Optical constants of silica glass from extreme ultraviolet ...pilon/Publications/AO2007-1.pdf ·...

Optical constants of silica glass from extreme ultravioletto far infrared at near room temperature

Rei Kitamura,1 Laurent Pilon,1,* and Miroslaw Jonasz2

1Department of Mechanical and Aerospace Engineering, Henry Samueli School of Engineering and Applied Science,University of California, Los Angeles, Los Angeles, California 90095, USA

2MJC Optical Technology, 217 Cadillac Street, Beaconsfield, QC H9W 2W7, Canada

*Corresponding author: [email protected]

Received 30 July 2007; accepted 31 August 2007;posted 25 September 2007 (Doc. ID 85883); published 19 November 2007

We thoroughly and critically review studies reporting the real (refractive index) and imaginary (absorp-tion index) parts of the complex refractive index of silica glass over the spectral range from 30 nm to1000 �m. The general features of the optical constants over the electromagnetic spectrum are relativelyconsistent throughout the literature. In particular, silica glass is effectively opaque for wavelengthsshorter than 200 nm and larger than 3.5–4.0 �m. Strong absorption bands are observed (i) below 160 nmdue to the interaction with electrons, absorption by impurities, and the presence of OH groups and pointdefects; (ii) at �2.73–2.85, 3.5, and 4.3 �m also caused by OH groups; and (iii) at �9–9.5, 12.5, and21–23 �m due to SiOOOSi resonance modes of vibration. However, the actual values of the refractiveand absorption indices can vary significantly due to the glass manufacturing process, crystallinity,wavelength, and temperature and to the presence of impurities, point defects, inclusions, and bubbles,as well as to the experimental uncertainties and approximations in the retrieval methods. Moreover,new formulas providing comprehensive approximations of the optical properties of silica glass areproposed between 7 and 50 �m. These formulas are consistent with experimental data and substantiallyextend the spectral range of 0.21–7 �m covered by existing formulas and can be used in various engi-neering applications. © 2007 Optical Society of America

OCIS codes: 160.6030, 160.4760, 290.3030, 160.4670, 050.5298, 110.5220.

1. Introduction

Silicon dioxide (SiO2 or silica) has many forms includ-ing three main crystalline varieties: quartz, tridym-ite, and cristobalite [1,2]. Silica can also exist innoncrystalline form as silica glass or vitreous silica[1], also referred to as amorphous silica and glassysilica. There are four basic types of commercial silicaglasses [3,4]: (1) Type I is obtained by the electricmelting of natural quartz crystal in vacuum or in aninert gas at low pressure, (2) Type II is produced fromquartz crystal powder by flame fusion, (3) Type III issynthetic and produced by the hydrolyzation of SiCl4when sprayed into an OH flame, and (4) Type IV isalso synthetic and produced from SiCl4 in a watervapor-free plasma flame.

Each type of silica glass has its own impurity leveland optical properties. For example, Type I silicaglasses tend to contain metallic impurities [3,4]. Onthe other hand, Types III and IV are much purer thanType I and feature greater ultraviolet (UV) transmis-sion [3,4]. However, Type III silica glasses have, ingeneral, higher water content, and infrared (IR)transmission is limited by strong water absorptionpeaks at wavelengths around 2.2 and 2.7 �m [3–5].Type IV is similar to Type III but contains less waterand thus has better IR transmission. Suppliers pro-vide various product grades for different optical ap-plications [3,4,6]. Trade names for Type I silica glassare Infrasil, IR-Vitreosil, Pursil, and GE 105, for ex-ample. On the other hand, KU, Herosil, Homosil,Vitrasil, and OG Vitreosil are Type II. Moreover,KU-1, KV, Suprasil I, Tetrasil, Spectrosil, and Corn-ing 7940 are known as synthetic fused silica andclassified as Type III. Finally, KI, Suprasil W, Spec-

0003-6935/07/338118-16$15.00/0© 2007 Optical Society of America

20 November 2007 � Vol. 46, No. 33 � APPLIED OPTICS 8118

trosil WF, and Corning 7943 are Type IV silica glass.The term “fused quartz” is usually used for Type Iand “fused silica” is used for synthetic fused silica,i.e., Types III or IV. In practice, however, those termsare often used interchangeably.

