Patent Publication Number: US-11656403-B2

Title: Single mode optical fibers with low cutoff wavelength high mechanical reliability

Description:
This application claims priority under 35 USC § 119(e) from U.S. Provisional Patent Application Ser. No. 63/023,428 filed on May 12, 2020 which is incorporated by reference herein in its entirety. 
    
    
     FIELD 
     The present disclosure relates to optical fibers and in particular relates to single mode optical fibers with low cutoff wavelength and high mechanical reliability. 
     BACKGROUND 
     There is an increasing demand for greater bandwidth and higher data transmission capacity for an optical fiber—driven by the rapid growth of internet traffic. The need for greater bandwidth and higher data transmission rates has motivated efforts to develop next-generation platforms for information storage and delivery. It is widely believed that optical information systems can provide superior performance as compared to present-day, microelectronics-based systems. 
     Data centers commonly employ vertical-cavity surface-emitting lasers (VCSEL) optical information transmission. However, chromatic dispersion is a limiting factor for high data rates and/or long-reach links because of the large transceiver linewidth associated with VCSELs. An example of so affected VCSELs are those that, for example, transmit over single mode fibers using a wavelength of approximately 850 nm. Integrated optical systems based on silicon photonics or long wavelength VCSELs are a leading replacement technology for microelectronic systems. Silicon photonics may be interfaced with standard CMOS technologies and WDM (wavelength division multiplexing) to provide numerous functions—for example, convert electrical signals to optical signals, transmit optical signals, and reconvert optical signals to electrical signals. 
     To increase the transmission capacity, WDM and spatial division multiplexing (SDM) have been used to increase the number of transmission channels. Advanced modulation formats have been developed to increase the data rate per channel. However, the number of channels and the channel data rate are nearly at the practical limits and increasing the number of fibers is unavoidable. A method to overcome this bottleneck is to upscale the hardware (e.g., more fibers). However, limited space availability at data centers and increased costs discourage such implementations. 
     Thus, optical fibers are routed through tight bend configurations or bend in a small radius arc inside a fiber array unit that couples an array of fibers to arrays of lasers and photodiodes. It is therefore desirable to design a single mode optical fiber having low attenuation, low bend losses, a G.657-compliant mode field diameter, and a low cutoff wavelength. 
     BRIEF SUMMARY 
     The present disclosure is directed to optical fiber cables having high transmission capacity, low transmission loss, low microbending loss, high puncture resistance, and high overall mechanical reliability. 
     Aspect 1 of the description discloses: A single mode optical fiber, comprising: a core region, the core region having a radius r 1  in a range from 3.0 microns to 6.0 microns and a core volume V 1  less than 6.0%-micron 2 ; a cladding region surrounding and directly adjacent to the core region, the cladding region including a first outer cladding region and a second outer cladding region surrounding and directly adjacent to the first outer cladding region, the first outer cladding region having a radius r 4a , the second outer cladding region having a radius r 4b  less than or equal to 65 microns and comprising silica based glass doped with titania; wherein the single mode optical fiber has a fiber cutoff wavelength λ CF  less than 1530 nm. 
     Aspect 2 of the description discloses: The single mode optical fiber of Aspect 1, wherein the radius r 1  is in a range from 3.5 microns to 5.5 microns. 
     Aspect 3 of the description discloses: The single mode optical fiber of Aspect 1 or 2, wherein the core volume V 1  is greater than the 3.0%-micron 2 . 
     Aspect 4 of the description discloses: The single mode optical fiber of any of Aspects 1-3, wherein the core region has a maximum relative refractive index Δ 1MAX  in a range from 0.25% to 0.40%. 
     Aspect 5 of the description discloses: The single mode optical fiber of any of Aspects 1-4, wherein the core region has a graded-index relative refractive index profile. 
     Aspect 6 of the description discloses: The single mode optical fiber of Aspect 5, wherein the graded-index relative refractive index profile is an α-profile with a value of a in a range from 2 to 20. 
     Aspect 7 of the description discloses: The single mode optical fiber of any of Aspects 1=6, wherein the cladding region includes an inner cladding region surrounding and directly adjacent to the core region, the inner cladding region having a radius r 2 , a thickness (r 2 −r 1 ) in a range from 1 micron to 5 microns and a relative refractive index Δ 2  in a range from −0.10% to 0.10%. 
     Aspect 8 of the description discloses: The single mode optical fiber of Aspect 7, wherein the cladding region further comprises a depressed-index cladding region surrounding and directly adjacent to the inner cladding region, the depressed-index cladding region having a radius r 3 , a thickness (r 3 −r 2 ) in a range from 3 microns to 10 microns, and a relative refractive index Δ 3  in a range from −0.50% to −0.20%. 
     Aspect 9 of the description discloses: The single mode optical fiber of Aspect 8, wherein the thickness (r 3 −r 2 ) in a range from 4 microns to 8 microns. 
     Aspect 10 of the description discloses: The single mode optical fiber of Aspect 8 or 9, wherein the ratio r 2 /r 1 &gt;0.4. 
     Aspect 11 of the description discloses: The single mode optical fiber of Aspect 10, wherein the ratio r 2 /r 1 &lt;0.7. 
     Aspect 12 of the description discloses: The single mode optical fiber of any of Aspects 8-11, wherein the depressed-index cladding region has a has a trench volume V 3  in a range from 20% Δ-microns 2  to 70% Δ-microns 2 . 
     Aspect 13 of the description discloses: The single mode optical fiber of any of Aspects 8-12, wherein the first outer cladding region surrounds and is directly adjacent to the depressed-index cladding region. 
     Aspect 14 of the description discloses: The single mode optical fiber of any of Aspects 1-13, wherein the first outer cladding region has a relative refractive index Δ 4a  that is in the range from −0.10% to 0.10% and the second outer cladding region has a relative refractive index Δ 4b  greater than 0.20%. 
     Aspect 15 of the description discloses: The single mode optical fiber of any of Aspects 1-14, wherein the second outer cladding region has a titania concentration in a range from 5 wt % to 20 wt %. 
     Aspect 16 of the description discloses: The single mode optical fiber of any of Aspects 1-15, wherein the second outer cladding region has a thickness (r 4b −r 4a ) in a range from 2 microns to 30 microns. 
     Aspect 17 of the description discloses: The single mode optical fiber of any of Aspects 1-16, wherein the single mode optical fiber has a fiber cutoff wavelength λ CF  less than 1260 nm. 
     Aspect 18 of the description discloses: The single mode optical fiber of Aspect 17, wherein the single mode optical fiber has a bend loss at 1310 nm, as determined by a mandrel wrap test using a mandrel with a diameter of 15 mm, less than 1.0 dB/turn. 
     Aspect 19 of the description discloses: The single mode optical fiber of any of Aspects 1-18, wherein the single mode optical fiber has a cable cutoff wavelength λ CC  less than 1060 nm. 
     Aspect 20 of the description discloses: The single mode optical fiber of Aspect 19, wherein the single mode optical fiber has a bend loss at 1310 nm, as determined by a mandrel wrap test using a mandrel with a diameter of 15 mm, less than 1.0 dB/turn. 
     Aspect 21 of the description discloses: The single mode optical fiber of any of Aspects 1-18, wherein the single mode optical fiber has a cable cutoff wavelength λ CC  less than 1000 nm. 
     Aspect 22 of the description discloses: The single mode optical fiber of Aspect 21, wherein the single mode optical fiber has a bend loss at 1310 nm, as determined by a mandrel wrap test using a mandrel with a diameter of 15 mm, less than 1.0 dB/turn 
     Aspect 23 of the description discloses: The single mode optical fiber of any of Aspects 1-22, wherein the single mode optical fiber has a mode field diameter (MFD) greater than 8.2 microns at 1310 nm. 
     Aspect 24 of the description discloses: The single mode optical fiber of any of Aspects 1-22, wherein the single mode optical fiber has a mode field diameter (MFD) in a range from 8.6 microns to 9.5 microns at 1310 nm. 
     Aspect 25 of the description discloses: The single mode optical fiber of any of Aspects 1-24, wherein the single mode optical fiber has a bend loss at 1550 nm, as determined by a mandrel wrap test using a mandrel with a diameter of 10 mm, less than 1.0 dB/turn. 
     Aspect 26 of the description discloses: The single mode optical fiber of any of Aspects 1-25, wherein the single mode optical fiber has a zero dispersion wavelength λ 0  in a range from 1300 nm to 1324 nm. 
     Additional features and advantages are set forth in the Detailed Description that follows, and in part will be apparent to those skilled in the art from the description or recognized by practicing the embodiments as described in the written description and claims hereof, as well as the appended drawings. It is to be understood that both the foregoing general description and the following Detailed Description are merely exemplary, and are intended to provide an overview or framework to understand the nature and character of the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The accompanying figures, which are incorporated herein, form part of the specification and illustrate embodiments of the present disclosure. Together with the description, the figures further serve to explain the principles of and to enable a person skilled in the relevant art(s) to make and use the disclosed embodiments. These figures are intended to be illustrative, not limiting. Although the disclosure is generally described in the context of these embodiments, it should be understood that it is not intended to limit the scope of the disclosure to these particular embodiments. In the drawings, like reference numbers indicate identical or functionally similar elements. 
         FIG.  1    is a side elevated view of a section of an exemplary optical fiber, according to some embodiments. 
         FIG.  2    is a cross-sectional view of an exemplary optical fiber, according to some embodiments. 
         FIG.  3    is a schematic view of a representative optical fiber ribbon, according to some embodiments. 
         FIG.  4    is a schematic view of a representative optical fiber cable, according to some embodiments. 
         FIG.  5    shows an optical fiber coupled to a Si photonic waveguide through a ferrule connector with a curved hole for supporting a section of the optical fiber. 
         FIG.  6    shows the relationship between minimum bend radius and fiber cladding diameter. 
         FIGS.  7 A through  7 E  are exemplary relative refractive index profiles of an optical fiber, according to some embodiments. 
         FIGS.  8 A through  8 D  are exemplary axial stress profiles of an optical fiber, according to some embodiments. 
         FIG.  9    shows the dependence of puncture load on cross-sectional area for three secondary coatings. 
         FIG.  10    is a schematic diagram of an exemplary optical fiber drawing system illustrating fabrication of an optical fiber, according to some embodiments. 
     
    
    
     DETAILED DESCRIPTION 
     Embodiments of the present disclosure are described in detail herein with reference to embodiments thereof as illustrated in the accompanying drawings, in which like reference numerals are used to indicate identical or functionally similar elements. References to “one embodiment,” “an embodiment,” “some embodiments,” “in certain embodiments,” etc., indicate that the embodiment described may include a particular feature, structure, or characteristic, but every embodiment may not necessarily include the particular feature, structure, or characteristic. Moreover, such phrases are not necessarily referring to the same embodiment. Further, when a particular feature, structure, or characteristic is described in connection with an embodiment, it is submitted that it is within the knowledge of one skilled in the art to affect such feature, structure, or characteristic in connection with other embodiments whether or not explicitly described. 
     The following examples are illustrative, but not limiting, of the present disclosure. Other suitable modifications and adaptations of the variety of conditions and parameters normally encountered in the field, and which would be apparent to those skilled in the art, are within the spirit and scope of the disclosure. 
     Where a range of numerical values is recited herein, comprising upper and lower values, unless otherwise stated in specific circumstances, the range is intended to include the endpoints thereof, and all integers and fractions within the range. It is not intended that the scope of the claims be limited to the specific values recited when defining a range. Further, when an amount, concentration, or other value or parameter is given as a range, one or more preferred ranges or a list of upper preferable values and lower preferable values, this is to be understood as specifically disclosing all ranges formed from any pair of any upper range limit or preferred value and any lower range limit or preferred value, regardless of whether such pairs are separately disclosed. Finally, when the term “about” is used in describing a value or an end-point of a range, the disclosure should be understood to include the specific value or end-point referred to. When a numerical value or end-point of a range does not recite “about,” the numerical value or end-point of a range is intended to include two embodiments: one modified by “about,” and one not modified by “about.” 
     As used herein, the term “about” means that amounts, sizes, formulations, parameters, and other quantities and characteristics are not and need not be exact, but may be approximate and/or larger or smaller, as desired, reflecting tolerances, conversion factors, rounding off, measurement error and the like, and other factors known to those of skill in the art. 
     As used herein, “comprising” is an open-ended transitional phrase. A list of elements following the transitional phrase “comprising” is a non-exclusive list, such that elements in addition to those specifically recited in the list may also be present. 
     The term “or,” as used herein, is inclusive; more specifically, the phrase “A or B” means “A, B, or both A and B.” Exclusive “or” is designated herein by terms such as “either A or B” and “one of A or B,” for example. 
     The indefinite articles “a” and “an” to describe an element or component means that one or at least one of these elements or components is present. Although these articles are conventionally employed to signify that the modified noun is a singular noun, as used herein the articles “a” and “an” also include the plural, unless otherwise stated in specific instances. Similarly, the definite article “the,” as used herein, also signifies that the modified noun may be singular or plural, again unless otherwise stated in specific instances. 
     The term “wherein” is used as an open-ended transitional phrase, to introduce a recitation of a series of characteristics of the structure. 
     Cartesian coordinates are used in some of the Figures for the sake of reference and ease of illustration and are not intended to be limiting as to direction or orientation. The z-direction is taken as the axial direction of the optical fiber. 
     The term “fiber” as used herein is shorthand for optical fiber. 
     The coordinate r is a radial coordinate, where r=0 corresponds to the centerline of the fiber. The term “radius” refers to a value of the radial coordinate. The term “outer radius”, when used in reference to a region of a fiber, refers to the largest radial coordinate included in the region. The term “diameter”, when used in reference to a region of a fiber, refers to twice the outer radius of the region. 
     The term “core” as used herein is a core region of an optical fiber, representing a cylinder of material, centered at r=0, that runs along the optical fiber&#39;s length. The core is characterized by its radius of cross-sectional area corresponding to confinement (e.g. 90%) of optical intensity in the optical fiber. The core is surrounded by a medium with a lower index of refraction, typically a cladding region. Light travelling in the core reflects from the core-cladding boundary due to total internal reflection, as long as the angle between the light and the boundary is greater than the critical angle. As a result, the optical fiber transmits all rays that enter the fiber with a sufficiently small angle relative to the optical fiber&#39;s axis. 
     The term “core radius” as used herein is referred to geometric core radius which is determined from the refractive index profile. 
     The symbol “μm” is used as shorthand for “micron,” which is a micrometer, i.e., 1×10 −6  meter. 
     The symbol “nm” is used as shorthand for “nanometer,” which is 1×10 −9  meter. 
     The limits on any ranges cited herein are inclusive and thus to lie within the range, unless otherwise specified. 
     The terms “comprising,” and “comprises,” e.g., “A comprises B,” is intended to include as a special case the concept of “consisting,” as in “A consists of B.” 
     The phrase “bare optical fiber” or “bare fiber” as used herein means an optical fiber directly drawn from a heated glass source (i.e., a “preform”) and prior to applying a protective coating layer to its outer surface (e.g., prior to the bare optical fiber being coated with a polymeric-based material). 
     “Refractive index” refers to the refractive index at a wavelength of 1550 nm. 
     The “refractive index profile” is the relationship between refractive index or relative refractive index and radius. For relative refractive index profiles depicted herein as having step boundaries between adjacent core and/or cladding regions, normal variations in processing conditions may preclude obtaining sharp step boundaries at the interface of adjacent regions. It is to be understood that although boundaries of refractive index profiles may be depicted herein as step changes in refractive index, the boundaries in practice may be rounded or otherwise deviate from perfect step function characteristics. It is further understood that the value of the relative refractive index may vary with radial position within the core region and/or any of the cladding regions. When relative refractive index varies with radial position in a particular region of the fiber (e.g. core region and/or any of the cladding regions), it is expressed in terms of its actual or approximate functional dependence, or its value at a particular position within the region, or in terms of an average value applicable to the region as a whole. Unless otherwise specified, if the relative refractive index of a region (e.g. core region and/or any of the cladding regions) is expressed as a single value or parameter (e.g. Δ or Δ %) applicable to the region as a whole, it is understood that the relative refractive index in the region is constant, or approximately constant, and corresponds to the single value, or that the single value or parameter represents an average value of a non-constant relative refractive index dependence with radial position in the region. For example, if “i” is a region of the glass fiber, the parameter Δ i  refers to the average value (Δ ave ) of relative refractive index in the region as defined below, unless otherwise specified. Whether by design or a consequence of normal manufacturing variability, the dependence of relative refractive index on radial position may be sloped, curved, or otherwise non-constant. 
     The average relative refractive index (Δ ave ) of a region of the fiber is determined from: 
               Δ   ave     =       ∫     r   inner       r   outer       ⁢         Δ   ⁡     (   r   )       ⁢   dr       (       r   outer     -     r   inner       )               
where r inner  is the inner radius of the region, r outer  is the outer radius of the region, and Δ(r) is the relative refractive index of the region.
 
     The “relative refractive index” as used herein is defined as: 
                 Δ   ⁡     (   r   )       ⁢   %     =     100   ⁢           ⁢           n   2     ⁢     (   r   )       -     n   cl   2         2   ⁢           ⁢       n   2     ⁡     (   r   )                   
where n(r) is the refractive index of the fiber at the radial distance r from the fiber&#39;s centerline AC (r=0) at a wavelength of 1550 nm, unless otherwise specified, and n c1  is a reference refractive index corresponding to the refractive index of pure silica, for which n c1 =1.444 at a wavelength of 1550 nm.
 
     In the description that follows, the relative refractive index (also referred herein as the “relative refractive index percent” for short) is represented by Δ (or “delta”), Δ % (or “delta %”), or %, all of which can be used interchangeably, and its values are given in units of percent or %, unless otherwise specified. Relative refractive index is also expressed as Δ(r) or Δ(r) %. 
     In some embodiments, the bare optical fiber includes a region with a refractive index less than the reference index n c1 , which means that the relative refractive index percent of the region is negative. In other embodiments, the bare optical fiber includes a region with a refractive index greater than the reference index n c1 , which means that the relative refractive index percent of the region is positive. In some embodiments, the bare optical fiber includes a region with a negative relative refractive index percent and a region with a positive relative refractive index percent. The minimum relative refractive index of a region corresponds to the point in the region at which the relative refractive index is lowest. The maximum relative refractive index of a region corresponds to the point in the region at which the relative refractive index is greatest. Depending on the relative refractive index profile and on the region, each of the minimum relative refractive index and the maximum relative refractive index may be positive or negative. 
     In some embodiments, the term “dopant” refers to a substance that changes the relative refractive index of glass relative to pure un-doped SiO 2 . In some embodiments, one or more other substances that are not updopants may be present in a region of an optical fiber (e.g., the core) having a positive relative refractive index Δ. That is, in some embodiments, the core includes an updopant and a downdopant and has a net relative refractive index that is positive. In some embodiments, the dopants used to form the core of the optical fiber disclosed herein include GeO 2  (germania), Al 2 O 3  (alumina), and the like. Dopants that increase the relative refractive index of glass relative to pure un-doped SiO 2  are referred to as up-dopants and dopants that decrease the relative refractive index of glass relative to pure un-doped SiO 2  are referred to as down-dopants. Glass that contains an up-dopant is said to be “up-doped” relative to pure un-doped SiO 2  and glass that contains a down-dopant is said to be “down-doped” relative to pure un-doped SiO 2 . When the undoped glass is silica glass, updopants include Cl, Br, Ge, Al, P, Ti, Zr, Nb, Ta, and oxides thereof, and downdopants include F and B. When comparing two doped glasses (or two doped regions of a bare optical fiber), the doped glass (or glass region) having the higher relative refractive index is said to be up-doped relative to the doped glass (or glass region) having a lower relative refractive index and the doped glass (or glass region) having the lower relative refractive index is said to be down-doped relative to the doped glass (or glass region) having the higher relative refractive index. Glass regions of constant refractive index may be formed by not doping or by doping at a uniform concentration over the thickness of the region. Regions of variable refractive index are formed through non-uniform spatial distributions of dopants over the thickness of a region and/or through incorporation of different dopants in different portions of a region. 
     In some embodiments, the relative refractive index of a region is an α-profile defined by a parameter α. The parameter α (also called the “profile parameter” or “alpha parameter”) relates to the relative refractive Δ(r) of a region through the equation:
 
Δ( r )=Δ 0 {1−[( r−r   m )/( r   0   −r   m )] α }
 
where r m  is the point where Δ(r) is the maximum Δ 0 , r 0  is the point at which Δ(r) reaches a minimum value and r is in the range r i  to r f , where Δ(r) is defined above, r i  is the initial point of the α-profile, r f  is the final point of the α-profile and α is an exponent that is a real number. In one example, r 0  is the point at which Δ(r)=0. At low values of α, the α-profile is a graded-index profile. As the value of α increases, the α-profile more closely resembles a step-index profile. For purposes of the present disclosure, an α-profile with α≥10 is regarded as a step-index relative refractive profile and an α-profile with α&lt;10 is regarded as a graded relative refractive index profile.
 
     “Core volume” V 1  is defined as:
 
 V   1 =2∫ 0   r     1   Δ( r ) rdr  
 
where r 1  is the radius of the core region and Δ 1 (r) is the relative refractive index of the core region. The core volume V 1  is a positive quantity and is expressed herein in units of % Δ-μm 2 , or equivalently in units of % Δ-micron 2 , %-μm 2 , or %-micron 2 .
 
     The “trench volume” is denoted by V Trench  and is defined as
 
 V   Trench =|2∫ r     Trench,inner     r     Trench,outer   Δ Trench ( r ) rdr| 
 
where r Trench,inner  is the inner radius of the trench region of the refractive index profile, r Trench,outer  is the outer radius of the trench region of the refractive index profile, Δ Trench (r) is the relative refractive index of the trench region of the refractive index profile, and r is radial position in the fiber. In one embodiment, the trench region is an inner cladding region and r Trench,inner  is r 1 , r Trench,outer  is r 2 , and Δ Trench  is Δ 2 . In another embodiment, the trench region is a depressed-index cladding region and r Trench,inner  is r 2 , r Trench,outer  is r 3 , and Δ Trench  is Δ 3 . Trench volume is defined as an absolute value and has a positive value. Trench volume is expressed herein in units of % Δμ 2 , %-sq. %Δ-μm 2 , or %-μm 2 , whereby these units can be used interchangeably herein.
 
     The “mode field diameter” or “MFD” of an optical fiber is determined using the Peterman II method, which is the current international standard measurement technique for measuring the MFD of an optical fiber. The MFD is given by: 
             MFD   =     2   ⁢   w                 w   =       [     2   ⁢           ⁢         ∫   0   ∞     ⁢         (     f   ⁡     (   r   )       )     2     ⁢   rdr           ∫   0   ∞     ⁢         (       df   ⁡     (   r   )       dr     )     2     ⁢   rdr           ]       1   ⁢     /     ⁢   2             
where f(r) is the transverse component of the electric field distribution of the guided optical signal and r is radial position in the fiber. The MFD depends on the wavelength of the optical signal and is reported for selected embodiments herein at wavelengths of 1310 nm and/or 1550 nm.
 
     The “effective area” of an optical fiber is defined as: 
               A   eff     =       2   ⁢           ⁢       π   ⁡     [       ∫   0   ∞     ⁢         (     f   ⁡     (   r   )       )     2     ⁢   rdr       ]       2           ∫   0   ∞     ⁢         (     f   ⁡     (   r   )       )     4     ⁢   rdr               
where f(r) is the transverse component of the electric field of the guided optical signal and r is radial position in the fiber. In some embodiment, the “effective area” or “A eff ” depends on the wavelength of the optical signal and is understood to refer to wavelengths of 1310 nm and 1550 nm as indicated herein.
 
     The zero-dispersion wavelength is denoted λ 0  and is the wavelength for a single mode fiber at which material dispersion and waveguide dispersion cancel. In some embodiments, in silica-based optical fibers, the zero-dispersion wavelength is about 1310 nm, e.g., between 1300 nm and 1320 nm, depending on the dopants and refractive index profile used to form the optical fiber. Dispersion slope is the rate of change of dispersion with respect to wavelength. Dispersion and dispersion slope are reported for selected embodiments herein at a wavelength of 1310 nm and/or 1550 nm. Dispersion and dispersion slope are expressed in units of ps/nm-km and ps/nm 2 -km, respectively. 
     The cutoff wavelength of an optical fiber is the minimum wavelength at which the optical fiber will support only one propagating mode. For wavelengths below (less than) the cutoff wavelength, multimode transmission may occur and an additional source of dispersion (intermodal dispersion) may arise to limit the fiber&#39;s information carrying capacity. Cutoff wavelength will be reported herein as a fiber cutoff wavelength (λ CF,2  or λ CF,1 ) or a cable cutoff wavelength (λ CC ). The cable cutoff wavelength is typically less than the fiber cutoff wavelength due to higher levels of bending and mechanical pressure in the cable environment. The fiber cutoff wavelength λ CF,1  is based on a 1-meter fiber length, the fiber cutoff wavelength λ CF,2  is based on a 2-meter fiber length, and the cable cutoff wavelength λ CC  is based on a 22-meter cabled fiber length as specified in TIA-455-80: FOTP-80 IEC-60793-1-44 Optical Fibres—Part 1-44: Measurement Methods and Test Procedures—Cut-off Wavelength (21 May 2003), by Telecommunications Industry Association (TIA). 
     The “spring constant” χ p  of a primary coating is computed from the following equation: 
               χ   p     =         E   p     ⁢     d   g         t   p             
where E p  is the in situ modulus of the primary coating, t p  is the thickness of the primary coating, and d g  is the diameter 2r 4b  of the glass fiber. The spring constant is a phenomenological parameter that describes the extent to which the primary coating mitigates coupling of the secondary coating to the glass fiber. (See “Relationship of Mechanical Characteristics of Dual Coated Single Mode Optical Fibers and Microbending Loss”, by J. Baldauf et al., IEICE Transactions on Communications, Vol. E76-B, No. 4, pp. 352-357 (1993)) In the phenomenological model, the buffering effect of the primary coating is modeled as a spring with the spring constant given in the above equation. A low spring constant leads to greater resistance (lower sensitivity) to microbending. The tradeoff of in situ modulus and thickness in establishing the resistance of the primary coating to microbending is reflected in the spring constant.
 
