text:
The Novaquatre is a car first presented in the Autumn of 1937 by Renault and produced until 1940. It was presented in 1937 as an economy version of the Primaquatre, combining the body of the little Renault Celtaquatre, with the 2383 cc 4-cylinder water-cooled engine of the larger Renault Primaquatre.

Supporting its credentials as an economy model, the engine in the Novaquatre came with a listed maximum output of  at 3,000 rpm as opposed the  at 3,300 rpm for the Primaquatre with what was, on paper, much the same engine.   The Primaquatre also came with a much reduced thirst for fuel, with consumption quoted at 9/10 litres per 100 km as against 12/13 litres for the more powerful car.

A top speed of 110 km/h (69 mph) was claimed for the car in its original form, although this reduced to 100 km/h (63 mph) at the end of 1939 when a smaller engine replaced the original unit. Power was transmitted to the rear wheels via a conventional three-speed manual transmission.

The Celtaquatre had ceased production in April 1938 and was effectively replaced by both the (slightly smaller) Renault Juvaquatre and by the Novaquatre.  With the disappearance of the Celtaquatre the  wheelbase also disappeared from the range, and the Novaquatre was slightly lengthened. The standard four-door saloon-bodied Novaquatre now sat on a wheelbase of , which had already been used for the "Commerciale" version of the Celtaquatre.

In September 1939, France, like Britain, declared war in Germany and in June 1940 the German Army invaded and rapidly occupied northern France. These seismic military events give context to the changes that the model underwent, and explain the shortness of its production run. Late in 1939, the engine was replaced with a smaller, 1813 cc unit, and the car's cable brakes were replaced by Lockheed hydraulic brakes. The new Novaquatre BFH1 would presumably have been presented formally in October at the 1939 Motor Show had the event not been cancelled.

Relatively few Novaquatres were produced. Taking the April 1940 factory output of 716 cars, 441 were Juvaquatres, 240 were Primaquatres and just 35 were Novaquatres. A couple of months later, probably before the German invasion, Novaquatre production came to an end.

Types
BDR1/2 till late 1939
BFH1 from 1940

References
 Automobilia , n°50, mai 2001

External links
 Picture of the Novaquatre

Novaquatre
Cars introduced in 1938

redpajama_set_name: RedPajamaWikipedia


If You Could Only See tops the entire chart this week for Tonic. Foo Fighters finds the number two position this week with Monkey Wrench. Semi-Charmed Life by Third Eye Blind is at number three on the chart.


text:
The Global Salon: Cities in Egypt
In collaboration with PEN World Voices Festival
Thursday October 27 2011 • 7:00pm - 8:30pm ET
The Global Salon: EGYPT provoked the spirit of the series. Our Egyptian salon transcended an energy of advocacy with the uprisings in Cairo's Tahrir Square as its nucleus and expanded this focal force to surrounding areas of the region and the evolving political and cultural transformations. Through extraordinary perspectives in music and culture, Host Eddie Robinson facilitated dialogue that featured an astounding look at the role of Egypt's women and youth. We examined the restoration of a "new" country in its transition to democracy and the reverberations experienced throughout the Northern regions of Africa and beyond.
Conversation and Performance with
Mona Eltahawy Award-winning Journalist, activist, former correspondent for Reuters, UK's The Guardian, and U.S. News and World Report
Omar Cheta Researcher; Ph.D. Candidate — Middle Eastern Studies, NYU
Malika Zarra Singer/composer/producer with the velvety, sinuous mezzo-soprano voice has demonstrated a rare ability to communicate both powerful and subtle ideas and feelings in Berber, Moroccan Arabic, French and English
Rula Jebreal Italo-Palestinian journalist, novelist, and screenwriter. Jebreal earned the highest journalism award, the International Ischia Award for Best Journalist of the Year
Basem Fathy Egyptian activist; 'Human Rights First' Awards Recipient; Co-founder of the April 6 Youth Movement, the Egyptian Facebook group which served as a precursor to the historic Tahrir Square uprisings in Cairo