Because of its favorable physical, chemical, andoptical characteristics, silica glass has been used innumerous applications: (1) as laboratory glassware,(2) in optics, as lenses or beam splitters, (3) for light-ing and IR heating, (4) in telecommunications, asfiber optics, (5) in micro- and optoelectronics, as adielectric insulator, a waveguide, photonic crystal fi-bers, or projection masks for photolithography, and(6) in thermal protection systems, as fibrous thermalinsulation. In all these applications, optical proper-ties are essential to predict and optimize the opticaland thermal radiation performances of this material.Silica fiber optics, for example, are used in thenear-IR at �1.31 and 1.55 �m due to their low opticalattenuation and optical dispersion [7,8]. In lens de-signing, one often needs to fit and interpolate refrac-tive index data that are reported or measured atdiscrete wavelengths over certain spectral regions[9,10]. On the other hand, astronomers and atmo-spheric scientists are interested in the optical prop-erties of interstellar and atmospheric silica particlesin the mid- and far-IR region of the spectrum [11–13].

The complex refractive index, m�, of silica glass atwavelength � is defined as

m� � n� � ik�, (1)

where n� is the refractive index and k� is the absorp-tion index. The wavelength � is related to otherquantities such as frequency � and wavenumber �according to

� �c�

��

1�

, (2)

where c� is the speed of light at wavelength � invacuum. Therefore, the refractive and absorption in-dices can also be expressed as functions of frequencyand denoted by n� and k� or as functions of wavenum-ber and denoted by n� and k�.

The experimental data for the refractive and ab-sorption indices vary in precision depending on themeasurement techniques used and on the approxi-mations made in retrieving the intrinsic optical prop-erties. One should also keep in mind that theseoptical properties may be sensitive to the presence ofimpurities, crystallinity, point defects, inclusions,bubbles, wavelength, temperature, and the glassmanufacturing process. In addition, when consider-ing the literature it is often difficult to choose the setof experimental data or formula to use and to assessits validity. The objectives of this paper are (1) tocritically review and compare the experimental datareported in the literature for the complex refractiveindex of silica glass and (2) to develop formulas thatprovide a comprehensive approximation of the mea-

sured data near room temperature. Given the widerange of engineering and scientific applications, thespectral range from 30 nm to 1000 �m is considered.

2. Experimental Methods

A. Refractive Index, n�

Various experimental techniques and procedureshave been used to retrieve the real part of the com-plex refractive index n�. The most accurate is theminimum deviation angle method [14], which relieson measuring the minimum deviation angle �min of anisosceles triangular prism made of the silica glassplaced in air. This method is based on Snell’s law [14],and the refractive index n� can be estimated by

n� �

sin��min � �

2 �sin��

2�nair, (3)

where � is an apex angle of the prism sample and nairis the refractive index of air �nair � 1�. This method isoften used to accurately measure the refractive indexof highly transparent glass for which the absorptionindex k�, i.e., the imaginary part of the complex re-fractive index, is negligibly small.

Alternatively, the interferometric method is alsoused to measure n�. It is based on observing the in-terference fringes created when light is incidentnormally upon a silica glass plate [15,16]. Other tech-niques include the Abbe’s or the Pulfrich’s refractom-eters whose accuracy on the index of refraction iswithin 2 10�3 and 5 10�5, respectively [17].

Moreover, when absorption cannot be ignored, bothn� and k� can be retrieved from the directional orhemispherical reflectance and�or emittance of a slabof known thickness. Electromagnetic wave theory canbe used to retrieve both n� and k�, assuming opticallysmooth surfaces and accounting for internal reflec-tion [18]. Finally, the Kramers–Krönig relations [18]can also be used to predict either the refractive indexfrom the absorption index, or vice versa, at frequency� [18,19]:

n� � 1 �2�

P�0

��k��

��2 � �2 d��, (4)

k� ��2�

�P�

0

n��

��2 � �2 d��, (5)

where P denotes the Cauchy principle value of theintegral [18]. Alternatively, the refractive and ab-sorption indices can be simultaneously obtained fromreflectance. First, the phase angle of the complex re-flection coefficient �, at frequency �0 can be expressedby the Kramers–Krönig relations [20]:

8119 APPLIED OPTICS � Vol. 46, No. 33 � 20 November 2007

�� 2�

�P�

0

d ln� R��2 � �2 d��, (6)

where R�� is the normal–normal reflectivity ex-pressed as a function of �:

R�� r�2 ��n� � 1�2 � k�

2

�n� � 1�2 � k�2. (7)

The Fresnel reflection coefficient r is defined as

r �1 � n� � ik�

1 � n� � ik�� r�e

i�. (8)

From Eqs. (7) and (8), the refractive and absorptionindices can be expressed as

n� �1 � R��

1 � R�� 2�R�� cos �, (9)

k� ��2�R�� sin �

1 � R�� 2�R�� cos �. (10)

However, due to the infinite bound of the integrals inEqs. (4)–(6), these techniques require extrapolationsinto spectral regions where data are not always avail-able. Furthermore, the integrals need to be computednumerically. Practical limitations and possible errorsof Kramers–Krönig relations have been discussed byRiu and Lapaz [19]. These authors concluded that theKramers–Krönig relations were practically applica-ble in almost every experimental situation.