     The “operating wavelength”, λ op , of an optical fiber is the wavelength at which the optical fiber is operated. The operating wavelength corresponds to the wavelength of a guided mode. Representative operating wavelengths include 850 nm, 1064 nm, 1310 nm and 1550 nm, which are commonly used in telecommunications systems, optical data links, and data centers. Although a particular operating wavelength may be specified for an optical fiber, it is understood that a particular optical fiber can operate at multiple operating wavelengths and/or over a continuous range of operating wavelengths. Characteristics such as modal bandwidth may vary with the operating wavelength and the relative refractive index profile of a particular optical fiber may be designed to provide optimal performance at a particular operating wavelength, a particular combination of operating wavelengths, or particular continuous range of operating wavelengths. 
     “Bandwidth” at a specified wavelength refers to overfilled launch (OFL) bandwidth at the specified wavelength as specified in the TIA/EIA 455-204 and IEC 60793-1-41 standards. 
     The optical fibers disclosed herein include a core region, a cladding region directly adjacent to and surrounding the core region, and a coating directly adjacent to and surrounding the cladding region. The cladding region is a single homogeneous region or multiple regions that differ in relative refractive index. The multiple cladding regions are preferably concentric regions. 
     In some embodiments, the cladding region includes an inner cladding region, a first outer cladding region directly adjacent to and surrounding the inner cladding region, and a second outer cladding region directly adjacent to and surrounding the first outer cladding region. The relative refractive index of the inner cladding region may be greater than, equal to, or less than the relative refractive index of the first outer cladding region. In embodiments without a depressed-index cladding region, an inner cladding region having a lower refractive index than the first outer cladding region is referred to herein as a trench or trench region. 
     In some embodiments, the cladding region includes a depressed-index cladding region between and directly adjacent to an inner cladding region and an outer cladding region, where the outer cladding region includes a first outer cladding region that surrounds and is directly adjacent to the depressed-index cladding region and a second outer cladding region that surrounds and is directly adjacent to the first outer cladding region. The depressed-index cladding region is a cladding region having a lower relative refractive index than the inner cladding and the first outer cladding region. The depressed-index cladding region may also be referred to herein as a trench or trench region. The depressed-index cladding region surrounds and is directly adjacent to the inner cladding region. The depressed-index cladding region is surrounded by and directly adjacent to a first outer cladding region. The first outer cladding region is surrounded by and directly adjacent to a second outer cladding region. The depressed-index cladding region may contribute to a reduction in bending losses. 
     The core region, inner cladding region, depressed-index cladding region, first outer cladding region and second outer cladding region are also referred to as core, inner cladding, depressed-index cladding, first outer cladding, and second outer cladding, respectively. The inner cladding, depressed-index cladding, first outer cladding, and second outer cladding may independently have a positive or negative relative refractive index. The core preferably has a positive relative refractive index. Preferred values for relative refractive index for each of the regions are given below. 
     Whenever used herein, radial position r 1  and relative refractive index Δ 1  or Δ 1 (r) refer to the core region, radial position r 2  and relative refractive index Δ 2  or Δ 2 (r) refer to the inner cladding region, radial position r 3  and relative refractive index Δ 3  or Δ 3 (r) refer to the depressed-index cladding region, radial position r 4a  and relative refractive index Δ 4a  or Δ 4a (r) refer to the first outer cladding region, radial position r 4b  and relative refractive index Δ 4b  or Δ 4b (r) refer to the second outer cladding region, radial position r 5  refers to the primary coating, radial position r 6  refers to the secondary coating, and radial position r 7  refers to the optional tertiary coating. 
     The relative refractive index Δ 1 (r) has a maximum value Δ 1max  and a minimum value Δ 1min . The relative refractive index Δ 2 (r) has a maximum value Δ 2max  and a minimum value Δ 2min . The relative refractive index Δ 3 (r) has a maximum value Δ 3max  and a minimum value Δ 3min . The relative refractive index Δ 4a (r) has a maximum value Δ 4amax  and a minimum value Δ 4amin . The relative refractive index Δ 4b (r) has a maximum value Δ 4bmax  and a minimum value Δ 4bmin . In embodiments in which the relative refractive index is constant or approximately constant over a region, the maximum and minimum values of the relative refractive index are equal or approximately equal. Unless otherwise specified, if a single value is reported for the relative refractive index of a region, the single value corresponds to an average value for the region. 
     It is understood that the central core region is substantially cylindrical in shape and that a surrounding inner cladding region, a surrounding depressed-index cladding region, a surrounding outer cladding region, a surrounding primary coating, a surrounding secondary coating, and a surrounding tertiary coating are substantially annular in shape. Annular regions are characterized in terms of an inner radius and an outer radius. Radial positions r 1 , r 2 , r 3 , r 4a , r 4b , r 5 , r 6 , and r 7  refer herein to the outermost radii of the core, inner cladding, depressed-index cladding, first outer cladding, second outer cladding, primary coating, secondary coating, and tertiary coating, respectively. Radius r 4b  corresponds to the outer radius of the glass portion of the optical fiber and is also referred to as the “cladding radius”. “Cladding diameter” refers to twice the cladding radius and corresponds to 2r 4b . The radius r 6  corresponds to the outer radius of the optical fiber in embodiments without a tertiary coating. When a tertiary coating is present, the radius r 7  corresponds to the outer radius of the optical fiber. 
     When two regions are directly adjacent to each other, the outer radius of the inner of the two regions coincides with the inner radius of the outer of the two regions. In one embodiment, for example, the fiber includes a depressed-index cladding region surrounded by and directly adjacent to a first outer cladding region. In such an embodiment, the radius r 3  corresponds to the outer radius of the depressed-index cladding region and the inner radius of the first outer cladding region. In embodiments in which the relative refractive index profile includes a depressed-index cladding region surrounding and directly adjacent to an inner cladding region, the radial position r 2  corresponds to the outer radius of the inner cladding region and the inner radius of the depressed-index cladding region. In embodiments in which the relative refractive index profile includes a depressed-index cladding region surrounding and directly adjacent to the core, the radial position r 1  corresponds to the outer radius of the core and the inner radius of the depressed-index cladding region. 
     The following terminology applies to embodiments in which the relative refractive index profile includes an inner cladding region surrounding and directly adjacent to the core, a depressed-index cladding region surrounding and directly adjacent to the inner cladding region, a first outer cladding region surrounding and directly adjacent to the depressed-index cladding region, a second outer cladding region surrounding and directly adjacent to the first outer cladding region, a primary coating surrounding and directly adjacent to the second outer cladding region, and a secondary coating surrounding and directly adjacent to the primary coating. The difference between radial position r 2  and radial position r 1  is referred to herein as the thickness of the inner cladding region. The difference between radial position r 3  and radial position r 2  is referred to herein as the thickness of the depressed-index cladding region. The difference between radial position r 4a  and radial position r 3  is referred to herein as the thickness of the first outer cladding region. The difference between radial position r 4b  and radial position r 4a  is referred to herein as the thickness of the second outer cladding region. The difference between radial position r 5  and radial position r 4b  is referred to herein as the thickness of the primary coating. The difference between radial position r 6  and radial position r 5  is referred to herein as the thickness of the secondary coating. 
     The following terminology applies to embodiments in which an inner cladding region is directly adjacent to a core region, a first outer cladding region directly adjacent to the core region, and a second outer cladding region is directly adjacent The first outer cladding region. The difference between radial position r 2  and radial position r 1  is referred to herein as the thickness of the inner cladding region. The difference between radial position r 4a  and radial position r 2  is referred to herein as the thickness of the first outer cladding region. The difference between radial position r 4b  and radial position r 4a  is referred to herein as the thickness of the second outer cladding region. The difference between radial position r 5  and radial position r 4  is referred to herein as the thickness of the primary coating. The difference between radial position r 6  and radial position r 5  is referred to herein as the thickness of the secondary coating. 
     The following terminology applies to embodiments in which the relative refractive index profile lacks both an inner cladding region and a depressed-index cladding region. The difference between radial position r 4a  and radial position r 1  is referred to herein as the thickness of the first outer cladding region. The difference between radial position r 4b  and radial position r 4a  is referred to herein as the thickness of the second outer cladding region. The difference between radial position r 5  and radial position r 4b  is referred to herein as the thickness of the primary coating. The difference between radial position r 6  and radial position r 5  is referred to herein as the thickness of the secondary coating. 
     The coatings described herein are formed from curable coating compositions. Curable coating compositions include one or more curable components. As used herein, the term “curable” is intended to mean that the component, when exposed to a suitable source of curing energy, includes one or more curable functional groups capable of forming covalent bonds that participate in linking the component to itself or to other components of the coating composition. The product obtained by curing a curable coating composition is referred to herein as the cured product of the composition. The cured product is preferably a polymer. The curing process is induced by energy. Forms of energy include radiation or thermal energy. In a preferred embodiment, curing occurs with radiation, where radiation refers to electromagnetic radiation. Curing induced by radiation is referred to herein as radiation curing or photocuring. A radiation-curable component is a component that can be induced to undergo a curing reaction when exposed to radiation of a suitable wavelength at a suitable intensity for a sufficient period of time. Suitable wavelengths include wavelengths in the infrared, visible, or ultraviolet portion of the electromagnetic spectrum. The radiation curing reaction occurs in the presence of a photoinitiator. A radiation-curable component may also be thermally curable. Similarly, a thermally curable component is a component that can be induced to undergo a curing reaction when exposed to thermal energy of sufficient intensity for a sufficient period of time. A thermally curable component may also be radiation curable. 
     A curable component includes one or more curable functional groups. A curable component with only one curable functional group is referred to herein as a monofunctional curable component. A curable component having two or more curable functional groups is referred to herein as a multifunctional curable component. Multifunctional curable components include two or more functional groups capable of forming covalent bonds during the curing process and can introduce crosslinks into the polymeric network formed during the curing process. Multifunctional curable components may also be referred to herein as “crosslinkers” or “curable crosslinkers”. Curable components include curable monomers and curable oligomers. Examples of functional groups that participate in covalent bond formation during the curing process are identified hereinafter. 
     The term “molecular weight” when applied to polyols means number average molecular weight (M n ). 
     The term “(meth)acrylate” means methacrylate, acrylate, or a combination of methacrylate and acrylate. 
     Values of in situ modulus, Young&#39;s modulus,% elongation, and tear strength refer to values as determined under the measurement conditions by the procedures described herein. 
     Embodiments of the present description relate to a single mode optical fiber with good microbending properties and good resistance to puncture. The disclosed single mode optical fiber is suitable for data center applications and features high modal bandwidth, low attenuation, low microbending sensitivity, and puncture resistance in a compact form factor. The disclosed single mode optical fibers have an increased reliability of being routed through extremely tight bend configurations or bend in a small radius arc inside a fiber array unit that couples an array of fibers to arrays of lasers and photodiodes. 
     In some embodiments, the disclosed single mode optical fibers comprise a silica based core region comprising an outer radius r 1  in a range from 3.6 μm to 5.4 μm, a maximum relative refractive index Δ 1MAX  in a range between 0.25% to 0.4%, and a core volume V 1  between 3.0%-μm 2  and 6.0%-μm 2 . 
     In some embodiments, the disclosed single mode optical fibers comprise a depressed-index cladding region surrounding the core region. The depressed-index cladding region comprises an inner radius r 2  such that the ratio r 1 /r 2  is greater than 0.4, an outer radius r 3  and a relative refractive index Δ 3MIN  less than −0.2%, and a trench volume V 3  between 20%-μm 2  and 70%-μm 2 . 
     In some embodiments, the disclosed single mode optical fibers comprise a first outer cladding region surrounding the depressed-index cladding region. The first outer cladding region has a relative refractive index Δ 4a  and an outer radius r 4a . The disclosed single mode optical fibers comprise a second outer cladding region doped with titanium to improve mechanical integrity by increasing the value of the fatigue constant. The second outer cladding region has a relative refractive index Δ 4b , and comprises silica based glass doped with 5 wt % to 20 wt % titania and has a thickness T M  in a range from 2 μm to 30 μm. The second outer layer has an outer radius r 4b  greater than 55 μm, or in a range from 55 μm to 70 μm, or in a range from 60 μm to 65 μm, or about 62.5 μm. 
     In some embodiments, the disclosed single mode optical fibers comprise a primary coating and a secondary coating. The thicknesses t P  of the primary coating and the thickness is of the secondary coating are each at least about 10 μm. In some embodiments, the relative coating thickness, t P /t S , is in the range 0.5≤t P /t S ≤1.5. 
     In some embodiments, the disclosed single mode optical fibers also possess large mode field diameters (MFD) without experiencing significant bending-induced signal degradation. The MFD of the disclosed single mode optical fibers at 1310 nm can be in a range between 8.6 and 9.5 μm. 
     In some embodiments, the disclosed single mode optical fibers have a 2 m fiber cutoff wavelength less than or equal to 1260 nm (e.g., a 1 m fiber cutoff wavelength ≤1260 nm, a 2 m fiber cutoff wavelength ≤1160 nm, a 1 m fiber cutoff wavelength ≤1160 nm, a 2 m fiber cutoff wavelength ≤1060 nm, a 1 m fiber cutoff wavelength ≤1060 nm, etc.) 
     In some embodiments, the disclosed single mode optical fibers have a bending loss at 1310 nm, as determined by the mandrel wrap test using a mandrel comprising a diameter of 15 mm, of less than or equal to 1.0 dB/turn. The disclosed single mode optical fibers have a zero dispersion wavelength (λ 0 ) in a range between 1300 nm and 1324 nm. The single mode optical fiber can be deployed in tight bends, at an effective bend radii less than 4.0 μm, less than 3.0 μm or even less than 2.5 μm for an 82 degree bend. 
     In some embodiments, the disclosed single mode optical fibers have an overfilled launch (OFL) bandwidth greater than 200 MHz-km (e.g., greater than 300 MHz-km, greater than 400 MHz-km, greater than 500 MHz-km, greater than 1000 MHz-km, greater than 2000 MHz-km, greater than 4000 MHz-km, etc.) at a wavelength 850 nm, 980 nm, 1064 nm and/or 1300 nm. 
     Reference will now be made in detail to illustrative embodiments of the present description. 
     One aspect of the present disclosure relates to a single mode optical fiber.  FIG.  1    is a side elevated view of a section of an exemplary single mode optical fiber, according to some embodiments. As shown, the single mode optical fiber  100  comprises a glass core region (“core”)  10  that is centered on the centerline AC. The core  10  can be immediately surrounded by a glass cladding region (“cladding”)  50 . The cladding  50  can be immediately surrounded by a protective coating  70  made of a non-glass material, such as a polymeric material. 
     Referring to  FIG.  2   , a cross-sectional view of an exemplary single mode optical fiber  100  is shown in a cross sectional view in x-y plane, in accordance with some embodiments of the present disclosure. 
     As shown, the single mode optical fiber  100  includes a waveguiding glass fiber surrounded by a protective coating  70  (hereinafter “coating”). The glass fiber includes a higher index core region  10  (hereinafter “core”) surrounded by a lower index cladding region  50  (hereinafter “cladding”). In some embodiments, the coating  70  typically includes a primary coating with low modulus in contact with the glass fiber and a secondary coating with high modulus that surrounds and contacts the primary coating. The secondary coating provides mechanical integrity and allows the single mode optical fiber  100  to be handled for processing and installation in cables, while the primary coating acts to dissipate external forces to prevent The external forces from being transferred to the glass fiber. By dampening the external forces, the primary coating prevents damage to the glass fiber and minimizes attenuation of optical signals caused by microbending. 
     As illustrated, the core  10  has a radius r 1  and a relative refractive index Δ 1 . The cladding  50  can be directly adjacent to and surrounding the core  10 . In some embodiments, the cladding  50  includes an inner cladding region (“inner cladding”)  52  directly adjacent to and surrounding the core  10 , a depressed-index cladding region (“depressed-index cladding”)  54  directly adjacent to and surrounding the inner cladding  52 , and a first outer cladding region (“first outer cladding”)  56  directly adjacent to and surrounding the depressed-index cladding  54 , and a second outer cladding region (“second outer cladding”)  58  directly adjacent to and surrounding the first outer cladding  56 . The inner cladding  52  extends from the radius r 1  to a radius r 2  and has a relative refractive index Δ 2 . The depressed-index cladding  54  extends from the radius r 2  to a radius r 3  and has a relative refractive index Δ 3 . The first outer cladding  56  extends from the radius r 3  to a radius r 4a  and has a relative refractive index Δ 4a  while the second outer cladding  58  extends from the radius r 4a  to a radius r 4b  and has a relative refractive index Δ 4b . 
     In some alternative embodiments not shown in the figures, the depressed-index cladding  54  can be omitted. That is, the cladding  50  includes only three cladding regions. The inner cladding  52  is directly adjacent to and surrounds the core  10 , extends from the radius r 1  to a radius r 2 , and has a relative refractive index Δ 2 . The first outer cladding  56  is directly adjacent to and surrounds the inner cladding  52 , extends from the radius r 2  to a radius r 4a , and has a relative refractive index Δ 4a . The second outer cladding  58  is directly adjacent to and surrounds the first outer cladding  56 , extends from the radius r 4a  to a radius r 4b , and has a relative refractive index Δ 4b . 
     In some other alternative embodiments not shown in the figures, the inner cladding  52  and the depressed-index cladding  54  can be omitted. That is, the cladding  50  includes only two outer cladding regions. The first outer cladding  56  is directly adjacent to and surrounding the core  10 , extends from the radius r 1  to a radius r 4a , and has a relative refractive index Δ 4a . The second outer cladding  58  is directly adjacent to and surrounds the first outer cladding  56 , extends from the radius r 4a  to a radius r 4b , and has a relative refractive index Δ 4b . 
     Protective coating  70  is directly adjacent to and surrounds the cladding  50 . In some embodiments, the protective coating  70  includes a primary coating  72  and a secondary coating  74 . The primary coating  72  and the secondary coating  74  are typically formed by applying a curable coating composition to the glass fiber as a viscous liquid and curing. In some embodiments, the protective coating  70  may also include a tertiary coating  76  that surrounds the secondary coating  74 . 
     The secondary coating  74  is a harder material (higher Young&#39;s modulus) than the primary coating  72  and is designed to protect the glass fiber from damage caused by abrasion or external forces that arise during processing, handling, and installation of the optical fiber. The primary coating  72  is a softer material (lower Young&#39;s modulus) than the secondary coating  74  and is designed to buffer or dissipates stresses that result from forces applied to the outer surface of the secondary coating  74 . 
     The primary coating  72  dissipates shear forces and minimizes the stress that reaches the glass fiber (which includes the core  10  and the cladding  50 ). The primary coating  72  is especially important in dissipating shear forces that arise due to the microbends that the optical fiber encounters when deployed in a cable. The primary coating  72  should maintain adequate adhesion to the glass fiber during thermal and hydrolytic aging, yet be strippable from the glass fiber for splicing purposes. 
     The optional tertiary coating  76  may include pigments, inks or other coloring agents to mark the optical fiber for identification purposes and typically has a Young&#39;s modulus similar to the Young&#39;s modulus of the secondary coating  74 . 
     Another aspect of the present disclosure relates to an optical fiber ribbon.  FIG.  3    illustrates a cross-sectional view of an exemplary optical fiber ribbon  300  in accordance with some embodiments of the present disclosure. As illustrated, the optical fiber ribbon  300  includes a plurality of optical fibers  100  and a matrix  320  encapsulating the plurality of optical fibers  100 . Each optical fiber  100  includes a core, a cladding, and a protective coating as described above. The ribbon matrix  320  can be formed from the same composition used to prepare a secondary coating, or the ribbon matrix  320  can be formed from a different composition that is otherwise compatible for use. 
     The optical fibers  100  are aligned relative to one another in a substantially planar and parallel relationship. The optical fibers  100  in the optical fiber ribbon  300  are encapsulated by the ribbon matrix  320  in any suitable configuration (e.g., edge-bonded ribbon, thin-encapsulated ribbon, thick-encapsulated ribbon, or multi-layer ribbon) by any suitable fabricating methods. In  FIG.  3   , the fiber optic ribbon  300  contains twelve (12) optical fibers  100 ; however, it should be apparent to those skilled in the art that any number of optical fibers  100  (e.g., two or more) may be employed to form fiber optic ribbon  300  disposed for a particular use. 
     Another aspect of the present disclosure relates to an optical fiber cable.  FIG.  4    illustrates a cross-sectional view of an exemplary optical fiber cable  400 . Cable  400  includes a plurality of optical fibers  100  surrounded by jacket  430 . Optical fibers  100  may be densely or loosely packed into a conduit enclosed by inner surface of jacket  430 . The number of fibers placed in the jacket  430  is referred to as the “fiber count” of optical fiber cable  400 . The jacket  430  is formed from an extruded polymer material and may include multiple concentric layers of polymers or other materials. Optical fiber cable  400  may include one or more strengthening members (not shown) embedded within jacket  430  or placed within the conduit defined by the inner surface of jacket  430 . Strengthening members include fibers or rods that are more rigid than jacket  430 . The strengthening member is made from metal, braided steel, glass-reinforced plastic, fiberglass, or other suitable material. Optical fiber cable  400  may include other layers surrounded by jacket  430  (e.g. armor layers, moisture barrier layers, rip cords, etc.). Optical fiber cable  400  may have a stranded, loose tube core or other fiber optic cable construction. 
     Fiber Coupling and Minimum Bend Radius 
     One challenging problem is to couple light from a silicon photonic device (e.g., a waveguide or VCSEL) to the single mode optical fiber with low cost. Referring to  FIG.  5   , an optical coupling  500  is shown, according to some embodiments. Elements of  FIG.  5    that share numbers with those of  FIGS.  1 - 4    may have the same structure and function as described herein in reference to  FIGS.  1 - 4   . In some embodiments, optical coupling  500  comprises a single mode optical fiber  100 , a waveguide  502  on a substrate  508 , and a grating  504 . Optical coupling  500  may further comprise a connector structure  506 . 
     In some embodiments, grating  504  may allow light to transmit from waveguide  502  (e.g., silicon waveguide) to single mode optical fiber  100  and/or vice versa (e.g., optical coupling). Connector structure  506  may be used to protect, support, and/or guide single mode optical fiber  100  close to where single mode optical fiber  100  interfaces with waveguide  502 . Depending on volume requirements (e.g., tight spaces at data centers, the space above waveguide  502  being about 4-5 mm or less), single mode optical fiber  100  may be bent with approximately a quarter of turn (e.g., approximately 82-degree arc) at bend radius of approximately 2.5 mm or less, for example ≤1.5 mm. Connector structure  506  may be, for example, a glass or ceramic ferrule with curved hole. The fiber-guiding geometry of connector structure  506  may be curved such that it matches a desired bend radius, r b , for single mode optical fiber  100 . It should be appreciated that the bend radius r b  may represent an average radius (e.g., if the fiber-guiding geometry has a non-constant radius of curvature). Single mode optical fiber  100  may be stripped of the coating down to the glass cladding, and the stripped portion inserted into the hole of connector structure  506  and glued with an epoxy. Typically, ordinary comparative optical fibers fiber may become easily damaged during stripping. Subsequently, the fiber insertion process may cause single mode optical fiber  100  to break (e.g., insertion failure). Insertion failure may result from flow stress (e.g., the stress at which a wire structure deforms plastically) induced by a small bend radius r b . Under stress, surface defects imposed on ordinary comparative optical fibers may propagate deeper into the glass of the optical fiber, causing mechanical failure (fiber break) and/or shortened life cycle of the product. However, embodiments disclosed herein provide optical fibers that can be coupled to photonic devices even when bent to radii of 3 mm or less while providing robustness against mechanical failure or degradation. By reducing the bend radius r b , the volume of optical coupling  500  may be made smaller (e.g., connector structure  506  may be made smaller), which in turn increases the number of fiber connections capable of being housed in a given space of a data center. 
     In some embodiments, the single mode optical fibers  100  disclosed herein can be coupled to silicon photonic device and having sufficient reliability when deployed inside an approximately 90-degree bend fiber array unit (FAU), even when bent to radii of 2.5 mm or less without mechanical failure due or fiber break, which have a minimum bend radius in the 1.5-2 mm range, even tighter bends are possible from a reliability standpoint if the cladding also includes a second outer cladding doped with titanium. The disclosed single mode optical fibers  100  can be advantageously inserted through the hole in the coupling structure  506  that has a bend radius r b  of approximately 2.5 mm or less (e.g., 1.5 mm≤r b ≤2.5 mm), without succumbing to mechanical failure or degradation, and thus can be bent to a such a small diameter without substantial loss of strength or significant loss of lifetime. Single mode optical fibers  100  advantageously have improved surface damage resistance and low bending loss. 
     It is noted that, strategies for reducing the diameter of the single mode optical fiber  100  include reducing the diameter of glass fiber and reducing the thickness of the primary and/or secondary coating. In some situations, these strategies may lead to compromises in the performance of the single mode optical fiber  100 . For example, a smaller cladding diameter tends to increase the microbending sensitivity and to increase attenuation of the optical signal propagating in the core. As another example, decreasing the cladding diameter from 125 μm to 90 μm may lead to an increase in microbending sensitivity by about an order of magnitude in current optical fiber designs. As yet another example, thinner primary coatings may be more susceptible to shear-induced defects during processing, while thinner secondary coatings may be more susceptible to punctures. 
     The disclosed single mode optical fiber  100  can have smaller outer coating diameters (2r 6  or 2r 7 ≤180 μm) while maintaining a conventional cladding diameter (2r 4b &gt;110 μm), which enable the single mode optical fiber  100  to be deployed in tight bends, at a radius as small as 1.5 μm for an 82-degree bend. 
     Further, a titanium-doped outer cladding can be used for more increased reliability requirement (resistance to fracture or fatigue), such as even smaller bend radii for a same arc length, or multiple bend conditions at a same bend radius, or a longer lifetime at a given arc length or given bend radius, etc. Therefore, the disclosed single mode optical fiber  100  can have a high transmission capacity, a small diameter, a low transmission loss, good microbending properties, high puncture resistance, and high reliability. 
       FIG.  6    illustrates a plot of the relationship between minimum bend radius (mm) and cladding diameter (μm) for an 82-degree bend condition. The vertical axis of the graph represents a bend radius r b  (see  FIG.  5   ), in mm, of an optical fiber. The horizontal axis represents an outermost diameter, in microns, of the cladding of the optical fiber (e.g., 2r 4b  for fibers with first and second outer cladding regions as described herein). The data that makes up the plots includes modeled and experimentally derived data and extrapolations. The plot in  FIG.  6    regards configurations, for a single mode optical fiber  100  (e.g.,  FIGS.  1 - 4   ) being bent in a single 82-degree (approximately) arc, of the optical fiber in which the probability of failure of the optical fiber over a 5-year lifetime is 10 −10  or less (i.e., the expectant lifetime of the bent optical fiber  100  is at least 5 years). 
     For describing embodiments with respect to  FIG.  6   , as a non-limiting example, reference is made to structures of single mode optical fiber  100  in  FIG.  2   . In some embodiments, plot  610  indicates the long term reliability limits of a comparative optical fiber having no titania-doped outer cladding (e.g., a comparative fiber in which the first outer cladding  56  is undoped silica and the second outer cladding  58  is undoped silica) based on both modelling and experiment. Plot  610  shows that, when the comparative optical fiber has an outermost diameter of cladding  50  of approximately 125 μm, the comparative optical fiber may be bent to a bend radius of approximately 2.3 mm with 10 −10  probability of failure over a 5-year lifetime. Bending of the comparative optical fiber with an outermost cladding diameter of 125 μm to a bend radius of less than approximately 2.3 mm reduces the lifetime of the comparative optical fiber to less than 5 years. Plot  610  shows that the relationship between bend radius and outermost cladding diameter is linear for the given conditions (i.e., approximately 82-degree bend arc and tolerance 10 −10  probability of failure over a 5-year). Following the linear relationship, plot  610  shows that, when the comparative optical fiber has an outer diameter of cladding  50  of approximately 100 μm, the comparative optical fiber may be bent to a bend radius of approximately 1.9 mm with 10 −10  probability of failure over a 5-year lifetime. Plot  610  shows that, when the comparative optical fiber has an outer diameter of cladding  50  of approximately 80 μm, the comparative optical fiber may be bent to a bend radius of approximately 1.5 mm (with 10 −10  probability of failure over a 5-year lifetime). Plot  610  shows that, when the comparative optical fiber has an outer diameter of cladding  50  of approximately 62.5 μm, the comparative optical fiber may be bent to a bend radius of approximately 1.2 mm (with 10 −10  probability of failure over a 5-year lifetime). Plot  610  shows that, when the comparative optical fiber has an outer diameter of cladding  50  of approximately 53 μm, the comparative optical fiber may be bent to a bend radius of approximately 1 mm (with 10 −10  probability of failure over a 5-year lifetime). 
     Embodiments described herein employ titania-doped silica as a second outer cladding to increase robustness of an optical fiber at smaller bend radii. In some embodiments, plot  620  indicates the long term reliability limits of optical fiber  100  having a titania-doped outer cladding with a titania concentration about 8 wt %. Plot  620  shows that, when optical fiber  100  has an outer diameter of cladding  50  of approximately 125 μm, optical fiber  100  may be bent to a bend radius of approximately 1.5 mm (with 10 −10  probability of failure over a 5-year lifetime). Plot  620  shows that, when optical fiber  100  has an outer diameter of cladding  50  of approximately 80 μm, optical fiber  100  may be bent to a bend radius of approximately 1 mm (with 10 −10  probability of failure over a 5-year lifetime). Such decreasing in the minimum allowable deployment bend radius (consistent with an expected 5-year lifetime) relative to the undoped silica-clad comparative fiber is due to titania doping of the second outer cladding region, which provide higher breaking stresses and increased fatigue resistance value (e.g., in a range from 26 to 32 instead of about 20 for undoped silica), which is a measure of a material&#39;s susceptibility to subcritical crack growth under stress. 
     The disclosed single mode optical fibers  100  has an improved fiber reliability, which is important for being able to deploy the single mode optical fibers  100  in short reach (e.g., fiber length less than 10 m, or less than 1 m, or between 1 cm and 1 m, or between 1 cm to 50 cm, or between 1 cm to 25 cm, etc.) interconnects within data centers, especially when the single mode optical fiber  100  is routed in configurations with effective bend radii r b  less than 4.0 mm, or less than 3.0 mm, or less than 2.5 mm, etc. The short reach interconnects typically have a fairly short usage lifetime (e.g., 3-5 years) in practice, which is the same lifetime as the electronic equipmen They will be connected to. These very short reach interconnects can be deployed within a rack or even within a server. 