redpajama_set_name: RedPajamaCommonCrawl


Every year the world uses 500 billion plastic bags!
The focus of the UN Environment's campaign for this World Environment Day (June 5) is reducing single-use and disposable plastic, which accounts for 50 per cent of plastic we use- including plastic bags, containers, forks and spoons, cups, straws, sanitary pads etc.
Nilgiris is the first supermarket in Pondicherry to have introduced cloth bags for shoppers at the checkout counter. It was a bold decision that we made for environmental reasons eight years ago. At one point, we also gave away jute bags worth Rs. 100 free of cost and encouraged customers to bring them back later so that they could get a 5% off on their purchase.
Now we sell bio-degradable non-woven bags for a minimal cost at the billing counters. We save up to 1 to 1.5 tons of plastic per month.
Supermarkets are great centres for spreading awareness. I always endorse and educate customers on the importance of switching to alternatives of single-use plastic. While some are eager to pick fights with us for charging for these bags, others who know the environmental reasons behind our decision to not provide plastic bags for purchases, buy these bags from us.


text:
An Analysis of Three Concepts of Psychodynamic, Adlerian and Jungian Psychological Perspectives
Weston (1998), states that one of the postulates fundamental to contemporary psychodynamic theory is that a large amount of mental existence, involving thoughts, emotions, and motivation, are unconscious. As a result, people are capable of acting in fashions or acquiring symptoms that are incomprehensible to them (Weston, 1998; Erdelyi, 2001). Weston asserts that research not only corroborates the theory that emotional and motivational processes are unconscious and that they are in fact unconscious' attempts to manage uncertain emotional events; but also gives credence to Freud's theory of a dynamically unconscious process, which more explicitly, suggests that there is a purpose for keeping some issues unconscious. Similar to defence mechanism (Erdelyi, 2001), through the avoidance of a painful stimulus because of negative reinforcement, people may learn to evade concentrating on specific cognitive or affective processes since it is linked to disgrace, culpability, or unhappiness (Weston, 1998).
Luborsky, O 'Reilly-Landry & Arlow (2008), spoke of explicit memory, which suggests the deliberate recovery of information, while implicit memory suggests that memory does not occur by way of the mind, but is instead, is exhibited by way of ones' behaviour. Implicit memory is also associated with transference patterns exhibited via the behaviour in new relationships. Associative memory makes connections to things through their resemblance. This process is similar to Freud's dynamic inquiry, with regard to unconscious meanings, for instance, when a counsellor pursues a line of thinking by its illogical, emotional associations.
The technique of free association is a prime example of the use of the unconscious in psychodynamic counselling. Customarily when a client exhibits manners of avoidance a psychodynamic counsellor will probably investigate particular examples and track the path of associations, until patterns start to appear which might offer insight into the cognitive-affective associations fundamental to the avoidance. Since these connections are not obtainable through introspection, the sole manner to plot them would be to observe what ideas, emotions, recollections, images etc. surface, after the client calms their consciousness and essentially states what ever comes into their heads' (Weston, 1998).
Early Recollections as Part of a Lifestyle Assessment
Examinations of family constellation, the responsibilities of life and early recollections are likely to generate life stories that when combined, bear patterns of how one lives and copes (Bitter & Nicoll, 2000). Bettner& Lew (1993) used these examinations in Connexions Focusing Technique in couple counselling, which allowed the counsellor to rapidly distinguish both partner's life-styles and to explain this information to the their clients. Early recollections transpire in the time prior to uninterrupted memory and is possibly inexact or a total falsehood. It symbolizes a solitary occurrence instead of a cluster of occurrences (Mosak & Maniacci, 2008). Taking note of the client's understanding of their position they hold within the family, assist the counsellor to grasping the client's overall perception of their position in the world. Paying attention to the person's encounters with life's stress, aids the counsellor in uncovering the client's strong points, perceived failing, and coping manners (Sweeney, Myers, & Stephan, 2006). Listening to early recollections may disclose the client's beliefs about themselves, other people, existence, the universe, as well as their principles; they may also divulge the client's attitude with regard to the counselling session, the counselling relationship and their obstructing thoughts, which undoubtedly will influence the subject matter and anxieties that are conveyed in the counselling session (Bitter & Nicoll, 2000).