B. Absorption Index, k�

The value of the absorption index k� was not alwaysdirectly available from the literature and was some-times recovered from the normal spectral transmit-tance or emittance data. Indeed, the value of k� can berecovered from the normal spectral transmittanceT0,�, accounting for multiple reflections and expressedas [21]

T0,��L� ��1 � ��2e��L

1 � ��2e�2��L, (11)

where L is the thickness of the sample and �� and ��

are the spectral reflectivity of the interface and thespectral absorption coefficient of silica glass, respec-tively, and are given by

�� n� � 1�2 � k�

2

�n� � 1�2 � k�2, (12)

�� 4�k�

�. (13)

Equations (11)–(13) can be solved as a quadratic inthe exponential factor in terms of k�. After some al-gebraic manipulation, one obtains the following ex-pression for k� as a function of the refractive index, n�,the sample thickness, L, and the spectral normaltransmittance, T0,�:

k� � ��

4�L�ln��1 � ��4 � 4��2T0,� � �1 � ��

2��2T0,�

.

(14)

Alternatively, the absorption index, k�, can also bedetermined from measurements of the spectral nor-mal emittance, ��,0, by using the following expression[22]:

k� � � �

4�L�ln1 � �� ,0

1 � �� ,0. (15)

Note that the above expressions for the absorptionindex k� given by Eqs. (14) and (15) are valid if both�n� � 1� and �n� � 1� are much larger than k�. Ineither case, an expression for n� is necessary to esti-mate ��. As discussed later in this paper, theSellmeier equation proposed by Malitson [23] can beused for that purpose between 0.21 and 6.7 �m.

In addition, in the UV and IR regions of the spec-trum when silica glass is strongly absorbing, mostreported values of n� and k� were retrieved from near-normal reflectance measurements in combinationwith the Kramers–Krönig relations [24].

3. Experimental Data and Discussion

Table 1 summarizes representative references re-porting experimental values of the complex refractiveindex of silica glass at room temperature for the spec-tral range from 30 nm to 1000 �m. For each study,the measurement method, the spectral range, as wellas the sample thicknesses, compositions, and temper-atures investigated are also reported when available.In addition, the absorption index k� was derived fromtransmittance measurements using Eq. (14) if it wasnot directly reported. Then, computation from trans-mittance and emittance data sometimes leads to neg-ative values, particularly in the spectral region from0.2 to 4.0 �m where silica glass is very weakly ab-sorbing. This was the case for transmittance datafrom [25,26]. Hence, in this region, the experimentaldata should be used with care since the uncertaintyfor k� is very large and k� effectively vanishes.

Figure 1 shows the real and imaginary parts of therefractive index of silica glass, n� and k�, as a functionof wavelength � over the spectral range from 30 nm to1000 �m as reported in the references listed in Table1. Because of the density of data points in some partsof the spectrum and for the sake of clarity, Figs. 2–5show details of both the real n� and imaginary k�

parts of the complex index of refraction of silica glassfor wavelengths between 30 nm and 1 �m, 1 and


Tab

le1.

Sum

mar

yo

fth

eE

xper

imen

talD

ata

Rep

ort

ing

the

Co

mp

lex

Ind

exo

fR

efra

ctio

no

fS

ilica

Gla

ssat

Ro

om

Tem

per

atur

ea

Ref

eren

ceW

avel

engt

hR

ange

(�m

)M

easu

rem

ent

Met

hod

Rep

orte

dD

ata

Tem

pera

ture

Sam

ple

Thi

ckne

ssT

ype

Com

men

ts

[11]

7.14

–25

Refl

ecti

on,t

rans

mis

sion

,K

ram

ers–

Krö

nig

rela

tion

n,k

RT

N�A

IV

itre

osil

[12]

2–50

0R

eflec

tion

,Kra

mer

s–K

röni

gm

etho

dR

,n,k

300,

200,

100,

and

10K

N�A

III

Sup

rasi

l,U

Vgr

ade

synt

heti

cfu

sed

SiO

2

[13]

7–30

0T

rans

mis

sion

n,k

RT

N�A

N�A

[15]

110–

550

Inte

rfer

omet

ric

T,n

,kR

T2.

1409

mm

IIn

fras

il(l

owH

2O

)[1

6]3–

6.7

Inte

rfer

omet

ric

n,T

RT

0.23

mm

III

Sup

rasi

l2

�10

00pp

mO

Hco

nten

t[2

3]0.

