WO1996024180A1

WO1996024180A1 - Apparatus for providing a phase shift in a pulse and method of use thereof

Info

Publication number: WO1996024180A1
Application number: PCT/US1996/001018
Authority: WO
Inventors: Jeffrey A. Squier; Steven J. Kane
Original assignee: The Regents Of The University Of Michigan
Priority date: 1995-02-01
Filing date: 1996-01-23
Publication date: 1996-08-08
Also published as: AU4965996A

Abstract

The invention provides a method an apparatus for imparting phase shift in one or more pulses. The apparatus comprises at least one optical diffraction grating element; where each one of the grating elements comprises a grating having a constant line spacing (d) and a dielectric body in light coupling arrangement with the grating. The dielectric body has a side through which the pulse traverses. The side is at an angle α (alpha) with respect to the grating. Alpha α defines a path of the pulse in the dielectric body corresponding to an incident angle Υi (theta incident) between the path and a plane normal to the grating. The incident angle υi and grating line spacing (d) cooperate to provide negative second-order dispersion (SOD) and negative third-order dispersion (TOD) so as to compensate for any positive SOD and any positive TOD initially present in the pulse.

Description

APPARATUS FOR PROVIDING A PHASE SHIFT

IN A PULSE AND METHOD OF USE THEREOF

Government Rights

This invention was made with government support provided by the National Science Foundation under the terms of No. STC PHY 8920108. The government has certain rights in the invention.

Field of the Invention This invention relates generally to an apparatus and a method for providing a phase shift in an optical pulse. This invention may be used in a variety of applications pertaining to optical pulses and more specifically to high speed optical devices and techniques, including lasers.

Background of the Invention

All optical materials, including crystals, glasses, liquids, and other dielectrics, possess the property of dispersion. Most optical material used in optical applications possess dispersion in three regimes: (1) the normal regime where such optical materials provide positive second-order dispersion (SOD) and positive third-order dispersion (TOD); (2) the zero dispersion regime providing zero SOD and positive TOD; and (3) the anomalous regime where the materials provide negative SOD and positive TOD. Dispersion can be most simply described as a frequency dependent transit time through the medium. All optical pulses are composed of multiple frequencies and, therefore, in the presence of dispersion, the different frequency components of the pulse will arrive at their destination at different times. This frequency dependent transit time will degrade the shape of an output pulse, providing second-order dispersion (SOD), wherein the transit time varies linearly with frequency, or (SOD), will cause a broadening of the pulse; and third-order dispersion (TOD) where the transit time varies with the square of the frequency, or (TOD), will result in asymmetric distortion of the pulse shape. In order to maintain the fidelity of a pulse after transit through an optical material, the SOD and TOD must be compensated.

A prism pair is known to compensate for small amounts of SOD and TOD. Prism pairs are used in three configurations. They are: (1) standard configuration where the prism pair provides negative SOD and negative TOD; (2) a first non-standard configuration where the prism pair provides zero SOD and negative TOD; and (3) a second non-standard configuration where the prism pair provides positive SOD, negative TOD. It is clear that prisms can be used to provide compensation of material dispersion in all three regimes, namely, normal, zero dispersion, and the anomalous regime. Optical materials used in many laser applications most typically and almost always provide positive SOD and positive TOD during generation, stretching, and amplification of the pulse. It is clear that prisms disposed in a standard configuration and providing negative SOD and negative TOD can compensate for such distortion. Accordingly, prisms can be used to provide compensation of material dispersion in all three regimes and importantly in the normal regime. However, the prisms can only be used to compensate for very small amounts of dispersion, as the physical dimensions of the prism sequence are much larger than the lengths of the material they are compensating. For example, a one centimeter piece of sapphire used in a laser system, which imparts SOD and TOD to a pulse, can be compensated by a pair prisms which are about fifty centimeters apart (see Figure 1). The prism separation scales linearly with the length of the dispersive material; therefore, to compensate the SOD and the TOD of a pulse which has travelled through a short optical fiber, for example, one hundred meters, the prism space needed to compensate SOD and TOD would be more than two miles. Clearly, prisms are not practical to compensate for large amounts of SOD and TOD, and even for laser sources which only have one centimeter of material, a fifty centimeter prism sequence is not a very compact way to compensate for dispersion.

As an alternative to prisms, there has been suggested compensating devices referred to as grating pairs which can provide very large amounts of dispersion and compensation. Unfortunately, such grating pairs provide compensation in three modes, none of which can compensate for both SOD and TOD in the normal regime, which is of greatest interest, and in the zero dispersion regime. For example, Treacy in the IEEE Journal of Quantum Electronics, Volume QE-5, No. 9, September 1969 describes a grating pair which provides negative SOD and positive TOD. Therefore, Treacy is not able to provide negative SOD and negative TOD compensation. In a paper printed in Electronics Letters, 5th, Volume 29, No. 16, August 1993 Tournois describes a grating pair which provides negative SOD but zero TOD. Tournois cannot provide compensation for normal regime positive SOD and positive TOD of optical materials. Likewise, Martinez in a paper published in IEEE Journal of Quantum Electronics, Volume QE-23, No. 1, January 1987 describes a grating pair which provides positive SOD and negative TOD. Again, Martinez cannot provide negative SOD and negative TOD to compensate for the positive SOD and positive TOD generated by most, if not all, optical materials used in generating optical pulses.

Accordingly, what is needed is a new compressor system for compressing optical pulses which provide large amounts of dispersion with the following signs: negative SOD, negative TOD which can be used to compensate for very large amounts of material dispersion in a normal regime, and which is adaptable for use in femtosecond optics.

The technology of femtosecond lasers has grown rapidly, however, the systems are becoming more and more complicated and require a huge amount of space making them somewhat impractical and inaccessible to many users. Part of the complexity arises from the pulse stretcher portion of the system where, in order to obtain high energy pulses, the pulse is stretched in time by a factor of two to three thousand. This is presently accomplished in a complicated arrangement of components which occupies five to eight square feet and the components are very difficult to align. Therefore, what is needed is a replacement for a conventional stretcher. The replacement should be compact, and not require delicate alignment and significant adjustment and realignment.

It is a general object to provide grating elements having novel dispersion properties, and more specifically reflection grating elements and transmission grating elements with third-order dispersion opposite to that of conventional grating elements and pairs. Another object is to provide an apparatus and method for stretching and compressing pulses utilizing the novel grating elements of the invention. Still another object is to provide a single grating element adaptable for use as both the stretching means and compressing means in combination with suitably arranged reflectors.

