.20 and the number of zero-intensity lines with the mode numbers.

2.Extend the modes using H 3(x) and H 3(y) and complete all graphs up to indices 0, 1, 2, and 3.

3.Convert surface to contour plots.

308

7. BLACKBODY RADIATION, ATOMIC EMISSION, AND LASERS

 

 

 

 

7.6.4.2 Confocal Cavity and Circular Mirrors (Cylindrical Coordinates)

 

 

For circular mirrors, one uses cylindrical coordinates (r, φ, z). After rewriting

 

the wave equation in these coordinates, one has the solutions

 

 

√

 

 

l

l

2

 

2

 

2

 

 

 

2r/w)

/w

) · exp{−i(P (z) + kr

/2q(z))},

(7.84)

 

ψ (

 

· Lp(2r

 

 

 

where Lpl (2r2/w2) is a generalized Laguerre polynomial. The Llp(2r2/w2) are solutions of the differential equation written in the variable u,

ud2Lpl /du2 + (l + 1 − u)dLpl /du + pLpl 0.

(7.85)

The index p is the radial mode number and l the angular mode number. For

p 0 to 2 we have for the polynomials

 

L0l (u) 1

(7.86)

L1l (u) l + 1 − u

 

L2l (u) (1/2)(l + 1)(l + 2) − (l + 2)u + (1/2)(u2).

(7.87)

The fundamental mode has the numbers l 0 and p 0. The modes in confocal cavities with round mirrors are labeled by TEMlpq , where l is the angular mode number corresponding to the number of angular zero-intensity lines in the transversal pattern (see the schematic presentation in Figure 7.21). The radial mode number correspond to the number of zero-intensity rings. The longitudinal mode is characterized by the large number q. The resonance condition, which is the condition that the phase is the same after one round trip, is

2L/(λlpq (1/2)(l + 2p + 1) + q.

(7.88)

There are degenerate modes if l+2p+2q l +2p +2q . The mode separation

is again independent of the Fresnel number and one has again ν00 q+1 − ν00q c /2L.

FIGURE 7.21 Schematic of modes for a cavity with round mirrors.

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