21–3

.71

Min

imum

devi

atio

nm

etho

dn

20°C

N�A

III

Cor

ning

code

7940

,Dyn

asil

high

-pur

ity

synt

heti

cS

iO2

glas

san

dG

Ety

pe15

1[2

5]0.

80–2

.60

kfr

omE

q.(1

4)T

298

K1.

6m

mN

�A[2

6]0.

19–0

.42

kfr

omE

q.(1

4)T

298

K2

mm

N�A

[27]

0.05

–0.7

Refl

ecti

onan

dK

ram

ers–

Krö

nig

n,�

RT

N�A

III

Sup

rasi

l(U

Vgr

ade)

[28]

7.19

–9.0

6In

terf

erom

etri

cn,

T,R

RT

0.19

56m

mIV

Sup

rasi

lW

2[2

9]0.

2–3

Refl

ecti

onan

dtr

ansm

issi

onn,

k,R

,T24

°CN

�AN

�A[3

0]16

.7–2

5R

eflec

tion

R,n

,kR

TN

�AN

�A[4

4]0.

35–3

.51

Min

imum

devi

atio

nm

etho

dn

24°C

N�A

N�A

Sam

ples

from

Gen

eral

Ele

ctri

c,H

erae

us,N

iede

rF

used

Qua

rtz,

and

Cor

ning

Gla

ssW

orks

[45]

1.44

–4.7

7In

terf

erom

etri

cn,

T23

.5°C

–481

°C0.

1994

mm

IVS

upra

sil

W2

[49]

7.14

–50

Refl

ecti

onn,

kR

TN

�AII

KU

and

KI

Kra

mer

s–K

röni

gre

lati

onIV

Impu

rity

did

not

exce

ed0.

007%

[50]

0.00

06–5

00n,

kR

TN

�AN

�AC

ompi

lati

onof

data

[51]

1.00

–7.5

Tra

nsm

issi

onT

,R

T5

�m

to3.

06m

N�A

[54]

0.10

–0.1

6R

eflec

tion

and

Kra

mer

s–K

röni

gn,

kR

TN

�AN

�AS

ampl

espr

oduc

edby

Ele

ctro

-Qua

rtz

[55]

1.35

–4.8

5In

terf

erom

etri

cn,

T23

.5°C

0.23

45m

mIV

Sup

rasi

lW

2[5

6]8.

13–9

.63

Refl

ecti

onR

,n,k

RT

N�A

[57]

7.84

–12.

9R

eflec

tion

n,k

RT

N�A

[58]

0.23

–3.3

7M

inim

umde

viat

ion

met

hod

n26

°C,4

71°C

,and

828

°CN

�AII

IC

orni

ng79

40

[59]

7.14

–11.

11R

eflec

tion

,tra

nsm

issi

on,a

ndK

ram

ers–

Krö

nig

rela

tion

n,k

RT

N�A

IIK

UII

IK

VIV

KI

IIn

fras

il[6

0]7.

14–1

4.29

Refl

ecti

onR

,kR

TN

�AN

�A[6

1]0.

5–4.

5M

inim

umde

viat

ion

nR

TN

�AN

�AS

ampl

eha

ving

low

wat

erco

nten

tsu

ppli

edby

Gen

eral

Ele

ctri

cC

ompa

ny[6

2]0.

37,0

.44,

0.55

,1.

01,a

nd1.

53M

inim

umde

viat

ion

n29

4,24

0,18

0,an

d12

0K

N�A

N�A

[63]

0.06

–40

n,k

RT

N�A

N�A

[64]

0.2–

3.5

Tra

nsm

issi

onk,

TR

T—

1500

°C0.

953

mm

I,II

IG

EIR

type

105,

GE

UV

type

151,

and

Cor

ning

Vyc

orIR

Gla

ssN

umbe

r79

05[6

5]2.

00–6

.00

kfr

omE

q.(1

4)T

25°C

and

400

°C2.

8m

mI

GE

Typ

e10

5[6

6]0.

31–3

.97

kfr

omE

q.(1

4)T

RT

5.45

mm

N�A


15 �m, 15 and 100 �m, and 100 and 1000 �m, re-spectively.

Overall, the reported values for both the real andimaginary parts of the complex index of refractionagree relatively well. Studies showing large devia-tions from other studies suggest that the data areunreliable [24]. For example, data reported by Elliset al. [27] strongly disagree with all other data be-tween 60 nm and the visible. Locations of extrema ofn� are consistent among all experimental data ex-cept for those reported by Tan [28] �7.19 �m � �� 9.06 �m�, Khashan and Nassif [29] �0.2 �m � �� 3.0 �m�, and Reitzel [30] �16.7 �m � � � 25 �m�.The data for wavelengths below 9 �m as reported byvarious authors agree well with one another. How-ever, the data agree considerably less for wave-lengths between 9 and 50 �m. Beyond 50 �m, asmaller number of values for n� have been reported,but the data agree relatively well.