Another object is to provide an apparatus for the production of one or more pulses which comprises one or more of the grating elements of the invention.

Summary of the Invention

In one aspect, the invention provides an apparatus for imparting a phase shift, typically a phase delay, in one or more pulses. The apparatus comprises a pair of optical diffraction grating elements. Each one of the grating elements comprises a grating in light coupling arrangement with a dielectric body to accept a pulse which is transmitted therethrough. The grating has a constant line spacing (d). The dielectric body has a side through which the pulse traverses and this side through which the pulse traverses is at an angle α (alpha) with respect to the grating. The angle α (alpha) defines or identifies the path of the pulse transmitted through the dielectric body corresponding to an incident angle θ (theta) between the path of the pulse and a plane normal to the grating. Grating elements so constructed and arranged provide negative second-order dispersion (SOD) and negative third- order dispersion (TOD) so as to compensate for any positive SOD and any positive TOD initially present in the pulse. The relationship between the angle α (alpha) and the incident angle θ_i (theta), is as follows: θ_i (theta incident) is related to α (alpha) according to the expression: θ_i - θ_c ≤ α < θ_i + θ_c, where θ_c = arcsin 1/n and n is index of refraction of the dielectric.

In one embodiment, the grating is in the form of a pattern on the surface of the dielectric body.

In another embodiment, the grating is a patterned surface on a substrate which is affixed to the dielectric body in light coupling arrangement or light transmission arrangement. The attachment is by a dielectric glue, cement, or the like.

In another embodiment, the grating is written, etched, or engraved on a surface of the dielectric body. The grating may be in the form of a pattern formed by holographic means.

In another embodiment, the apparatus for imparting a phase shift, as described immediately above, is used as a compressor as a part of a larger system for producing laser pulses. Such a system comprises means to generate the optical pulse, means to stretch the pulse, and means to amplify the pulse. The compressor comprises the grating elements, as described immediately above, for providing negative SOD and negative TOD so as to compensate for any positive SOD and any positive TOD induced by the generating, stretching, and amplifying of the pulse preceding its entry into the compressor. Accordingly, the apparatus for imparting a phase shift may be used in combination with an optical fiber stretcher, direct injection, or intralaser pulse formation. The apparatus for imparting a phase shift may be used as a compressor where the means to generate, stretch, and amplify a pulse comprises an oscillator, fiber optic stretcher, and regenerative amplifier. In still another embodiment, an apparatus is provided for producing one or more pulses which comprises means to stretch the pulse in time comprising a pair of optical diffraction grating elements where each one of the grating elements has a grating in light coupling arrangement with a dielectric body to accept the pulse. The grating elements are spaced apart with optical imaging system disposed between the grating elements that optically displaces the apparent placement of the grating elements with respect to one another, thereby providing positive SOD and positive TOD in the pulse. This is advantageously used in combination with a compressor which comprises a pair of optical diffraction grating elements where each one of the grating elements comprises a grating in light coupling arrangement with the dielectric body to accept the stretched pulse and to provide negative SOD and negative TOD so as to compensate for the positive SOD and positive TOD induced by the stretching. In this configuration, when the imaging system of the stretcher is negative magnification, the gratings of the stretcher are arranged anti-parallel. Alternatively, the imaging system is positive magnification and the gratings of the stretcher are arranged in parallel configuration with respect to one another.

The relationship between α (alpha) and θ_i

(theta), as described above, is followed in both the stretcher and compressor so that the angle α (alpha) defines or identifies the path of the pulse in the dielectric body to provide or correspond to an incident angle θ (theta) between the path of the pulse and a plane normal to the grating. Here θ_i (theta) is related to α (alpha) as described above. In another embodiment, an apparatus for stretching and compressing one or more pulses comprises an optical diffraction grating element comprising a grating in light coupling arrangement with the dielectric body; and a mirror spaced from the grating element. A lens is disposed between the mirror and the grating element. A corner or dihedral reflector is also spaced from the grating element. The grating element, lens, and mirror are all arranged to define a first beam path which traverses through the grating element, then through the lens, and then through the mirror. The grating element and dihedral reflector are arranged with respect to one another so as to define a second beam path which traverses through the grating element and to the reflector. The dielectric body has a side through which the pulse traverses and this side is at an angle α (alpha) with respect to the grating where α (alpha) defines the path of the pulse in the dielectric body to provide an incident angle θ (theta) between the path and a plane normal to the grating to provide positive SOD and positive TOD in the stretcher and negative SOD and negative TOD in the compressor. In this embodiment, the grating element, lens, and mirror which define the first beam path provide positive SOD and positive TOD as they function as the stretcher. The grating element and corner reflector, in this configuration, provide negative SOD and negative TOD to compensate for the positive SOD and positive TOD induced in the aforesaid stretcher.

The relationship between α (alpha) and θ_i (theta incident) is as described above.

The apparatus for imparting a phase shift which comprises one or more of the above described grating elements having an angle α (alpha) which identifies the path of the pulse in the dielectric body corresponding to an incident angle θ (theta) between the beam path and a plane normal to the grating, is particularly useful when incorporated into a chirped-pulse amplification laser system. It very effectively provides negative SOD and positive TOD so as to compensate for any positive SOD and any positive TOD present in the pulse as a result of other components of the CPA system.

In the method of the invention, it is necessary to characterize the condition of the pulse entering the apparatus of the invention for imparting a phase shift. That characterization is represented by ξ (xi) which is defined as the third-order dispersion (TOD) divided by the absolute value of the second-order dispersion (SOD). When ξ (xi) provided by the phase shift apparatus of the invention is equal in magnitude and opposite in sign to the ξ (xi) characterizing the condition of the pulse entering such apparatus then cubic phase compensation is possible. In a typical laser system, a value ξ (xi) is characterized for a stretcher and compressor with the condition that ξ (xi) of the stretcher and amplification system is equal to and opposite in sign to ξ (xi) of the compressor. To fabricate a set of grating elements or a grating pair to function as a compressor with equal and opposite ξ (xi) there are several parameters which may be varied, including, the grating group spacing d, the index of refraction of the dielectric n, and the angle of incidents θ_i (theta). The angle of incidents θ_i (theta) is in turn dependent on the angle α (alpha) between the grating or the surface which carries the grating pattern, and the side of the dielectric body through which the light is transmitted. The potential for commercial application of these optical diffraction grating elements is considerable, including the elimination of conventional stretchers and compressors in laser systems which would result in greater ease of use and reduce the size of these systems. The optical diffraction grating elements when used as a compressor and/or stretcher in laser systems provide the advantage of ease of set-up and alignability. The optical diffraction grating elements of the invention provide essentially perfect compression of a pulse close to, and within a few percent of, the transform limit.