     Relative Refractive Index Profiles 
     In some embodiments, the single mode optical fiber  100  can have a number of different physical configurations defined by way of example as a relative refractive index profile. One type of the disclosed optical fiber is a graded-index fiber, which has a core region with a refractive index that varies with distance from the fiber center. Examples of graded-index fibers include fibers having a core with a relative refractive index having an α-profile defined above or a super-Gaussian relative refractive index profile. Examples of the graded-index fibers are set forth below in connection with  FIGS.  7 A- 7 E . 
     Referring to  FIG.  7 A , a plot of the relative refractive index Δ % (r) versus the radial coordinate illustrating a first exemplary physical configuration of a graded-index fiber, according to some embodiments. As shown, the core  10  having a radius r 1  can have a graded refractive index defined by described by an α-profile. The radial position r 0  (corresponding to Δ 1max ) of the α-profile corresponds to the centerline AC (r=0) of the fiber and the radial position r z  of the α-profile corresponds to the core radius r 1 . In some embodiments with a centerline dip, the radial position r 0  is slightly offset from the centerline AC of the fiber. 
     In some embodiments, the core  10  is immediately surrounded by the cladding  50  that extends from the radius r 1  to a radius r 4b . The cladding  50  can include a first outer cladding region  56  surrounding the core  10 , and a second outer cladding region  58  surrounding the first outer cladding region  56 . The first outer cladding region  56  can be a pure silica cladding or a SiO 2  cladding extending from the radius r 1  to a radius r 4a  and having a relative refractive index Δ 4a  that is substantially zero. The second outer cladding region  58  can be an up-doped cladding extending from the radius r 4a  to the radius r 4b , and having a relative refractive index Δ 4b . In some embodiments, the second outer cladding region  58  is a silica based glass doped with titania. In some embodiments, Δ 4b  is larger than Δ 1max . 
       FIG.  7 B  is a plot of the relative refractive index Δ % (r) versus the radial coordinate illustrating a second exemplary physical configuration of a graded-index fiber, according to some embodiments. The core  10  having a radius r 1  can have a graded refractive index defined by described by an α-profile. The core  10  is immediately surrounded by the cladding  50  that extends from the radius r 1  to a radius r 4b . The cladding  50  includes an inner cladding region  52 , a first outer cladding region  56  surrounding the core  10 , and a second outer cladding region  58  surrounding the first outer cladding region  56 . The inner cladding region  52  can be a down-doped cladding extending from the radius r 1  to a radius r 2  and having a negative relative refractive index Δ 2 . The first outer cladding region  56  can be a pure silica cladding or a doped SiO 2  cladding extending from the radius r 2  to a radius r 4a  and having a relative refractive index Δ 4a , which in one embodiment is substantially zero. The second outer cladding region  58  can be an up-doped cladding extending from the radius r 4a  to the radius r 4b , and having a relative refractive index Δ 4b . In some embodiments, the second outer cladding region  58  is a silica based glass doped with titania. In some embodiments, Δ 4b  is larger than Δ 1max . 
       FIG.  7 C  is a plot of the relative refractive index Δ % (r) versus the radial coordinate illustrating a third exemplary physical configuration of a graded-index fiber, according to some embodiments. The core  10  having a radius r 1  can have a graded refractive index defined by described by an α-profile. The core  10  is immediately surrounded by the cladding  50  that extends from the radius r 1  to a radius r 4b . The cladding  50  includes an inner cladding region  52 , a depressed-index cladding  54  directly adjacent to and surrounding the inner cladding  52 , a first outer cladding region  56  directly adjacent to and surrounding the depressed-index cladding  54 , and a second outer cladding region  58  directly adjacent to and surrounding the first outer cladding region  56 . The inner cladding region  52  can be a pure silica cladding or a SiO 2  cladding extending from the radius r 1  to a radius r 2  and having a relative refractive index Δ 2 . The depressed-index cladding  54  can be a down-doped silica cladding extending from the radius r 2  to a radius r 3  and having a negative relative refractive index Δ 3 . The first outer cladding region  56  can be a pure silica cladding extending from the radius r 1  to a radius r 4a  and having a relative refractive index Δ 4a  that is substantially zero. The second outer cladding region  58  can be an up-doped cladding extending from the radius r 4a  to the radius r 4b , and having a relative refractive index Δ 4b . In some embodiments, the second outer cladding region  58  is a silica based glass doped with titania. In some embodiments, Δ 4b  is larger than Δ 1max . The relative refractive index Δ 3  is less than the relative refractive index Δ 2  and less than the relative refractive index Δ 4a . The relative refractive index Δ 4b  is greater than the relative refractive index Δ 4a . In some embodiments, the relative refractive index Δ 2  is less than the relative refractive index Δ 4a  and in other embodiments, the relative refractive index Δ 2  is greater than or equal to the relative refractive index Δ 4a . 
       FIG.  7 D  is a plot of the relative refractive index Δ % (r) versus the radial coordinate illustrating a fourth exemplary physical configuration of a step-index fiber, according to some embodiments. The core  10  having a radius r 1  can have a step refractive index Δ 1 . The core  10  is immediately surrounded by the cladding  50  that extends from the radius r 1  to a radius r 4b . The cladding  50  includes an inner cladding region  52 , a depressed-index cladding  54  directly adjacent to and surrounding the inner cladding  52 , a first outer cladding region  56  directly adjacent to and surrounding the depressed-index cladding  54 , and a second outer cladding region  58  directly adjacent to and surrounding the first outer cladding region  56 . The inner cladding region  52  can be a pure silica cladding or a SiO 2  cladding extending from the radius r 1  to a radius r 2  and having a relative refractive index Δ 2 . The boundary between Δ 1  and Δ 2  may be a step boundary. It should be appreciated that a step change is an idealization and that a change in relative refractive index at an interface between two materials may not be strictly vertical in practice. Instead, a change in relative refractive index at an interface between two materials may have a slope or curvature. The depressed-index cladding  54  can be a down-doped silica cladding extending from the radius r 2  to a radius r 3  and having a negative relative refractive index Δ 3 . The first outer cladding region  56  can be a pure silica cladding extending from the radius r 1  to a radius r 4a  and having a relative refractive index Δ 4a  that is substantially zero. The second outer cladding region  58  can be an up-doped cladding extending from the radius r 4a  to the radius r 4b , and having a relative refractive index Δ 4b . In some embodiments, the second outer cladding region  58  is a silica based glass doped with titania. In some embodiments, Δ 4b  is larger than Δ 1max . The relative refractive index Δ 3  is less than the relative refractive index Δ 2  and less than the relative refractive index Δ 4a . The relative refractive index Δ 4b  is greater than the relative refractive index Δ 4a . In some embodiments, the relative refractive index Δ 2  is less than the relative refractive index Δ 4a  and in other embodiments, the relative refractive index Δ 2  is greater than or equal to the relative refractive index Δ 4a . 
       FIG.  7 E  is a plot of the relative refractive index Δ % (r) versus the radial coordinate illustrating a fifth exemplary physical configuration of a graded-index fiber, according to some embodiments. The core  10  having a radius r 1  can have a graded refractive index defined by described by an α-profile. The core  10  is immediately surrounded by the cladding  50  that extends from the radius r 1  to a radius r 4b . The cladding  50  includes an inner cladding region  52 , a depressed-index cladding  54  directly adjacent to and surrounding the inner cladding  52 , a first outer cladding region  56  directly adjacent to and surrounding the depressed-index cladding  54 , and a second outer cladding region  58  directly adjacent to and surrounding the first outer cladding region  56 . The inner cladding region  52  can be a down-doped silica cladding or a SiO 2  cladding extending from the radius r 1  to a radius r 2  and having a negative relative refractive index Δ 2 . The depressed-index cladding  54  can be a down-doped silica cladding extending from the radius r 2  to a radius n and having a negative relative refractive index Δ 3  which is less than the relative refractive index Δ 2 . The first outer cladding region  56  can be a pure silica cladding extending from the radius r 1  to a radius r 4a  and having a relative refractive index Δ 4a  that is substantially zero. The second outer cladding region  58  can be an up-doped cladding extending from the radius r 4a  to the radius r 4b , and having a relative refractive index Δ 4b . In some embodiments, the second outer cladding region  58  is a silica based glass doped with titania. In some embodiments, the relative refractive index Δ 4a  is substantially zero. The relative refractive index Δ 4b  is greater than the relative refractive index Δ 1max . 
     Parameter Specifications of Properties 
     In some embodiments, various fiber parameters described herein can be properly designed. In the following, details of various designs of the fiber parameters are described below. It is noted that, the following numerical values for fiber parameters can be applied to fibers with either step-index cores or graded-index cores described, for example, above in connection with  FIGS.  7 A- 7 E . 
     Referring again optical fiber  100  in  FIG.  2    as a non-limiting example, in some embodiments, core  10  comprises silica glass. The silica glass of the core region may be Ge-free; that is the core region comprises silica glass that lacks Ge. The silica glass of core  10  may be undoped silica glass, updoped silica glass, and/or downdoped silica glass. Updoped silica glass may include silica glass doped with an alkali metal oxide (e.g. Na 2 O, K 2 O, Li 2 O, Cs 2 O, or Rb 2 O). Downdoped silica glass may include silica glass doped with F. In some embodiments, core  10  is co-doped with alkali metal oxide and fluorine. The concentration of K 2 O in core  10 , expressed in terms of the amount of K, may be in the range from 20 ppm to 1000 ppm, or 35 ppm to 500 ppm, or 50 ppm to 300 ppm, where ppm refers to parts per million by weight. Alkali metal oxides other than K 2 O may be present in amounts corresponding to the equivalent molar amount of K 2 O as determined from the amount of K indicated above 
     In some embodiments, core  10  may include an updopant and/or a downdopant. The concentration of updopant may be highest at the centerline (r=0) and lowest at the radius r 1 . In some embodiments, the concentration of downdopant may be lowest at the centerline (r=0) and highest at the radius r 1 . In some embodiments, the relative refractive index Δ 1  can have a positive value near the centerline (r=0) and decrease to a negative value at the radius r 1 . 
     In some embodiment, core  10  may be a segmented core that includes an inner core region surrounded by an outer core region, where the inner core region comprises updoped silica glass and has a positive maximum relative refractive index Δ 1max  and the outer core region comprises downdoped silica glass and has a negative minimum relative refractive index Δ 1min . The updoped silica glass of the inner core region includes an updopant or a combination of an updopant and downdopant. 
     In embodiments in which the inner core region includes a combination an updopant and downdopant, the relative concentrations of updopant and downdopant may be adjusted to provide a net positive value of the maximum relative refractive index. 
     In embodiments in which the outer core region includes a combination an updopant and downdopant, the relative concentrations of updopant and downdopant may be to increase transmission capacity, puncture resistance and microbending performance. 
     In embodiments in which the outer core region includes a combination an updopant and downdopant, the relative concentrations of updopant and downdopant may be adjusted to provide a net negative value of the relative refractive index. 
     In embodiments with a segmented core, Δ 1  (and Δ 1max  and Δ 1min ) refer to the entirety of the core region, including the inner core region and the outer core region, r 1  corresponds to the outer radius of the outer core region, and r 1a  corresponds to the outer radius of the inner core region. The boundary between the inner core region and outer core region occurs at radial position r 1a , where r 1a &lt;r 1 . 
     In various embodiments, the outer radius r 1  of the core is in a range from 2.0 μm to 15 μm, or in a range from 2.0 μm to 10 μm, or in a range from 2.0 μm to 8.0 μm, or in a range from 3.0 μm to 7.0 μm, or in a range from 3.0 μm to 6.5 μm, or in a range from 3.0 μm to 6.0 μm, or in a range from 3.6 μm to 5.4 μm, or in a range from 3.6 μm to 5.0 μm, or in a range from 4.0 μm to 5.0 μm. In some embodiments, the relative refractive index of the core is described by a step-index profile (e.g. α-profile with α≥10) having a constant or approximately constant value corresponding to Δ 1  in a range from 0.15% to 0.50%, or in a range from 0.20% to 0.45%, or in a range from 0.25% to 0.40%, or in a range from 0.25% to 0.38%, or in a range from 0.27% to 0.35%. In some other embodiments, the relative refractive index of the core is described by graded-index profile (e.g. α-profile with α&lt;10). The corresponding maximum relative refractive index Δ 1max  of the core region is in a range from 0.15% to 0.50%, or in a range from 0.20% to 0.45%, or in a range from 0.25% to 0.40%, or in a range from 0.25% to 0.38%, or in a range from 0.27% to 0.35%. 
     Cores with a relative refractive index having or that can be modelled as an α-profile have values of α in a range from 1 to 200, or in a range from 1 to 100, or in a range from 1 to 20, or in a range from 1 to 10, or in a range from 2 to 8, or in a range from 3 to 7, or in a range from 5 to 15, or in a range from 7 to 13, or in a range from 6 to 12, or in a range from 10 to 100, or in a range from 20 to 100. 
     In some embodiments, a core volume V 1  of the core region is in a range from 2.0%-μm 2  to 10.0%-μm 2 , or in a range from 3.0%-μm 2  to 4.6%-μm 2 , or in a range from 3.0%-μm 2  to 6.0%-μm 2 . 
     In some embodiments, the core can be doped with chlorine (“Cl-doped” hereinafter), with the chlorine concentration in a range from 1.5 wt % to 6.0 wt %, or in a range from 2.0 wt % to 5.5 wt %, or in a range from 2.5 wt % to 5.0 wt %, or in a range from 3.0 wt % to 4.5 wt %, or greater than or equal to 1.5 wt % (e.g., ≥2 wt %, ≥2.5 wt %, ≥3 wt %, ≥3.5 wt %, ≥4 wt %, ≥4.5 wt %, ≥5 wt %, etc.) It is noted that, the notation “wt %” used herein means a weight percentage. It is also noted that, the core is GeO 2  free, or a concentration of GeO 2  in the cores is less than 1.0 wt %. 
     In various embodiments in which the cladding includes an inner cladding surrounded by and directly adjacent to a depressed-index cladding region. In some embodiments, the outer radius r 2  of the inner cladding is in a range from 6.0 μm to 18.0 μm, or in a range from 6.0 μm to 15.0 μm, or in a range from 6.0 μm to 12.0 μm, or in a range from 6.5 μm to 10.0 μm, or in a range from 7.0 μm to 9.0 μm, or in a range from 5.0 μm to 11.0 μm, or in a range from 7.0 μm to 10.0 μm. A thickness (r 2 −r 1 ) of the inner cladding is in a range from 2.0 μm to 9.0 μm, or in a range from 2.5 μm to 8.0 μm, or in a range from 3.0 μm to 7.0 μm, or in a range from 3.5 μm to 6.5 μm. In some embodiments, the ratio r 1 /r 2  is greater than 0.2, or greater than 0.3, or greater than 0.4, or greater than 0.5, or in a range from 0.2 to 0.9, or in a range from 0.3 to 0.8, or in a range from 0.4 to 0.7. In some embodiments, when the α value of the core is greater than 5, the ratio r 1 /r 2  is in a range from 0.4 to 0.5, or in a range from 0.42 to 0.48. In some other embodiments, when the α value of the core is less than 5, the ratio r 1 /r 2  is in a range from 0.5 to 0.65, or in a range from 0.55 to 0.65. 
     The relative refractive index Δ 2  of the inner cladding is in a range from −0.15% to 0.15%, or in a range from −0.10% to 0.10%, or in a range from −0.05 to 0.05%. In some embodiments, the inner cladding is a pure silica cladding. In some other embodiments, the inner cladding is downdoped. For example, the inner cladding can be doped with fluorine (“F-doped” hereinafter) with a fluorine concentration in a range from 0.01 wt % to 0.20 wt %, or in a range from 0.05 wt % to 0.15 wt %. In some other embodiments, the inner cladding is updoped. For example, the inner cladding can be Cl-doped with a chlorine concentration in a range from 0.01 wt % to 0.50 wt %, or in a range from 0.05 wt % to 0.40 wt %, or in a range from 0.10 wt % to 0.30 wt %. 
     In embodiments in which the cladding includes an inner cladding region surrounded by and directly adjacent to a depressed-index cladding region, the inner radius of the depressed-index cladding region is r 2  and has the values specified above. In various embodiments, the outer radius r 3  of the depressed-index cladding is less than 18 μm, or less than 16 μm, or less than 14 μm, or less than 12 μm, or greater than 9 μm, or greater than 10 μm, or greater than 11 μm, or in a range from 10 μm to 17 μm, or in a range from 11 μm to 15 μm. In various embodiments, a thickness (r 3 −r 2 ) of the depressed-index cladding is in a range from 3 μm to 10 μm, or in a range from 3.5 μm to 9 μm, or in a range from 4 μm to 8 μm, or in a range from 4.5 μm to 6.5 μm. The relative refractive index Δ 3  of the depressed-index cladding is in a range from −0.8% to 0.0%, or in a range from −0.7% to −0.1%, or in a range from −0.6% to −0.2%. As described above, the depressed-index cladding is downdoped, such as being F-doped with a fluorine concentration in a range from 0.10 wt % to 0.50 wt %, or in a range from 0.15 wt % to 0.45 wt %, or in a range from 0.20 wt % to 0.40 wt %, or greater than 0.10 wt %, or greater than 0.15 wt %, or greater than 0.20 wt %. A trench volume of the depressed-index cladding is greater than 20%-μm 2 , or greater than 30%-μm 2 , or greater than 40%-μm 2 , or greater than 50%-μm 2 , or greater than 60%-μm 2 , or in a range from 20%-μm 2  to 70%-μm 2 , or in a range from 20%-μm 2  to 100%-μm 2 , or in a range from 20%-μm 2  to 80%-μm 2 , or in a range from 30%-μm 2  to 70%-μm 2 , or in a range from 30%-μm 2  to 60%-μm 2 . 
     It is noted that, in some embodiments, the cladding includes only the inner cladding and outer cladding without the intervening depressed-index cladding, as described above in connection with  FIG.  7 B . In such embodiments, the inner cladding surrounds and is directly adjacent to the core and the first outer cladding surrounds and is directly adjacent to the inner cladding. In these embodiments, the inner cladding functions as a trench region. 
     In embodiments in which the cladding lacks an intervening depressed-index cladding and has a first outer cladding surrounding and directly adjacent to an inner cladding that surrounds and is directly adjacent to a core region, the inner radius of the inner cladding is r 1  with the values specified above, a thickness (r 2 −r 1 ) of the inner cladding is in a range from 3 μm to 10 μm, or in a range from 3.5 μm to 9 μm, or in a range from 4 μm to 8 μm, or in a range from 4.5 μm to 6.5 μm, and the relative refractive index Δ 2  of the inner cladding is in a range from −0.6% to 0.0%, or in a range from −0.55% to −0.1%, or in a range from −0.5% to −0.2%. In various embodiments, the outer radius r 2  of the inner cladding is less than 26 μm, or less than 23 μm, or less than 20 μm, or less than 16 μm, or less than 14 μm, or less than 12 μm, or less than 8 μm, or less than 5 μm, or greater than 3 μm, or greater than 5 μm, or greater than 7 μm, or greater than 10 μm, or greater than 12 μm, or greater than 15 μm, or greater than 20 μm, or in a range from 3 μm to 26 μm, or in a range from 5 μm to 23 μm, or in a range from 7 μm to 20 μm, or in a range from 10 μm to 18 μm, or in a range from 12 μm to 17 μm, or in a range from 11 μm to 15 μm. In various embodiments, a thickness (r 2 −r 1 ) of the inner cladding is in a range from 3 μm to 25 μm, or in a range from 5 μm to 23 μm, or in a range from 7 μm to 20 μm, or in a range from 9 μm to 17 μm, or in a range from 3 μm to 10 μm, or in a range from 3.5 μm to 9 μm, or in a range from 4 μm to 8 μm, or in a range from 4.5 μm to 6.5 μm. The relative refractive index Δ 2  of the inner cladding is in a range from −0.8% to 0.0%, or in a range from −0.7% to −0.1%, or in a range from −0.6% to −0.2%. In these embodiments, the inner cladding is downdoped, such as being F-doped with a fluorine concentration in a range from 0.10 wt % to 0.50 wt %, or in a range from 0.15 wt % to 0.45 wt %, or in a range from 0.20 wt % to 0.40 wt %, or greater than 0.10 wt %, or greater than 0.15 wt %, or greater than 0.20 wt %. A trench volume of the inner cladding is greater than 20%-μm 2 , or greater than 30%-μm 2 , or greater than 40%-μm 2 , or greater than 50%-μm 2 , or greater than 60%-μm 2 , or in a range from 20%-μm 2  to 70%-μm 2 , or in a range from 20%-μm 2  to 150%-μm 2 , or in a range from 20%-μm 2  to 200%-μm 2 , or in a range from 20%-μm 2  to 100%-μm 2 , or in a range from 20%-μm 2  to 80%-μm 2 , or in a range from 30%-μm 2  to 70%-μm 2 , or in a range from 30%-μm 2  to 60%-μm 2 . 
     The inner radius of the first outer cladding region is r 3  (in embodiments in which the first outer cladding is directly adjacent to a depressed-index cladding that is directly adjacent to an inner cladding that is directly adjacent to a core) or r 2  (in embodiments in which the first outer cladding is directly adjacent to an inner cladding that is directly adjacent to a core) and has the values specified above. In various embodiments, a thickness (r 4a −r 3 ) or (r 4a −r 2 ) of the first outer cladding is in a range from 5 μm to 55 μm, or in a range from 10 μm to 50 μm, or in a range from 15 μm to 45 μm, or in a range from 20 μm to 45 μm, or in a range from 20 μm to 40 μm, or in a range from 20 μm to 35 μm. The radius r 4a  of the first outer cladding region is in a range from 25 μm to 68 μm, or in a range from 27.5 μm to 65 μm, or in a range from 32.5 μm to 60.5 μm, or in a range from 37.5 μm to 58.5 μm, or in a range from 42.5 μm to 56.5 μm, or in a range from 47.5 μm to 52.5 μm. The relative refractive index Δ 4a  of the first outer cladding is in a range from −0.15% to 0.15%, or in a range from −0.10% to 0.10%, or in a range from −0.05 to 0.05%, or in a range from −0.15% to 0.0%, or in a range from −0.10% to 0.0%. In some embodiments, the first outer cladding is a pure silica cladding. In some other embodiments, the first outer cladding is downdoped. For example, the first outer cladding can be doped with fluorine (“F-doped” hereinafter) with a fluorine concentration in a range from 0.01 wt % to 0.20 wt %, or in a range from 0.05 wt % to 0.15 wt %. In some other embodiments, the first outer cladding is updoped. For example, the first outer cladding can be Cl-doped with a chlorine concentration in a range from 0.01 wt % to 0.50 wt %, or in a range from 0.05 wt % to 0.40 wt %, or in a range from 0.10 wt % to 0.30 wt %. 
     The inner radius of the second outer cladding region is r 4a  and has the values specified above. In various embodiments, a thickness (r 4b −r 4a ) of the second outer cladding is in a range from 2 μm to 30 μm, or in a range from 2 μm to 25 μm, or in a range from 2 μm to 20 μm, or in a range from 2 μm to 15 μm, or in a range from 2 μm to 10 μm or in or in a range from 5 μm to 25 μm, or in a range from 8 μm to 20 μm, or in a range from 10 μm to 15 μm. In various embodiments, the outer radius r 4b  of the second outer cladding is greater than 55 μm, or in a range from 55 μm to 70 μm, or in a range from 60 μm to 65 μm, or about 62.5 μm. The relative refractive index Δ 4b  of the second outer cladding is in a range from 0.2% to 2%, or in a range from 0.4% to 1.8%, or in a range from 0.6% to 1.6%, or in a range from 0.8% to 1.4%, or in a range from 0.9% to 1.2%. In some embodiments, the relative refractive index Δ 4b  of the second outer cladding is greater than the maximum relative refractive index Δ 1max  of the core region. In some other embodiments, the second outer cladding is updoped. For example, the second outer cladding can be doped with a titania (TiO 2 ) concentration in a range from 1 wt % to 20 wt %, or in a range from 4 wt % to 20 wt %, or in a range from 6 wt % to 15 wt %, or in a range from 8 wt % to 12 wt %. 
     The mode field diameter of the single mode optical fibers disclosed herein is greater than or equal to 8.2 μm, or greater than or equal to 8.4 μm, or greater than or equal to 8.6 μm, or greater than or equal to 8.8 μm, or less than or equal to 9.8 μm, or less than or equal to 9.6 μm, or less than or equal to 9.4 μm, or less than or equal to 9.2 μm, or in a range from 8.2 μm to 9.8 μm, or in a range from 8.3 μm to 9.6 μm, or in a range from 8.4 μm to 9.5 μm, or in a range from 8.3 μm to 9.4 μm at a wavelength of 1310 nm. 
     The mode field diameter of the single mode optical fibers disclosed herein is greater than or equal to 9.0 μm, or greater than or equal to 9.2 μm, or greater than or equal to 9.4 μm, or greater than or equal to 9.6 μm, or greater than or equal to 9.8 μm, or less than or equal to 11.0 μm, or less than or equal to 10.8 μm, or less than or equal to 10.6 μm, or less than or equal to 10.4 μm, or in a range from 9.0 μm to 11.0 μm, or in a range from 9.1 μm to 10.8 μm, or in a range from 9.2 μm to 10.6 μm, or in a range from 9.3 μm to 10.4 μm at a wavelength of 1550 nm. 
     The effective area A eff  of the single mode optical fibers disclosed herein is greater than 50 μm 2 , or greater than 55 μm 2 , or greater than 60 μm 2 , or greater than 65 μm 2 , or greater than 70 μm 2 , or less than 130 μm 2 , or less than 115 μm 2 , or less than 100 μm 2 , or in the range from 50 μm 2  to 100 μm 2 , or in the range from 55 μm 2  to 90 μm 2 , or in the range from 60 μm 2  to 80 μm 2  at a wavelength of 1310 nm. 
     The effective area Δ eff  of the single mode optical fibers disclosed herein is greater than 100 μm 2 , or greater than 110 μm 2 , or greater than 120 μm 2 , or greater than 130 μm 2 , or greater than 140 μm 2 , or greater than 150 μm 2 , or in the range from 100 μm 2  to 180 μm 2 , or in the range from 110 μm 2  to 165 μm 2 , or in the range from 120 μm 2  to 155 μm 2  at a wavelength of 1550 nm. 
     The attenuation of the single mode optical fibers disclosed herein is less than or equal to 0.170 dB/km, or less than or equal to 0.165 dB/km, or less than or equal to 0.160 dB/km, or less than or equal to 0.155 dB/km, or less than or equal to 0.150 dB/km at a wavelength of 1550 nm. 
     It is noted that, macrobending loss can be determined using a mandrel wrap test specified in standard IEC 60793-1-47. In the mandrel wrap test, the optical fiber is wrapped one or more times around a cylindrical mandrel having a specified diameter, and the increase in attenuation at a specified wavelength due to the bending is determined. Attenuation in the mandrel wrap test is expressed in units of dB/turn, where one turn refers to one revolution of the optical fiber about the mandrel. Macrobending losses at a wavelength of 1310 nm or 1550 nm were determined for selected examples described below with the mandrel wrap test using mandrels with diameters of ranging from 10 mm to 60 mm. In some embodiments, the bending loss of the optical fibers at a wavelength of 1310 nm as determined by the mandrel wrap test using a mandrel having a diameter of 15 mm can be less than 3.0 dB/turn, or less than 2.5 dB/turn, or less than 2.0 dB/turn, or less than 1.5 dB/turn, or less than 1.0 dB/turn. In some embodiments, the bending loss of the optical fibers at a wavelength of 1550 nm as determined by the mandrel wrap test using a mandrel having a diameter of 10 mm can be less than 3.0 dB/turn, or less than 2.5 dB/turn, or less than 2.0 dB/turn, or less than 1.5 dB/turn, or less than 1.0 dB/turn. 
     The cable cutoff λ CC  of the optical fibers disclosed herein is less than 1530 nm, or less than 1520 nm, or less than 1500 nm, or less than 1450 nm, or less than 1400 nm, or less than 1350 nm, less than 1300 nm, or less than 1260 nm. In some embodiments, a cutoff wavelength λ CF,2  is equal to or less than 1530 nm, or equal to or less than 1510 nm, or equal to or less than 1460 nm, or equal to or less than 1410 nm, or equal to or less than 1360 nm, or equal to or less than 1310 nm, equal to or less than 1260 nm, or equal to or less than 1210 nm, or equal to or less than 1160 nm, or equal to or less than 1110 nm, or equal to or less than 1060 nm. In some embodiments, a fiber cutoff wavelength λ CF,1  is equal to or less than 1530 nm, or equal to or less than 1510 nm, or equal to or less than 1460 nm, or equal to or less than 1410 nm, or equal to or less than 1360 nm, or equal to or less than 1310 nm, or equal to or less than 1260 nm, or equal to or less than 1210 nm, or equal to or less than 1160 nm, or equal to or less than 1110 nm, or equal to or less than 1060 nm. 
     In some embodiments, the zero dispersion wavelength (λ 0 ) of the single mode optical fibers disclosed herein is in a range 1200 nm to 1500 nm, in a range 1240 nm to 1400 nm, or in a range 1280 nm to 1360 nm, or in a range 1300 nm to 1324 nm. 
     In some embodiments, the overfilled launch (OFL) bandwidth of the single mode optical fibers disclosed herein is greater than 200 MHz-km (e.g., &gt;300 MHz-km, &gt;400 MHz-km, &gt;500 MHz-km, greater than 1000 MHz-km, greater than 2000 MHz-km, greater than 4000 MHz-km, etc.) at a wavelength of 850 nm, 980 nm, 1064 nm, and/or 1300 nm (that is, the overfilled launch (OFL) bandwidth has the value specified at at least one of the wavelengths of 850 nm, 980 nm, 1064 nm, and 1300 nm). 
     In some embodiments, the puncture resistance of the secondary coating of the single mode optical fibers disclosed herein can be greater than 30 g (e.g., &gt;40 g, &gt;50 g, etc.). In some embodiments, the normalized puncture load of the optical fiber  100  is greater than 3.6×10 −3  g/μm 2  (e.g., &gt;3.8×10 −3  g/μm 2 , &gt;4.0×10 −3  g/μm 2 , etc.). 
     In some embodiments, proper combination of the fiber parameters in the ranges described above can result in optical fiber properties that meet the requirements of a high transmission capacity, a small diameter, a low transmission loss, good microbending properties, and high puncture resistance. The profiles designs shown in  FIGS.  7 A to  7 E , other profiles as described herein, and the various fiber parameters described above can be compliant with various optical fibers, such as ITU G.652.D, G.657.A1, G.657.A2, G.654, etc. 
     Optical Fiber Coatings 
     The transmissivity of light through an optical fiber is highly dependent on the properties of the coatings applied to the glass fiber. The coatings typically include a primary coating and a secondary coating, where the secondary coating surrounds the primary coating and the primary coating contacts the glass fiber (which includes a central core region surrounded by a cladding region). The secondary coating is a harder material (higher in situ modulus) than the primary coating and is designed to protect the glass fiber from damage caused by abrasion or external forces that arise during processing, handling, and installation of the optical fiber. The primary coating is a softer material (lower in situ modulus) than the secondary coating and is designed to buffer or dissipates stresses that result from lateral forces applied to the outer surface of the secondary coating. The primary coating is especially important in dissipating stresses that arise due to the microbends that the optical fiber encounters when deployed in a cable. The primary coating should maintain adequate adhesion to the glass fiber during thermal and hydrolytic aging, yet be strippable from the glass fiber for splicing purposes. 
     Primary and secondary coatings are typically formed by applying a curable coating composition to the glass fiber as a viscous liquid and curing. The optical fiber may also include a tertiary coating (not shown) that surrounds the secondary coating. The tertiary coating may include pigments, inks or other coloring agents to mark the optical fiber for identification purposes and typically has an in situ modulus similar to the in situ modulus of the secondary coating. 
     As described above, the design of the refractive index profile of the cladding may include a refractive index trench that diminishes the sensitivity of the coated fiber to bending, and a titania-doped outer cladding, which improves the mechanical reliability of the fiber and provides a substantial increase in resistance to mechanical abrasion. This cladding structure may enable use of a primary coating and a secondary coating with reduced thickness relative to commercially available fibers. The thinner coating thickness of the optical fiber embodiments described herein advantageously provides compact coated fibers that can be densely packed and/or readily installed in existing fiber infrastructures. The mechanical properties of the primary coating are selected such that good microbending performance of the coated fiber is achieved, even when the thickness of the primary coating is reduced. The mechanical properties of the secondary coating are selected such that good puncture resistance of the coated fiber is achieved, even when the thickness of the secondary coating is reduced. 
     Primary Coating—Compositions. The primary coating is a cured product of a curable primary coating composition. The curable primary coating compositions provide a primary coating for optical fibers that exhibits low Young&#39;s modulus, low pullout force, and strong cohesion. The curable primary coating compositions further enable formation of a primary coating that features clean strippability and high resistance to defect formation during the stripping operation. Low pullout force facilitates clean stripping of the primary coating with minimal residue and strong cohesion inhibits initiation and propagation of defects in the primary coating when it is subjected to stripping forces. Even for optical fibers with reduced primary coating thicknesses, the optical fibers are expected to have low loss and low microbend loss performance. The primary coatings exhibit these advantages even at reduced thickness. 
     The primary coating is a cured product of a radiation-curable primary coating composition that includes an oligomer, a monomer, a photoinitiator and, optionally, an additive. The following disclosure describes oligomers for the radiation-curable primary coating compositions, radiation-curable primary coating compositions containing at least one of the oligomers, cured products of the radiation-curable primary coating compositions that include at least one of the oligomers, glass fibers coated with a radiation-curable primary coating composition containing at least one of the oligomers, and glass fibers coated with the cured product of a radiation-curable primary coating composition containing at least one of the oligomers. 
     The oligomer preferably includes a polyether urethane diacrylate compound and a di-adduct compound. In one embodiment, the polyether urethane diacrylate compound has a linear molecular structure. In one embodiment, the oligomer is formed from a reaction between a diisocyanate compound, a polyol compound, and a hydroxy acrylate compound, where the reaction produces a polyether urethane diacrylate compound as a primary product (majority product) and a di-adduct compound as a byproduct (minority product). The reaction forms a urethane linkage upon reaction of an isocyanate group of the diisocyanate compound and an alcohol group of the polyol. The hydroxy acrylate compound reacts to quench residual isocyanate groups that are present in the composition formed from reaction of the diisocyanate compound and polyol compound. As used herein, the term “quench” refers to conversion of isocyanate groups through a chemical reaction with hydroxyl groups of the hydroxy acrylate compound. Quenching of residual isocyanate groups with a hydroxy acrylate compound converts terminal isocyanate groups to terminal acrylate groups. 
     A preferred diisocyanate compound is represented by formula (I):
 
