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Fundamental Optics Material Properties Optical Specifications Gaussian Beam Optics Suppose instead that we decide to reduce the divergence by directing the laser into a beam expander (reversed telescope) of angular magnification m = 10, such as Melles Griot model 09 LBM 013 (figure 2.3). Consider the case in which the expander is focused to form a waist of radius w 0 = 4.0 mm at the expander output lens. Since v ∝ 1/w 0 , by definition, v is reduced by a factor of 10; therefore, for z = 100 m, For the expanded beam, the ratio w(z)/w 0 is only a factor of 12.6 for a distance of 100 m, but it is a factor of 126 for the same distance when the laser is used alone. OPTIMUM COLLIMATION Typically, one has a fixed value for w 0 and uses the previously given expression to calculate w(z) for an input value of z. However, one can also utilize this equation to see how final beam radius varies with starting beam radius at a fixed distance, z. Figure 2.4 shows the Gaussian beam propagation equation plotted as a function of w 0 , with the particular values of l = 632.8 nm and z = 100 m. The beam radius at 100 m reaches a minimum value for a starting beam radius of about 4.5 mm. Therefore, if we wanted to achieve the best combination of minimum beam diameter and minimum beam spread (or best collimation) over a distance of 100 m, our optimum starting beam radius would be 4.5 mm. Any other starting value would result in a larger beam at z = 100 m. We can find the general expression for the optimum starting beam radius for a given distance, z. Doing so yields w (optimum) = 0 5 5 4 w(z) = (10 )( 504 . × 10 ) = 5.04 mm. 10 1/2 ⎛ lz ⎞ ⎜ ⎝ p ⎟ . (2.6) ⎠ Using this optimum value of w 0 will provide the best combination of minimum starting beam diameter and minimum beam spread (ratio of w(z)/w 0 ) over the distance z. The previous example of z = 100 and l=632.8 nm gives w 0 (optimum) = 4.48 mm, shown graphically in figure 2.4. If we put this value for w 0 (optimum) back into the expression for w(z), w(z) = √}}2 w 0 . Thus, for this example, w(100) = √}}2 (4.48) = 6.3 mm. By turning this previous equation around, we can define a distance, called the Rayleigh range (z R ), over which the beam radius spreads by a factor of √}}2 as z R = with If we use beam-expanding optics (such as the 09 LBC, 09 LBX, 09 LBZ, or 09 LCM series), which allow us to adjust the position of the beam waist, we can actually double the distance over which beam divergence is minimized. Figure 2.5 illustrates this situation, in which the beam starts off at a value of w(z R ) = (2lz/p) 1/2 , goes through a minimum value of w 0 = w(z R )/√}}2 , and then returns to w(z R ). By focusing the beam-expanding optics to place the beam waist at the midpoint, we can restrict beam spread to a factor of √}}2 over a distance of 2z R , as opposed to just z R . This result can now be used in the problem of finding the starting beam radius that yields the minimum beam diameter and beam spread over 100 m. Using 2z R = 100, or z R = 50, and l = 632.8 nm, we get a value of w(z R ) = (2lz/p) 1/2 = 4.5 mm, and w 0 = 3.2 mm. Thus, the optimum starting beam radius is the same as previously calculated. However, by focusing the expander we achieve a final beam radius that is no larger than our starting beam radius, while still maintaining the √}}2 factor in overall variation. Alternately, if we started off with a beam radius of 6.3 mm (√}}2w 0 ), we could focus the expander to provide a beam waist of w 0 = 4.5 mm at 100 m, and a final beam radius of 6.3 mm at 200 m. FINAL BEAM RADIUS (mm) 100 80 60 40 20 pw l 2 0 w(z ) = 2w . R 0 0 0 1 2 3 4 5 6 7 8 9 10 STARTING BEAM RADIUS w 0 (mm) (2.7) Optical Coatings Figure 2.3 telescope) Laser beam expander 09 LBM 013 (reversed Figure 2.4 Beam radius at 100 m as a function of starting beam radius for a HeNe laser at 632.8 nm 2.4 1 Visit Us OnLine! www.mellesgriot.com