These and other objects, features, and advantages of the invention will become apparent from the following description of the preferred embodiments, claims, and accompanying drawings.

Brief Description of the Drawings

Figure 1 is a schematic of a standard titanium sapphire mode-lock laser system in which compensating prisms occupy a significant portion of the total area occupied by the system.

Figure 2 is an illustration of an optical diffraction grating element, dispersive device, having a transmission grating.

Figure 3 is an illustration of an optical diffraction grating element, dispersive device, having a reflection grating.

Figure 4 illustrates a pair of optical diffraction grating elements where each of the elements is a transmission grating element as described with respect to Figure 2 and having all angles determined by the orientation of the center wavelength with respect to the grating normal. Figure 5 illustrates a pair of optical diffraction grating elements where each of the elements is a reflection grating element as described with respect to Figure 3 and having an incident angle measured with respect to the grating normal and a diffraction angle subtended by the central wavelength and the normal of the dielectric's face from which the beam is emitted.

Figures 6 and 7 are schematics of chirped pulse amplification (CPA) laser systems for providing a laser beam where the stretcher of the system is a fiber stretcher and the compression means comprises a pair of transmission gratings (Figure 6) or a pair of reflection gratings (Figure 7), respectively.

Figures 8 and 9 are schematics of direct injection type CPA laser systems with compression means comprising transmission grating elements (Figure 8) and reflection grating elements (Figure 9), respectively.

Figure 10 is a schematic of CPA laser system having stretcher means comprising a pair of transmission grating elements with optical imaging device and compression means comprising a pair of transmission grating elements.

Figure 11 is similar to Figure 10 except that the stretching means comprises reflection grating elements and the compression means comprises reflection grating elements.

Figure 12 illustrates a pair of grating elements disposed in a parallel configuration.

Figure 13 illustrates a pair of grating elements disposed in an anti-parallel configuration. Figures 12 (A) and 13 (A), respectively, show negative magnification and positive magnification optical imaging systems for use with grating elements of respective Figures 12 and 13. In Figure 12 (A) the angle of a ray is not preserved in a negative magnification system; a ray exiting this system is anti-parallel to the ray which entered the system. In Figure 13 (A) the angle of a ray is preserved in a positive magnification system; a ray exiting this system is parallel to the ray which entered the system.

Figure 14 is a 3-dimensional illustration of a unitary grating stretcher and compressor. A portion of the grating element illustrated as the top half is the stretching means and the bottom half is the compressor means. The single grating element is used to provide the same incident angle for both the stretching and compressing.

Figure 15 is a top view according to Figure 14.

Figure 16 shows the results of modeling a 100 femtosecond pulse which was stretched to 250 picoseconds in a fiber and compressed with three different devices: devices (a) and (b) are conventional compressors and device

(c) is according to the invention.

Figure 17 shows the auto-correlation of a 60 femtosecond pulse stretched to 60 picoseconds in a fiber and compressed according to the apparatus and method of the invention.

Figure 18 is a schematic of a modified titanium sapphire mode-lock laser modified as compared to Figure 1. The compression means comprises grating elements as shown in Figure 18 which are a fraction of the size of a conventional prism sequence previously shown in Figure 1.

Figure 19 shows the monolithic device comprised of a pair of transmission grating elements (10, 12) and a dielectric body (200). Figure 20 shows the monolithic device comprised of a pair of reflective grating elements (10, 14) and a dielectric (200). Detailed Description of the Preferred Embodiments

Figures 2 and 3 each show an apparatus for imparting phase shift in one or more pulses of a pulse beam (5). The apparatus comprises at least one optical diffraction grating element (10). Each one of the grating elements comprises a grating (15) having an essentially constant line spacing (d) and a dielectric body (20) in light coupling arrangement with the grating (15). The dielectric body has a side (25) through which the pulse traverses. The side (25) is at an angle α (alpha) with respect to the grating (15). The angle α (alpha) defines a path (30) of the pulse in the dielectric body (20) corresponding to an incident angle θ_i (theta incident) between the path (30) and a plane (35) normal to the grating (15). The incident angle θ_i (theta incident) and the grating line spacing (d) cooperate to provide negative second-order dispersion and negative third-order dispersion so as to compensate for any positive second-order dispersion and any positive third-order dispersion initially present in the pulsed beam (5).

More specifically, the grating element (10) is characterized by an angle of diffraction (θ_d) between a path (40) of the pulse emitted from a side (45) of the grating element (10) and a plane (50) normal to the side (45) of the grating element (10) from which the pulse is emitted. The angle of diffraction (θ_d) is related to θ_i (incident angle) by the expression: Sin θ_d = m λ/d - n Sin θ_i, where m is the diffraction order, λ is wavelength, d is the grating line (groove) spacing expressed in units of length, 1/d is the grating line density expressed as number of lines per unit of length, and n is the index of refraction of the dielectric. Figure 2 shows a grating element (10) which is a transmission grating element (12) having a grating (15) which is transmission grating (16). Figure 3 shows a grating element (10) which is a reflection grating element (14) having a grating (15) which is a reflection grating (18).

The pair of transmission grating elements (12) shown in Figure 4, is each characterized by an angle of diffraction θ_d between a path of the pulse emitted from the grating element (12) and the plane (35) normal to the grating (15, 16) from which the pulse is emitted. θ_d is related to θ₄ as described in connection with Figure 2. The plane (35) normal to the transmission grating (15, 16) defines one of the legs by which θ_i is subtended, plane (35) is the same plane as plane (50) that defines one of the legs by which θ_d is subtended; that is plane (35) and plane (50) are the same. Accordingly, in the case of a pair of transmission grating elements (12), all angles are determined by the orientation of the center wavelength (7) of the beam (5) with respect to the grating normal (35, 50) or the plane normal (35, 50) to the grating (15, 16).

