DIFFRACTIVE OPTICAL ELEMENTS
4.2 DIFFRACTION GRATINGS
Gratings [49] are periodic structures used extensively in military laser systems: waveg- uide filters, optical fiber filters, communications, and spectrum analyzers (Chapter 15).
Periodic structures [63, 147, 176] in more dimensions, photonic crystals and pho- tonic crystal fibers [14], are important for future integrated optics. Figure 4.1 shows sinusoidal (a) and blazed (b) gratings. The grating is typically constructed by rul- ing or etching (by lithography) lines on a smooth glass substrate or by holographic
DIFFRACTION GRATINGS 63
(a)
(b)
FIGURE 4.1 Gratings: (a) sinusoidal and (b) blazed.
techniques (for sinusoidal gratings) [83]. The grating can be transmissive or reflective for which it is coated with a reflecting surface. For a reflective grating, the blazed grat- ing emphasizes certain reflection orders relative to the sinusoidal grating; therefore, these orders have a higher reflectance and are more efficient.
4.2.1 Bending Light with Diffraction Gratings and Grating Equation A beam of light can be bent by both diffraction (Figure 4.2c) and refraction (Fig- ure 4.2a and b). In Figure 4.2a, a beam from left strikes an interface with a material of lower refractive index so that the side of the beam that strikes the interface first is speeded up causing the beam to bend, similar to the bending of a squad of soldiers marching at an angle out of a muddy field into a parade ground. This explains the trapping of light in optical fiber by total internal reflection. The reverse direction for the beam also shows bending. The amount of bend is specified by Snell’s law (Sec- tion 1.2.2). In Figure 4.2b, a prism has a longer path in glass for one side of the beam than the other side, which slows down that side causing the beam to bend.
In Figure 4.2c, bending by diffraction, one side of the beam, labeledC, travels farther than the other sideAif a bent plane wave front is desired at the output. Hence, the slowing, of the light of one side of the beam causes the beam to bend and for a grating period ofd, the extra distance traveled isd sin θ1+d sin θ2. For the output to have a plane wave front as shown, the extra distance traveled must be an integer multiplemof a wavelengthλ. This gives a grating equation
d sin θ1+d sinθ2=mλ (4.1)
(a) (b)
(c) d
d sinθ1 d sinθ2
A
A A
B B θ1
θ2 θ1 θ2
θ1 θ2
B C
n1 n1
n2
n2
FIGURE 4.2 Bending a light beam: (a) by an interface with different refractive indices by refraction, (b) by a prism by refraction, and (c) by a grating by diffraction.
64 DIFFRACTIVE OPTICAL ELEMENTS
0
0
2
1 2f0
1 2f0
2W
2W
Horizontal grating
Mean
Intensity 1
x Spatial
frequency f0
x
y
(a) (b)
FIGURE 4.3 Cosinusoidal grating shape.
A major difference between diffractive and refractive bending (Figure 4.2c and a or b, is that in diffractive elements long wavelengths such as infrared make the grating look finer and are bent more than shorter wavelengths, while in refractive elements the reverse occurs; that is, the shorter wavelengths are bent more.
4.2.2 Cosinusoidal Grating
Figure 4.3a shows a cosinusoidal transmission grating of size 2W×2Wand grating frequencyf0for which the cross-sectional transmittance profile inx (Figure 4.3b) may be written as
Uin(x, y)= 1
2 +1
2cos(2πf0x1)
rect x0
2W
rect y0
2W
(4.2)
A blazed grating is analyzed in Section 15.3.3. The first 12 in equation (4.2) arises because intensity cannot be negative, so a zero spatial frequency or a DC component exists to produce the zeroth-order component (Figure 4.4). The far-field diffraction (called Fraunhofer field) may be written using the Fourier transform of the intensity immediately after the grating:
Uout(x0, y0)=F{Uin(x1, y1)}
=F 1
2+1
2cos(2πf0x1)
∗F rect
x0
2W
rect x0
2W
= 1
2δ(fxfy)+1
4δ(fx+f0, fy)+1
4δ(fx−f0, fy)
∗ (2W)2sinc(2Wfx)sinc(2Wfy)
DIFFRACTION GRATINGS 65
FIGURE 4.4 Fraunhofer field from a cosinusoidal grating.
=1
2(2W)2sinc(2Wfy)
sinc(2Wfx)+1
2sinc{2W(fx+f0)}
+1
2sinc{2W(fx−f0)}
(4.3) where we used the Fourier transform property that the convolution in one domain becomes multiplication in the other and cosθ=(exp{jθ} +exp{−jθ})/2. The Fraunhofer diffraction field may be written from equation (3.53) (Section 3.3.5), using the Fourier transform from equation (4.3) and scaling withfx=x0/(λz) and fy=y0/(λz):
U(x0, y0)=exp(jkz) jλz exp
jk
2z{x20+y02} 1
2(2W2)sinc 2Wy0
λz sinc
2Wx0
λz +1 2sinc
2W
λz(x0+λzf0)
+ 1 2sinc
2W
λz(x0−λzf0)
(4.4) The three orders in the square brackets correspond with those shown in Figure 4.4.
The far-field shape in thexdirection from a cosinusoidal grating, equation (4.4), at a far-field plane is plotted in Figure 4.5. For sufficient spacing to avoid overlap, as shown in Figure 4.4, of diffraction orders, we can write far-field intensity at the output by multiplying by its conjugate:
Iout(x0y0)= 1 (λz)2
1 2(2W)2
2
sinc2 2Wy0
λz sinc2
2Wx0
λz +1 4sinc2
2W
λz(x0+λzf0)
+ 1 4sinc2
2W
λz(x0−λzf0)
(4.5)
66 DIFFRACTIVE OPTICAL ELEMENTS
*
=1 4
1 4 1
2
f0 f0
(2W)2
(2W)2/4 (2W)2/4 (2W)2/2
fx fx fx
2x0
FIGURE 4.5 Far field from a cosinusoidal grating shows the three orders and their shape.
4.2.3 Performance of Grating
The separation of the−1 and+1 orders from the zeroth order, the direction the light would travel in the absence of the grating, is proportional to the spatial frequencyf0
of the grating. For diffraction in optics, the output is scaled byλz, the wavelength λand the distance to the output plane z(Figure 4.4), equation (4.5). Note that the more dense the grating, that is, lines close to each other, the larger thef0, and the diffraction orders spread more. Also, the shorter the wavelength of the light, (toward UV), the smaller the λ, and the diffraction orders are bent farther away from the zeroth order. This is used for spectral analysis in Chapter 15 to identify chemical and biological weapons. As mentioned earlier, light is bent in the direction opposite to that in a prism, making diffractive–refractive spherical elements possible that, for example, will focus different colors to the same position, useful for eye surgery or weapon systems where burning is conducted with infrared while a tracking red light is observed. An aspheric lens to perform this function without a diffractive element cannot be made with commonly used spherical lens grinding systems.
The half-width of the diffraction orderx0seen in Figure 4.5 is obtained by setting the sinc term to zero in equation (4.4):
sinc 2W
λzx0 = sin
π2Wx0/λz
π2Wx0/λz =0 (4.6)
which occurs for
π2W
λzx0=π or x0= λz
2W (4.7)
Equation (4.7) shows that the width becomes narrower as the grating width 2W is increased, an anticipated outcome from Fourier transform inverse relations between output and input. The scaling λzindicates that width increases with distance and longer wavelengths.
ZONE PLATE DESIGN AND SIMULATION 67