eam expander INCORPORATING M 2 INTO THE BASIC EQUATIONS The following discussion is taken from the analysis by Sun [Haiyin Sun, “Thin Lens Equation for a Real Laser Beam with Weak Lens Aperture Truncation,” Opt. Eng. 37, no. 11 (November 1998)]. From equation 2.5 we see that, for a theoretical Gaussian beam, the smallest possible value of the radius-divergence product is w 0 v = l/p. For a real laser beam, we have w 0M v M = M 2 l/p >l/p where w 0M and v M are the 1/e 2 intensity waist radius and the farfield half-divergent angle of the real laser beam, respectively, and M 2 factors into equations 2.1 and 2.2 as follows: w M (z) = w 0M [1+(zlM 2 /pw 0M 2 ) 2 ] 1/2 R M (z) = z[1+(pw 0M 2 /zlM 2 ) 2 ] where w M and R M are the 1/e 2 intensity radius of the beam and the beam wavefront radius at z, respectively. The definition for the Rayleigh range (equation 2.7) remains the same for a real laser beam and becomes z R = pw 0R 2 /l. z R Together, equations 2.9, 2.10, and 2.11 form a complete set to denote the input of a real laser beam into a thin lens. w 0 w(–z R ) = √2w 0 w(z R ) = √2w 0 Figure 2.5 Focusing a beam expander to minimize beam radius and spread over a specified distance z R (2.8) (2.9) (2.10) (2.11) LASERS AND LASER SYSTEMS Melles Griot manufactures many types of lasers and laser systems for laboratory and OEM applications. These, along with a wide variety of laser accessories, are found in Chapter 41 through 47. Laser types include helium neon (HeNe) and helium cadmium (HeCd) lasers; argon, krypton, and mixed gas (argon/krypton) ion lasers; diode lasers, and diode-pumped solid-state (DPSS) lasers. Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings Visit Us Online! www.mellesgriot.com 1 2.5
Page 1 and 2: Fundamental Optics 1 Introduction 1
Page 3 and 4: Paraxial Formulas SIGN CONVENTIONS
Page 5 and 6: object F 2 Figure 1.4 image Example
Page 7 and 8: Figure 1.6 Figure 1.7 f f 2 object
Page 9 and 10: Figure 1.8 Figure 1.9 INDIVIDUAL EL
Page 11 and 12: Performance Factors After paraxial
Page 13 and 14: third-order, monochromatic, spheric
Page 15 and 16: Positive lens elements usually have
Page 17 and 18: Lens Shape Aberrations described in
Page 19 and 20: ay f-numbers 2.7 3.3 4.4 6.7 13.3 S
Page 21 and 22: AIRY DISC DIAMETER = 2.44 l f/# Fig
Page 23 and 24: Lens Selection Having discussed the
Page 25 and 26: Example 4: Diffraction-Limited Perf
Page 27 and 28: Aberration Balancing To improve sys
Page 29 and 30: Definition of Terms FOCAL LENGTH (f
Page 31 and 32: infinity (which is a comfortable vi
Page 33 and 34: For symmetric lenses (r 2 = 4r 1 ),
Page 35 and 36: Separation of Nodal Point from Corr
Page 37 and 38: Gaussian Beam Optics 2 Introduction
Page 39: Even if a Gaussian TEM 00 laser-bea
Page 43 and 44: M 2 AND THE LENS EQUATION For real-
Page 45 and 46: employed when laser power must be c
Page 47 and 48: Figure 2.13 Keplerian beam expander
Page 49 and 50: Optical Specifications 3 Wavefront
Page 51 and 52: Centration The mechanical axis and
Page 53 and 54: PERFECT RECTANGULAR LENS The MDMTF
Page 55 and 56: The permissible number of maximum-s
Page 57 and 58: Material Properties 4 Material Prop
Page 59 and 60: Introduction Glass manufacturers pr
Page 61 and 62: maximum value for homogeneity withi
Page 63 and 64: Melles Griot Lens Materials Melles
Page 65 and 66: Refractive Index of Five Schott Gla
Page 67 and 68: Synthetic Fused Silica Fused silica
Page 69 and 70: PERCENT INTERNAL TRANSMITTANCE 99.9
Page 71 and 72: Low-Expansion Borosilicate Glass Th
Page 73 and 74: ZERODUR ® Many optical application
Page 75 and 76: Optical Coatings 5 Optical Coatings
Page 77 and 78: OEM and Special Coatings Melles Gri
Page 79 and 80: PERCENT REFLECTANCE 100 90 80 70 60
Page 81 and 82: will depend on the ratio of optical
Page 83 and 84: v = angle of incidence PERCENT REFL
Page 85 and 86: PERCENT REFLECTANCE PER SURFACE 2.0
Page 87 and 88: Two-Layer Coatings of Arbitrary Thi
Page 89 and 90: ION-ASSISTED BOMBARDMENT Ion-assist
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Single-Layer MgF 2 Antireflection C
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PERCENT REFLECTANCE 5 4 3 2 1 norma
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EXTENDED-RANGE HEBBAR ANTIREFLECTIO
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V-Coatings V-coatings are multilaye
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Metallic High-Reflection Coatings W
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INTERNAL SILVER (/036) $ For intern
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Dielectric High-Reflection Coatings
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PERCENT REFLECTANCE 100 80 60 40 20
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MAXBRIte Coatings MAXBRIte (multi
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Laser-Line MAX-R Coatings These mu
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Ultrafast Coating Melles Griot has
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