In Life-Style assessments, the client's family is examined, followed by the interpretation the patient's early recollections. Recollections are dealt with as a projective modus operandi. In comprehending early recollections, we are able to comprehend the client's life story, since people chose to recollect events that are agreeable with their life-styles (Mosak & Maniacci, 2008).
The summarization of early recollections for life style assessments demonstrates the origins of the client's "basic mistakes". Ones' way of life may be regarded as an individuals' fundamental myths, that they accept as true. People will conduct themselves as if the myths were factual since, for them, they are. Consequently, there are "truths" and "partial truths" in myths, and there are myths we confuse as truth. Lastly, for the assessment, the counsellor is curious about how the client perceives' their own positive features (Mosak & Maniacci, 2008).
Jung's Theory of Psychological Types Used in Counselling
Analytical psychology differentiates a number of psychological types. They address inherent distinctions in disposition; the reason why people sense and respond to the world in different ways. Therapy, undoubtedly, is affected by one's "personal equation", this can be categorized using Jungian typology, the benefits being able to think back to dynamics of type which can be ascertained in the connection with the client. Additionally, with regard to this relationship, the counsellor may reflect on what is often highlighted in analytical listening, such as, imagery, verbal communication, physical experience, etc. (Dehing, 1992).
Counsellors frequently give types assessments for those in couple or family counselling. By way of its' reading, people may comprehend that some of their difficulties may have to do with their differences in type. Differences can be acknowledged and dealt with, with less difficulties when understand as conflicts in type, and the understanding of other peoples specific combination of attitude and function types (introversion and extraversion, thinking, feeling, sensation, and intuition) which may result in enhanced communication with others (Douglas, 2008).
Downey (1924) points out that speed of movement and speed of decision are consequences of typology. Your personality can affect your decision making process. Decision making principles offer guidance for making effective decisions by utilizing shared set of assumptions that allow us to comprehend or predict behaviour (Downey, 1924). A modified form of cognitive styles can be assessed through the use of the Myers-Briggs Type Indicator (MBTI). The fundamental premise for the MBTI is that people have preferences for doing one thing over another. Additionally, what people do and how they choose to do it demonstrates the exercising of their true preference (Douglas, 2008; Bradway, 1964).
Bettner, B. L. & Lew, A. (1993). The connexions focusing technique for couple therapy:
A model for understanding life-style and complementarity in couples. The Journal of Adlerian Theory, Research & Practice, 49, 3/4, 372-391.
Bitter, J. R., & Nicoll, W. G. (2000). Adlerian brief therapy with individuals: Process and
practice. Journal of Individual Psychology, 56, 31-44.
Bradway, K. (1964). Jung's psychological types. Journal of Analytical Psychology, 9, 2, 129-
Dehing, Jef (1992), The therapist's interventions in Jungian analysis. Journal of Analytical
Psychology, 37, 1, 29-47.
Douglas, C. (2008). Analytical psychotherapy. In R.J. Corsini & D. Wedding (Eds.). Current
psychotherapies (8th ed., pp.107-140). Belmont, CA: Thomson Higher Education.
Downey, J. E. (1924). Jung's 'Psychological Types' and Will-Temperament Patterns. The Journal
of Abnormal Psychology and Social Psychology, 18,4, 345-349.
Erdelyi, M. H. (2001). Defense processes can be conscious or unconscious. American
Psychologist, 56, 9, 761-762.
Luborsky, E., O'Reilly-Landry M., & ArlowJ. A. (2008). Psychoanalysis. In R. Corsini, & D.
Wedding (Eds.), Current psychotherapies. (8th ed., pp.15-62). Belmont, CA: The Thomson Corporation.
Mosak, H.H., & Maniacci, M. (2008). Adlerian psychotherapy. In R.J. Corsini & D. Wedding
(Eds.), Current psychotherapies (8th ed., pp.63-106). Belmont, CA: Thomson Higher Education.
Russell-Chapin, L.A. & Rybak, C.J. (1996). The art of teaching Jungian analysis. Journal
of Humanistic Education & Development, 34,4, 171-182.
Sweeney, T. J., Myers, J. E., & Stephan, J. B. (2006). Integrating Developmental Counseling and
Therapy Assessment with Adlerian Early Recollections. Journal of Individual Psychology, 62, 3, 251-269.
Westen, D. (1998). The scientific legacy of Sigmund Freud: Toward a psychodynamically
informed psychological science. Psychological Bulletin, 124, 3, 333-37.
Research Papers on Three Concepts of Psychodynamic
The Project Managment Office System
Never Been Kicked Out of a Place This Nice
Bionic Assembly System: A New Concept of Self…
Incorporating Risk and Uncertainty Factor in Capital…
Hip-Hop is Art
The Hockey Game
← Genetic Engineering
The Mechanics of Grading & Grading Systems →