Furthermore, trends and the locations of extremain the measured absorption index are consistent fromone study to another. However, discrepancies largerthan those for n� can be observed in reported data fork� in some parts of the spectrum. They are most likelydue to (i) the impurity of the sample (e.g., OH group,alkali, metallic content), (ii) the presence of inclu-sions, bubbles, or point defects, (iii) the sample prep-aration and surface optical quality, and�or (iv) theuncertainty in the measurement and retrieval tech-niques. Note that the flatness of the sample surfacebecomes a critical parameter in the visible and UV[31,32]. In this wavelength range, the surface rough-ness must remain much smaller than the wavelengthto avoid surface scattering and consider the surfaceas optically smooth.

The imaginary part of the complex refractive indexof silica glass k� is small from the near UV to thenear-IR part of the spectrum. Practically, silica glassis transparent from 200 nm up to 3.5–4.0 �m. In theextreme UV (for wavelengths below 200 nm) and inthe IR and far-IR (beyond 4.0) silica glass can beconsidered opaque. In the UV region of the spectrumbelow 200 nm, the strong absorption of silica glass iscaused by the interaction of the electromagnetic ra-diation with electrons of SiOO bonds [17] and withstructural imperfections or point defects such as OHgroups, SiOSi bonds, and strained SiOOOSi bonds[33]. This results in a sharp UV cutoff (also called theabsorption edge) at �160 nm [17,34]. The location ofthe absorption edge depends on the glass composi-tion, impurity level, and point defects formed duringthe manufacturing process [17,33] as well as on tem-perature [35]. For example, it is shifted toward thevisible wavelengths due to the presence of impuritiesin particular ions Fe3�, Cr3�, and Ti3� [36]. Similareffects are observed when increasing the alkali con-tents [34] or the temperature [35]. On the contrary,the absorption edge is slightly shifted to lower wave-lengths for crystal quartz [34]. Shifting the absorp-tion edge to lower wavelengths (even slightly) hasbeen the subject of intense studies to enable the use

[67]

0.16

–0.3

0k

from

Eq.

(14)

TR

T2.

04–3

.29

mm

III

Cor

ning

code

7940

and

Dyn

asil

[68]

1.00

–4.6

2k

from

Eq.

(14)

TR

T3.

18m

m[6

9]7.

69–1

1.11

Refl

ecti

onan

dK

ram

ers–

Krö

nig

n,k

RT

1.0

mm

IIK

UII

IO

ptic

algr

ade

fuse

dqu

artz

IVK

I[7

0]7.

41–5

0R

eflec

tion

and

disp

ersi

onan

alys

isb

n,k

RT

N�A

IVK

I[7

1]60

–560

Tra

nsm

issi

onan

dre

flect

ion

R,n

,kR

T0.

258,

1.05

,2.0

3,4.

07,1

2.35

,12.

35,

and

25m

m

N�A

Al 2

O3

�4.

0�

10�

3%

,Fe 2

O3

�4.

0�

10�

3%

,C

aO�

2.0

�10

�3%

,Na 2

O�

1.0

�10

�3%

,C

uO,T

iO2,M

gO,M

n 3O

4

0.00

1%[7

2]83

.3–5

00T

rans

mis

sion

n,

300

K1.

340

�0.

001

mm

Opt

ical

lypo

lish

ed[7

3]50

–100

0R

eflec

tion

and

tran

smis

sion

n,k

RT

N�A

N�A

Impu

rity

�3

�10

�5

[74]

100–

1000

Tra

nsm

issi

on

RT

N�A

GE

type

s10

1an

d10

6[7

5]0.

029–

1.77

Tra

nsm

issi

on,r

eflec

tion

,and

Kra

mer

s–K

röni

gT

,R,n

,kR

TN

�AII

IS

upra

sil

I

[76]

100–

667

Tra

nsm

issi

on

100

and

300

KG

Ety

pe10

1[7

7]7.

69–1

1.1

Inte

rnal

refle

ctio

nan

dK

ram

ers–

Krö

nig

n,k

RT

N�A

III

KU

-1gl

ass,

brok

ensu

rfac

e

[78]

2–35

Refl

ecti

onan

dK

ram

ers–

Krö

nig

T,R

,n,k

RT

N�A

N�A

aR

Tis

room

tem

pera

ture

.bR

efer

ence

[24]

.


Fig. 1. Real n� and imaginary k� parts of the complex refractive index of silica glass reported in the literature and summarized in Table1. The solid curve (present study) was obtained with Eqs. (21)–(24) by using coefficients listed in Table 2.


