Abstract

We show that a gradient-index element designed from a blend of three materials allows a designer to specify independently the element’s refractive index and its change in refractive index with respect to wavelength. We show further the effectiveness of this approach by comparing modeled chromatic performance of deflectors consisting of a single material, a binary blend of materials, and a ternary blend.

© 2016 Optical Society of America

1. Introduction

The desire to create an arbitrary distribution of refractive indices within a volume has intrigued designers of optical elements for over a century [1]. The first practical methods for fabricating elements with graded indices were developed in the late 1960s and relied on diffusing salts into glass [2]. More recently, researchers have fabricated graded-index (GRIN) elements by layering nanometer-thick polymer films of differing indices to produce a material with a desired effective index [3]. This technique mixes two optical materials to create arbitrary refractive index distributions.

As we show here, a blend of two materials, i.e., a binary blend, implies that the index and the change in index with respect to wavelength are dependent. Specifying one determines the other. As we also show here, blending three materials allows us to break this dependency. To highlight the power of this separation, we examine optical elements for deflection.

We begin in Sec. 2 with a discussion of the optical properties of GRIN materials formed by blending pairs and triplets of homogeneous materials. In Sec. 3, we compare the modeled chromatic performance of deflectors consisting of a single material, material pairs, and material triplets. We provide some discussion and concluding remarks in Sec. 4.

2. Graded-index materials

In this section we consider optical materials whose refractive index varies along one spatial dimension n(x, λ). One can fabricate such materials using the nanolayer-based approach described in [3].

2.1. Binary material blend

A binary material blend is fabricated from two materials a and b. If we assume at the design wavelength λ0 that nb(λ0) > na(λ0), the square of the effective refractive index for TE-polarized illumination is [4]

n2(x,λ)=γa(x)na2(λ)+γb(x)nb2(λ),=na2(λ)+γb(x)[nb2(λ)na2(λ)],
where the functions γa(x) and γb(x) indicate the volume fraction of material in a unit volume. They are, therefore, bounded 1 ≥ γ(x) ≥ 0 and constrained,
γa(x)+γb(x)=1.

We used Eq. (2) to arrive at the final form of Eq. (1). Note in Eq. (1) that spatial structure and the chromatic properties of the optical materials are separable.

Given our objective for achromatic design, we assess chromatic behavior by the change in index as a function of wavelength at a specific wavelength. For the blended material this is

dn(x,λ)dλ=[1n(x,λ)]{na(λ)dna(λ)dλ+γb(x)[nb(λ)dnb(λ)dλna(λ)dna(λ)dλ]}.

Although the quantity dn(λ)/ is related to dispersion, i.e., index as a function of wavelength, to insure clarity in this work, we refer to it as the index slope.

Specifying a desired index profile ñ(x) at the operating wavelength λ0 determines the filling factor γb(x) through the relationship

γb(x)=n˜2(x)na2(λ0)nb2(λ0)na2(λ0).

Although Eqs. (1) and (4) are related, they differ in intent. We refer to Eq. (1) as an analysis equation and Eq. (4), synthesis.

Note that specifying the volume fraction γb(x) at a single wavelength, determines the index for all wavelengths [see Eq. (1)]. Given that the index slope of the blended material is determined completely by n(x, λ) and γb(x) [see Eq. (3)], specifying ñ(x) in a binary blend determines both the index and the index slope.

2.2. Ternary material blend

We now consider blending three materials a, b, and c with refractive properties na(λ), nb(λ), and nc(λ), such that nc(λ0) > nb(λ0) > na(λ0). Under these conditions and assuming TE-polarized illumination, the square of the refractive index is

n2(x,λ)=na2(λ)+γb(x)[nb2(λ)na2(λ)]+γc(x)[nc2(λ)na2(λ)],
where, given that the material volume fractions are again bounded and constrained, the material a volume fraction γa(x) is
γa(x)=1γb(x)γc(x),
and the index slope is
dn(x,λ)dλ=[1n(x,λ)]{na(λ)dna(λ)dλ+γb(x)[nb(λ)dnb(λ)dλna(λ)dna(λ)dλ]+γc(x)[nc(λ)dnc(λ)dλna(λ)dna(λ)dλ]}.