text:
describe "ProMotion::Screen functionality" do
  tests PM::Screen

  before { UIView.setAnimationDuration 0.01 }

  # Override controller to properly instantiate
  def controller
    rotate_device to: :portrait, button: :bottom
    @controller = FunctionalScreen.new(nav_bar: true)
    @controller.navigationController
  end

  it "should call the on_back method on the root controller when navigating back" do
    presented_screen = PresentScreen.new
    @controller.open presented_screen, animated: false
    @controller.navigationController.viewControllers.should == [ @controller, presented_screen ]
    presented_screen.close animated: false
    @controller.on_back_fired.should == true
  end

  it "should call the correct on_back method when nesting screens" do
    child_screen = @controller.open NavigationScreen.new, animated: false
    grandchild_screen = child_screen.open NavigationScreen.new, animated: false

    # start closing
    grandchild_screen.close animated: false
    child_screen.on_back_fired.should == true
    child_screen.close animated: false
    @controller.on_back_fired.should == true
  end

  it "should fire the will_present, on_present, will_dismiss, and on_dismiss_methods" do
    @presented_screen = PresentScreen.new
    @controller.open @presented_screen, animated: false

    wait 0.01 do
      @presented_screen.will_present_fired.should == true
      @presented_screen.on_present_fired.should == true

      @presented_screen.will_dismiss_fired.should.not == true
      @presented_screen.on_dismiss_fired.should.not == true

      @presented_screen.reset
      @presented_screen.close animated: false

      wait 0.01 do
        @presented_screen.will_dismiss_fired.should == true
        @presented_screen.on_dismiss_fired.should == true

        @presented_screen.will_present_fired.should.not == true
        @presented_screen.on_present_fired.should.not == true

        @presented_screen = nil
      end
    end
  end

  it "should pop to the root view controller" do
    @root_vc = @controller.navigationController.visibleViewController
    @controller.navigationController.viewControllers.count.should == 1
    @controller.open BasicScreen.new, animated: false
    wait 0.01 do
      @controller.open BasicScreen.new, animated: false
      wait 0.01 do
        @controller.open BasicScreen.new, animated: false
        wait 0.01 do
          @controller.navigationController.viewControllers.count.should == 4
          @controller.close to_screen: :root, animated: false
          wait 0.01 do
            @controller.navigationController.viewControllers.count.should == 1
            @controller.navigationController.topViewController.should == @root_vc
          end
        end
      end
    end
  end
end

redpajama_set_name: RedPajamaGithub


credit history is our lifetime. for Uv Desk Light , Home Desk Light , uv desk light , and provide goods with competitive prices and high quality! Customer satisfaction is our priority! You can let us know your idea to develop unique design for your own model to prevent too much similar parts in the market! We will offer our best service to satisfy all your needs! Please contact us right away!
"As a way to finest meet up with client's wants, all of our operations are strictly performed in line with our motto ""High Quality, Aggressive Price, Fast Service"" for Uv Desk Light , Home Desk Light , uv desk light , Why we can do these? Because: A We've been honest and reliable. Our goods have high quality attractive price sufficient supply capacity and perfect service. B Our geographical position has a big advantage . C Various types: Welcome your inquiry It will likely be highly appreciated.