Fig. 2. Real n� and imaginary k� parts of the complex refractive index of silica glass between 30 nm and 1 �m as reported in the literatureand summarized in Table 1.


Fig. 3. Real n� and imaginary k� parts of the complex refractive index of silica glass between 1 and 15 �m as reported in the literatureand summarized in Table 1. The solid curve (present study) was obtained with Eqs. (21)–(24) by using coefficients listed in Table 2.


Fig. 4. Real n� and imaginary k� parts of the complex refractive index of silica glass between 15 and 100 �m as reported in the literatureand summarized in Table 1. The solid curve (present study) was obtained with Eqs. (21)–(24) by using coefficients listed in Table 2.


Fig. 5. Real n� and imaginary k� parts of the complex refractive index of silica glass between 100 and 1000 �m as reported in the literatureand summarized in Table 1.


of silica glass for photomask material in 157 nmphotolithography using F2 excimer lasers [32,37,38].An acceptable transmittance at �157 nm has beenachieved by minimizing the OH content of silica [37]or by doping silica glass with network modifiers suchas fluorine, which relaxes the glass structure andeliminates strained SiOOOSi bonds [33]. Experi-mental measurements and theoretical calculations ofthe electronic structure of SiO2 has been reviewed byGriscom [39] and spectroscopic data for wavelengthsbetween 90 and 350 nm have been discussed by Sigel[34]. An interaction between the UV radiation andelectrons and point defects is also responsible forthe steep increase of n� for wavelengths less than300 nm.

In the IR part of the spectrum, silica glass iseffectively opaque for a wavelength larger than3.5–4.0 �m. Beyond this wavelength, three major ab-sorption bands can be observed (Fig. 1) due to theresonance of SiOOOSi vibrations. The absorptionpeak between 9.0 and 9.5 �m can be attributed tothe asymmetric stretching vibration of SiOOOSibridges [17,24]. The absorption band at �12.5 �m isdue to the symmetric vibration stretching of theSiOOOSi bridge involving the displacement of the Oatom perpendicular to the SiOSi direction in theSiOOOSi plane [24]. The third band between 21 and23 �m is the consequence of the OOSiOO bendingvibration but has also been attributed to the “rocking”mode of SiOOOSi bonds caused by the displacementof an oxygen atom out of the SiOOOSi plane [24].The resonance of SiOOOSi vibrations are also re-sponsible for the sharp decreases in n� around theresonance wavelengths [17]. The reader is referredto [24, pp. 63–77] for a detailed discussion on vi-brational spectroscopy of silica glass at these wave-lengths. Moreover, smaller absorption bands at� wavelengths 2.73–2.85, 3.5, and 4.3 �m correspondto the presence of OH groups in the structure of theglass [17,40–42]. The magnitude of the absorptiondepends on the melting technology and in particularon the partial pressure of water vapor above the meltduring the melting process [17]. The concentration ofOH groups in silica glass can be computed from theabsorption band at � wavelength 2.73–2.85 �m [17].To the best of our knowledge, no model or approxi-mate equation has been proposed for the absorptionindex of silica glass. This is the subject of Section 4.

4. Optical Constant Theory

The complex index of refraction, m� � n� � ik�,and the complex relative dielectric permittivity,�� i�� are related by the expression�� m�

2, i.e., [21]

�� n�2 � k�

2, � �� 2n�k�. (16)

Numerous physical models such as the Lorentzmodel, the Drude model, and the Debye relaxationmodel have been proposed to predict the optical prop-erties of solids [18]. The Lorentz model assumes

that electrons and ions in the material are harmonicoscillators subject to the force applied by time-dependent electromagnetic fields. Then the complexrelative dielectric permittivity can be expressed interms of frequency � as follows [21]:

�� 1 � �j

�pj2

�j2 � �2 � i�j�

� 1 � �j

�pj2��j

2 � �2� � i�j�pj2�

��j2 � �2�2 � �j

2�2 , (17)

where �pj and �j are the plasma and resonance fre-quencies, respectively. The parameter �j is the damp-ing factor of the oscillators. Only when � is very closeto one of the resonance frequencies �j, the imaginaryterms in Eq. (17) are important [43]. Thus, �j� arenegligibly small compared with ��j

2 � �2� for silicaglass for a wavelength below 7 �m and �� is virtuallyequal to 0.0. Hence, after substituting Eq. (2) intoEq. (17), � can be expressed in terms of � as follows:

�� 1 � �j

Aj2�2

��2 � �j2�

, (18)

where Aj � �pj�j�c� with �j being the resonance wave-length. Moreover, as �� vanishes, the medium isweakly absorbing and k� is negligibly small comparedwith n�. Then �� is equal to n�

2 and given by theSellmeier dispersion formula:

�� n�2 � 1 � �

j

Aj2�2

��2 � �j2�

. (19)

Different formulas for the refractive index of silicaglass as a function of wavelength and based on theSellmeier dispersion formula have been proposed inthe literature [23,44,45] for different spectral regions.Rodney and Spindler [44] suggested a formula for n�

over a spectral range from 0.347 to 3.508 �m at 31 °Cwhile Tan and Arndt [45] proposed another equationin the spectral region from 1.44 to 4.77 �m at tem-peratures ranging from 23.5 to 481 °C. In addition,for the spectral range from 0.21 to 3.71 �m at 20 °C,Malitson [23] fitted experimental data with the fol-lowing three-term Sellmeier equation:

n�2 � 1 �

0.6961663�2

�2 � �0.0684043�2�

0.4079426�2

�2 � �0.1162414�2

�0.8974794�2

�2 � �9.896161�2. (20)

Tan [16] confirmed the validity of Eq. (20) for wave-lengths up to 6.7 �m. Furthermore, for a spectralrange over 8 �m, an approximate piecewise linear fitwas given by Dombrovsky [46]. However, no physics-


based formulas have been developed for the spectralrange beyond 8 �m.

In parts of the spectrum where k� cannot be ne-glected or when the frequency � is very close to theresonance frequencies, the Sellmeier equation for n�

is no longer valid and an alternative model must beused. Recently, Meneses et al. [47] proposed a newdielectric function model based on the causal versionof the Voigt function. The model was validated byfitting the IR spectra of two different glasses and

confirmed to be more appropriate than the Lorentzmodel [47]. Moreover, the authors proposed anothersimplified model based on Gaussian functions [48].Then, the dielectric constant can be written as

�� i� �� j

�gcjkkg�� igcj�� ,

(21)

where the high frequency dielectric constant is de-noted by � . In addition, the Gaussian functions gcj��and gcj

kkg�� are defined as

gcj�� j exp�4 ln 2�� 0j

�j�2� �j

exp�4 ln 2�� 0j

�j�2, (22)

gcjkkg��

2�j

� D�2�ln 2� � �0j

�j�� D�2�ln 2

� � �0j

�j�.

(23)

Here, �j is the amplitude, �0j is the peak position, �j isthe full width at half-maximum, and D�x� is an oper-ator defined as

Fig. 6. Residuals between experimental [49] and predicted values of n� and k�. The predicted values are based on Eqs. (21)–(24) withcoefficients listed in Table 2.

Table 2. Parameters Used to Interpolate the Refractive Index n� andAbsorption Index k� of Silica Glass by Using Eqs. (21)–(24)a,b

j �j �0j �j

1 3.7998 1089.7 31.4542 0.46089 1187.7 100.463 1.2520 797.78 91.6014 7.8147 1058.2 63.1535 1.0313 446.13 275.1116 5.3757 443.00 45.2207 6.3305 465.80 22.6808 1.2948 1026.7 232.14

a�� 2.1232.bThese parameters were obtained by fitting the equations to data

of Popova et al. [49].


Fig. 7. Residuals between the experimental data on the refractive index and absorption index and values predicted in this work by usingEqs. (21)–(24) along with coefficients listed in Table 2.


D�x� � e�x2�0

x

et2

dt. (24)

In the present study, this model is used to interpolatethe refractive index n� and absorption index k� forwavelengths � between 7 and 50 �m. It enables one todescribe the experimental data with a reduced set ofparameters [48] over a wide spectral range, includingthe spectral range where k� may be large. Note that theabove simplified model satisfies the Kramers–Krönigrelation.

The practical procedure for fitting complex refrac-tive index data conducted in this paper is as follows:(i) the spectral reflectivity at normal incidence, R��,is computed from Eq. (7), (ii) parameters of �� inEqs. (21) and (22) are determined by curve fitting forR�� expressed as

R�� 1

�� 1�2

, (25)

and (iii) n� and k� are computed using Eq. (16). Theadvantage of this procedure is that (a) fitting thereflectivity is easier than fitting n� and k� indepen-dently, (b) both n� and k� can be derived from a singlecurve fitting, and (c) the result automatically satisfiesthe Kramers–Krönig relations.