O═C═N—R 1 ═N═C═O  (I)
 
which includes two terminal isocyanate groups separated by a linkage group R 1 . In one embodiment, the linkage group R 1  includes an alkylene group. The alkylene group of linkage group R 1  is linear (e.g. methylene or ethylene), branched (e.g. isopropylene), or cyclic (e.g. cyclohexylene, phenylene). The cyclic group is aromatic or non-aromatic. In some embodiments, the linkage group R 1  is 4,4′-methylene bis(cyclohexyl) group and the diisocyanate compound is 4,4′-methylene bis(cyclohexyl isocyanate). In some embodiments, the linkage group R 1  lacks an aromatic group, or lacks a phenylene group, or lacks an oxyphenylene group.
 
     The polyol is represented by molecular formula (II): 
                         
where R 2  includes an alkylene group, O—R 2 — is a repeating alkoxylene group, and x is an integer. Preferably, x is greater than 20, or greater than 40, or greater than 50, or greater than 75, or greater than 100, or greater than 125, or greater than 150, or in the range from 20 to 500, or in the range from 20 to 300, or in the range from 30 to 250, or in the range from 40 to 200, or in the range from 60 to 180, or in the range from 70 to 160, or in the range from 80 to 140. R 2  is preferably a linear or branched alkylene group, such as methylene, ethylene, propylene (normal, iso or a combination thereof), or butylene (normal, iso, secondary, tertiary, or a combination thereof). The polyol may be a polyalkylene oxide, such as polyethylene oxide, or a polyalkylene glycol, such as polypropylene glycol. Polypropylene glycol is a preferred polyol. The molecular weight of the polyol is greater than 1000 g/mol, or greater than 2500 g/mol, or greater than 5000 g/mol, or greater than 7500 g/mol, or greater than 10000 g/mol, or in the range from 1000 g/mol to 20000 g/mol, or in the range from 2000 g/mol to 15000 g/mol, or in the range from 2500 g/mol to 12500 g/mol, or in the range from 2500 g/mol to 10000 g/mol, or in the range from 3000 g/mol to 7500 g/mol, or in the range from 3000 g/mol to 6000 g/mol, or in the range from 3500 g/mol to 5500 g/mol. In some embodiments, the polyol is polydisperse and includes molecules spanning a range of molecular weights such that the totality of molecules combines to provide the number average molecular weight specified hereinabove.
 
     The unsaturation of the polyol is less than 0.25 meq/g, or less than 0.15 meq/g, or less than 0.10 meq/g, or less than 0.08 meq/g, or less than 0.06 meq/g, or less than 0.04 meq/g, or less than 0.02 meq/g, or less than 0.01 meq/g, or less than 0.005 meq/g, or in the range from 0.001 meq/g to 0.15 meq/g, or in the range from 0.005 meq/g to 0.10 meq/g, or in the range from 0.01 meq/g to 0.10 meq/g, or in the range from 0.01 meq/g to 0.05 meq/g, or in the range from 0.02 meq/g to 0.10 meq/g, or in the range from 0.02 meq/g to 0.05 meq/g. As used herein, unsaturation refers to the value determined by the standard method reported in ASTM D4671-16. In the method, the polyol is reacted with mercuric acetate and methanol in a methanolic solution to produce acetoxymercuricmethoxy compounds and acetic acid. The reaction of the polyol with mercuric acetate is equimolar and the amount of acetic acid released is determined by titration with alcoholic potassium hydroxide to provide the measure of unsaturation used herein. To prevent interference of excess mercuric acetate on the titration of acetic acid, sodium bromide is added to convert mercuric acetate to the bromide. 
     The reaction to form the oligomer further includes addition of a hydroxy acrylate compound to react with terminal isocyanate groups present in unreacted starting materials (e.g. the diisocyanate compound) or products formed in the reaction of the diisocyanate compound with the polyol (e.g. urethane compounds with terminal isocyanate groups). The hydroxy acrylate compound reacts with terminal isocyanate groups to provide terminal acrylate groups for one or more constituents of the oligomer. In some embodiments, the hydroxy acrylate compound is present in excess of the amount needed to fully convert terminal isocyanate groups to terminal acrylate groups. The oligomer includes a single polyether urethane acrylate compound or a combination of two or more polyether urethane acrylate compounds. 
     The hydroxy acrylate compound is represented by molecular formula (III): 
                         
where R 3  includes an alkylene group. The alkylene group of R 3  is linear (e.g. methylene or ethylene), branched (e.g. isopropylene), or cyclic (e.g. phenylene). In some embodiments, the hydroxy acrylate compound includes substitution of the ethylenically unsaturated group of the acrylate group. Substituents of the ethylenically unsaturated group include alkyl groups. An example of a hydroxy acrylate compound with a substituted ethylenically unsaturated group is a hydroxy methacrylate compound. The discussion that follows describes hydroxy acrylate compounds. It should be understood, however, that the discussion applies to substituted hydroxy acrylate compounds and in particular to hydroxy methacrylate compounds.
 
     In different embodiments, the hydroxy acrylate compound is a hydroxyalkyl acrylate, such as 2-hydroxyethyl acrylate. 
     In the foregoing exemplary molecular formulas (I), II), and (III), the groups R 1 , R 2 , and R 3  independently are all the same, are all different, or include two groups that are the same and one group that is different. 
     The diisocyanate compound, hydroxy acrylate compound and polyol are combined simultaneously and reacted, or are combined sequentially (in any order) and reacted. In one embodiment, the oligomer is formed by reacting a diisocyanate compound with a hydroxy acrylate compound and reacting the resulting product composition with a polyol. In another embodiment, the oligomer is formed by reacting a diisocyanate compound with a polyol compound and reacting the resulting product composition with a hydroxy acrylate compound. 
     The oligomer is formed from a reaction of a diisocyanate compound, a hydroxy acrylate compound, and a polyol, where the molar ratio of the diisocyanate compound to the hydroxy acrylate compound to the polyol in the reaction process is n:m:p, n, m, and p are referred to herein as mole numbers or molar proportions of diisocyanate, hydroxy acrylate, and polyol; respectively. The mole numbers n, m and p are positive integer or positive non-integer numbers. In embodiments, when p is 2.0, n is in the range from 3.0 to 5.0, or in the range from 3.2 to 4.8, or in the range from 3.4 to 4.6, or in the range from 3.5 to 4.4, or in the range from 3.6 to 4.2, or in the range from 3.7 to 4.0; and m is in the range from 1.5 to 4.0, or in the range from 1.6 to 3.6, or in the range from 1.7 to 3.2, or in the range from 1.8 to 2.8, or in the range from 1.9 to 2.4. For values of p other than 2.0, the molar ratio n:m:p scales proportionally. For example, the molar ratio n:m:p=4.0:3.0:2.0 is equivalent to the molar ratio n:m:p=2.0:1.5:1.0. 
     In one embodiment, the oligomer is formed from a reaction mixture that includes 4,4′-methylene bis(cyclohexyl isocyanate), 2-hydroxyethyl acrylate, and polypropylene glycol in the molar ratios n:m:p as specified above, where the polypropylene glycol has a number average molecular weight in the range from 2500 g/mol to 6500 g/mol, or in the range from 3000 g/mol to 6000 g/mol, or in the range from 3500 g/mol to 5500 g/mol. 
     The oligomer preferably includes two components. The first component is a polyether urethane diacrylate compound having the molecular formula (IV): 
                         
and the second component is a di-adduct compound having the molecular formula (V):
 
                         
where the groups R 1 , R 2 , R 3 , and the integer x are as described hereinabove, y is a positive integer, and it is understood that the group R 1  in molecular formulas (IV) and (V) is the same as group R 1  in molecular formula (I), the group R 2  in molecular formula (IV) is the same as group R 2  in molecular formula (II), and the group R 3  in molecular formulas (IV) and (V) is the same as group R 3  in molecular formula (III). The di-adduct compound corresponds to the compound formed by reaction of both terminal isocyanate groups of the diisocyanate compound of molecular formula (I) with the hydroxy acrylate compound of molecular formula (II) where the diisocyanate compound has undergone no reaction with the polyol of molecular formula (II).
 
     The di-adduct compound is formed from a reaction of the diisocyanate compound with the hydroxy acrylate compound during the reaction used to form the oligomer. Alternatively, the di-adduct compound is formed independent of the reaction used to form the oligomer and is added to the product of the reaction used to form the polyether urethane diacrylate compound or to a purified form of the polyether urethane diacrylate compound. The hydroxy group of the hydroxy acrylate compound reacts with an isocyanate group of the diisocyanate compound to provide a terminal acrylate group. The reaction occurs at each isocyanate group of the diisocyanate compound to form the di-adduct compound. The di-adduct compound is present in the oligomer in an amount of at least 1.0 wt %, or at least 1.5 wt %, or at least 2.0 wt %, or at least 2.25 wt %, or at least 2.5 wt %, or at least 3.0 wt %, or at least 3.5 wt %, or at least 4.0 wt %, or at least 4.5 wt %, or at least 5.0 wt %, or at least 7.0 wt % or at least 9.0 wt %, or in the range from 1.0 wt % to 10.0 wt %, or in the range from 2.0 wt % to 9.0 wt %, or in the range from 2.5 wt % to 6.0 wt %, or in the range from 3.0 wt % to 8.0 wt %, or in the range from 3.0 wt % to 5.0 wt %, or in the range from 3.0 wt % to 5.5 wt %, or in the range from 3.5 wt % to 5.0 wt %, or in the range from 3.5 wt % to 7.0 wt %. It is noted that the concentration of di-adduct is expressed in terms of wt % of the oligomer and not in terms of wt % in the coating composition. 
     An illustrative reaction for synthesizing an oligomer in accordance with the present disclosure includes reaction of a diisocyanate compound (4,4′-methylene bis(cyclohexyl isocyanate, which is also referred to herein as H12MDI) and a polyol (polypropylene glycol with M n ˜4000 g/mol, which is also referred to herein as PPG4000) to form a polyether urethane diisocyanate compound with formula (VI):
 
H12MDI˜PPG4000˜H12MDI˜PPG4000˜H12MDI  (VI)
 
where “˜” denotes a urethane linkage formed by the reaction of a terminal isocyanate group of H12MDI with a terminal alcohol group of PPG4000; and ˜H12MDI, ˜H12MDI˜, and ˜PPG4000˜ refer to residues of H12MDI and PPG4000 remaining after the reaction; and M n  refers to number average molecular weight. The polyether urethane diisocyanate compound has a repeat unit of the type ˜(H12MDI˜PPG4000)˜. The particular polyether urethane diisocyanate shown includes two PPG4000 units. The reaction may also provide products having one PPG4000 unit, or three or more PPG4000 units. The polyether urethane diisocyanate and any unreacted H12MDI include terminal isocyanate groups. In accordance with the present disclosure, a hydroxy acrylate compound (such as 2-hydroxyethyl acrylate, which is referred to herein as HEA) is included in the reaction to react with terminal isocyanate groups to conver Them to terminal acrylate groups. The conversion of terminal isocyanate groups to terminal acrylate groups effects a quenching of the isocyanate group. The amount of HEA included in the reaction may be an amount estimated to react stoichiometrically with the expected concentration of unreacted isocyanate groups or an amount in excess of the expected stoichiometric amount. Reaction of HEA with the polyether urethane diisocyanate compound forms the polyether urethane acrylate compound with formula (VII):
 
HEA˜H12MDI˜PPG4000˜H12MDI˜PPG4000˜H12MDI  (VII)
 
and/or the polyether urethane diacrylate compound with formula (VIII):
 
HEA˜H12MDI˜PPG4000˜H12MDI˜PPG4000˜H12MDI˜HEA  (VIII)
 
and reaction of HEA with unreacted H12MDI forms the di-adduct compound with formula (IX):
 
HEA˜H12MDI˜HEA  (IX)
 
where, as above, ˜ designates a urethane linkage and ˜HEA designates the residue of HEA remaining after reaction to form the urethane linkage (consistent with formulas (IV) and (V)). The combination of a polyether urethane diacrylate compound and a di-adduct compound in the product composition constitutes an oligomer in accordance with the present disclosure. As described more fully hereinbelow, when one or more oligomers are used in coating compositions, coatings having improved tear strength and critical stress characteristics result. In particular, it is demonstrated that oligomers having a high proportion of di-adduct compound provide coatings with high tear strengths and/or high critical stress values.
 