Meghan is a Tampa native who got her first exposure to Moffitt Cancer Center through the summer Teen Volunteer Program. She got her undergraduate degree from the University of Florida in Mechanical Engineering. During that time, she spent two of her summers developing her passion for biomedical research as a Moffitt SPARK Intern in the Biomedical Informatics Department with Dr. Steven Eschrich. Upon graduation, Meghan started her PhD in Biomedical Engineering at Georgia Tech and Emory University in Atlanta. There, under the mentorship of Dr. Manu Platt, she used computational methods to tease apart complex enzyme-on-enzyme interactions in intricate proteolytic networks upregulated in tissue destructive disease. In August 2017, Meghan returned to Moffitt as a Postdoctoral Fellow in the Integrated Mathematical Oncology Department under the supervision of Dr. Philipp Altrock. Now her research focuses on using mathematical modeling to study single cell heterogeneity and clonal hematopoiesis in cancer. Outside of her research, Meghan is an avid Florida Gators football fan and enjoys spending time playing with her poodles, crafting, and curling.
Born and raised in NYC, Gregory moved to Florida with his parents before his senior year of high school, and went to the University of Florida for undergrad. He then moved back to NYC for a year teaching chess and mathematics. Later, He went to Northwestern University, where he received his PhD in applied mathematics. Gregory now works in Dr. Altrock's lab at IMO, where his work focuses on developing mechanistic models of tumor ecology using evolutionary game theory, to explain cancer progression and cancer treatment. When Gregory is not working, he enjoys playing racketball, beach volleyball and chess.


text:
Feminist Fight Club: A Survival Manual For A Sexist Workplace
byJessica Bennett
September 5, 2017|
Audio Book (CD)
Ships within 1-2 weeks.Free shipping on orders over $35
A Wall Street Journal Bestseller! Named a best book of 2016 by: Chicago Tribune, Refinery 29, Forbes, Bust, CEO Reads.
It was a fight club—but without the fighting or the men. Every month, women would gather in a New York apartment to share sexist-job frustrations and trade strategies for how to tackle them. For years, these meetings were kept secret. But the time has come to talk about the club.
In Feminist Fight Club, acclaimed journalist Jessica Bennett blends the personal story of her real-life fight club with a studied assessment of the gender gap that continues to plague the American workplace. With equal measures wit and rigor, Bennett provides the tactical strategies—and the camaraderie—every woman needs to fight back, as well as tools for the men who support the cause.
Award-winning journalist Jessica Bennett is a contributing writer for the New York Times and a columnist for Time.com. She edits special projects for Sheryl Sandberg's nonprofit, LeanIn.org, where she is the co-founder and curator of the Lean In Collection with Getty Images, a photo initiative to make stock photography less sexis...
Title:Feminist Fight Club: A Survival Manual For A Sexist Workplace
Product dimensions:336 pages, 8 X 6 X 0.84 in
Shipping dimensions:336 pages, 8 X 6 X 0.84 in
Published:September 5, 2017
Publisher:HarperCollins


text:
Marcos Tébar Ramiro est un joueur de football espagnol né le  à Madrid.

Carrière de joueur

Ramiro a fait la plus grande partie de sa formation au Real Madrid, où il jouera en équipe B avant que le Real ne le prête au Rayo Vallecano.

Il rejoint pour la saison 2010-2011 l'équipe A du Real faisant ses débuts sous les couleurs de Real seulement vers la fin de la saison. Il se fait ensuite prêter au Girona FC avant que le Real ne le vende à ce dernier pour la modique somme de 100 €. Lors de la saison 2012-2013, il joue 36 matches pour le FC Girona, et inscrit 4 buts. Mais Tébar et son club, échoueront à la  du classement.