The experimental data of Popova et al. [49] wereselected to develop formulas for both n� and k� be-cause these data cover a wide spectral range from 7 to50 �m, and both the refractive and absorption indicesare reported at the same wavelength enabling thecalculation of R��. In the spectral region from 0.2 to7 �m, the absorption index of silica glass is verysmall and may be assumed to be zero for all practicalpurposes as suggested by Fig. 1. Moreover, reporteddata, including that of Popova et al. [49], indicatesthat the refractive index is satisfactorily predicted bythe Sellmeier formula reported by [16] and Malitson[23] between 0.2 and 7 �m and given by Eq. (20).Therefore, the present study focuses on the spectralrange between 7 and 50 �m.

To fit the model with experimental data, the FOCUS

software was used [47,48]. By adding terms in Eq.(21) one by one, eight terms were found to best fit theexperimental data. The fitting curves obtained in thisstudy are shown in Figs. 3 and 4, and the associatedparameters �j, �0j, and �j used in Eqs. (21)–(23) aresummarized in Table 2. The fitting curves for n� andk� obtained in the present study agree well with thedata of Popova et al. [49] and with most of other datashown in Figs. 3 and 4. The differences (or residuals)between the data of Popova et al. [49] and the modelpredictions for both n� and k� using the parametersgiven in Table 2 are shown in Fig. 6. It indicates thatthe residuals are less than 0.06 except near 9 and22.5 �m, where they reach up to 0.3 around 9 �m and0.14 around 22.5 �m. This can be attributed to the

fact that the refractive index changes greatly in thosewavelength regions. However, these residuals aresmall compared with much larger difference observedin the experimental data reported for n�. For exam-ple, one can see differences larger than 0.5 amongreported refractive index data in the spectral rangebetween 9 and 10 �m. Thus, the approximation ob-tained here is acceptable and can be useful in engi-neering applications.

Moreover, the residuals between the experimentaldata for the refractive and absorption indices listed inTable 1, and the model predictions using the param-eters given in Table 2 are shown in Fig. 7. Here again,the residuals are less than 0.3 except at �9.0 and22.5 �m. This indicates that the model predictionsusing the parameters given in Table 2 agree with alldata reported in the literature at the spectral rangeexcept at �9 and 22.5 �m. However, large discrep-ancies in reported experimental data sets can be ob-served around these two wavelengths. In general, themodel should be used with care when applied outsidethe spectral range from 0.2 to 50 �m.

Finally, given its widespread use, particular atten-tion was paid to the compilation of data reported byPhilipp [50] for the refractive and absorption indicesof silica glass over the spectral range from 0.006 to500 �m. The compilation consists of experimentaldata reported by various authors [15,20,51–54] aswell as unpublished data. Philipp [50] also retrievedoptical properties from the absorption coefficient aswell as computer generated values. Qualitatively,good agreement between the data compiled by Phil-ipp [50] and the other data is observed in Figs. 2–5.Moreover, residuals with the present model rangefrom �0.42 to 0.19 for n� and �0.25 to 0.37 for k� from7 to 50 �m.

5. Conclusions

A thorough review of experimental data for thecomplex refractive index of silica glass at near roomtemperatures over a spectral range from 30 nmto 1000 �m implies that the values reported in theliterature can vary significantly due to numeroussample features and experimental methods and con-ditions. Hence, it is essential to report the silica glasssynthesis method, composition, impurity, and defectslevel, sample thickness, surface roughness, and tem-peratures as well as the retrieval method and theunderlying assumptions when one reports opticalproperties of glass. However, the general features ofthe complex refractive index spectra are relativelyconsistent throughout the region of the spectrumconsidered. Silica glass is effectively opaque forwavelengths shorter than 200 nm and larger than3.5–4.0 �m. Strong absorption bands are observed (i)below 160 nm due to interaction with electrons, ab-sorption by impurities, and the presence of OHgroups and point defects, (ii) at �2.73–2.85, 3.5, and4.3 �m also caused by OH groups, (ii) at �9–9.5, 12.5,and 21–23 �m due to SiOOOSi resonance modes ofvibration. New formulas for both the real and imag-inary parts of the complex refractive index are pro-


posed over a wide spectral range between 7 and50 �m, thus complementing the existing analyticalformula for n� in the range of 0.21–7 �m [16,23]. Theimaginary part of the complex refractive index canbe neglected in much of this range �0.21–4 �m�. Thedifferences between various experimental data arecomparable and greater than the differences betweenthe results of these formulas and the experimentaldata used to develop them. Hence it is believed thatthe formulas proposed are useful for practical engi-neering applications such as simulations and optimi-zations of optical and thermal systems. The datacollected and presented in this study are available indigital form online [79] or directly from the corre-sponding author upon request.

The authors thank Asahi Glass Co., Ltd. Japan forfinancial support. They are grateful to D. De SousaMeneses for helpful discussion about FOCUS. The con-tribution of M. Jonasz was supported by MJC OpticalTechnology.

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