     The oligomer includes a compound that is a polyether urethane diacrylate compound with formula (X):
 
(hydroxy acrylate)˜(diisocyanate˜polyol) x ˜diisocyanate˜(hydroxy acrylate)  (X)
 
and a compound that is a di-adduct compound with formula (XI):
 
(hydroxy acrylate)˜diisocyanate˜(hydroxy acrylate)  (XI)
 
where the relative proportions of diisocyanate compound, hydroxy acrylate compound, and polyol used in the reaction to form the oligomer correspond to the mole numbers n, m, and p disclosed hereinabove.
 
     Compounds represented by molecular formulas (I) and (II) above, for example, react to form a polyether urethane diisocyanate compound represented by molecular formula (XII): 
                         
where y is the same as y in formula (IV) and is 1, or 2, or 3 or 4 or higher; and x is determined by the number of repeat units of the polyol (as described hereinabove).
 
     Further reaction of the polyether urethane isocyanate of molecular formula (VI) with the hydroxy acrylate of molecular formula (III) provides the polyether urethane diacrylate compound represented by molecular formula (IV) referred to hereinabove and repeated below: 
                         
where y is 1, or 2, or 3, or 4 or higher; and x is determined by the number of repeat units of the polyol (as described hereinabove).
 
     Variations in the mole numbers n, m, and p provide control over the relative proportions of polyether urethane diacrylate and di-adduct formed in the reaction. Increasing the mole number n relative to the mole number m or the mole number p, for example, may increase the amount of di-adduct compound formed in the reaction. Reaction of the diisocyanate compound, the hydroxy acrylate compound, and polyol compound in molar ratios n:m:p, where n is in the range from 3.0 to 5.0, m is in the range within ±15% of 2n−4 or within ±10% of 2n−4 or within ±5% of 2n−4, and p is 2.0, for example, produce amounts of the di-adduct compound in the oligomer sufficient to achieve the preferred primary coating properties. By way of example, the embodiment in which n=4.0, m is within ±15% of 2n−4, and p=2.0 means that n=4.0, m is within ±15% of 4, and p=2.0, which means that that n=4.0, m is in the range from 3.4 to 4.6, and p=2.0. 
     Variations in the relative proportions of di-adduct and polyether urethane diacrylate are obtained through changes in the mole numbers n, m, and p and through such variations, it is possible to precisely control the Young&#39;s modulus, in situ modulus, tear strength, critical stress, tensile toughness, and other mechanical properties of coatings formed from coating compositions that include the oligomer. 
     Improved fiber primary coatings result when utilizing a primary coating composition that incorporates an oligomer that includes a polyether urethane acrylate compound represented by molecular formula (IV) and a di-adduct compound represented by molecular formula (V), where concentration of the di-adduct compound in the oligomer is at least 1.0 wt %, or at least 1.5 wt %, or at least 2.0 wt %, or at least 2.25 wt %, or at least 2.5 wt %, or at least 3.0 wt %, or at least 3.5 wt %, or at least 4.0 wt %, or at least 4.5 wt %, or at least 5.0 wt %, or at least 7.0 wt % or at least 9.0 wt %, or in the range from 1.0 wt % to 10.0 wt %, or in the range from 2.0 wt % to 9.0 wt %, or in the range from 3.0 wt % to 8.0 wt %, or in the range from 3.5 wt % to 7.0 wt % or in the range from 2.5 wt % to 6.0 wt %, or in the range from 3.0 wt % to 5.5 wt %, or in the range from 3.5 wt % to 5.0 wt %. It is noted that the concentration of di-adduct is expressed in terms of wt % of the oligomer and not in terms of wt % in the coating composition. The concentration of the di-adduct compound is increased in one embodiment by varying the molar ratio n:m:p of diisocyanate:hydroxy acrylate:polyol. In one aspect, molar ratios n:m:p that are rich in diisocyanate relative to polyol promote the formation of the di-adduct compound. 
     The oligomer of the primary coating composition includes a polyether urethane diacrylate compound and di-adduct compound as described hereinabove. In some embodiments, the oligomer includes two or more polyether urethane diacrylate compounds and/or two or more di-adduct compounds. The oligomer content of the primary coating composition includes the combined amounts of the one or more polyether urethane diacrylate compound(s) and one or more di-adduct compound(s) and is greater than 20 wt %, or greater than 30 wt %, or greater than 40 wt %, or in the range from 20 wt % to 80 wt %, or in the range from 30 wt % to 70 wt %, or in the range from 40 wt % to 60 wt %, where the concentration of di-adduct compound within the oligomer content is as described above. 
     The curable primary coating composition further includes one or more monomers. The one or more monomers is/are selected to be compatible with the oligomer, to control the viscosity of the primary coating composition to facilitate processing, and/or to influence the physical or chemical properties of the coating formed as the cured product of the primary coating composition. The monomers include radiation-curable monomers such as ethylenically-unsaturated compounds, ethoxylated acrylates, ethoxylated alkylphenol monoacrylates, propylene oxide acrylates, n-propylene oxide acrylates, isopropylene oxide acrylates, monofunctional acrylates, monofunctional aliphatic epoxy acrylates, multifunctional acrylates, multifunctional aliphatic epoxy acrylates, and combinations thereof. 
     Representative radiation-curable ethylenically unsaturated monomers include alkoxylated monomers with one or more acrylate or methacrylate groups. An alkoxylated monomer is one that includes one or more alkoxylene groups, where an alkoxylene group has the form ˜O—R— and R is a linear or branched alkylene group. Examples of alkoxylene groups include ethoxylene (—O—CH 2 —CH 2 —), n-propoxylene (—O—CH 2 —CH 2 —CH 2 —), isopropoxylene (—O—CH 2 —CH(CH 3 )—, or —O—CH(CH 3 )—CH 2 —), etc. As used herein, the degree of alkoxylation refers to the number of alkoxylene groups in the monomer. In one embodiment, the alkoxylene groups are bonded consecutively in the monomer. 
     In some embodiments, the primary coating composition includes an alkoxylated monomer of the form R 4 —R 5 —O—(CH(CH 3 )CH 2 —O) q —C(O)CH═CH 2 , where R 4  and R 5  are aliphatic, aromatic, or a mixture of both, and q=1 to 10, or R 4 —O—(CH(CH 3 )CH 2 —O) q —C(O)CH═CH 2 , where C(O) is a carbonyl group, R 1  is aliphatic or aromatic, and q=1 to 10. 
     Representative examples of monomers include ethylenically unsaturated monomers such as lauryl acrylate (e.g., SR335 available from Sartomer Company, Inc., AGEFLEX FA12 available from BASF, and PHOTOMER 4812 available from IGM Resins), ethoxylated nonylphenol acrylate (e.g., SR504 available from Sartomer Company, Inc. and PHOTOMER 4066 available from IGM Resins), caprolactone acrylate (e.g., SR495 available from Sartomer Company, Inc., and TONE M-100 available from Dow Chemical), phenoxyethyl acrylate (e.g., SR339 available from Sartomer Company, Inc., AGEFLEX PEA available from BASF, and PHOTOMER 4035 available from IGM Resins), isooctyl acrylate (e.g., SR440 available from Sartomer Company, Inc. and AGEFLEX FA8 available from BASF), tridecyl acrylate (e.g., SR489 available from Sartomer Company, Inc.), isobornyl acrylate (e.g., SR506 available from Sartomer Company, Inc. and AGEFLEX IBOA available from CPS Chemical Co.), tetrahydrofurfuryl acrylate (e.g., SR285 available from Sartomer Company, Inc.), stearyl acrylate (e.g., SR257 available from Sartomer Company, Inc.), isodecyl acrylate (e.g., SR395 available from Sartomer Company, Inc. and AGEFLEX FA10 available from BASF), 2-(2-ethoxyethoxy)ethyl acrylate (e.g., SR256 available from Sartomer Company, Inc.), epoxy acrylate (e.g., CN120, available from Sartomer Company, and EBECRYL 3201 and 3604, available from Cytec Industries Inc.), lauryloxyglycidyl acrylate (e.g., CN130 available from Sartomer Company) and phenoxyglycidyl acrylate (e.g., CN131 available from Sartomer Company) and combinations thereof. 
     In some embodiments, the monomer component of the primary coating composition includes a multifunctional (meth)acrylate. Multifunctional ethylenically unsaturated monomers include multifunctional acrylate monomers and multifunctional methacrylate monomers. Multifunctional acrylates are acrylates having two or more polymerizable acrylate moieties per molecule, or three or more polymerizable acrylate moieties per molecule. Examples of multifunctional (meth)acrylates include dipentaerythritol monohydroxy pentaacrylate (e.g., PHOTOMER 4399 available from IGM Resins); methylolpropane polyacrylates with and without alkoxylation such as trimethylolpropane triacrylate, ditrimethylolpropane tetraacrylate (e.g., PHOTOMER 4355, IGM Resins); alkoxylated glyceryl triacrylates such as propoxylated glyceryl triacrylate with propoxylation being 3 or greater (e.g., PHOTOMER 4096, IGM Resins); and erythritol polyacrylates with and without alkoxylation, such as pentaerythritol tetraacrylate (e.g., SR295, available from Sartomer Company, Inc. (Westchester, Pa.)), ethoxylated pentaerythritol tetraacrylate (e.g., SR494, Sartomer Company, Inc.), dipentaerythritol pentaacrylate (e.g., PHOTOMER 4399, IGM Resins, and SR399, Sartomer Company, Inc.), tripropyleneglycol diacrylate, propoxylated hexanediol diacrylate, tetrapropyleneglycol diacrylate, pentapropyleneglycol diacrylate, methacrylate analogs of the foregoing, and combinations thereof. 
     In some embodiments, the primary coating composition includes an N-vinyl amide monomer such as an N-vinyl lactam, or N-vinyl pyrrolidinone, or N-vinyl caprolactam, where the N-vinyl amide monomer is present in the coating composition at a concentration greater than 1.0 wt %, or greater than 2.0 wt %, or greater than 3.0 wt %, or in the range from 1.0 wt % to 15.0 wt %, or in the range from 2.0 wt % to 10.0 wt %, or in the range from 3.0 wt % to 8.0 wt %. 
     In an embodiment, the primary coating composition includes one or more monofunctional acrylate or methacrylate monomers in an amount from 15 wt % to 90 wt %, or from 30 wt % to 75 wt %, or from 40 wt % to 65 wt %. In another embodiment, the primary coating composition may include one or more monofunctional aliphatic epoxy acrylate or methacrylate monomers in an amount from 5 wt % to 40 wt %, or from 10 wt % to 30 wt %. 
     In different embodiments, the total monomer content of the primary coating composition is between about 15 wt % and about 90 wt %, or between about 30 wt % and about 75 wt %, or between about 40 wt % and about 65 wt %. 
     In addition to a curable monomer and a curable oligomer, the curable primary coating composition also includes a polymerization initiator. The polymerization initiator facilitates initiation of the polymerization process associated with the curing of the coating composition to form the coating. Polymerization initiators include thermal initiators, chemical initiators, electron beam initiators, and photoinitiators. Photoinitiators include ketonic photoinitiators and/or phosphine oxide photoinitiators. 
     Representative photoinitiators include 1-hydroxycyclohexylphenyl ketone (e.g., IRGACURE 184 available from BASF)); bis(2,6-dimethoxybenzoyl)-2,4,4-trimethylpentylphosphine oxide (e.g., commercial blends IRGACURE 1800, 1850, and 1700 available from BASF); 2,2-dimethoxy-2-phenylacetophenone (e.g., IRGACURE 651, available from BASF); bis(2,4,6-trimethylbenzoyl)-phenylphosphine oxide (IRGACURE 819); (2,4,6-trimethylbenzoyl)diphenyl phosphine oxide (LUCIRIN TPO, available from BASF (Munich, Germany)); ethoxy(2,4,6-trimethylbenzoyl)-phenylphosphine oxide (LUCIRIN TPO-L from BASF); and combinations thereof. 
     The coating composition includes a single photoinitiator or a combination of two or more photoinitiators. The total photoinitiator content of the coating composition is up to about 10 wt %, or between about 0.5 wt % and about 6 wt %. 
     The curable primary coating composition optionally includes one or more additives. Additives include an adhesion promoter, a strength additive, an antioxidant, a catalyst, a stabilizer, an optical brightener, a property-enhancing additive, an amine synergist, a wax, a lubricant, and/or a slip agent. 
     An adhesion promoter is a compound that facilitates adhesion of the primary coating and/or primary composition to glass (e.g. the cladding portion of a glass fiber). Suitable adhesion promoters include alkoxysilanes, mercapto-functional silanes, organotitanates, and zirconates. Representative adhesion promoters include mercaptoalkyl silanes or mercaptoalkoxy silanes such as 3-mercaptopropyl-trialkoxysilane (e.g., 3-mercaptopropyl-trimethoxysilane, available from Gelest (Tullytown, Pa.)); bis(trialkoxysilyl-ethyl)benzene; acryloxypropyltrialkoxysilane (e.g., (3-acryloxypropyl)-trimethoxysilane, available from Gelest), methacryloxypropyltrialkoxysilane, vinyltrialkoxysilane, bis(trialkoxysilylethyl)hexane, allyltrialkoxysilane, styrylethyltrialkoxysilane, and bis(trimethoxysilylethyl)benzene (available from United Chemical Technologies (Bristol, Pa.)); see U.S. Pat. No. 6,316,516, the disclosure of which is hereby incorporated by reference in its entirety herein. 
     The adhesion promoter is present in the primary coating composition in an amount between 0.02 wt % and 10.0 wt %, or between 0.05 wt % and 4.0 wt %, or between 0.1 wt % and 4.0 wt %, or between 0.1 wt % and 3.0 wt %, or between 0.1 wt % and 2.0 wt %, or between 0.1 wt % and 1.0 wt %, or between 0.5 wt % and 4.0 wt %, or between 0.5 wt % and 3.0 wt %, or between 0.5 wt % and 2.0 wt %, or between 0.5 wt % and 1.0 wt %. 
     A representative antioxidant is thiodiethylene bis[3-(3,5-di-tert-butyl)-4-hydroxy-phenyl) propionate] (e.g., IRGANOX 1035, available from BASF). In some aspects, an antioxidant is present in the coating composition in an amount greater than 0.25 wt %, or greater than 0.50 wt %, or greater than 0.75 wt %, or greater than 1.0 wt %, or an amount in the range from 0.25 wt % to 3.0 wt %, or an amount in the range from 0.50 wt % to 2.0 wt %, or an amount in the range from 0.75 wt % to 1.5 wt %. 
     Primary Coating Properties. Relevant properties of the primary coating include radius, thickness, Young&#39;s modulus, and in situ modulus. 
     The radius r 5  of the primary coating is less than or equal to 120.0 μm, or less than or equal to 110.0 μm, or less than or equal to 100.0 μm, or less than or equal to 90.0 μm, or in a range from 65.0 μm to 120.0 μm, or in a range from 70.0 μm to 110.0 μm, or in the range from 75.0 μm to 100.0 μm, or in the range from 80.0 μm to 90.0 μm. To facilitate decreases in the diameter of the optical fiber, it is preferable to minimize the thickness r 5 −r 4  of the primary coating. The thickness r 5 −r 4  of the primary coating is greater than or equal to 8.0 μm (e.g., greater than or equal to 8.5 μm, greater than or equal to 9.0 μm, greater than or equal to 9.5 μm, greater than or equal to 10.0 μm, etc.), and less than or equal to 30.0 μm, or less than or equal to 25.0 μm, or less than or equal to 20.0 μm, or less than or equal to 15.0 μm, or less than or equal to 10.0 μm, or in the range from 8.0 μm to 25.0 μm, or in the range from 9.0 μm to 20.0 μm, or in the range from 10.0 μm to 17.0 μm. 
     In some embodiments, to facilitate effective buffering of stress and protection of the glass fiber, it is preferable for the primary coating to have a low Young&#39;s modulus and/or a low in situ modulus E P . The Young&#39;s modulus of the primary coating is less than or equal to 0.7 MPa, or less than or equal to 0.6 MPa, or less than or equal to 0.5 MPa, or less than or equal to 0.4 MPa, or in the range from 0.2 MPa to 0.7 MPa, or in the range from 0.3 MPa to 0.6 MPa. The in situ modulus E P  of the primary coating is less than or equal to 0.35 MPa, or less than or equal to 0.30 MPa, or less than or equal to 0.25 MPa, or less than or equal to 0.20 MPa, or less than or equal to 0.15 MPa, or less than or equal to 0.10 MPa, or in the range from 0.05 MPa to 0.25 MPa, or in the range from 0.10 MPa to 0.20 MPa. 
     In some other embodiments, the primary coating can act as a “spring” that couples the stiff glass portion (e.g., the second outer cladding region  58 ) to the relatively stiff secondary coating that has an in situ modulus E P  greater than 1200 MPa, or greater than 1400 MPa, or greater than 1500 MPa, or greater than 1800 MPa. The spring constant of the primary coating is defined as χ P =E P *d 4 /t P , where d 4  is the diameter of the glass portion of the fiber (i.e., d 4 =2r 4b ), and t P  and E P  are the thickness and in situ modulus, respectively, of the primary coating. In some embodiments, the spring constant of the primary coating has a value χ P ≤1.6 MPa, or χ P ≤1.5 MPa, or χ P ≤1.4 MPa, or χ P ≤1.3 MPa, or χ P ≤1.2 MPa, or χ P ≤1.1 MPa, or χ P ≤1.0 MPa, or χ P ≤0.9 MPa, or χ P ≤0.8 MPa, or χ P ≤0.7 MPa, or χ P ≤0.6 MPa, or 0.5 MPa≤χ P ≤1.5 MPa, or 0.5 MPa≤χ P ≤1.2 MPa, or 0.6 MPa≤χ P ≤1.0 MPa. Such design can reduce the microbending losses and improve microbending resistance, since a small spring constant provides lower degree of coupling between the glass portion of the fiber and the secondary coating. 
     Secondary Coating—Compositions. The secondary coating is a cured product of a curable secondary coating composition that includes a monomer, a photoinitiator, an optional oligomer, and an optional additive. The present disclosure describes optional oligomers for the radiation-curable secondary coating compositions, radiation-curable secondary coating compositions, cured products of the radiation-curable secondary coating compositions, optical fibers coated with a radiation-curable secondary coating composition, and optical fibers coated with the cured product of a radiation-curable secondary coating composition. 
     The secondary coating is formed as the cured product of a radiation-curable secondary coating composition that includes a monomer component with one or more monomers. The monomers preferably include ethylenically unsaturated compounds. The one or more monomers may be present in an amount of 50 wt % or greater, or in an amount from about 60 wt % to about 99 wt %, or in an amount from about 75 wt % to about 99 wt %, or in an amount from about 80 wt % to about 99 wt % or in an amount from about 85 wt % to about 99 wt %. In one embodiment, the secondary coating is the radiation-cured product of a secondary coating composition that contains urethane acrylate monomers. 
     The monomers include functional groups that are polymerizable groups and/or groups that facilitate or enable crosslinking. The monomers are monofunctional monomers or multifunctional monomers. In combinations of two or more monomers, the constituent monomers are monofunctional monomers, multifunctional monomers, or a combination of monofunctional monomers and multifunctional monomers. In one embodiment, the monomer component of the curable secondary coating composition includes ethylenically unsaturated monomers. Suitable functional groups for ethylenically unsaturated monomers include, without limitation, (meth)acrylates, acrylamides, N-vinyl amides, styrenes, vinyl ethers, vinyl esters, acid esters, and combinations thereof. 
     In one embodiment, the monomer component of the curable secondary coating composition includes ethylenically unsaturated monomers. The monomers include functional groups that are polymerizable groups and/or groups that facilitate or enable crosslinking. The monomers are monofunctional monomers or multifunctional monomers. In combinations of two or more monomers, the constituent monomers are monofunctional monomers, multifunctional monomers, or a combination of monofunctional monomers and multifunctional monomers. Suitable functional groups for ethylenically unsaturated monomers include, without limitation, (meth)acrylates, acrylamides, N-vinyl amides, styrenes, vinyl ethers, vinyl esters, acid esters, and combinations thereof. 
     Exemplary monofunctional ethylenically unsaturated monomers for the curable secondary coating composition include, without limitation, hydroxyalkyl acrylates such as 2-hydroxyethyl-acrylate, 2-hydroxypropyl-acrylate, and 2-hydroxybutyl-acrylate; long- and short-chain alkyl acrylates such as methyl acrylate, ethyl acrylate, propyl acrylate, isopropyl acrylate, butyl acrylate, amyl acrylate, isobutyl acrylate, t-butyl acrylate, pentyl acrylate, isoamyl acrylate, hexyl acrylate, heptyl acrylate, octyl acrylate, isooctyl acrylate, 2-ethylhexyl acrylate, nonyl acrylate, decyl acrylate, isodecyl acrylate, undecyl acrylate, dodecyl acrylate, lauryl acrylate, octadecyl acrylate, and stearyl acrylate; aminoalkyl acrylates such as dimethylaminoethyl acrylate, diethylaminoethyl acrylate, and 7-amino-3,7-dimethyloctyl acrylate; alkoxyalkyl acrylates such as butoxyethyl acrylate, phenoxyethyl acrylate (e.g., SR339, Sartomer Company, Inc.), and ethoxyethoxyethyl acrylate; single and multi-ring cyclic aromatic or non-aromatic acrylates such as cyclohexyl acrylate, benzyl acrylate, dicyclopentadiene acrylate, dicyclopentanyl acrylate, tricyclodecanyl acrylate, bomyl acrylate, isobornyl acrylate (e.g., SR423, Sartomer Company, Inc.), tetrahydrofiurfuryl acrylate (e.g., SR285, Sartomer Company, Inc.), caprolactone acrylate (e.g., SR495, Sartomer Company, Inc.), and acryloylmorpholine; alcohol-based acrylates such as polyethylene glycol monoacrylate, polypropylene glycol monoacrylate, methoxyethylene glycol acrylate, methoxypolypropylene glycol acrylate, methoxypolyethylene glycol acrylate, ethoxydiethylene glycol acrylate, and various alkoxylated alkylphenol acrylates such as ethoxylated(4) nonylphenol acrylate (e.g., Photomer 4066, IGM Resins); acrylamides such as diacetone acrylamide, isobutoxymethyl acrylamide, N,N′-dimethyl-aminopropyl acrylamide, N,N-dimethyl acrylamide, N,N diethyl acrylamide, and t-octyl acrylamide; vinylic compounds such as N-vinylpyrrolidone and N-vinylcaprolactam; and acid esters such as maleic acid ester and fumaric acid ester. With respect to the long and short chain alkyl acrylates listed above, a short chain alkyl acrylate is an alkyl group with 6 or less carbons and a long chain alkyl acrylate is an alkyl group with 7 or more carbons. 
     Representative radiation-curable ethylenically unsaturated monomers included alkoxylated monomers with one or more acrylate or methacrylate groups. An alkoxylated monomer is one that includes one or more alkoxylene groups, where an alkoxylene group has the form —O—R— and R is a linear or branched hydrocarbon. Examples of alkoxylene groups include ethoxylene (—O—CH 2 —CH 2 —), n-propoxylene (—O—CH 2 —CH 2 —CH 2 —), isopropoxylene (—O—CH 2 —CH(CH 3 )—), etc. As used herein, the degree of alkoxylation refers to the number of alkoxylene groups in the monomer. In one embodiment, the alkoxylene groups are bonded consecutively in the monomer. 
     Representative multifunctional ethylenically unsaturated monomers for the curable secondary coating composition include, without limitation, alkoxylated bisphenol-A diacrylates, such as ethoxylated bisphenol-A diacrylate, and alkoxylated trimethylolpropane triacrylates, such as ethoxylated trimethylolpropane triacrylate, with the degree of alkoxylation being 2 or greater, or 4 or greater, or 6 or greater, or less than 16 or less than 12, or less than 8, or less than 5, or in the range from 2 to 16, or in the range from 2 to 12, or in the range from 2 to 8, or in the range from 2 to 4, or in the range from 3 to 12, or in the range from 3 to 8, or in the range from 3 to 5, or in the range from 4 to 12, or in the range from 4 to 10, or in the range from 4 to 8. 
     Multifunctional ethylenically unsaturated monomers for the curable secondary coating composition include, without limitation, alkoxylated bisphenol A diacrylates, such as ethoxylated bisphenol A diacrylate, with the degree of alkoxylation being 2 or greater. The monomer component of the secondary coating composition may include ethoxylated bisphenol A diacrylate with a degree of ethoxylation ranging from 2 to about 30 (e.g. SR349, SR601, and SR602 available from Sartomer Company, Inc. West Chester, Pa. and Photomer 4025 and Photomer 4028, available from IGM Resins), or propoxylated bisphenol A diacrylate with the degree of propoxylation being 2 or greater; for example, ranging from 2 to about 30; methylolpropane polyacrylates with and without alkoxylation such as ethoxylated trimethylolpropane triacrylate with the degree of ethoxylation being 3 or greater; for example, ranging from 3 to about 30 (e.g., Photomer 4149, IGM Resins, and SR499, Sartomer Company, Inc.); propoxylated-trimethylolpropane triacrylate with the degree of propoxylation being 3 or greater; for example, ranging from 3 to 30 (e.g., Photomer 4072, IGM Resins and SR492, Sartomer); ditrimethylolpropane tetraacrylate (e.g., Photomer 4355, IGM Resins); alkoxylated glyceryl triacrylates such as propoxylated glyceryl triacrylate with the degree of propoxylation being 3 or greater (e.g., Photomer 4096, IGM Resins and SR9020, Sartomer); erythritol polyacrylates with and without alkoxylation, such as pentaerythritol tetraacrylate (e.g., SR295, available from Sartomer Company, Inc. (West Chester, Pa.)), ethoxylated pentaerythritol tetraacrylate (e.g., SR494, Sartomer Company, Inc.), and dipentaerythritol pentaacrylate (e.g., Photomer 4399, IGM Resins, and SR399, Sartomer Company, Inc.); isocyanurate polyacrylates formed by reacting an appropriate functional isocyanurate with an acrylic acid or acryloyl chloride, such as tris-(2-hydroxyethyl) isocyanurate triacrylate (e.g., SR368, Sartomer Company, Inc.) and tris-(2-hydroxyethyl) isocyanurate diacrylate; alcohol polyacrylates with and without alkoxylation such as tricyclodecane dimethanol diacrylate (e.g., CD406, Sartomer Company, Inc.) and ethoxylated polyethylene glycol diacrylate with the degree of ethoxylation being 2 or greater; for example, ranging from about 2 to 30; epoxy acrylates formed by adding acrylate to bisphenol A diglycidylether and the like (e.g., Photomer 3016, IGM Resins); and single and multi-ring cyclic aromatic or non-aromatic polyacrylates such as dicyclopentadiene diacrylate and dicyclopentane diacrylate. 
     In some embodiments, the curable secondary coating composition includes a multifunctional monomer with three or more curable functional groups in an amount greater than 2.0 wt %, or greater than 5.0 wt %, or greater than 7.5 wt %, or greater than 10 wt %, or greater than 15 wt %, or greater than 20 wt %, or in the range from 2.0 wt % to 25 wt %, or in the range from 5.0 wt % to 20 wt %, or in the range from 8.0 wt % to 15 wt %. In a preferred embodiment, each of the three or more curable functional groups is an acrylate group. 
     In some embodiments, the curable secondary coating composition includes a trifunctional monomer in an amount greater than 2.0 wt %, or greater than 5.0 wt %, or greater than 7.5 wt %, or greater than 10 wt %, or greater than 15 wt %, or greater than 20 wt %, or in the range from 2.0 wt % to 25 wt %, or in the range from 5.0 wt % to 20 wt %, or in the range from 8.0 wt % to 15 wt %. In a preferred embodiment, the trifunctional monomer is a triacrylate monomer. 
     In some embodiments, the curable secondary coating composition includes a difunctional monomer in an amount greater than 55 wt %, or greater than 60 wt %, or greater than 65 wt %, or greater than 70 wt %, or in the range from 55 wt % to 80 wt %, or in the range from 60 wt % to 75 wt %, and further includes a trifunctional monomer in an amount in the range from 2.0 wt % to 25 wt %, or in the range from 5.0 wt % to 20 wt %, or in the range from 8.0 wt % to 15 wt %. In a preferred embodiment, the difunctional monomer is a diacrylate monomer and the trifunctional monomer is a triacrylate monomer. Preferred diacrylate monomers include alkoxylated bisphenol-A diacrylates. Preferred triacrylate monomers include alkoxylated trimethylolpropane triacrylates and isocyanurate triacrylates. Preferably the curable secondary coating composition lacks an alkoxylated bisphenol-A diacrylate having a degree of alkoxylation greater than 17, or greater than 20, or greater than 25, or in the range from 15 to 40, or in the range from 20 to 35. 
     A preferred difunctional monomer is an alkoxylated bisphenol-A diacrylate. Alkoxylated bisphenol-A diacrylate has the general formula (XIII): 
                         
where R 1  and R 2  are alkylene groups, R 1 —O and R 2 —O are alkoxylene groups, and R 3  is H. Any two of the groups R 1 , R 2 , and R 3  are the same or different. In one embodiment, the groups R 1  and R 2  are the same. The number of carbons in each of the groups R 1  and R 2  is in the range from 1 to 8, or in the range from 2 to 6, or in the range from 2 to 4. The degree of alkoxylation is ½(x+y). The values of x and y are the same or different. In one embodiment, x and y are the same.
 