For analysis, we combine Eqs. (5) and (7) into a single matrix equation,

[n2(x,λ)n(x,λ)dn(x,λ)dλ]=N(λ)[γb(x)γc(x)]+[na2(λ)na(λ)dna(λ)dλ],
where
N(λ)=[nb2(λ)na2(λ)nb(λ)dnb(λ)dλna(λ)dna(λ)dλnc2(λ)na2(λ)nc(λ)dnc(λ)dλna(λ)dna(λ)dλ].

The analysis Eq. (8) once again separates spatial structure from material chromatic properties.

For synthesis, i.e., to determine the volume ratios γb(x) and γc(x), we invert Eq. (8) and insert the desired index ñ(x) and desired index slope (x) at the design wavelength λ0,

[γb(x)γc(x)]=N1(λ0)[n˜2(x)na2(λ0)n˜(x)dn˜(x)na(λ0)dna(λ0)dλ].

In the next section we compare the chromatic performance of GRIN elements for deflection, contrasting the use of material pairs and triplets.

3. Deflection

With reference to Fig. 1, a plane wave with wavelength λ normally incident upon a triangular prism made from a material a with index na(λ), width W, thickness d, and prism angle α = tan−1(d W) exits the prism tilted at an angle βR such that

tanβR(λ)=[na(λ)1]tanα.

 figure: Fig. 1

Fig. 1 Optical elements considered for deflection. (a) Triangular prism constructed from a single material. (b) Rectangular prism constructed from gradient index materials.

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Now consider a slab also with width W and thickness d but whose refractive index n(x, λ) varies spatially. The phase a normally incident plane wave accumulates when it propagates through this slab is

θG(x,λ)=(2πdλ)n(x,λ).

If the index is a linear gradient,

n(x,λ)=nmin(λ)+[nmax(λ)nmin(λ)](xW),
the slab tilts the incident plane wave by an angle βG,
tanβG(λ)=(dW)[nmax(λ)nmin(λ)].

Note that the chromatic behavior of the prism is dependent only upon the properties of the material a. The chromatic behavior of the GRIN slab is dependent upon the properties of the extremal indices nmin(λ) and nmax(λ).

We now consider the performance of the three deflectors listed in Table 1. For illustrative purposes, the elements are designed using the materials listed in Table 2, whose last column is the Abbe number V. All values for index, Abbe number, and index slope (used later) are from the Schott catalog [5]. We assume in both tables an operating wavelength λ0 = 0.55 µm.

Tables Icon

Table 1. Glass materials used to design deflectors in Table 1.

The first deflector is a 45°-refractive prism made from N-LASF31A (material b in Table 2). The other two are GRIN elements whose width and depth are equal (d = W). One is a binary blend of N-FK51A and N-SF66 (materials a and c) and the other a ternary blend of N-FK51A, N-LASF31A, and N-SF66 (materials a, b, and c).

Tables Icon

Table 2. Deflection elements.

Figure 2 is a representation of these materials, in which the material index n (λ0) at the design wavelength λ0 = 0.55 µm is plotted against its index slope dn(λ0)/dλ at the same wavelength. For both GRIN blends, we assume for consistency ñmin = 1.6615 and ñmax = 1.8830. For the ternary blend, the index slope is a constant, (x) = dnb(λ0)/dλ. The index-index slope relationship for the binary blend is represented by the dotted curve labeled path 2 (for two materials) and, for the ternary blend, by the dotted curve labeled path 3 (i.e., three materials). The spatial functions for the binary and ternary blend designs are represented in Fig. 3.

 figure: Fig. 2

Fig. 2 Index as a function of index slope for materials in Table 2 at λ0 = 0.55 µm. The circled numbers refer to the number of materials used to design the deflection elements in Table 1.

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 figure: Fig. 3

Fig. 3 Deflector designs using GRIN materials. For the binary blend (a) the index, (b) index slope, and (c) volume ratios as a function of space at the design wavelength. (d)–(f) As in (a)–(c) but for the ternary blend. Note that the index slope in (b) is dependent upon the desired index in (a), whereas the index slope in (e) is specified independent of the index in (d).

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Figures 2 and 3 represent the index and index slope at the design wavelength, λ0 = 0.55 µm. How the materials perform at other wavelengths is presented in Fig. 4, which plots index versus index slope for three wavelengths that define the waveband of interest: the extremal wavelengths at 0.40 and 0.70 µm and the operating wavelength 0.55 µm. The values presented in Fig. 2 are represented by the central figure in Fig. 4. Path 3 is shown as a dotted line for each wavelength. Only at the design wavelength is the index slope constant. (The green dotted line is vertical.) Thus, Fig. 4 provides insight into how the optical properties of a material blend are transformed as a function of wavelength.

 figure: Fig. 4

Fig. 4 Index as a function of index slope at three different wavelengths for a ternary blend of glasses. Blue represents 0.40 µm, green, 0.55 µm, and red, 0.70 µm.