Le , il rejoint le club d'Almería, club promu en  espagnole.

Le , il rejoint le club anglais Brentford.

Palmarès
 Finaliste de la coupe du monde des -17 ans avec l'Espagne.

Liens externes

Naissance à Madrid
Naissance en février 1986
Footballeur espagnol
Joueur du Real Madrid Castilla
Joueur du Rayo Vallecano
Joueur du Real Madrid
Joueur du Girona FC
Joueur du Brentford FC
Joueur de l'UD Almería
Joueur de l'UE Llagostera
Joueur du CF Reus Deportiu


Ok, here is a classical flash that has been done in many ways. This one is from my point of view.
I'm dying to tattoo this in original size ( 15 x 16 cm without background ).
Or even better come to see me (from wednesday till saturday between 12 and 12:30 o'clock) at Kynst to make directly an appointment!!!
I have to anounce that this design is no longer available, it has been reserved.
Thank you all for your interest and till the next project! Please keep an eye on it.
Check out if you like something from the latest post.


text:
package com.ironz.binaryprefs.serialization.serializer;

import org.junit.Test;

import static org.junit.Assert.*;

public final class ByteSerializerTest {

    private static final byte INCORRECT_FLAG = 0;

    private final ByteSerializer serializer = new ByteSerializer();

    @Test
    public void byteConvert() {
        byte value = 53;

        byte[] bytes = serializer.serialize(value);
        byte restored = serializer.deserialize(bytes);

        assertTrue(serializer.isMatches(bytes[0]));
        assertEquals(serializer.bytesLength(), bytes.length);
        assertEquals(value, restored);
    }

    @Test
    public void byteIncorrectFlag() {
        byte value = 53;
        byte[] bytes = serializer.serialize(value);
        bytes[0] = INCORRECT_FLAG;