     A preferred trifunctional monomer is an alkoxylated trimethylolpropane triacrylate. Alkoxylated trimethylolpropane triacrylate has the general formula (XIV): 
                         
where where R 1  and R 2  are alkylene groups, O—R 1 , O—R 2 , and O—R 3  are alkoxylene groups. Any two of the groups R 1 , R 2 , and R 3  are the same or different. In one embodiment, the groups R 1 , R 2 , and R 3  are the same. The number of carbons in the each of the groups R 1 , R 2 , and R 3  is in the range from 1 to 8, or in the range from 2 to 6, or in the range from 2 to 4. The degree of alkoxylation is ⅓(x+y+z). The values of any two of x, y and z are the same or different. In one embodiment, x, y, and z are the same.
 
     Another preferred trifunctional monomer is a tris[(acryloyloxy)alkyl] isocyanurate. Tris[(acryloyloxy)alkyl] isocyanurates are also referred to as tris[n-hydroxyalkyl) isocyanurate triacrylates. A representative tris[(acryloyloxy)alkyl] isocyanurate is tris[2-hydroxyethyl) isocyanurate triacrylate, which has the general formula (XV): 
                         
In formula (III), an ethylene linkage (—CH 2 —CH 2 —) bonds each acryloyloxy group to a nitrogen of the isocyanurate ring. In other embodiments of tris[(acryloyloxy)alkyl] isocyanurates, alkylene linkages other than ethylene bond the acryloyloxy groups to nitrogen atoms of the isocyanurate ring. The alkylene linkages for any two of the three alkylene linkages are the same or different. In one embodiment, the three alkylene linkages are the same. The number of carbons in each of the alkylene linkages is in the range from 1 to 8, or in the range from 2 to 6, or in the range from 2 to 4.
 
     In one embodiment, the curable secondary composition includes an alkoxylated bisphenol-A diacrylate monomer in an amount greater than 55 wt %, or greater than 60 wt %, or greater than 65 wt %, or greater than 70 wt %, or in the range from 55 wt % to 80 wt %, or in the range from 60 wt % to 75 wt %, and further includes an alkoxylated trimethylolpropane triacrylate monomer in an amount in the range from 2.0 wt % to 25 wt %, or in the range from 5.0 wt % to 20 wt %, or in the range from 8.0 wt % to 15 wt %. Preferably the curable secondary coating composition lacks an alkoxylated bisphenol-A diacrylate having a degree of alkoxylation greater than 17, or greater than 20, or greater than 25, or in the range from 15 to 40, or in the range from 20 to 35. 
     In one embodiment, the curable secondary composition includes an alkoxylated bisphenol-A diacrylate monomer in an amount greater than 55 wt %, or greater than 60 wt %, or greater than 65 wt %, or greater than 70 wt %, or in the range from 55 wt % to 80 wt %, or in the range from 60 wt % to 75 wt %, and further includes a tris[(acryloyloxy)alkyl] isocyanurate monomer in an amount in the range from 2.0 wt % to 25 wt %, or in the range from 5.0 wt % to 20 wt %, or in the range from 8.0 wt % to 15 wt %. Preferably the curable secondary coating composition lacks an alkoxylated bisphenol-A diacrylate having a degree of alkoxylation greater than 17, or greater than 20, or greater than 25, or in the range from 15 to 40, or in the range from 20 to 35. 
     In one embodiment, the curable secondary composition includes bisphenol-A epoxy diacrylate monomer in an amount greater than 5.0 wt %, or greater than 10 wt %, or greater than 15 wt %, or in the range from 5.0 wt % to 20 wt % or in the range from 8 wt % to 17 wt %, or in the range from 10 wt % to 15 wt %, and further includes an alkoxylated bisphenol-A diacrylate monomer in an amount greater than 55 wt %, or greater than 60 wt %, or greater than 65 wt %, or greater than 70 wt %, or in the range from 55 wt % to 80 wt %, or in the range from 60 wt % to 75 wt %, and further includes an alkoxylated trimethylolpropane triacrylate monomer in an amount in the range from 2.0 wt % to 25 wt %, or in the range from 5.0 wt % to 20 wt %, or in the range from 8.0 wt % to 15 wt %. Preferably the curable secondary coating composition lacks an alkoxylated bisphenol-A diacrylate having a degree of alkoxylation greater than 17, or greater than 20, or greater than 25, or in the range from 15 to 40, or in the range from 20 to 35. 
     In one embodiment, the curable secondary composition includes bisphenol-A epoxy diacrylate monomer in an amount greater than 5.0 wt %, or greater than 10 wt %, or greater than 15 wt %, or in the range from 5.0 wt % to 20 wt % or in the range from 8 wt % to 17 wt %, or in the range from 10 wt % to 15 wt %, and further includes an alkoxylated bisphenol-A diacrylate monomer in an amount greater than 55 wt %, or greater than 60 wt %, or greater than 65 wt %, or greater than 70 wt %, or in the range from 55 wt % to 80 wt %, or in the range from 60 wt % to 75 wt %, and further includes a tris[(acryloyloxy)alkyl] isocyanurate monomer in an amount in the range from 2.0 wt % to 25 wt %, or in the range from 5.0 wt % to 20 wt %, or in the range from 8.0 wt % to 15 wt %. Preferably the curable secondary coating composition lacks an alkoxylated bisphenol-A diacrylate having a degree of alkoxylation greater than 17, or greater than 20, or greater than 25, or in the range from 15 to 40, or in the range from 20 to 35. 
     In one embodiment, the curable secondary composition includes bisphenol-A epoxy diacrylate monomer in an amount greater than 5.0 wt %, or greater than 10 wt %, or greater than 15 wt %, or in the range from 5.0 wt % to 20 wt % or in the range from 8 wt % to 17 wt %, or in the range from 10 wt % to 15 wt %, and further includes an alkoxylated bisphenol-A diacrylate monomer in an amount greater than 55 wt %, or greater than 60 wt %, or greater than 65 wt %, or greater than 70 wt %, or in the range from 55 wt % to 80 wt %, or in the range from 60 wt % to 75 wt %, and further includes tris(2-hydroxyethyl) isocyanurate triacrylate monomer in an amount in the range from 2.0 wt % to 25 wt %, or in the range from 5.0 wt % to 20 wt %, or in the range from 8.0 wt % to 15 wt %. Preferably the curable secondary coating composition lacks an alkoxylated bisphenol-A diacrylate having a degree of alkoxylation greater than 17, or greater than 20, or greater than 25, or in the range from 15 to 40, or in the range from 20 to 35. 
     Secondary Coating—Properties. Relevant properties of the secondary coating include radius, thickness, Young&#39;s modulus, tensile strength, yield strength, elongation at yield, glass transition temperature, and puncture resistance 
     The radius r 6  of the secondary coating is less than or equal to 140.0 μm, or less than or equal to 130.0 μm, or less than or equal to 120.0 μm, or less than or equal to 110.0 μm, or less than or equal to 100.0 μm, or less than or equal to 90.0 μm, or less than or equal to 80.0 μm. To facilitate decreases in the diameter of the optical fiber, it is preferable to minimize the thickness (r 6 −r 5 ) of the secondary coating. The thickness (r 6 −r 5 ) of the secondary coating is greater than or equal to 7.0 μm (e.g., greater than or equal to 8.0 μm, greater than or equal to 9.0 μm, greater than or equal to 10.0 μm, etc.), and less than or equal to 30.0 μm, or less than or equal to 25.0 μm, or less than or equal to 20.0 μm, or less than or equal to 15.0 μm, or in the range from 7.0 μm to 25.0 μm, or in the range from 8.0 μm to 20.0 μm, or in the range from 9.0 μm to 18.0 μm, or in the range from 10.0 μm to 16.0 μm. 
     A factor promoting puncture resistance and low microbend loss is the ratio of the thickness (r 5 −r 4 ) of the primary coating to the thickness (r 6 −r 5 ) of the secondary coating. The ratio of the thickness (r 5 −r 4 ) of the primary coating to the thickness (r 6 −r 5 ) of the secondary coating is in a range from 0.3 to 1.7, or in a range from 0.5 to 1.5, or in a range from 0.7 to 1.2. 
     To facilitate puncture resistance and high protective function, it is preferable for the secondary coating to have a high Young&#39;s modulus and/or a high in situ modulus E S . The Young&#39;s modulus of the secondary coating is greater than or equal to 1600 MPa, or greater than or equal to 1800 MPa, or greater than or equal to 2000 MPa, or greater than or equal to 2200 MPa, or in the range from 1600 MPa to 2800 MPa, or in the range from 1800 MPa to 2600 MPa. In some embodiments, the in situ modulus E S  of the secondary coating is greater than or equal to 1200 MPa, or greater than or equal to 1500 MPa, or greater than or equal to 1800 MPa, or greater than or equal to 2000 MPa, or in the range from 1200 MPa to 2800 MPa, or in the range from 1500 MPa to 2600 MPa. 
     Tertiary Coating—Properties. As described above, the optical fiber can optionaly have a tertiary coating situated on top of the secondary coating. In some embodiments, the tertiary coating is an ink layer or a coating containing ink. A sum of the thicknesses of the secondary coating and the tertiary coating can be larger than or equal to 10 μm, or larger than or equal to 12 μm, or in a range from 12 μm to 30 μm. The combined cross-sectional areas of the secondary coating and optional tertiary coating can be larger than or equal to 20000 μm 2 , or larger than or equal to 25000 μm 2 , or larger than or equal to 30000 μm 2 , which advantageously ensures that the fiber has sufficient puncture resistance. 
     Design Examples—Glass Fiber 
     Eight modeled design examples Ex. 1 through Ex. 8 of the single mode optical fiber  100  with different core/cladding designs and optical attributes are set forth in Tables 1-2 below. Ex. 1 through Ex. 8 are graded-index fibers having a relative refractive index profile of the type shown in  FIG.  7 C . Ex. 1 through Ex. 8 each comprises an updoped graded-index core, an undoped inner cladding, a downdoped depressed-index cladding (corresponding to a trench), an undoped first outer cladding, and an updoped second outer cladding (TiO 2 -doped silica). 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Core/Cladding Design Examples of Glass Fibers 
               
            
           
           
               
               
               
               
               
            
               
                 Parameter 
                 Ex. 1 
                 Ex. 2 
                 Ex. 3 
                 Ex. 4 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Radius of core r 1  (μm) 
                 4.07 
                 4.04 
                 4.11 
                 4.00 
               
               
                 Core Alpha α 
                 11.06 
                 10.14 
                 11.38 
                 10.90 
               
               
                 Core Index maximum Δ 1max  (%) 
                 0.284 
                 0.284 
                 0.275 
                 0.288 
               
               
                 Core Volume V 1  (%-μm 2 ) 
                 3.97 
                 3.88 
                 3.94 
                 3.89 
               
               
                 Radius of inner cladding r 2  (μm) 
                 8.78 
                 8.96 
                 9.00 
                 7.94 
               
               
                 Index of inner cladding Δ 2  (%) 
                 0.00 
                 0.00 
                 0.00 
                 0.00 
               
               
                 Radius of depressed-index 
                 11.90 
                 13.75 
                 14.67 
                 13.63 
               
               
                 cladding r 3  (μm) 
                   
                   
                   
                   
               
               
                 Index of depressed-index 
                 −0.36 
                 −0.29 
                 −0.28 
                 −0.36 
               
               
                 cladding Δ 3  (%) 
                   
                   
                   
                   
               
               
                 Trench Volume V 3  (%-μm 2 ) 
                 23.0 
                 31.5 
                 37.2 
                 43.7 
               
               
                 Radius of first outer cladding r 4a    
                 59.5 
                 59.5 
                 59.5 
                 59.5 
               
               
                 (μm) 
                   
                   
                   
                   
               
               
                 Index of first outer cladding 
                 0.00 
                 0.00 
                 0.00 
                 0.00 
               
               
                 Δ 4a  (%) 
                   
                   
                   
                   
               
               
                 Radius of second outer cladding 
                 62.5 
                 62.5 
                 62.5 
                 62.5 
               
               
                 r 4b  (μm) 
                   
                   
                   
                   
               
               
                 Index of second outer cladding 
                 1.92 
                 1.92 
                 1.92 
                 1.92 
               
               
                 Δ 4b  (%) 
                   
                   
                   
                   
               
               
                 LP11 Theoretical Cutoff 
                 1.003 
                 0.995 
                 1.002 
                 0.984 
               
               
                 Wavelength (μm) 
                   
                   
                   
                   
               
               
                 MFD at 1064 nm (μm) 
                 8.10 
                 8.09 
                 8.23 
                 7.95 
               
               
                 MFD at 1310 nm (μm) 
                 9.28 
                 9.31 
                 9.46 
                 9.04 
               
               
                 A eff  at 1310 nm (sq. μm) 
                 66.5 
                 66.8 
                 68.9 
                 63.4 
               
               
                 Pin Array Bend Loss at 1310 nm 
                 14.91 
                 14.10 
                 14.38 
                 10.92 
               
               
                 (dB) 
                   
                   
                   
                   
               
               
                 Dispersion at 1310 nm 
                 −0.05 
                 −0.48 
                 −0.18 
                 0.62 
               
               
                 (ps/nm/km) 
                   
                   
                   
                   
               
               
                 Slope at 1310 nm (ps/nm 2 /km) 
                 0.092 
                 0.092 
                 0.092 
                 0.093 
               
               
                 Zero Dispersion Wavelength 
                 1311 
                 1315 
                 1312 
                 1303 
               
               
                 (nm) 
                   
                   
                   
                   
               
               
                 λ CC  (nm) 
                 976 
                 984 
                 1012 
                 1023 
               
               
                 λ CF,2  (nm) 
                 1136 
                 1144 
                 1172 
                 1183 
               
               
                 λ CF,1  (nm) 
                 1216 
                 1224 
                 1152 
                 1263 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 Core/Cladding Design Examples of Glass Fibers 
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 Parameter 
                 Ex. 5 
                 Ex. 6 
                 Ex. 7 
                 Ex. 8 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 Radius of core r 1   
                 4.74 
                 4.66 
                 4.59 
                 4.77 
               
               
                   
                 (μm) 
                   
                   
                   
                   
               
               
                   
                 Core Alpha α 
                 2.78 
                 2.33 
                 2.90 
                 2.41 
               
               
                   
                 Core Index 
                 0.315 
                 0.333 
                 0.325 
                 0.348 
               
               
                   
                 maximum Δ 1max   
                   
                   
                   
                   
               
               
                   
                 (%) 
                   
                   
                   
                   
               
               
                   
                 Core Volume V 1   
                 4.11 
                 3.89 
                 4.05 
                 4.32 
               
               
                   
                 (%-μm 2 ) 
                   
                   
                   
                   
               
               
                   
                 Radius of inner 
                 8.00 
                 7.73 
                 8.69 
                 7.63 
               
               
                   
                 cladding r 2  (μm) 
                   
                   
                   
                   
               
               
                   
                 Index of inner 
                 0.00 
                 0.00 
                 0.00 
                 0.00 
               
               
                   
                 cladding Δ 2  (%) 
                   
                   
                   
                   
               
               
                   
                 Radius of 
                 13.94 
                 14.69 
                 12.09 
                 12.73 
               
               
                   
                 depressed-index 
                   
                   
                   
                   
               
               
                   
                 cladding r 3  (μm) 
                   
                   
                   
                   
               
               
                   
                 r 1 /r 2   
                 0.59 
                 0.60 
                 0.53 
                 0.62 
               
               
                   
                 Index of 
                 −0.30 
                 −0.31 
                 −0.35 
                 −0.33 
               
               
                   
                 depressed-index 
                   
                   
                   
                   
               
               
                   
                 cladding Δ 3  (%) 
                   
                   
                   
                   
               
               
                   
                 Trench Volume 
                 −39.7 
                 −48.6 
                 −24.8 
                 −34.5 
               
               
                   
                 V 3  (%-μm 2 ) 
                   
                   
                   
                   
               
               
                   
                 Radius of first 
                 59.5 
                 59.5 
                 59.5 
                 59.5 
               
               
                   
                 outer cladding r 4a   
                   
                   
                   
                   
               
               
                   
                 (μm) 
                   
                   
                   
                   
               
               
                   
                 Index of first 
                 0.00 
                 0.00 
                 0.00 
                 0.00 
               
               
                   
                 outer cladding 
                   
                   
                   
                   
               
               
                   
                 Δ 4a  (%) 
                   
                   
                   
                   
               
               
                   
                 Radius of second 
                 62.5 
                 62.5 
                 62.5 
                 62.5 
               
               
                   
                 outer cladding r 4b   
                   
                   
                   
                   
               
               
                   
                 (μm) 
                   
                   
                   
                   
               
               
                   
                 Index of second 
                 1.92 
                 1.92 
                 1.92 
                 1.92 
               
               
                   
                 outer cladding Δ 4b   
                   
                   
                   
                   
               
               
                   
                 (%) 
                   
                   
                   
                   
               
               
                   
                 LP11 Theoretical 
                 0.985 
                 0.954 
                 0.990 
                 1.001 
               
               
                   
                 Cutoff 
                   
                   
                   
                   
               
               
                   
                 Wavelength (μm) 
                   
                   
                   
                   
               
               
                   
                 MFD at 1064 nm 
                 8.03 
                 7.85 
                 7.93 
                 7.77 
               
               
                   
                 (μm) 
                   
                   
                   
                   
               
               
                   
                 MFD at 1310 nm 
                 9.18 
                 9.00 
                 9.14 
                 8.85 
               
               
                   
                 (μm) 
                   
                   
                   
                   
               
               
                   
                 A eff  at 1310 nm 
                 64.70 
                 62.10 
                 63.90 
                 60.30 
               
               
                   
                 (sq. μm) 
                   
                   
                   
                   
               
               
                   
                 Pin Array Bend 
                 11.40 
                 11.66 
                 12.32 
                 6.48 
               
               
                   
                 Loss at 1310 nm 
                   
                   
                   
                   
               
               
                   
                 (dB) 
                   
                   
                   
                   
               
               
                   
                 Dispersion at 
                 0.30 
                 0.01 
                 −0.59 
                 0.32 
               
               
                   
                 1310 nm 
                   
                   
                   
                   
               
               
                   
                 (ps/nm/km) 
                   
                   
                   
                   
               
               
                   
                 Slope at 1310 nm 
                 0.094 
                 0.095 
                 0.094 
                 0.094 
               
               
                   
                 (ps/nm 2 /km) 
                   
                   
                   
                   
               
               
                   
                 Zero Dispersion 
                 1307 
                 1310 
                 1316 
                 1307 
               
               
                   
                 Wavelength (nm) 
                   
                   
                   
                   
               
               
                   
                 λ CC  (nm) 
                 1017 
                 1009 
                 969 
                 1026 
               
               
                   
                 λ CF,2  (nm) 
                 1177 
                 1169 
                 1139 
                 1186 
               
               
                   
                 λ CF,1  (nm) 
                 1257 
                 1249 
                 1219 
                 1266 
               
               
                   
                   
               
            
           
         
       
     
     Measurement of Stresses in Optical Fiber 
     Residual stresses in optical fibers are induced due to the viscosity and coefficient of thermal expansion differences from radial composition distribution. The mismatch in the properties result in thermal and mechanical stresses induced during the drawing process of manufacturing optical fibers. Residual stresses in optical fibers are measured using well-documented methods in literature and familiar to those skilled in the art. These methods include measuring stresses using a polariscope or using traverse interferometry. Details of method for measuring residual stresses in optical fibers using polariscope are described in Park et al., Applied Optics, 41 (1), 21-26 (2002) and Chu and Whitbread, Applied Optics, 21 (23), 4241 (1982). The method entails immersing the optical fiber in an index matching fluid and impinging a light from a light source onto the optical fiber laterally. The ray entering the optical fiber is split into two components due to stress-induced birefringence. The two components experience different refractive indices in the optical fiber. As the light exits the fiber, the two components have a phase shift called retardation, with the magnitude of the retardation being a function of the ray incident position. The residual stresses can be estimated from the measurements of retardation as a function of ray incident position. Alternate method of measuring residual stresses is using the transverse interferometry method, as described in A. Yablon, “Advanced Fiber Characterization Technologies for Fiber Lasers and Amplifiers”, Conference on Advanced Solid State Lasers, Paper #ATh2A.45, Shanghai, China, 16-21 Nov. 2014. The measurement of stresses using this method can be performed using the IFA instrument available from Interfiber Analysis LLC (Sharon, Mass., USA). 
       FIGS.  8 A- 8 D  illustrate axial stress of the single mode optical fiber  100  along the radial position with various designs of the second outer cladding, according to some embodiments of the present disclosure. 
     Referring to  FIG.  8 A , three lines indicating the axial stress vurses radial position of the five examples of the single mode optical fiber has a core/cladding designs and optical attributes as Ex. 1, Ex. 5, and Ex. 8 listed above in Table 1 and Table 2. Specifically, the second outer cladding has a thickness of 3 μm and a TiO 2  concentration 8 wt %. The five lines show the axial stress distribution of the five examplary single mode optical fibers along the radial position under a 100 g draw tension. The dashed line shows the axial stress distribution of the single mode optical fiber as Ex. 1 listed in Table 1, the dotted line shows the axial stress distribution of the single mode optical fiber as Ex. 5 listed in Table 2, and the solid line shows the axial stress distribution of the single mode optical fiber as Ex. 8 listed in Table 2. 
     Referring to  FIG.  8 B , three lines indicating the axial stress vurses radial position of the single mode optical fiber has a core/cladding designs and optical attributes as Ex. 1 listed above in Table 1. Speficifically, while keeping the thickness of the second outer cladding as 3 microns and the draw tension as 100 g, the TiO 2  concentration of the second outer cladding is varied to show the changes of the axial stress of the single mode optical fiber. The dotted line shows the axial stress distribution of the single mode optical fiber with a 4 wt % TiO 2  concentration of the second outer cladding, the dashed line shows the axial stress distribution of the single mode optical fiber with an 8 wt % TiO 2  concentration of the second outer cladding, and the solid line shows the axial stress distribution of the single mode optical fiber with a 12 wt % TiO 2  concentration of the second outer cladding. 
     Referring to  FIG.  8 C , three lines indicating the axial stress vurses radial position of the single mode optical fiber has a core/cladding designs and optical attributes as Ex. 1 listed above in Table 1. Speficifically, while keeping the TiO 2  concentration as 8 wt % and the draw tension as 100 g, the thickness of the second outer cladding doped with TiO 2  is varied to show the changes of the axial stress of the single mode optical fiber. The dotted line shows the axial stress distribution of the single mode optical fiber with a 3-μm-thick second outer cladding, the dashed line shows the axial stress distribution of the single mode optical fiber with a 5-μm-thick second outer cladding, and the solid line shows the axial stress distribution of the single mode optical fiber with a 10-μm-thick second outer cladding. 
     Referring to  FIG.  8 D , three lines indicating the axial stress vurses radial position of the single mode optical fiber has a core/cladding designs and optical attributes as Ex. 1 listed above in Table 1. Speficifically, while keeping the thickness of the second outer cladding as 3 microns and the TiO 2  concentration as 8 wt %, the draw tension is varied to show the changes of the axial stress of the single mode optical fiber. The dotted line shows the axial stress distribution of the single mode optical fiber under a 50 g draw tension, the dashed line shows the axial stress distribution of the single mode optical fiber under a 100 g draw tension, and the solid line shows the axial stress distribution of the single mode optical under a 150 g draw tension. 
     The following examples illustrate preparation of a representative primary and secondary coatings. Measurements of selected properties of the representative primary and secondary coatings are also described. In addition, modeled properties of glass fibers coated with primary and secondary coatings at different coating thickness and modulus are presented. 
     Design Examples—Primary Coating 
     Primary Coating—Oligomer. The primary coating composition included an oligomer. For purposes of illustration, preparation of exemplary oligomers from H12MDI (4,4′-methylene bis(cyclohexyl isocyanate), PPG4000 (polypropylene glycol with M n ˜4000 g/mol) and HEA (2-hydroxyethyl acrylate) in accordance with the reaction scheme hereinabove is described. All reagents were used as supplied by the manufacturer and were not subjected to further purification. H12MDI was obtained from ALDRICH. PPG4000 was obtained from COVESTRO and was certified to have an unsaturation of 0.004 meq/g as determined by the method described in the standard ASTM D4671-16. HEA was obtained from KOWA. 
     The relative amounts of the reactants and reaction conditions were varied to obtain a series of six oligomers. Oligomers with different initial molar ratios of the constituents were prepared with molar ratios of the reactants satisfying H12MDI:HEA:PPG4000=n:m:p, where n was in the range from 3.0 to 4.0, m was in the range from 1.5n to 3 to 2.5n to 5, and p=2. In the reactions used to form the oligomers materials, dibutyltin dilaurate was used as a catalyst (at a level of 160 ppm based on the mass of the initial reaction mixture) and 2,6-di-tert-butyl-4-methylphenol (BHT) was used as an inhibitor (at a level of 400 ppm based on the mass of the initial reaction mixture). 
     The amounts of the reactants used to prepare each of the six oligomers are summarized in Table 3 below. The six oligomers are identified by separate Sample numbers 1-6. Corresponding sample numbers will be used herein to refer to coating compositions and cured films formed from coating compositions that individually contain each of the six oligomers. The corresponding mole numbers used in the preparation of each of the six samples are listed in Table 4 below. The mole numbers are normalized to set the mole number p of PPG4000 to 2.0. 
     