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Figure 5 compares the deflection angle [Eqs. (11) and (14)] for all three deflecting elements as a function of wavelength across the waveband. Note that the binary blend GRIN element is more dispersive than the prism and the ternary blend GRIN element. The preponderance of the dispersive material N-SF66 in the binary blend [see Fig. 3(c)] yields a highly dispersive element. Although the prism and the ternary element have the same index slope at the operating wavelength, the dispersive properties of the prism are determined by the dispersion of material b, N-LASF31A. The chromatic behavior of the ternary blend is balanced by mixing all three materials.

 figure: Fig. 5

Fig. 5 Deflection angle as a function of wavelength for the elements in Table 1. Deflection angles for the prism are on the right and, for the GRIN elements, on the left. The scales are equal for both.

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A designer can generate alternative ternary GRIN deflectors by changing the design wavelength. See Fig. 6. Each curve represents an alternate design generated using a different design wavelength λ0 and the constraints

n˜max=nb(λ0),
n˜min=na2(λ0)+γac[nc2(λ0)na2(λ0)],
dn˜(x)=dnb(λ0)dλ.

 figure: Fig. 6

Fig. 6 Deflection angle as a function of wavelength for ternary designs with different design wavelengths.

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The index ñmin is that mixture of materials a and c whose index slope is the same as material b; thus,

γac=nb(λ0)dnb(λ0)/dλna(λ0)dna(λ0)/dλnc(λ0)dnc(λ0)/dλna(λ0)dna(λ0)/dλ.

Note that the deflector designed for nominal operation at λ0 = 0.40 µm produces an element that disperses light in a linear fashion. That is, a pencil beam of white light incident on the surface of this GRIN element produces a rainbow with equally spaced colors. White light incident on the surface of the element designed for operation at 0.55 µm produces almost no rainbow.

Figure 4 highlights a key issue in GRIN design. Whereas our desire is to insure that some optical function that is dependent upon chromatic behavior is nearly constant across the waveband of interest, the design freedom available to us exists in the spatial domain. This is a consequence of the separability noted previously. We, therefore, need some way to visualize the impact of spatial variations in index on chromatic behavior. This linkage dictates a careful consideration of design metrics, which is especially true for more complex optical behavior, such as focusing [6–8].

4. Discussion and concluding remarks

By examining optical deflection through a GRIN element, we demonstrated the advantage of optics blended from a triplet of materials to control and improve chromatic performance. Our preliminary work on achromatic lens design [6, 7] indicates it also benefits from this approach, but the complexities of the design do not render it as simple an introduction to the concept as the deflector application. For example, we continue to work on lens design metrics and metrics for material selection as reported in [8]. Our work is informed by previous work on material selection for achromatic design [9].

Nonetheless, considerable effort remains before this potential can be realized practically. Most important is not just the demonstration of a ternary material blend with prescribed index and dispersion profiles, but also the demonstration of a fabrication process with sufficient precision and robustness to control the volumetric ratio of three materials. This is imperative if the process is to be commercially viable. Further, one must demonstrate that the improvement in chromatic control provided by a ternary blend of materials is sufficiently significant to outweigh the costs of its manufacture.

If these can be demonstrated, one can contemplate the advantages provided by additional materials. In general, given M materials, a designer can control the spatio-chromatic behavior of a blended material from index n(x, λ) through dM−2n(x, λ)/M−2. This opens the possibility for the independent correction of geometric and chromatic aberrations. To correct geometric aberrations, a designer uses surface parameters, e.g., ones associated with freeform surfaces, and, to correct chromatic aberrations, a designer uses a blend of multiple materials.

References and links

1. R. W. Wood, Physical Optics (Macmillan, New York, 1905).

2. S. N. Houde-Walter and D. T. Moore, “Gradient-index profile control by field-assisted ion exchange in glass,” Appl. Opt. 24, 4326–4333 (1985). [CrossRef]   [PubMed]  

3. M. Ponting, A. Hiltner, and E. Baer, “Polymer Nanostructures by Forced Assembly: Process, Structure, and Properties,” Macromol. Symp. 294, 19–32 (2010). [CrossRef]  

4. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2003).