The pair of reflection grating elements (14) as shown in Figure 5 are each characterized by an angle of diffraction θ_d between a path of the pulse emitted from a side (45) of the dielectric (20) opposite the grating (15, 18) and a plane (50) normal to the side (45) from which the pulse is emitted as shown in Figures 3 and 5. θ_d is related to θ_d as identified in connection with Figure 2 and generally applicable to each of the optical diffraction grating elements (10) of the invention. In the case of the reflection grating element (14), plane (35) is normal to the grating (15, 18) and plane (50) is normal to a side (45) of the dielectric (20) opposite the grating (15, 18). Accordingly, the incident angle θ_i is defined with respect to the grating normal (35), and the diffracted angle θ_d is the angle subtended by a central wavelength (7) of the beam (5) and a normal (50) of the dielectric's face (45) from which the beam (7) is emitted.

It is preferred that the dielectric (20) be in the shape of a prism or a parallelopiped, but this is not necessary. The only requirement is that any side of the dielectric body (20) through which the beam traverses is essentially flat. That is, any surface forming a face of the dielectric through which the beam is transmitted should be flat so that in the case of the transmission mode two faces are required to be flat as shown in Figures 2 and 4 because light is transmitted through sides (25) and (45). In the reflection mode, it is preferred that each of the grating elements (14) have three flat surfaces since there are three different surfaces from which light may be reflected or transmitted. That is, one surface having the grating from which light is reflected, and two other surfaces through which light may be transmitted. It is preferred that each of the optical diffraction grating elements (10) be constructed and arranged so that θ_i (theta incident) is related to α (alpha) according to the expression: θ_i - θ_c ≤ α < θ_i + θ_c, where θ_c equals arcsin 1/n and n is the index of refraction of the dielectric (20). It is most preferred that α (alpha) is about equal to θ_i (theta incident). The gratings (15) may be provided in a variety of forms so long as the grating line spacing (d) and line density (1/d) are as defined above. The gratings may be in the form of a pattern on the surface of the dielectric body (20), or engraved, written, or inscribed on a surface of the dielectric body (20). In one embodiment, the gratings are in the form of grooves. In another embodiment, the grating (15) is formed as a holographic pattern on the surface of the dielectric body (20). The grating (15) may be provided on a substrate which is then placed in light coupling arrangement with the dielectric body (20). There are several dielectric adhesives or cement which permit the grating element (10) to be comprised of two separate pieces, the dielectric body (20) and the grating (15) carried on the substrate. In its most preferred form, however, the grating (15) is in the form of a pattern in or on the surface of the dielectric body (20) so that the grating element (10) is essentially a single unitary body.

In the preferred embodiments, as shown in Figures 4 and 5, it is preferred that each of the elements in the pair be either reflection gratings or transmission gratings; and that an optional retro-reflector (52) be used. Depending on the application, it is also possible to use a grating element (10) of the invention whether reflection (12) or transmission (14) in combination with another type of grating element. Accordingly, each of the embodiments of the grating element (10) as generally described in Figures 2 and 3 are useful either alone or in combination with a variety of other types of grating elements. The grating elements of Figures 2 through 5 are adaptable for a variety of uses, including but not limited to a combination in which one or more of the grating elements is used with an optical fiber; a combination in which one or more of the grating elements is used in a direct injection system for forming an optical pulse; a combination in which one or more of the grating elements, as defined herein, are used as a stretcher portion and as a compressor portion of an overall laser system; and a combination in which one or more of the grating elements, as defined herein, is used in combination with a laser medium in an intra-laser configuration.

The use of one or more of the grating elements, as defined herein, in a chirped pulse amplification (CPA) laser system will now be described. The specifics of the chirped pulse amplification laser system for providing a pulsed beam are known and will not be repeated here as they are described in U.S. Patent Number 5,235,606 which is incorporated herein by reference in its entirety. Such systems can be roughly divided into four categories. The first includes the high energy, low repetition system such as ND glass lasers with outputs of several joules but they may fire at less than 1 shot per minute. A second category are lasers that have an output of approximately one joule and repetition rates from 1 to 20 hertz. The third group consists of millijoule level lasers that operate at rates ranging from 1 to 10 kilohertz. A fourth group of lasers operates at 250 to 350 kilohertz and produces 1 to 2 microjoules per pulse. In U.S. Patent No. 5,235,606, several solid-state amplifying materials are identified and the invention is illustrated using Alexandrite. The examples which are described hereinbelow and are shown in prior art Figure 1 use Ti: sapphire. This is merely illustrative and other laser media such as glass, LiSAF,

Alexandrite, dyes, LiCAF, and the like may be used. In a basic scheme for CPA, first a chirped pulse is generated.

Ideally, the pulse from the oscillator is sufficiently short so that further pulse compression is not necessary. After the pulse is produced, it is stretched by a grating pair arranged to provide positive group velocity dispersion. The amount the pulses are stretched depends on the amount of amplification. Below a millijoule, tens of picoseconds are usually sufficient. A first stage of amplification typically takes place in either a regenerative or a multi-pass amplifier. In one configuration, this consists of an optical resonator that contains the gain media, a Pockels cell, and a polarizer. After the regenerative amplification stage, the pulse can either be recompressed or further amplified. The means of compression preferably consist of a grating or grating pair as defined by the invention, and as shown in Figures 6 and 7 , providing negative second-order dispersion and negative third-order dispersion in order to compensate for any positive second-order dispersion and any positive third-order dispersion which may have been present in the pulsed laser beam prior to compression. Figure 6 shows a schematic of a preferred chirped pulse amplification design which incorporates a pair of transmission grating elements (12) as a compressor (55) and an optical fiber (60) as the stretcher. The system also includes an oscillator (66) and an amplifier (67) as more particularly described in U.S. Patent No. 5,235,606. The oscillator (66), fiber stretcher (60), and amplifier (67) provide the means to generate, stretch, and amplify an optical pulse and the transmission grating compressor (55) provides the means to induce negative second-order dispersion and negative third-order dispersion so as to compensate for any positive second- order dispersion and any positive third-order dispersion induced by the generating, stretching, and/or amplifying of the pulse. Figure 7 is similar to Figure 6 except that the compression means is a reflection grating compressor (65), utilizing a pair of reflection grating elements (14).

In a typical system, as in Figures 6 and 7, the oscillator produces a beam of laser pulses having a frequency of 100 femtoseconds and 1 nanojoule. In the fiber stretcher the pulse is stretched to 250 picoseconds with 0.5 Nanojoules and the amplifier produces a pulse at

250 picoseconds, 1.0 millijoules. After compression the condition of the pulse is typically 100 femtoseconds and

0.5 to 1.0 millijoules. The compression means (c) prepared according to the invention was: a dielectric having an index of refraction of 1.51, a grating having 800 lines per millimeter, and a θ_i of 74. With a center wavelength of λ_o = 790 nanometers.