       
         
           
               
             
               
                 TABLE 3 
               
             
            
               
                   
               
               
                 Reactants and Amounts for Exemplary 
               
               
                 Oligomer Samples 1-6 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Sample 
                 H12MDI (g) 
                 HEA (g) 
                 PPG4000 (g) 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 1 
                 22 
                 6.5 
                 221.5 
               
               
                   
                 2 
                 26.1 
                 10.6 
                 213.3 
               
               
                   
                 3 
                 26.1 
                 10.6 
                 213.3 
               
               
                   
                 4 
                 27.8 
                 12.3 
                 209.9 
               
               
                   
                 5 
                 27.8 
                 12.3 
                 209.9 
               
               
                   
                 6 
                 22 
                 6.5 
                 221.5 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 4 
               
             
            
               
                   
               
               
                 Mole Numbers for Oligomer Samples 1-6 
               
            
           
           
               
               
               
               
               
            
               
                   
                 H12MDI 
                 HEA 
                 PPG4000 
                 Di- 
               
               
                   
                 Mole 
                 Mole 
                 Mole 
                 adduct 
               
               
                 Sample 
                 Number (n) 
                 Number (m) 
                 Number (p) 
                 (wt %) 
               
               
                   
               
               
                 1 
                 3.0 
                 2.0 
                 2.0 
                 1.3 
               
               
                 2 
                 3.7 
                 3.4 
                 2.0 
                 3.7 
               
               
                 3 
                 3.7 
                 3.4 
                 2.0 
                 3.7 
               
               
                 4 
                 4.0 
                 4.0 
                 2.0 
                 5.0 
               
               
                 5 
                 4.0 
                 4.0 
                 2.0 
                 5.0 
               
               
                 6 
                 3.0 
                 2.0 
                 2.0 
                 1.3 
               
               
                   
               
            
           
         
       
     
     The oligomers were prepared by mixing 4,4′-methylene bis(cyclohexyl isocyanate), dibutyltin dilaurate and 2,6-di-tert-butyl-4 methylphenol at room temperature in a 500 mL flask. The 500 mL flask was equipped with a thermometer, a CaCl 2  drying tube, and a stirrer. While continuously stirring the contents of the flask, PPG4000 was added over a time period of 30-40 minutes using an addition funnel. The internal temperature of the reaction mixture was monitored as the PPG4000 was added and the introduction of PPG4000 was controlled to prevent excess heating (arising from the exothermic nature of the reaction). After the PPG4000 was added, the reaction mixture was heated in an oil bath at about 70° C. to 75° C. for about 1 to 1½ hours. At various intervals, samples of the reaction mixture were retrieved for analysis by infrared spectroscopy (FTIR) to monitor the progress of the reaction by determining the concentration of unreacted isocyanate groups. The concentration of unreacted isocyanate groups was assessed based on the intensity of a characteristic isocyanate stretching mode near 2265 cm −1 . The flask was removed from the oil bath and its contents were allowed to cool to below 65° C. Addition of supplemental HEA was conducted to insure complete quenching of isocyanate groups. The supplemental HEA was added dropwise over 2-5 minutes using an addition funnel. After addition of the supplemental HEA, the flask was returned to the oil bath and its contents were again heated to about 70° C. to 75° C. for about 1 to 1½ hours. FTIR analysis was conducted on the reaction mixture to assess the presence of isocyanate groups and the process was repeated until enough supplemental HEA was added to fully react any unreacted isocyanate groups. The reaction was deemed complete when no appreciable isocyanate stretching intensity was detected in the FTIR measurement. The HEA amounts listed in Table 3 and Table 4 include the initial amount of HEA in the composition and any amount of supplemental HEA needed to quench unreacted isocyanate groups. 
     The concentration (wt %) of di-adduct compound in each oligomer was determined by gel permeation chromatography (GPC). A Waters Alliance 2690 GPC instrument was used to determine the di-adduct concentration. The mobile phase was THF. The instrument included a series of three Polymer Labs columns. Each column had a length of 300 mm and an inside diameter of 7.5 mm. Two of the columns (columns 1 and 2) were sold under Part No. PL1110-6504 by Agilent Technologies and were packed with PLgel Mixed D stationary phase (polystyrene divinyl benzene copolymer, average particle size=5 μm, specified molecular weight range=200 g/mol to 400,000 g/mol). The third column (column 3) was sold under Part No. PL1110-6520 by Agilent Technologies and was packed with PLgel 100A stationary phase (polystyrene divinyl benzene copolymer, average particle size=5 μm, specified molecular weight range=up to 4,000 g/mol). The columns were calibrated with polystyrene standards ranging from 162 g/mol to 6,980,000 g/mol using EasiCal PS-1 &amp; 2 polymer calibrant kits (Agilent Technologies Part Nos. PL2010-505 and PL2010-0601). The GPC instrument was operated under the following conditions: flow rate=1.0 mL/min, column temperature=40° C., injection volume=100 μL, and run time=35 min (isocratic conditions). The detector was a Waters Alliance 2410 differential refractometer operated at 40° C. and sensitivity level  4 . The samples were injected twice along with a THF+0.05% toluene blank. 
     The amount (wt %) of di-adduct in the oligomers was quantified using the preceding GPC system and technique. A calibration curve was obtained using standard solutions containing known amounts of the di-adduct compound (HEA˜H12MDI-HEA) in THF. Standard solutions with di-adduct concentrations of 115.2 μg/g, 462.6 μg/g, 825.1 μg/g, and 4180 μg/g were prepared. (As used herein, the dimension “μg/g” refers to μg of di-adduct per gram of total solution (di-adduct+THF)). Two 100 μL aliquots of each di-adduct standard solution were injected into the column to obtain the calibration curve. The retention time of the di-adduct was approximately 23 min and the area of the GPC peak of the di-adduct was measured and correlated with di-adduct concentration. A linear correlation of peak area as a function of di-adduct concentration was obtained (correlation coefficient (R 2 )=0.999564). 
     The di-adduct concentration in the oligomers was determined using the calibration. Samples were prepared by diluting ˜0.10 g of oligomeric material in THF to obtain a ˜1.5 g test solution. The test solution was run through the GPC instrument and the area of the peak associated with the di-adduct compound was determined. The di-adduct concentration in units of μg/g was obtained from the peak area and the calibration curve, and was converted to wt % by multiplying by the weight (g) of the test solution and dividing by the weight of the sample of oligomeric material before dilution with THF. The wt % of di-adduct compound present in each of the six oligomers prepared in this example are reported in Table 3. 
     Through variation in the relative mole ratios of H12MDI, HEA, and PPG4000, the illustrative oligomers include a polyether urethane compound of the type shown in molecular formula (IV) hereinabove and an enhanced concentration of di-adduct compound of the type shown in molecular formula (V) hereinabove. 
     Primary Coating—Compositions. Oligomers corresponding to Samples 1-6 were separately combined with other components to form a series of six coating compositions. The amount of each component in the coating composition is listed in Table 5 below. The entry in Table 5 for the oligomer includes the combined amount of polyether urethane acrylate compound and di-adduct compound present in the oligomer. A separate coating composition was made for each of the six exemplary oligomers corresponding to Samples 1-6, where the amount of di-adduct compound in the oligomeric material corresponded to the amount listed in Table 4. 
     
       
         
           
               
             
               
                 TABLE 5 
               
             
            
               
                   
               
               
                 Coating Composition 
               
            
           
           
               
               
               
            
               
                   
                 Component 
                 Amount 
               
               
                   
                   
               
               
                   
                 Oligomeric Material 
                 49.10 wt % 
               
               
                   
                 Sartomer SR504 
                 45.66 wt % 
               
               
                   
                 V-CAP/RC 
                  1.96 wt % 
               
               
                   
                 TPO 
                  1.47 wt % 
               
               
                   
                 Irganox 1035 
                  0.98 wt % 
               
               
                   
                 adhesion promoter 
                  0.79 wt % 
               
               
                   
                 Tetrathiol 
                  0.03 wt % 
               
               
                   
                   
               
            
           
         
       
     
     Sartomer SR504 is ethoxylated(4)nonylphenol acrylate (available from Sartomer). V-CAP/RC is N-vinylcaprolactam (available from ISP Technologies). TPO is 2,4,6-trimethylbenzoyl)diphenyl phosphine oxide (available from BASF under the trade name Lucirin and functions as a photoinitiator). Irganox 1035 is thiodiethylene bis[3-(3,5-di-tert-butyl)-4-hydroxy-phenyl) propionate] (available from BASF) and functions as an antioxidant. The adhesion promoters were 3-acryloxypropyl trimethoxysilane (available from Gelest) and 3-mercaptopropyl trimethoxysilane (available from Aldrich). 3-acryloxypropyl trimethoxysilane was used for Samples 1, 3, and 5. 3-mercaptopropyl trimethoxysilane was used for Samples 2, 4, and 6. Tetrathiol is a catalyst quencher. 
     The coating compositions of Table 5 were each formulated using a high-speed mixer in an appropriate container heated to 60° C., with a heating band or heating mantle. In each case, the components were weighed into the container using a balance and allowed to mix until the solid components were thoroughly dissolved and the mixture appeared homogeneous. The oligomer and monomers (SR504, NVC) of each composition were blended together for at least 10 minutes at 55° C. to 60° C. The photoinitiator, antioxidant, and catalyst quencher were then added, and blending was continued for one hour while maintaining a temperature of 55° C. to 60° C. Finally, the adhesion promoter was added, and blending was continued for 30 minutes at 55° C. to 60° C. to form the coating compositions. 
     Primary Coating—Properties—Tensile Properties. Tensile properties (Young&#39;s modulus, tensile strength at yield, and elongation at yield) were measured on films formed by curing the six coating compositions. Separate films were formed from each coating composition. Wet films of the coating composition were cast on silicone release paper with the aid of a draw-down box having a gap thickness of about 0.005″. The wet films were cured with a UV dose of 1.2 J/cm 2  (measured over a wavelength range of 225 to 424 nm by a Light Bug model IL490 from International Light) by a Fusion Systems UV curing apparatus with a 600 W/in D-bulb (50% Power and approximately 12 ft/min belt speed) to yield cured coatings in film form. Cured film thickness was between about 0.0030″ and 0.0035″. 
     The films were aged (23° C., 50% relative humidity) for at least 16 hours prior to testing. Film samples were cut to dimensions of 12.5 cm×13 mm using a cutting template and a scalpel. Young&#39;s modulus, tensile strength at yield, and elongation at yield were measured at room temperature (approximately 20° C.) on the film samples using a MTS Sintech tensile test instrument following procedures set forth in ASTM Standard D882-97. Young&#39;s modulus is defined as the steepest slope of the beginning of the stress-strain curve. Films were tested at an elongation rate of 2.5 cm/min with the initial gauge length of 5.1 cm. The results are shown in Table 6. 
     
       
         
           
               
             
               
                 TABLE 6 
               
             
            
               
                   
               
               
                 Young&#39;s Modulus, Tensile Strength, and 
               
               
                 Elongation of Film Samples 
               
            
           
           
               
               
               
               
               
            
               
                   
                   
                 Young&#39;s 
                 Tensile 
                   
               
               
                   
                   
                 Modulus 
                 Strength 
                 Elongation 
               
               
                   
                 Sample 
                 (MPa) 
                 (MPa) 
                 (%) 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 1 
                 0.72 
                 0.51 
                 137.9 
               
               
                   
                 2 
                 0.57 
                 0.44 
                 173 
               
               
                   
                 3 
                 1.0 
                 0.86 
                 132.8 
               
               
                   
                 4 
                 0.71 
                 0.45 
                 122.3 
               
               
                   
                 5 
                 0.72 
                 0.56 
                 157.4 
               
               
                   
                 6 
                 0.33 
                 0.33 
                 311.9 
               
               
                   
                   
               
            
           
         
       
     
     Primary Coating—Properties—In Situ Modulus. In situ modulus measurements of primary coating composition Samples 2, 3, and 5 were completed. In situ modulus measurements require forming the primary coatings on a glass fiber having a diameter of 125 μm. Each of Samples 2, 3, and 5 was separately applied as a primary coating composition to a glass fiber as the glass fiber was being drawn. The fiber draw speed was 50 m/s. The primary coating compositions were cured using a stack of five LED sources. Each LED source was operated at 395 nm and had an intensity of 12 W/cm 2 . Subsequent to application and curing of the primary coating compositions, a secondary coating composition was applied to each of the cured primary coatings and cured using UV sources to form a secondary coating layer. The thickness of the primary coating was 32.5 μm and the thickness of the secondary coating was 26.0 μm. 
     The in situ modulus was measured using the following procedure. A six-inch sample of fiber was obtained and a one-inch section from the center of the fiber was window stripped and wiped with isopropyl alcohol. The window-stripped fiber was mounted on a sample holder/alignment stage equipped with 10 mm×5 mm rectangular aluminum tabs that were used to affix the fiber. Two tabs were oriented horizontally and positioned so that the short 5 mm sides were facing each other and separated by a 5 mm gap. The window-stripped fiber was laid horizontally on the sample holder across the tabs and over the gap separating the tabs. The coated end of one side of the window-stripped region of the fiber was positioned on one tab and extended halfway into the 5 mm gap between the tabs. The one-inch window-stripped region extended over the remaining half of the gap and across the opposing tab. After alignment, the sample was moved and a small dot of glue was applied to the half of each tab closest to the 5 mm gap. The fiber was then returned to position and the alignment stage was raised until the glue just touched the fiber. The coated end was then pulled away from the gap and through the glue such that the majority of the 5 mm gap between the tabs was occupied by the window-stripped region of the fiber. The portion of the window-stripped region remaining on the opposing tab was in contact with the glue. The very tip of the coated end was left to extend beyond the tab and into the gap between the tabs. This portion of the coated end was not embedded in the glue and was the object of the in situ modulus measurement. The glue was allowed to dry with the fiber sample in this configuration to affix the fiber to the tabs. After drying, the length of fiber fixed to each of the tabs was trimmed to 5 mm. The coated length embedded in glue, the non-embedded coated length (the portion extending into the gap between the tabs), and the primary diameter were measured. 
     The in situ modulus measurements were performed on a Rheometrics DMTA IV dynamic mechanical testing apparatus at a constant strain of 9e-6 l/s for a time of forty-five minutes at room temperature (21° C.). The gauge length was 15 mm. Force and the change in length were recorded and used to calculate the in situ modulus of the primary coating. The tab-mounted fiber samples were prepared by removing any epoxy from the tabs that would interfere with the 15 mm clamping length of the testing apparatus to ensure that there was no contact of the clamps with the fiber and that the sample was secured squarely to the clamps. The instrument force was zeroed out. The tab to which the non-coated end of the fiber was affixed was then mounted to the lower clamp (measurement probe) of the testing apparatus and the tab to which the coated end of the fiber was affixed was mounted to the upper (fixed) clamp of the testing apparatus. The test was then executed and the sample was removed once the analysis was completed. 
     The in situ modulus of primary coating Samples 2, 3, and 5 are listed in Table 7. 
     
       
         
           
               
             
               
                 TABLE 7 
               
             
            
               
                   
               
               
                 In Situ Modulus of Selected Primary Coatings 
               
            
           
           
               
               
               
            
               
                   
                 Sample 
                 In-Situ Modulus (MPa) 
               
               
                   
                   
               
            
           
           
               
               
               
            
               
                   
                 2 
                 0.27 
               
               
                   
                 3 
                 0.33 
               
               
                   
                 5 
                 0.3 
               
               
                   
                   
               
            
           
         
       
     
     Design Examples—Secondary Coating 
     Secondary Coating Compositions. Representative curable secondary coating compositions are listed in Table 8. 
     
       
         
           
               
             
               
                 TABLE 8 
               
             
            
               
                   
               
               
                 Secondary Coating Compositions 
               
            
           
           
               
               
               
            
               
                   
                   
                 Composition 
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 Component 
                 KA 
                 KB 
                 KC 
                 KD 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 SR601 (wt %) 
                 72.0 
                 30.0 
                 30.0 
                 30.0 
               
               
                   
                 SR602 (wt %) 
                   
                 37.0 
                 37.0 
                 37.0 
               
               
                   
                 SR349 (wt %) 
                   
                 30.0 
                 15.0 
                   
               
               
                   
                 SR399 (wt %) 
                   
                   
                 15.0 
                   
               
               
                   
                 SR499 (wt %) 
                   
                   
                   
                 30.0 
               
               
                   
                 CD9038 (wt %) 
                 10.0 
                   
                   
                   
               
               
                   
                 Photomer 3016 (wt %) 
                 15.0 
                   
                   
                   
               
               
                   
                 TPO (wt %) 
                 1.5 
                   
                   
                   
               
               
                   
                 Irgacure 184 (wt %) 
                 1.5 
                   
                   
                   
               
               
                   
                 Irgacure 1850 (wt %) 
                   
                 3.0 
                 3.0 
                 3.0 
               
               
                   
                 Irganox 1035 (pph) 
                 0.5 
                   
                   
                   
               
               
                   
                 DC-190 (pph) 
                 1.0 
               
               
                   
                   
               
            
           
         
       
     
     SR601 is ethoxylated (4) bisphenol A diacrylate (a monomer). SR602 is ethoxylated (10) bisphenol A diacrylate (a monomer). SR349 is ethoxylated (2) bisphenol A diacrylate (a monomer). SR399 is dipentaerythritol pentaacrylate. SR499 is ethoxylated (6) trimethylolpropane triacrylate. CD9038 is ethoxylated (30) bisphenol A diacrylate (a monomer). Photomer 3016 is bisphenol A epoxy diacrylate (a monomer). TPO is a photoinitiator. Irgacure 184 is 1-hydroxycyclohexylphenyl ketone (a photoinitiator). Irgacure 1850 is bis(2,6-dimethoxybenzoyl)-2,4,4-trimethylpentylphosphine oxide (a photoinitiator). Irganox 1035 is thiodiethylene bis(3,5-di-tert-butyl)-4-hydroxyhydrocinnamate (an antioxidant). DC190 is silicone-ethylene oxide/propylene oxide copolymer (a slip agent). The concentration unit “pph” refers to an amount relative to a base composition that includes all monomers, oligomers, and photoinitiators. For example, for secondary coating composition KA, a concentration of 1.0 pph for DC-190 corresponds to 1 g DC-190 per 100 g combined of SR601, CD9038, Photomer 3016, TPO, and Irgacure 184. 
     A comparative curable secondary coating composition (A) and three representative curable secondary coating compositions (SB, SC, and SD) within the scope of the disclosure are listed in Table 9. 
                     TABLE 9                  Secondary Coating Compositions                         Composition                                 Component   A   SB   SC   SD                                         PE210 (wt %)   15.0   15.0   15.0   15.0       M240 (wt %)   72.0   72.0   72.0   62.0       M2300 (wt %)   10.0   —   —   —       M3130 (wt %)   —   10.0   —   —       M370 (wt %)   —   —   10.0   10.0       TPO (wt %)   1.5   1.5   1.5   1.5       Irgacure 184 (wt %)   1.5   1.5   1.5   1.5       Irganox 1035 (pph)   0.5   0.5   0.5   0.5       DC-190 (pph)   1.0   1.0   1.0   1.0                    
PE210 is bisphenol-A epoxy diacrylate (available from Miwon Specialty Chemical, Korea), M240 is ethoxylated (4) bisphenol-A diacrylate (available from Miwon Specialty Chemical, Korea), M2300 is ethoxylated (30) bisphenol-A diacrylate (available from Miwon Specialty Chemical, Korea), M3130 is ethoxylated (3) trimethylolpropane triacrylate (available from Miwon Specialty Chemical, Korea), TPO (a photoinitiator) is (2,4,6-trimethylbenzoyl)diphenyl phosphine oxide (available from BASF), Irgacure 184 (a photoinitiator) is 1-hydroxycyclohexyl-phenyl ketone (available from BASF), Irganox 1035 (an antioxidant) is benzenepropanoic acid, 3,5-bis(1,1-dimethylethyl)-4-hydroxythiodi-2,1-ethanediyl ester (available from BASF). DC190 (a slip agent) is silicone-ethylene oxide/propylene oxide copolymer (available from Dow Chemical). The concentration unit “pph” refers to an amount relative to a base composition that includes all monomers and photoinitiators. For example, for secondary coating composition A, a concentration of 1.0 pph for DC-190 corresponds to 1 g DC-190 per 100 g combined of PE210, M240, M2300, TPO, and Irgacure 184.
 
     Secondary Coating—Properties. The Young&#39;s modulus, tensile strength at break, and elongation at break of secondary coatings made from secondary compositions A, KA, KB, KC, KD, SB, SC and SD were measured. 
     Secondary Coating—Properties—Measurement Techniques. Properties of secondary coatings were determined using the measurement techniques described below: 
     Tensile Properties. The curable secondary coating compositions were cured and configured in the form of cured rod samples for measurement of Young&#39;s modulus, tensile strength at yield, yield strength, and elongation at yield. The cured rods were prepared by injecting the curable secondary composition into Teflon® tubing having an inner diameter of about 0.025″. The rod samples were cured using a Fusion D bulb at a dose of about 2.4 J/cm 2  (measured over a wavelength range of 225-424 nm by a Light Bug model IL390 from International Light). After curing, the Teflon® tubing was stripped away to provide a cured rod sample of the secondary coating composition. The cured rods were allowed to condition for 18-24 hours at 23° C. and 50% relative humidity before testing. Young&#39;s modulus, tensile strength at break, yield strength, and elongation at yield were measured using a Sintech MTS Tensile Tester on defect-free rod samples with a gauge length of 51 mm, and a test speed of 250 mm/min. Tensile properties were measured according to ASTM Standard D882-97. The properties were determined as an average of at least five samples, with defective samples being excluded from the average. 
     In Situ Modulus of Secondary Coating. For secondary coatings, the in situ modulus was measured using fiber tube-off samples prepared from the fiber samples. A 0.0055 inch Miller stripper was clamped down approximately 1 inch from the end of the fiber sample. This one-inch region of fiber sample was immersed into a stream of liquid nitrogen and held for 3 seconds. The fiber sample was then removed and quickly stripped. The stripped end of the fiber sample was then inspected. If coating remained on the glass portion of the fiber sample, the tube-off sample was deemed defective and a new tube-off sample was prepared. A proper tube-off sample is one that stripped clean from the glass and consisted of a hollow tube with primary and secondary coating. The glass, primary and secondary coating diameter were measured from the end-face of the un-stripped fiber sample. 
     The fiber tube-off samples were run using a Rheometrics DMTA IV instrument at a sample gauge length 11 mm to obtain the in situ modulus of the secondary coating. The width, thickness, and length were determined and provided as input to the operating software of the instrument. The sample was mounted and run using a time sweep program at ambient temperature (21° C.) using the following parameters:
         Frequency: 1 Rad/sec   Strain: 0.3%   Total Time=120 sec.   Time Per Measurement=1 sec   Initial Static Force=15.0 g   Static&gt;Dynamic Force by=10.0%
 
Once completed, the last five E′ (storage modulus) data points were averaged. Each sample was run three times (fresh sample for each run) for a total of fifteen data points. The averaged value of the three runs was reported.
       