5. http://www.schott.com/advanced_optics/english/download/schott-optical-glass-pocket-catalog-january-2014-row.pdf. (Accessed March 5, 2015).

6. R. A. Flynn, E. F. Fleet, G. Beadie, and J. S. Shirk, “Achromatic GRIN singlet lens design,” Opt. Express 21, 4970–4978 (2013). [CrossRef]   [PubMed]  

7. J. N. Mait, G. Beadie, P. Milojkovic, and R. A. Flynn, “Chromatic Analysis of a First-order Radial GRIN Lens,” Opt. Express 23, 22069–22086 (2015). [CrossRef]   [PubMed]  

8. G. Beadie, R. A. Flynn, J. N. Mait, and P. Milojkovic, “Materials figure of merit for achromatic gradient index (GRIN) optics,” in Advanced Optics for Defense Applications: UV through LWIR, J. N. Vizgaitis, B. F. Andresen, P. L. Marasco, J. S. Sanghera, and M. P. Snyder, eds., Proc. SPIE 9822, 98220Q (2016).

9. P. N. Robb, “Selection of optical glasses. 1: Two materials,” Appl. Opt. 24, 1864–1877 (1985). [CrossRef]   [PubMed]  

References

  • View by:

  1. R. W. Wood, Physical Optics (Macmillan, New York, 1905).
  2. S. N. Houde-Walter and D. T. Moore, “Gradient-index profile control by field-assisted ion exchange in glass,” Appl. Opt. 24, 4326–4333 (1985).
    [Crossref] [PubMed]
  3. M. Ponting, A. Hiltner, and E. Baer, “Polymer Nanostructures by Forced Assembly: Process, Structure, and Properties,” Macromol. Symp. 294, 19–32 (2010).
    [Crossref]
  4. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2003).
  5. http://www.schott.com/advanced_optics/english/download/schott-optical-glass-pocket-catalog-january-2014-row.pdf . (Accessed March 5, 2015).
  6. R. A. Flynn, E. F. Fleet, G. Beadie, and J. S. Shirk, “Achromatic GRIN singlet lens design,” Opt. Express 21, 4970–4978 (2013).
    [Crossref] [PubMed]
  7. J. N. Mait, G. Beadie, P. Milojkovic, and R. A. Flynn, “Chromatic Analysis of a First-order Radial GRIN Lens,” Opt. Express 23, 22069–22086 (2015).
    [Crossref] [PubMed]
  8. G. Beadie, R. A. Flynn, J. N. Mait, and P. Milojkovic, “Materials figure of merit for achromatic gradient index (GRIN) optics,” in Advanced Optics for Defense Applications: UV through LWIR, J. N. Vizgaitis, B. F. Andresen, P. L. Marasco, J. S. Sanghera, and M. P. Snyder, eds., Proc. SPIE9822, 98220Q (2016).
  9. P. N. Robb, “Selection of optical glasses. 1: Two materials,” Appl. Opt. 24, 1864–1877 (1985).
    [Crossref] [PubMed]

2015 (1)

2013 (1)

2010 (1)

M. Ponting, A. Hiltner, and E. Baer, “Polymer Nanostructures by Forced Assembly: Process, Structure, and Properties,” Macromol. Symp. 294, 19–32 (2010).
[Crossref]

1985 (2)

Baer, E.

M. Ponting, A. Hiltner, and E. Baer, “Polymer Nanostructures by Forced Assembly: Process, Structure, and Properties,” Macromol. Symp. 294, 19–32 (2010).
[Crossref]

Beadie, G.

J. N. Mait, G. Beadie, P. Milojkovic, and R. A. Flynn, “Chromatic Analysis of a First-order Radial GRIN Lens,” Opt. Express 23, 22069–22086 (2015).
[Crossref] [PubMed]

R. A. Flynn, E. F. Fleet, G. Beadie, and J. S. Shirk, “Achromatic GRIN singlet lens design,” Opt. Express 21, 4970–4978 (2013).
[Crossref] [PubMed]

G. Beadie, R. A. Flynn, J. N. Mait, and P. Milojkovic, “Materials figure of merit for achromatic gradient index (GRIN) optics,” in Advanced Optics for Defense Applications: UV through LWIR, J. N. Vizgaitis, B. F. Andresen, P. L. Marasco, J. S. Sanghera, and M. P. Snyder, eds., Proc. SPIE9822, 98220Q (2016).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2003).