Figure 8 illustrates a schematic of a direct injection CPA or chirped pulse amplification laser system which is of a direct injection design and comprises a laser medium (68) and means for dispersion compensation. In the embodiment, as shown in Figure 8, the system comprises an oscillator (66), an amplifying medium (68), a dispersive medium (70), and a pair of transmission grating elements (12) forming a transmission compressor (55). Figure 9 is the same as Figure 8 except that the transmission grating compressor (55) has been replaced by reflection grating compressor (65) in the direct injection CPA system.

In the direct injection microjoule level systems of Figures 8 and 9 the short pulse is directly injected into the amplifier and is stretched during each round trip by a dispersive intracavity element. The condition of the pulse in such a system is illustrated by the following values: oscillator produces a 50 femtosecond pulse at 1 nanojoule; regenerative amplifier after 20 round trips produces a 40 picosecond pulse at 2 to 5 microjoules; and after compression by the optical diffraction grating elements of the invention the output pulse is at 50 femtoseconds and 1 to 5 microjoules.

In still another embodiment of the invention, an apparatus for producing a beam comprising one or more laser pulses comprises a pair of grating elements (10) prepared and arranged in accordance with the invention to function as a stretcher (80) and a second pair constructed and arranged to function as a compressor (85). Figures 10 and 11. Figure 11 is the same as Figure 10 except that in Figure 11 the stretcher (82) and compressor (86) are reflection grating elements (14), whereas in Figure 10 the stretcher (80) and compressor (85) are transmission grating elements (12). In this configuration, the means to stretch the pulse in time comprises a pair of optical diffraction grating elements (10), where each one of the grating elements (10) comprises a grating (15) in light couple arrangement with the dielectric body (20) to accept the pulse as described hereinabove. The grating elements (10) are spaced apart with optical imaging system (90) disposed between the grating elements (10). The optical imaging system (90) optically displaces the apparent placement of the grating elements (10) with respect to one another. The optical imaging system (90) in combination with the grating elements (10) provides positive second- order dispersion and positive third-order dispersion in the pulse. The system further comprises grating compressor (85) comprising a pair of optical diffraction grating elements (10) constructed and arranged as described hereinabove. The imaging system (90) is negative magnification (90A of Figure 12 (A)) when the grating elements (10) of the stretcher (80) are arranged anti- parallel as shown in Figure 12. The imaging system (90) is positive magnification (90B of Figure 13 (A)) when the grating elements (10) of the stretcher (80) are arranged in parallel configuration as shown in Figure 13. Figure 10 shows the stretcher (80) and compressor (85) being formed of transmission grating elements (12).

In the parallel configuration, (Figure 12) the grating pattern surface (15) of the grating elements (10) are arranged such that the surfaces (15) are parallel to one another. In an anti-parallel configuration (Figure 13) the pattern surfaces (15) are not parallel to one another. The lenses (105a, 105b) of Figure 12 (A) provide negative magnification system (90A) and the lenses (105c, 105d, 105e) of Figure 13 (A) provide positive magnification system (90B).

In still another embodiment of the invention, as shown in Figure 14, an apparatus is provided for stretching and compressing one or more pulses and comprises an optical diffraction grating element (10) in light coupling arrangement with a dielectric body (20). A mirror (100) is spaced from the grating element (10). A lens (105) is between the mirror (100) and the grating element (10). The grating element (10), lens (105), and mirror (100) are arranged to define a first beam path (110) with a first central wavelength (112) which traverses through the grating element (10), through the lens (105), and to the mirror (100). There is also a dihedral corner reflector (120) spaced from the grating element (10). The grating element (10) and dihedral reflector (120) are arranged to define a second beam path (130) with a second central wavelength (132) which traverses through the grating element (10) and to the corner reflector (120). The dielectric body (20) has a side (a) through which the pulse traverses. Both the first beam path (110) and the second beam path (130) traverse through side (a). Side (a) is at an angle α (alpha) so that α (alpha) defines the path of the pulse in the dielectric body (20) corresponding to an incident angle θ_i between the beam path (110, 112) and a plane normal to the grating, which in cooperation with lens (105) provides positive second-order dispersion and positive third-order dispersion as the beam traverses in the first beam path (110). Lens (105) and mirror (100) comprise a negative magnification optical imaging system. Negative second-order dispersion and negative third-order dispersion is provided as the beam travels in the second beam path (130, 132), as shown in Figure 14. As can be seen in Figure 14 which is in the form of a 3-D sketch, a single grating stretcher and compressor (200) provides, respectively, positive second-order dispersion, positive third-order dispersion; and negative second-order dispersion, negative third-order dispersion. One portion is the stretcher and the other portion is the compressor. Without being held to any particular designation of orientation as shown in Figure 14, the stretcher is illustrated as the top half and the compressor is illustrated as the bottom half. The same grating is used at the same incident angle θ_i for the stretcher and the compressor.

Method of use of the grating elements of the invention will now be described in accordance with their use to compensate for ϕ₂ second-order dispersion (SOD) and ϕ₃ third-order dispersion (TOD) in a pulse having an initial state (I). First, determine the SOD (ϕ_2(I)) and TOD (ϕ_3(I)) of the pulse in its initial state and determine a numerical value ξ = ϕ_3(I)/| ϕ_2(I)| by: (i) computing a phase delay from each part of a system for generating the pulse in its initial state according to ϕ(w) = n(w)Lw/ c where L is the length of the dispersive material in each part of the system and c is the speed of light in vacuum; (ii) summing the phase delay for each such part of the system; (iii) successively differentiating the total phase with respect to frequency by evaluating the derivatives at a central frequency of the pulse to obtain numerical values for SOD ϕ_2(I) and TOD ϕ_3(I); and (iv) computing ξ for the system for generating the pulse according to ξ_(I) = ϕ_3(I)/| ϕ_2(I)| ; (b) providing a compressor having a pair of grating elements providing ξ_(II) with a sign opposite to ξ_(I) and close to or equal to in absolute numerical value to ξ_(I) according to:

where Sin θ_d = m λ/d - Sin θ_i and where θ_d is an angle of diffraction between a path of the pulse emitted from a side of the grating element and a plane normal to the side of the grating element from which the pulse is emitted; the θ_d being related to θ_i by the expression: Sin θ_d = m λ/d - n Sin θ_i, where m is diffraction order, λ is wavelength, d is the grating line spacing, n is index of refraction of the dielectric, and θ_i is the incident angle between the path of the pulse in the grating element and a plane normal to the grating; and (c) transmitting the pulse in its initial state through the compressor which provides negative SOD and negative TOD to compensate for the positive SOD (ϕ_2(I)) and positive TOD (ϕ_3(I)) determined in step (a). The above method is based on the more specific derivations which will now be explained.