     Puncture Resistance of Secondary Coating. Puncture resistance measurements were made on samples that included a glass fiber, a primary coating, and a secondary coating. The glass fiber had a diameter of 125 μm. The primary coating was formed from the reference primary coating composition listed in Table 10 below. Samples with various secondary coatings were prepared as described below. The thicknesses of the primary coating and secondary coating were adjusted to vary the cross-sectional area of the secondary coating as described below. The ratio of the thickness of the secondary coating to the thickness of the primary coating was maintained at about 0.8 for all samples. 
     The puncture resistance was measured using the technique described in the article entitled “Quantifying the Puncture Resistance of Optical Fiber Coatings”, by G. Scott Glaesemann and Donald A. Clark, published in the Proceedings of the 52 nd  International Wire &amp; Cable Symposium, pp. 237-245 (2003). A summary of the method is provided here. The method is an indentation method. A 4-centimeter length of optical fiber was placed on a 3 mm-thick glass slide. One end of the optical fiber was attached to a device that permitted rotation of the optical fiber in a controlled fashion. The optical fiber was examined in transmission under 100× magnification and rotated until the secondary coating thickness was equivalent on both sides of the glass fiber in a direction parallel to the glass slide. In this position, the thickness of the secondary coating was equal on both sides of the optical fiber in a direction parallel to the glass slide. The thickness of the secondary coating in the directions normal to the glass slide and above or below the glass fiber differed from the thickness of the secondary coating in the direction parallel to the glass slide. One of the thicknesses in the direction normal to the glass slide was greater and the other of the thicknesses in the direction normal to the glass slide was less than the thickness in the direction parallel to the glass slide. This position of the optical fiber was fixed by taping the optical fiber to the glass slide at both ends and is the position of the optical fiber used for the indentation test. 
     Indentation was carried out using a universal testing machine (Instron model 5500R or equivalent). An inverted microscope was placed beneath the crosshead of the testing machine. The objective of the microscope was positioned directly beneath a 75° diamond wedge indenter that was installed in the testing machine. The glass slide with taped fiber was placed on the microscope stage and positioned directly beneath the indenter such that the width of the indenter wedge was orthogonal to the direction of the optical fiber. With the optical fiber in place, the diamond wedge was lowered until it contacted the surface of the secondary coating. The diamond wedge was then driven into the secondary coating at a rate of 0.1 mm/min and the load on the secondary coating was measured. The load on the secondary coating increased as the diamond wedge was driven deeper into the secondary coating until puncture occurred, at which point a precipitous decrease in load was observed. The indentation load at which puncture was observed was recorded and is reported herein as grams of force. The experiment was repeated with the optical fiber in the same orientation to obtain ten measurement points, which were averaged to determine a puncture resistance for the orientation. A second set of ten measurement points was taken by rotating the orientation of the optical fiber by 180°. 
     Microbending. In the wire mesh covered drum test, the attenuation of light at wavelength of 1550 nm through a coated fiber having a length of 750 m was determined at room temperature. The microbend induced attenuation was determined by the difference between a zero-tension deployment and a high-tension deployment on the wire mesh drum. Separate measurements were made for two winding configurations. In the first configuration, the fiber was wound in a zero-tension configuration on an aluminum drum having a smooth surface and a diameter of approximately 400 mm. The zero-tension winding configuration provided a stress-free reference attenuation for light passing through the fiber. After sufficient dwell time, an initial attenuation measurement was performed. In the second winding configuration, the fiber sample was wound to an aluminun drum that was wrapped with fine wire mesh. For this deployment, the barrel surface of the aluminum drum was covered with wire mesh and the fiber was wrapped around the wire mesh. The mesh was wrapped tightly around the barrel without stretching and was kept intact without holes, dips, tearing, or damage. The wire mesh material used in the measurements was made from corrosion-resistant type 304 stainless steel woven wire cloth and had the following characteristics: mesh per linear inch: 165×165, wire diameter: 0.0019″, width opening: 0.0041″, and open area %: 44.0. A 750 m length of coated fiber was wound at 1 m/s on the wire mesh covered drum at 0.050 cm take-up pitch while applying 80 (+/−1) grams of tension. The ends of the fiber were taped to maintain tension and there were no fiber crossovers. The points of contact of the wound fiber with the mesh impart stress to the fiber and the attenuation of light through the wound fiber is a measure of stress-induced (microbending) losses of the fiber. The wire drum measurement was performed after a dwell time of 1-hour. The increase in fiber attenuation (in dB/km) in the measurement performed in the second configuration (wire mesh covered drum) relative to the first configuration (smooth drum) was determined for each wavelength. The average of three trials was determined at each wavelength and is reported as the wire mesh microbend loss. 
     Reference Primary Coating. In measurements of in situ glass transition temperature (T g ), puncture resistance, and wire mesh covered drum microbending attenuation, the measurement samples included a primary coating between the glass fiber and a secondary coating. The primary coating composition had the formulation given in Table 10 and is typical of commercially available primary coating compositions. 
                     TABLE 10                  Reference Primary Coating Composition                             Component   Amount                       Oligomeric Material   50.0 wt %           SR504   46.5 wt %           NVC    2.0 wt %           TPO    1.5 wt %           Irganox 1035    1.0 pph           3-Acryloxypropyl    0.8 pph           trimethoxysilane               Pentaerythritol   0.032 pph           tetrakis (3-mercapto               propionate)                        
where the oligomeric material was prepared as described above from H12MDI, HEA, and PPG4000 using a molar ratio n:m:p=3.5:3.0:2.0, SR504 is ethoxylated(4)nonylphenol acrylate (available from Sartomer), NVC is N-vinylcaprolactam (available from Aldrich), TPO (a photoinitiator) is (2,4,6-trimethylbenzoyl)-diphenyl phosphine oxide (available from BASF), Irganox 1035 (an antioxidant) is benzenepropanoic acid, 3,5-bis(1,1-dimethylethyl)-4-hydroxythiodi-2,1-ethanediyl ester (available from BASF), 3-acryloxypropyl trimethoxysilane is an adhesion promoter (available from Gelest), and pentaerythritol tetrakis(3-mercaptopropionate) (also known as tetrathiol, available from Aldrich) is a chain transfer agent. The concentration unit “pph” refers to an amount relative to a base composition that includes all monomers, oligomers, and photoinitiators. For example, a concentration of 1.0 pph for Irganox 1035 corresponds to 1 g Irganox 1035 per 100 g combined of oligomeric material, SR504, NVC, and TPO.
 
     Secondary Coatings—Properties—Tensile Properties. The results of tensile property measurements prepared from the curable secondary compositions are shown in Table 11. 
     
       
         
           
               
             
               
                 TABLE 11 
               
             
            
               
                   
               
               
                 Tensile Properties of Secondary Coatings 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Tensile 
                   
                 Yield 
                 Young&#39;s 
               
               
                 Compo- 
                 Strength 
                 Elongation 
                 Strength 
                 Modulus 
               
               
                 sition 
                 (MPa) 
                 at yield (%) 
                 (MPa) 
                 (MPa) 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 KA 
                 54.3 
                 39.0 
                   
                 1528 
               
               
                 KB 
                 63.1 
                 24.1 
                   
                 1703 
               
               
                 KC 
                 45.7 
                 28.4 
                   
                 1242 
               
               
                 KD 
                 61.8 
                 32.5 
                   
                 1837 
               
               
                 A 
                 86.09 
                 4.60 
                 48.21 
                 2049 
               
               
                 SB 
                 75.56 
                 4.53 
                 61.23 
                 2532 
               
               
                 SC 
                 82.02 
                 4.76 
                 66.37 
                 2653 
               
               
                 SD 
                 86.08 
                 4.87 
                 70.05 
                 2776 
               
               
                   
               
            
           
         
       
     
     The results show that secondary coatings prepared from compositions SB, SC, and SD exhibited higher Young&#39;s modulus, and higher yield strength than the secondary coating prepared from comparative composition A. The higher values represent improvements that make secondary coatings prepared for the curable coating compositions disclosed herein better suited for small diameter optical fibers. More specifically, the higher values enable use of thinner secondary coatings on optical fibers without sacrificing performance. Thinner secondary coatings reduce the overall diameter of the optical fiber and provide higher fiber counts in cables of a given cross-sectional area. 
     The Young&#39;s modulus of secondary coatings prepared as cured products from the curable secondary coating compositions disclosed herein is greater than 2400 MPa, or greater than 2500 MPa, or greater than 2600 MPa, or greater than 2700 MPa, or in the range from 2400 MPa to 3000 MPa, or in the range from 2600 MPa to 2800 MPa. 
     The yield strength of secondary coatings prepared as cured products from the curable secondary coating compositions disclosed herein is greater than 55 MPa, or greater than 60 MPa, or greater than 65 MPa, or greater than 70 MPa, or in the range from 55 MPa to 75 MPa, or in the range from 60 MPa to 70 MPa. 
     Secondary Coatings—Properties—Puncture Resistance. The puncture resistance of secondary coatings made from comparative curable secondary coating composition A, a commercial curable secondary coating composition (CPC6e) from a commercial vendor (DSM Desotech) having a proprietary composition, and curable secondary coating composition SD was determined according to the method described above. Several fiber samples with each of the three secondary coatings were prepared. Each fiber sample included a glass fiber with a diameter of 125 μm, a primary coating formed from the reference primary coating composition listed in Table 10, and one of the three secondary coatings. Samples with various secondary coatings were prepared. The thicknesses of the primary coating and secondary coating were adjusted to vary the cross-sectional area of the secondary coating as shown in  FIG.  9   . The ratio of the thickness of the secondary coating to the thickness of the primary coating was maintained at about 0.8 for all samples. 
     Fiber samples with a range of thicknesses were prepared for each of the secondary coatings to determine the dependence of puncture load on the thickness of the secondary coating. One strategy for achieving higher fiber count in cables is to reduce the thickness of the secondary coating. As the thickness of the secondary coating is decreased, however, its performance diminishes and its protective function is compromised. Puncture resistance is a measure of the protective function of a secondary coating. A secondary coating with a high puncture resistance withstands greater impact without failing and provides better protection for the glass fiber. 
     The puncture load as a function of cross-sectional area for the three coatings is shown in  FIG.  9   . Cross-sectional area is selected as a parameter for reporting puncture load because an approximately linear correlation of puncture load with cross-sectional area of the secondary coating was observed. Traces 92, 94, and 96 shows the approximate linear dependence of puncture load on cross-sectional area for the comparative secondary coatings obtained by curing the comparative CPC6e secondary coating composition, the comparative curable secondary coating composition A, and curable secondary coating composition SD; respectively. The vertical dashed lines are provided as guides to the eye at cross-sectional areas of 10000 μm 2 , 15000 μm 2 , and 20000 μm 2  as indicated. 
     The CPC6e secondary coating depicted in Trace 92 corresponds to a conventional secondary coating known in the art. The comparative secondary coating A depicted in Trace 94 shows an improvement in puncture load for high cross-sectional areas. The improvement, however, diminishes as the cross-sectional area decreases. This indicates that a secondary coating obtained as a cured product from comparative curable secondary coating composition A is unlikely to be suitable for low diameter, high fiber count applications. Trace 96, in contrast, shows a significant increase in puncture load for the secondary coating obtained as a cured product from curable secondary coating composition SD. At a cross-sectional area of 7000 μm 2 , for example, the puncture load of the secondary coating obtained from curable secondary coating composition SD is 50% or more greater than the puncture load of either of the other two secondary coatings. 
     The puncture load of secondary coatings formed as cured products of the curable secondary coating compositions disclosed herein at a cross-sectional area of 10000 μm 2  is greater than 36 g, or greater than 40 g, or greater than 44 g, or greater than 48 g, or in the range from 36 g to 52 g, or in the range from 40 g to 48 g. The puncture load of secondary coatings formed as cured products of the curable secondary coating compositions disclosed herein at a cross-sectional area of 15000 μm 2  is greater than 56 g, or greater than 60 g, or greater than 64 g, or greater than 68 g, or in the range from 56 g to 72 g, or in the range from 60 g to 68 g. The puncture load of secondary coatings formed as cured products of the curable secondary coating compositions disclosed herein at a cross-sectional area of 20000 μm 2  is greater than 68 g, or greater than 72 g, or greater than 76 g, or greater than 80 g, or in the range from 68 g to 92 g, or in the range from 72 g to 88 g. Embodiments include secondary coatings having any combination of the foregoing puncture loads. 
     As used herein, normalized puncture load refers to the ratio of puncture load to cross-sectional area. The puncture load of secondary coatings formed as cured products of the curable secondary coating compositions disclosed herein have a normalized puncture load greater than 3.2×10 −3  g/μm 2 , or greater than 3.6×10 −3  g/μm 2 , or greater than 4.0×10 −3  g/μm 2 , or greater than 4.4×10 −3  g/μm 2 , or greater than 4.8×10 −3  g/μm 2 , or in the range from 3.2×10 −3  g/μm 2  to 5.6×10 −3  g/μm 2 , or in the range from 3.6×10 −3  g/μm 2  to 5.2×10 −3  g/μm 2 , or in the range from 4.0×10 −3  g/μm 2  to 4.8×10 −3  g/μm 2 . 
     Design Examples—Coated Optical Fibers 
     Modeled Results. The experimental examples and principles disclosed herein indicate that by varying the mole numbers n, m, and p, it is possible to control the relative amount of di-adduct compound in the oligomer as well as the properties of cured films formed from the primary coating compositions over a wide range, including the ranges specified herein for Young&#39;s modulus and in situ modulus. Similarly, variations in the type and concentration of different monomers in the secondary composition leads to variations in the Young&#39;s modulus over the range disclosed herein. Curing dose is another parameter that can be used to vary modulus of primary and secondary coatings formed from the curable compositions disclosed herein. 
     To examine the effect of the thickness and modulus of the primary and secondary coatings on transmission of a radial force to a glass fiber, a series of modeled examples was considered. In the model, a radial external load P was applied to the surface of the secondary coating of an optical fiber and the resulting load at the surface of the glass fiber was calculated. The glass fiber was modeled with a Young&#39;s modulus of 73.1 GPa (consistent with silica glass) and a diameter of 125 μm. The Poisson ratios ν p  and ν s  of the primary and secondary coatings were fixed at 0.48 and 0.33, respectively. A comparative sample Cl and six samples M1-M6 in accordance with the present disclosure were considered. The comparative sample included primary and secondary coatings with thicknesses and moduli consistent with optical fibers known in the art. Samples M1-M6 are examples with reduced thicknesses of the primary and secondary coatings. Parameters describing the configurations of the primary and secondary coatings are summarized in Table 12. 
     
       
         
           
               
             
               
                 TABLE 12 
               
             
            
               
                   
               
               
                 Coating Properties of Modeled Optical Fibers 
               
            
           
           
               
               
               
            
               
                   
                 Primary Coating 
                 Secondary Coating 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                 In Situ 
                   
                 Thick- 
                 Young&#39;s  
                   
                 Thick- 
               
               
                   
                 Modulus 
                 Diameter 
                 ness 
                 Modulus 
                 Diameter 
                 ness 
               
               
                 Sample 
                 (MPa) 
                 (μm)  
                 (μm)  
                 (MPa) 
                 (μm)  
                 (μm) 
               
               
                   
               
               
                 C1 
                 0.20 
                 190 
                 32.5 
                 1600 
                 242 
                 26.0 
               
               
                 M1 
                 0.14 
                 167 
                 21.0 
                 1900 
                 200 
                 16.5 
               
               
                 M2 
                 0.12 
                 161 
                 18.0 
                 1900 
                 190 
                 14.5 
               
               
                 M3 
                 0.10 
                 155 
                 15.0 
                 2000 
                 180 
                 12.5 
               
               
                 M4 
                 0.09 
                 150 
                 12.5 
                 2300 
                 170 
                 10.0 
               
               
                 M5 
                 0.12 
                 145 
                 15.0 
                 2200 
                 170 
                 12.5 
               
               
                 M6 
                 0.11 
                 138 
                 14.0 
                 2200 
                 160 
                 11.0 
               
               
                   
               
            
           
         
       
     
     Table 13 summarizes the load P1 at the outer surface of the glass fiber as a fraction of load P applied to the surface of the secondary coating. The ratio P1/P is referred to herein as the load transfer parameter and corresponds to the fraction of external load P transmitted through the primary and secondary coatings to the surface of the glass fiber. The load P is a radial load and the load transfer parameter P1/P was calculated from a model based on the equations below: 
     
       
         
           
             
               
                 P 
                 1 
               
               P 
             
             = 
             
               
                 4 
                 ⁢ 
                 
                   ( 
                   
                     1 
                     - 
                     
                       v 
                       p 
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   ( 
                   
                     1 
                     - 
                     
                       v 
                       s 
                     
                   
                   ) 
                 
               
               
                 { 
                 
                   A 
                   + 
                   B 
                 
                 } 
               
             
           
         
       
       
         
           where 
         
       
       
         
           
             A 
             = 
             
               ( 
               
                 
                   
                     
                       E 
                       s 
                     
                     ⁡ 
                     
                       ( 
                       
                         1 
                         + 
                         
                           v 
                           p 
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       1 
                       - 
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           v 
                           p 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     ( 
                     
                       1 
                       - 
                       
                         
                           ( 
                           
                             
                               r 
                               4 
                             
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             
                               r 
                               5 
                             
                           
                           ) 
                         
                         2 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     ( 
                     
                       1 
                       - 
                       
                         
                           ( 
                           
                             
                               r 
                               5 
                             
                             ⁢ 
                             
                               / 
                             
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     In the equations, ν p  and ν s  are the Poisson&#39;s ratios of the primary and secondary coatings, r 4  is the outer radius of the glass fiber, r 5  is the outer radius of the primary coating, r 6  is the outer radius of the secondary coating, E p  is the in situ modulus of the primary coating, and E s  is the Young&#39;s modulus of the secondary coating. The scaled load transfer parameter P1/P (scaled) in Table 13 corresponds to the ratio P1/P for each sample relative to comparative sample Cl. 
     
       
         
           
               
             
               
                 TABLE 13 
               
             
            
               
                   
               
               
                 Load Transfer Parameter (P1/P) 
               
               
                 at Surface of Glass Fiber 
               
            
           
           
               
               
               
               
            
               
                   
                 Sample 
                 P1/P 
                 P1/P (scaled) 
               
               
                   
                   
               
               
                   
                 C1 
                 0.0178 
                 1.00 
               
               
                   
                 M1 
                 0.0171 
                 0.97 
               
               
                   
                 M2 
                 0.0175 
                 0.98 
               
               
                   
                 M3 
                 0.0172 
                 0.97 
               
               
                   
                 M4 
                 0.0170 
                 0.95 
               
               
                   
                 M5 
                 0.0167 
                 0.94 
               
               
                   
                 M6 
                 0.0166 
                 0.94 
               
               
                   
                   
               
            
           
         
       
     
     The modeled examples show that despite smaller coating thicknesses, optical fibers having primary and secondary coatings as described herein exhibit a reduction in the force experienced by a glass fiber relative to a comparative optical fiber having conventional primary and secondary coatings with conventional thicknesses. The resulting reduction in overall size of the optical fibers described herein enables higher fiber count in cables of a given size (or smaller cable diameters for a given fiber count) without increasing the risk of damage to the glass fiber caused by external forces. 
     The scaled load transfer parameter P 1 /P (scaled) of the secondary coating is less than 0.99, or less than 0.97, or less than 0.95. The load transfer parameter P 1 /P of the secondary coating is less than 0.0200, or less than 0.0180, or less than 0.0178, or less than 0.0176, or less than 0.0174, or less than 0.0172, or less than 0.0170, or less than 0.0168, or in the range from 0.0160-0.0180, or in the range from 0.0162-0.0179, or in the range from 0.0164-0.0178, or in the range from 0.0166-0.0177, or in the range from 0.0168-0.0176. 
     Fabricating the Optical Fibers 
     The optical fibers described here can be made using the standards methods of making optical fiber. A core cane comprising the core and the trench region can be made using processes such as OVD, VAD, MCVD and PCVD. In some embodiments, the first outer cladding layer and the second outer cladding layer (e.g., titania-doped layer) can be put on the core cane in a single step to make the optical fiber preform. The first outer cladding layer and the second outer cladding layer (e.g., titania-doped layer) can be deposited on the core cane in laydown as soot in a single step, with the silica precursor introduced to the laydown burner during the deposition of layers corresponding to the first outer cladding layer, and silica and titania precursors introduced to the laydown burner during the deposition of layers corresponding to the second outer cladding layer (e.g., titania-doped layer). Silica precursors include SiCl 4  and OMCTS (octamethylcyclotetrasiloxane). Titania precursors include TiCl 4  and titanium alkoxides (e.g. Ti(isopropoxide) 4 ). The produced cane-soot layer can then be moved to a onsolidation furnace where it can be first dehydrated using a dehydration agent and then the temperature of the assembly can be increased to between 1350° C. to 1500° C. to sinter the soot layer to a void free densified preforms. In some embodiments, oxygen can be introduced during the sintering of the preform process to have the titania oxidation state of 4+ that results in the titania-doped layer to have higher transparency. In some embodiments, the dehydration agent can be chosen from chlorine, thionyl chlorine, silicon tetrachloride or combinations thereof. In some embodiments, the titania precursor can be titanium tetrachloride, titanium isopropoxide, or combination thereof. The formed sintered optical fiber can then be drawn into an optical fiber. 
       FIG.  10    is a schematic diagram of an example optical fiber drawing system (“drawing system”)  200  for drawing a glass preform  100 P into the optical fiber  100 , according to some embodiments. The single mode optical fiber  100  can be fabricated using the drawing system  200  and fiber drawing techniques known in the art. 
     The core and cladding layers of the glass preform can be produced in a single-step process or multi-step process using chemical vapor deposition (CVD) methods which are well known in the art. A variety of CVD processes are known and are suitable for producing the core and cladding layers used in the optical fibers of the present invention. They include outside vapor deposition process (OVD) process, vapor axial deposition (VAD) process, modified CVD (MCVD), and plasma-enhanced CVD (PECVD). 
     As shown in  FIG.  10   , the exemplary drawing system  200  can include a draw furnace (“furnace”)  102  for heating the glass preform  100 P to the glass melt temperature. In an example, the fiber draw process is carried out a glass melt temperature, which in an example is in the range from 1800° C. to 1900° C. A preform holder  116  is used to hold the glass preform  100 P. 
     In some embodiments, the drawing system  200  also includes non-contact measurement sensors  104 A and  104 B for measuring the size of a drawn (bare) optical fiber  100 G that exits the draw furnace  102  for size (diameter) control. A cooling station  106  can reside downstream of the measurement sensors  104 A and  104 B and is configured to cool the bare optical fiber  100 G. A coating station  107  can reside downstream of the cooling station  106  and can be configured to deposit one or more protective coating materials  71  onto the bare optical fiber  100 G to form the protective coating  70  including the primary coating  72 , the secondary coating  74 , and optionally the tertiary coating  76 . A tensioner  220  can reside downstream of the coating station  107 . The tensioner  220  can have a surface  222  that pulls (draws) the coated optical fiber  100 . A set of guide wheels  230  with respective surfaces  232  resides downstream of the tensioner  220 . The guide wheels  230  can serve to guide the coated optical fiber  100  to a fiber take-up spool (“spool”)  250  for storage. 
     In some embodiments, the close-up inset I 1  of  FIG.  10    shows a cross-sectional view of the glass preform  100 P used to fabricate the single mode optical fiber  100 . The glass preform  100 P includes a preform core  10 P, and a preform cladding  50 P comprising a preform inner cladding region, a preform depressed-index cladding, a preform first outer cladding, and a preform second outer cladding (not shown). In some embodiments, the preform core  10 P can be a graded refractive index core. The preform  100 P can be fabricated using known techniques, such as an outside vapor deposition (OVD) process. The cross-sectional view of the coated single mode optical fiber  100  can be referred to the descriptions above in connection with  FIGS.  1  and  2   . 
     The disclosed single mode optical fibers are capable of being routed through extremely tight bend configurations or bending in a small radius arc inside a fiber array unit that couples an array of fibers to arrays of lasers and photodiodes. The disclosed single mode optical fibers comprise an outer cladding layer doped with titanium to increase the value of the fatigue constant, n d . The disclosed single mode optical fibers fibers also have low bend loss, which is achieved through the addition of a trench in the inner cladding. The dimensions of the core and the trench of the fibers can be engineered to ensure that the fibers are single moded at wavelengths greater than or equal to 1260 nm for deployment lengths of 2 meters or less, with fiber cutoff wavelengths which are as low as 1260 nm for silicon photonics transceivers or in the 1000-1300 nm range for long wavelength VCSELs. Advantages of the disclosed single mode optical fibers include: low bend loss, e.g., less than 1 dB/turn at a wavelength of 1550 nm when wrapped around a mandrel having a diameter of 10 mm; low short-length cutoff wavelength, e.g., a fiber cutoff wavelength less than 1310 nm; G.652-compatible MFD, e.g., MFD≥8.2 μm at a wavelength of 1310 nm; increased reliability for extremely tight bends achieved with a reduced cladding diameter that includes an outer layer comprising titania-doped silica; and improved resistance to mechanical abrasion, which facilitates connectorization. 
     While various embodiments have been described herein, they have been presented by way of example only, and not limitation. It should be apparent that adaptations and modifications are intended to be within the meaning and range of equivalents of the disclosed embodiments, based on the teaching and guidance presented herein. It therefore will be apparent to one skilled in the art that various changes in form and detail can be made to the embodiments disclosed herein without departing from the spirit and scope of the present disclosure. The elements of the embodiments presented herein are not necessarily mutually exclusive, but may be interchanged to meet various needs as would be appreciated by one of skill in the art. 
     It is to be understood that the phraseology or terminology used herein is for the purpose of description and not of limitation. The breadth and scope of the present disclosure should not be limited by any of the above-described exemplary embodiments, but should be defined only in accordance with the following claims and their equivalents.