Fleet, E. F.

Flynn, R. A.

J. N. Mait, G. Beadie, P. Milojkovic, and R. A. Flynn, “Chromatic Analysis of a First-order Radial GRIN Lens,” Opt. Express 23, 22069–22086 (2015).
[Crossref] [PubMed]

R. A. Flynn, E. F. Fleet, G. Beadie, and J. S. Shirk, “Achromatic GRIN singlet lens design,” Opt. Express 21, 4970–4978 (2013).
[Crossref] [PubMed]

G. Beadie, R. A. Flynn, J. N. Mait, and P. Milojkovic, “Materials figure of merit for achromatic gradient index (GRIN) optics,” in Advanced Optics for Defense Applications: UV through LWIR, J. N. Vizgaitis, B. F. Andresen, P. L. Marasco, J. S. Sanghera, and M. P. Snyder, eds., Proc. SPIE9822, 98220Q (2016).

Hiltner, A.

M. Ponting, A. Hiltner, and E. Baer, “Polymer Nanostructures by Forced Assembly: Process, Structure, and Properties,” Macromol. Symp. 294, 19–32 (2010).
[Crossref]

Houde-Walter, S. N.

Mait, J. N.

J. N. Mait, G. Beadie, P. Milojkovic, and R. A. Flynn, “Chromatic Analysis of a First-order Radial GRIN Lens,” Opt. Express 23, 22069–22086 (2015).
[Crossref] [PubMed]

G. Beadie, R. A. Flynn, J. N. Mait, and P. Milojkovic, “Materials figure of merit for achromatic gradient index (GRIN) optics,” in Advanced Optics for Defense Applications: UV through LWIR, J. N. Vizgaitis, B. F. Andresen, P. L. Marasco, J. S. Sanghera, and M. P. Snyder, eds., Proc. SPIE9822, 98220Q (2016).

Milojkovic, P.

J. N. Mait, G. Beadie, P. Milojkovic, and R. A. Flynn, “Chromatic Analysis of a First-order Radial GRIN Lens,” Opt. Express 23, 22069–22086 (2015).
[Crossref] [PubMed]

G. Beadie, R. A. Flynn, J. N. Mait, and P. Milojkovic, “Materials figure of merit for achromatic gradient index (GRIN) optics,” in Advanced Optics for Defense Applications: UV through LWIR, J. N. Vizgaitis, B. F. Andresen, P. L. Marasco, J. S. Sanghera, and M. P. Snyder, eds., Proc. SPIE9822, 98220Q (2016).

Moore, D. T.

Ponting, M.

M. Ponting, A. Hiltner, and E. Baer, “Polymer Nanostructures by Forced Assembly: Process, Structure, and Properties,” Macromol. Symp. 294, 19–32 (2010).
[Crossref]

Robb, P. N.

Shirk, J. S.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2003).

Wood, R. W.

R. W. Wood, Physical Optics (Macmillan, New York, 1905).

Appl. Opt. (2)

Macromol. Symp. (1)

M. Ponting, A. Hiltner, and E. Baer, “Polymer Nanostructures by Forced Assembly: Process, Structure, and Properties,” Macromol. Symp. 294, 19–32 (2010).
[Crossref]

Opt. Express (2)

Other (4)

G. Beadie, R. A. Flynn, J. N. Mait, and P. Milojkovic, “Materials figure of merit for achromatic gradient index (GRIN) optics,” in Advanced Optics for Defense Applications: UV through LWIR, J. N. Vizgaitis, B. F. Andresen, P. L. Marasco, J. S. Sanghera, and M. P. Snyder, eds., Proc. SPIE9822, 98220Q (2016).

R. W. Wood, Physical Optics (Macmillan, New York, 1905).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2003).

http://www.schott.com/advanced_optics/english/download/schott-optical-glass-pocket-catalog-january-2014-row.pdf . (Accessed March 5, 2015).