To stretch a pulse it is necessary to impart a frequency dependent phase shift ϕ(w), which can express as a Taylor series about a central frequency w_o:

where the coefficients ϕ_n are given by:

A quantity is defined to conveniently characterize a stretcher (either a grating pair or dispersive material, like a fiber) or compressor:

If ξ in the stretcher and compressor are equal in magnitude and opposite in sign (with the requirement that ϕ₂ ^(stretcher) = -ϕ₂ ^(compressor)) then cubic phase compensated compression is possible. If £^(stretcher) ≠ -ξ^(compressor), the resulting compressed pulse will have wings due to uncompensated dispersion.

EXAMPLE In dispersive media (such as a fiber stretcher or intracavity material), the phase ϕ(w) is given by ϕ(w) = n(w)Lw/c, where L is the length of the material. Using a Sellmier or Cauchy formula, compute the index n(w), to successively differentiate the phase with respect to w to obtain the quadratic and cubic coefficients ϕ₂ and ϕ₃, and therefore, determine the parameter ξ. For example, fused silica possesses a value ξ = +0.247 fs at λ = 800 nm. In order to compress a pulse which was stretched in a fused silica fiber or by intracavity material, it requires a compressor with negative ϕ₂ and negative ϕ₃, with a ratio ξ = -0.247 fs.

The expression for ξ of the new grating pair is given by:

where n_p is index of the dielectric, λ is the wavelength of light, and m is the order of diffraction.

It is possible to obtain for this compressor a value of ξ^(g-p) = -0.247 (which is necessary to match a fused silica fiber) with standard optical materials and components.

To design this compressor for a general system, the following is done:

1) Using a Sellmier, Cauchy, or other dispersion equation, compute the index of refraction for all of the elements in the system (stretcher, amplifier, other optics, etc.);

2) Compute and sum the phase delay from each part of the system, given by ϕ(w) = n(w)Lw/c, where L is the length of the material. Successively differentiate the total phase with respect to frequency, and obtain numerical values for the SOD and TOD of the system by evaluating these derivatives at the central frequency of the pulse.

3) Obtain ξ for the system by dividing the TOD by the absolute value of the SOD.

4) To fabricate the compressor with equal and opposite ξ there are several parameters which can be varied: the grating groove spacing d, the index of the glass n, and the angle of incidence θ_i. There exists a family of compressors which will satisfy the ξ requirement.

For example, choose a glass which is easily obtainable: BK-10 glass from Schott. The index of this glass is approximately n - 1.51 at the wavelength λ = 790 nm. The value of m is equal to one. Then choose an incident angle on the grating 74 degrees, for example.

Note that this θ_i is dependent on the angle α between the grating and the side of the dielectric body through which the light transverses. It is preferred that α = θ_i . α is related to θ_i by the expression: θ_i - θ_c ≤ α ≤ θ_i + θ_c, where θ_c = arcsin 1/n and n is index of refraction of the dielectric. Then use these values for n, θ_i, and λ into equation (4), and using the desired value of ξ it is apparent that the only variable in the equation is the grating groove spacing d. Now solve for d, which for this example, d = 1 micron.

Any two of the grating parameters can be specified as given, and the third parameter can be obtained by solving equation (4). For common glasses and incident angles in the 60 - 80 degree range, the gratings which satisfy (4) possess groove spacings ranging from d = 0.8 to d = 2.0 microns, and line density 1/d of 100 1/mm to 2400 1/mm uniformly spaced, which are easily manufactured by holography, as exemplified in Table I.

The phase ϕ(w) is given by ϕ(w) = n(w)Lw/c, where L is the length of the material. Figure 16 shows the results of modeling a 100 femtosecond pulse which was stretched to 250 picoseconds in a fiber and compressed with three different devices: (a) conventional grating that provides positive TOD, opposite to that desired; (b) conventional grating that cannot provide any third-order dispersion, of no practical use to provide the desired dispersion; and (c) a pair of optical diffraction grating elements prepared according to the specification of the invention which provides complete compensation of dispersion as per Figure 6. It can be seen from Figure 16 that the optical diffraction grating elements of the invention provide near perfect compression and compensation while the others leave huge wings on the pulse.

Figure 17 shows the auto-correlation of a 60 femtosecond pulse stretched to 60 picoseconds in a fiber and compressed according to the apparatus and method of the invention demonstrating the utility of the invention in short pulse microjoule lasers.

While not wishing to be held to any particular theory, it is thought that the phase delay in a grating pair, and the reversal of the sign of the cubic phase in this new design, might be understood by deriving the angular dispersion θ' and the rate of change of angular dispersion θ":

and expressing ϕ₂ and ϕ₃ in terms of these physical quantities:

Recall that ξ ≡ ϕ₃/| ϕ₂| and that ϕ₃/ϕ₂ is expresses as equation 4 but with a negative sign. A physical picture of phase delay in a grating pair is the quadratic phase ϕ₂ results from the angular dispersion θ', and the cubic phase ϕ₃ results from the rate of change of angular dispersion θ". For gratings used in the m = 1 order (the most common usage), the dispersion θ' is always positive, and the rate of change θ" is generally positive (though it can be zero or even slightly negative). A standard grating pair can never achieve the large negative values of θ" necessary for operation in the negative-cubic-phase regime; however, in the new compressor of the invention, the index change across the grating reverses the sign of θ", and allows for the rate of change to be of sufficient magnitude such that the cubic phase can flip sign.

Fourth-order dispersion ultimately limits the pulse durations from these stretcher-compressor schemes. The fiber stretcher and grating-prism compressor both impart negative quartic phase, and therefore cannot compensate each other beyond third order. This leftover dispersion, however, is small and should not affect low- contrast applications.

As described earlier, the Figure 1 mode-locked Ti:Al_aO₃ and rare-earth-doped fiber lasers now utilize prisms to cancel the second- and third-order dispersion of the gain medium. Prism sequences, however, are not very compact; the prisms in Ti:Al₂O₃ oscillators is typically 50 centimeters apart (Figure 1), and the six- prism sequence in some fiber lasers can be as long as 1.5 meters. The invention provides compact femtosecond sources, by providing optical diffraction grating elements (10) as in Figure 18. Such intracavity grating elements provide the same cubic-phase compensation as a prism sequence, but require separation of only millimeters. (Figure 18.) This scheme can be used in any laser cavity requiring SOD and TOD compensation. There is less than one centimeter distance between gratings (10), with end mirror pushed against or coated onto grating glass.

In order to operate in the negative-θ₃ regime, this device relies on a large index change across the grating; in the CPA compressor, this index difference is about 0.7 (glass to air). Also consider a monolithic (Figures 19 and 20) geometry, where the space between the grating elements (10) was not air, but rather a dielectric (200) with index n_s. Figure 19 shows the monolithic device comprised of a pair of transmission grating elements (10, 12) and a dielectric (200). Figure 20 shows the monolithic device comprised of a pair of reflection grating elements (10, 14) and a dielectric (200). The equations governing this device would be:

Operation in the negative-ϕ₃ regime is therefore possible if n_p/n_s is sufficiently large. A compressor with high-index flint glass (n_P ~ 1.9) on the outside of the gratings and a low-index material like magnesium fluoride (n_s ~ 1.37) between the grating elements (10) will behave like an air-spaced compressor with an index n_p of 1.4. This one-piece (unitary) device would be very easy to implement, and would be useful for compensating fixed amounts of material dispersion. Interestingly, high-index materials generally contribute large amounts of positive dispersion, and the dispersion of the high-index prisms in this device can no longer be neglected. However, the compressor can be self-compensating, cancelling not only the dispersion of the laser medium but also the dispersion of its own prisms.

The invention provides novel optical diffraction grating elements and more specifically transmission and reflection optical diffraction grating elements with third-order dispersion opposite to that of a traditional grating pair. The invention also provides apparatus and method for easily alignable and easily tunable stretchers and compressors utilizing the novel grating elements. The angle θ_i in the stretcher and compressor are preferably identical; and the stretcher and compressor are preferably identical. Further, the compressor angle is independent of the amplifier length. With proper choice of index of refraction, grating spacing, and θ_i , it is possible to configure a stretcher or compressor such that third-order dispersion and second-order dispersion are of the same sign. In the case where the grating elements are used as both the stretcher and the compressor, the stretcher utilizes the exact same grating elements as the compressor but incorporates a one to one telescope. The grating elements may be made as a single, unitary structure having the gratings patterned or written onto the surface of a dielectric or the gratings may be written onto a flat substrate which is then attached by optical cement to a dielectric. Further, it is possible to construct a grating element in accordance with the invention where the same element is used for both the stretcher and the compressor where suitable reflector mirror lenses define a beam path based on the design of the grating element which satisfies the requirement of positive SOD, positive TOD in the stretcher and negative SOD, negative TOD in the compressor. The potential for commercial application of these optical diffraction grating elements is considerable, including the elimination of conventional stretchers and compressors in laser systems which would result in greater ease of use and reduce the size of these systems. The optical diffraction grating elements when used as a compressor and/or stretcher in laser systems provide the advantage of ease of set-up and alignability. The optical diffraction grating elements of the invention provide essentially perfect compression of a pulse close to, and within a few percent of, the transform limit.

While this invention has been described in terms of certain embodiments thereof, it is not intended that it be limited to the above description, but rather only to the extent set forth in the following claims.

The embodiments of the invention in which an exclusive property or privilege is claimed are defined in the following claims.

Claims

CLAIMS:

1.

An apparatus for imparting phase shift in one or more pulses comprising at least one optical diffraction grating element; each one of said grating elements comprising a grating having a constant line spacing (d) and a dielectric body in light coupling arrangement with said grating; said dielectric body having a side through which the pulse traverses, said side being at an angle α (alpha) with respect to said grating where α (alpha) defines a path of the pulse in said dielectric body corresponding to an incident angle θ_i (theta incident) between the path and a plane normal to said grating, whereby said incident angle θ_i, and said grating line spacing (d) cooperate to provide negative second-order dispersion (SOD) and negative third-order dispersion (TOD) so as to compensate for any positive SOD and any positive TOD initially present in the pulse.

2.

The apparatus according to claim 1 wherein one or more of said grating elements is characterized by an angle of diffraction θ_d between a path of the pulse emitted from a side of said grating element and a plane normal to said side of said grating element from which the pulse is emitted; said θ_d being related to θ_i by the expression: Sin θ_d = m λ/d - n Sin θ_i where m is diffraction order, λ is wavelength, d is the grating line spacing, and n is index of refraction of the dielectric.

3.

The apparatus according to claim 1 wherein one or more of said grating elements comprises a transmission grating characterized by an angle of diffraction θ_d (theta diffraction) between a path of the pulse emitted from said grating element and said plane normal to said grating from which the pulse is emitted, said θ_d being related to θ_i by the expression: Sin θ_d = m λ/d - n Sin θ_i where m is diffraction order, λ is wavelength, d is the grating line spacing, and n is index of refraction of the dielectric.

4.

The apparatus according to claim 1 wherein one or more of said grating elements comprises a reflection grating characterized by an angle of diffraction θ_d (theta diffraction) between a path of the pulse emitted from a side of the dielectric opposite the grating and a plane normal to said side from which the pulse is emitted; said θ_d being related to θ_i by the expression: Sin θ_d = m λ/d - n Sin θ_i where m is diffraction order, λ is wavelength, d is the grating line spacing, and n is index of refraction of the dielectric.

5.

The apparatus according to claim 1 further characterized in that θ_i (theta incident) is related to α

(alpha) according to the expression: θ_i - θ_c ≤ α ≤ θ_i + θ_c, where θ_c = arcsin 1/n and n is index of refraction of the dielectric.

6.

The apparatus according to claim 1 wherein α (alpha) is about equal to θ_i (theta).

7.

The apparatus according to claim 1 wherein each one of said gratings is engraved on a surface of said dielectric body.

8.

The apparatus according to claim 1 wherein each one of said gratings is in the form of a pattern on a surface of said dielectric body.

9.