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Figures (6)

Fig. 1
Fig. 1 Optical elements considered for deflection. (a) Triangular prism constructed from a single material. (b) Rectangular prism constructed from gradient index materials.
Fig. 2
Fig. 2 Index as a function of index slope for materials in Table 2 at λ0 = 0.55 µm. The circled numbers refer to the number of materials used to design the deflection elements in Table 1.
Fig. 3
Fig. 3 Deflector designs using GRIN materials. For the binary blend (a) the index, (b) index slope, and (c) volume ratios as a function of space at the design wavelength. (d)–(f) As in (a)–(c) but for the ternary blend. Note that the index slope in (b) is dependent upon the desired index in (a), whereas the index slope in (e) is specified independent of the index in (d).
Fig. 4
Fig. 4 Index as a function of index slope at three different wavelengths for a ternary blend of glasses. Blue represents 0.40 µm, green, 0.55 µm, and red, 0.70 µm.
Fig. 5
Fig. 5 Deflection angle as a function of wavelength for the elements in Table 1. Deflection angles for the prism are on the right and, for the GRIN elements, on the left. The scales are equal for both.
Fig. 6
Fig. 6 Deflection angle as a function of wavelength for ternary designs with different design wavelengths.

Tables (2)

Tables Icon

Table 1 Glass materials used to design deflectors in Table 1.

Tables Icon

Table 2 Deflection elements.

Equations (18)

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n 2 ( x , λ ) = γ a ( x ) n a 2 ( λ ) + γ b ( x ) n b 2 ( λ ) , = n a 2 ( λ ) + γ b ( x ) [ n b 2 ( λ ) n a 2 ( λ ) ] ,
γ a ( x ) + γ b ( x ) = 1 .
d n ( x , λ ) d λ = [ 1 n ( x , λ ) ] { n a ( λ ) d n a ( λ ) d λ + γ b ( x ) [ n b ( λ ) d n b ( λ ) d λ n a ( λ ) d n a ( λ ) d λ ] } .
γ b ( x ) = n ˜ 2 ( x ) n a 2 ( λ 0 ) n b 2 ( λ 0 ) n a 2 ( λ 0 ) .
n 2 ( x , λ ) = n a 2 ( λ ) + γ b ( x ) [ n b 2 ( λ ) n a 2 ( λ ) ] + γ c ( x ) [ n c 2 ( λ ) n a 2 ( λ ) ] ,
γ a ( x ) = 1 γ b ( x ) γ c ( x ) ,
d n ( x , λ ) d λ = [ 1 n ( x , λ ) ] { n a ( λ ) d n a ( λ ) d λ + γ b ( x ) [ n b ( λ ) d n b ( λ ) d λ n a ( λ ) d n a ( λ ) d λ ] + γ c ( x ) [ n c ( λ ) d n c ( λ ) d λ n a ( λ ) d n a ( λ ) d λ ] } .
[ n 2 ( x , λ ) n ( x , λ ) d n ( x , λ ) d λ ] = N ( λ ) [ γ b ( x ) γ c ( x ) ] + [ n a 2 ( λ ) n a ( λ ) d n a ( λ ) d λ ] ,
N ( λ ) = [ n b 2 ( λ ) n a 2 ( λ ) n b ( λ ) d n b ( λ ) d λ n a ( λ ) d n a ( λ ) d λ n c 2 ( λ ) n a 2 ( λ ) n c ( λ ) d n c ( λ ) d λ n a ( λ ) d n a ( λ ) d λ ] .
[ γ b ( x ) γ c ( x ) ] = N 1 ( λ 0 ) [ n ˜ 2 ( x ) n a 2 ( λ 0 ) n ˜ ( x ) d n ˜ ( x ) n a ( λ 0 ) d n a ( λ 0 ) d λ ] .
tan β R ( λ ) = [ n a ( λ ) 1 ] tan α .
θ G ( x , λ ) = ( 2 π d λ ) n ( x , λ ) .
n ( x , λ ) = n min ( λ ) + [ n max ( λ ) n min ( λ ) ] ( x W ) ,
tan β G ( λ ) = ( d W ) [ n max ( λ ) n min ( λ ) ] .
n ˜ max = n b ( λ 0 ) ,
n ˜ min = n a 2 ( λ 0 ) + γ a c [ n c 2 ( λ 0 ) n a 2 ( λ 0 ) ] ,
d n ˜ ( x ) = d n b ( λ 0 ) d λ .
γ a c = n b ( λ 0 ) d n b ( λ 0 ) / d λ n a ( λ 0 ) d n a ( λ 0 ) / d λ n c ( λ 0 ) d n c ( λ 0 ) / d λ n a ( λ 0 ) d n a ( λ 0 ) / d λ .

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