An apparatus for producing one or more pulses comprising means to generate, stretch, and amplify an optical pulse; and means to compress the stretched and amplified pulse comprising at least one optical diffraction grating element; each one of said grating elements comprising a grating having a constant line spacing (d) and a dielectric body in light coupling arrangement with said grating to accept the stretched and amplified pulse; said dielectric body having a side through which the pulse traverses, said side being at an angle α (alpha) with respect to said grating where α (alpha) defines a path of the pulse in said dielectric body corresponding to an incident angle θ_i (theta incident) between the path and a plane normal to said grating, where θ_i is related to α according to the expression: θ_i - θ_c ≤ α ≤ θ_i + θ_c, where θ_c = arcsin 1/n and n is index of refraction of the dielectric, whereby said incident angle and said grating density cooperate to provide negative second-order dispersion (SOD) and negative third-order dispersion (TOD) so as to compensate for any positive SOD and any positive TOD induced by the generating, stretching, and/or amplifying of the pulse.

10.

The apparatus according to claim 9 wherein one or more of said grating elements is characterized by an angle of diffraction θ_d between a path of the pulse emitted from a side of said grating element and a plane normal to said side of said grating element from which the pulse is emitted; said θ_d being related to θ_i by the expression: Sin θ_d = m λ/d - n Sin θ_i where m is diffraction order, λ is wavelength, d is the grating line spacing, and n is index of refraction of the dielectric.

11.

The apparatus according to claim 9 wherein the means to stretch the pulse comprise an optical fiber.

12.

The apparatus according to claim 9 wherein the means to generate, stretch, and amplify comprise an oscillator and amplifier.

13.

The apparatus according to claim 9 having a pair of said grating elements and where at least one of said gratings is a reflection grating.

14.

The apparatus according to claim 9 having a pair of said grating elements and where at least one of said gratings is a transmission grating.

15.

An apparatus for stretching and compressing one or more pulses comprising an optical diffraction grating element comprising a grating in light coupling arrangement with a dielectric body; a mirror spaced from the grating element; a lens between the mirror and the grating element; and a dihedral reflector spaced from the grating element; the grating element, lens, and mirror arranged to define a first beam path which traverses through said grating element, through said lens, and to said mirror; and said grating element and dihedral reflector arranged to define a second beam path which traverses through said grating element and to said corner reflector; said dielectric body having a side through which the pulse traverses, said side being at an angle α (alpha) with respect to said grating where α (alpha) defines the path of the pulse in said dielectric body corresponding to an incident angle θ_i (theta incident) between the path and a plane normal to said grating, thereby providing positive SOD and positive TOD in said stretcher and negative SOD and negative TOD in said compressor.

16.

An apparatus for producing one or more pulses comprising: a. means to stretch the pulse in time comprising a pair of optical diffraction grating elements, each one of the grating elements comprising a grating in light coupling arrangement with a dielectric body to accept the pulse, said grating elements being spaced apart with optical imaging system disposed between said grating elements that optically displaces the apparent placement of the grating elements with respect to one another, and which provides positive SOD and positive TOD in the pulse; and

b. means to compress the pulse comprising a pair of optical diffraction grating elements, each one of the grating elements comprising a grating in light coupling arrangement with a dielectric body to accept the stretched pulse and to provide negative SOD and negative TOD so as to compensate for the positive SOD and positive TOD induced by the stretching means.

17.

The apparatus according to claim 16 wherein each of said gratings is engraved on a surface of a said dielectric body.

18.

The apparatus according to claim 16 wherein each one of said gratings is in the form of a pattern on a surface of the dielectric body.

19.

The apparatus according to claim 16 wherein said imaging system is negative magnification and the gratings of the stretcher are arranged anti-parallel.

20.

The apparatus according to claim 16 wherein said imaging system is positive magnification and the gratings of the stretcher are arranged in a parallel configuration.

21.

The apparatus according to claim 16 wherein said gratings each have grooves that are essentially uniformly spaced apart by a distance (d) and have essentially the same grating line density (1/d) as measured in number of grating grooves per unit of distance.

22.

An apparatus for producing one or more pulses comprising laser medium and means for dispersion compensation arranged in a single cavity; said dispersion compensation comprising at least one optical diffraction grating element; each one of said grating elements comprising a grating having a constant line spacing (d) and a dielectric body in light coupling arrangement with said grating to accept the stretched and amplified pulse; said dielectric body having a side through which the pulse traverses, said side being at an angle α (alpha) with respect to said grating where α (alpha) defines a path of the pulse in said dielectric body corresponding to an incident angle θ_i (theta incident) between the path and a plane normal to said grating, where θ_i is related to α according to the expression: θ_i - θ_c ≤ α ≤ θ_i + θ_c, where θ_c = arcsin 1/n and n is index of refraction of the dielectric, whereby said incident angle and said grating line spacing cooperate to provide negative second-order dispersion (SOD) and negative third-order dispersion (TOD) so as to compensate for any positive SOD and any positive TOD induced by the generation of the pulse.

23.

A method to compensate for ϕ₂ second-order dispersion (SOD) and ϕ₃ third-order dispersion(TOD) in a pulse having an initial state (I):

a. determining the SOD (ϕ_2(I)) and TOD (ϕ_3(I)) of the pulse in its initial state and determining a numerical value ξ = ϕ_3(I)/| ϕ_2(I)| by:

i. computing a phase delay from each part of a system for generating the pulse in its initial state according to ϕ(w) = n(w)Lw/c where L is the length of the dispersive material in each part of the system and c is the speed of light in a vacuum;

ii. summing the phase delay for each such part of the system;

iii. successively differentiating the total phase with respect to frequency by evaluating the derivatives at a central frequency of the pulse to obtain numerical values for SOD ϕ_2(I) and TOD ϕ_3(I); and

iv. computing ξ for the system for generating the pulse according to ξ_(I)

= ϕ_3(I)/|ϕ_2(I)| ; b. providing a compressor having a pair of grating elements providing ξ_(II) with a sign opposite to ξ_(II) and close to or equal to in absolute numerical value according to:

where Sin θ_d = m λ| d - Sin θ_i and where θ_d is an angle of diffraction between a path of the pulse emitted from a side of said grating element and a plane normal to said side of said grating element from which the pulse is emitted; said θ_d being related to θ_i by the expression: Sin θ_d = m λ/d - n Sin θ_i, where m is diffraction order, λ is wavelength, d is the grating line spacing, n is index of refraction of the dielectric, and θ_i is the incident angle between the path of the pulse in said grating element and a plane normal to said grating; and c. transmitting the pulse in its initial state through said compressor which provides negative SOD and negative TOD to compensate for the positive SOD (ϕ_2(I)) and positive TOD (ϕ_3(I)) determined in step (a).