        assertFalse(serializer.isMatches(bytes[0]));
    }
}


text:
\section{Introduction}
  Electron-positron ($e^-e^+$) pair production is one of the important phenomena for studying the non-linear processes in the presence of strong field interacting with the massive (fermionic) vacuum state in the realm of quantum electrodynamics (QED) \cite{RevModPhys.84.1177DiPiazza}. In QED, the vacuum is not an empty space but is a full of virtual $e^-e^+$ pairs. The word virtual means that the lifetime and the separation between electron and positron of these pair are shorter than the Compton time and length scales respectively so as to satisfy the Heisenberg uncertainty principle. If  these virtual pairs are separated up to a length scale $\lambdabar (= \hbar/m_ec)$ and over a time interval $\tau (= \hbar/m_ec^2)$, they become the real pairs.  The strength of electric field needed to have a real pair is the characteristic field of QED. It is also known as the Schwinger limit and its value is  $1.32\times 10^{16}\textnormal V/cm$ \cite{Sauter}. In the presence of such a field the vacuum becomes unstable and is depleted to $e^-e^+$ pairs. The process in which the $e^-e^+$ pairs are generated from vacuum in the presence of such constant electric field, is known as the Schwinger mechanism \cite{PhysRev.82.664}. This mechanism is very interesting because it is a non-perturbative process. From the theoretical point of view the imaginary part of the Heisenberg-Euler Lagrangian density \cite{Heisenberg} of the electromagnetic (EM) field interacting with the vacuum of the charged particles with spins $0$ and $1/2$ gives rise to particle-antiparticle pair generation which was obtained in an explicit form  in  \cite{PhysRev.82.664} wherein electric field was taken to be uniform in space and time.  The value of the uniform electric field strength is so huge that this process can not be realized experimentally because of the unavailability of such fields in laboratory.  The only way of studying such a process is by using a time-varying fields  of the ultrafast and ultraintense lasers \cite{PhysRevSTAB.5.031301}.\\
At present the available laser intensity is of the order of $\approx 10^{22}\textnormal W/cm^2$, which is still far below the critical intensity $I_{cr}=\frac{c}{4\pi}E_S^2 \approx 4.6\times 10^{29}\textnormal W/cm^2$. Several projects have been undertaken throughout the world to achieve intensities of the order of  $I\approx I_{cr}$. The SLAC (Standford Linear Accelerator Centre) \cite{PhysRevLett.79.1626SLAC} team carried out an experiment to investigate the non-linear QED processes accompanying the interaction of high energy electrons and photons with laser pulses. The experiment on  the non-linear Compton scattering of $46.6$-GeV electrons by a laser pulse with intensity of $10^{18}\textnormal W/cm^2$ was performed \cite{CbulaPhysRevLett.76.3116}.   $e^-e^+$ pairs were observed when laser photons which were backscattered up to several GeVs by the $46.6$-GeV electron beam interacted with a pulse of the second laser beam\cite{PhysRevLett.79.1626SLAC}.\\
Many theoretical studies on particle production via Schwinger mechanism have been carried out for both space and time varying fields \cite{PhysRevD.2.1191Itzykson} as given in \cite{PhysRevE.69.036408,PhysRevLett.104.220404} and the cited papers therein.
These studies have explored the pair production mechanism by using focused laser pulses having both circular and linear polarizations. It has been investigated how the pair creation mechanism depends on the electric field intensity for a single, two counter-propagating, and  multiple-colliding laser pulses \cite{PhysRevLett.104.220404}. It has been established now that it is possible to have pair creation for laser intensities much smaller than the critical intensity. These investigations have shown further that the threshold value of the intensity of the electric field of two pulses for producing a pair is about two orders less than that for a single laser pulse. The reported electric field intensity threshold value for the  two counter-propagating ultra-short laser pulse is $0.033E_S$  for laser wavelength $\lambda = 1\mu m$ and pulse duration $\tau = 10\textnormal fs$ \cite{bula}.\\

Bell et al. \cite{PhysRevLett.101.200403Bell} have demonstrated a mechanism of $e^+e^-$ pair production due to the interaction of high intensity ($\approx 10^{24}W/cm^2$) counterpropagating laser beams with the accelerated $e^-$ beam.
In \cite{PhysRevLett.102.080402Ruf} Ruf et al. have studied a scheme of $e^+e^-$ pair production by the counterpropagating laser pulses by solving the Dirac equation numerically. They reported a characteristic modifications of the particle spectra and the Rabi oscillation dynamics. The narrow peak splitting of the resonant pair production probability served as a sensitive probe of the quasi-energy band structure.
 Hebenstriet et al. \cite{PhysRevD.82.105026Hebenstreit} have used Dirac-Heisenberg-Wigner formalism to investigate  space-and time characteristics of nonperturbative $e^+e^-$ pair generation process by various types of electric fields.
 The dynamics of the electron-positron pair plasma has been studied in \cite{PhysRevLett.106.035001Nerush} by Nerush et al. in a strong laser field. It has been shown that QED effects can be experimentally studied with soon-coming laser facilities like Extreme Light Infrastructure  (ELI)  and  High Power laser Energy Research (HiPER) \cite{NaturePublishingGroupDunne}. Su et al. in \cite{PhysRevA2012Su}  have investigated the particle creation process in the presence of magnetic field perpendicular to the electric field.
 It has also been shown \cite{PhysRevLett.109.253202Su} that the magnetic field can diminish the pair production.
Gonoskov et al. \cite{PhysRevLett.111.060404Gonoskov}  have employed e-dipole field for investigating the pair production process. It has been shown that it has maximum field strength conversion efficiency compared to other focused field models (Narzhony-Fofanov \cite{Narozhny2000} and Fedotov \cite{fedotov2009electron}).
 Using the quantum kinetic theory, Kohlf\"urst et al. have studied \cite{PhysRevD.88.045028Kohlfurst} the dynamically assisted Schwinger pair-creation process  for the electric field dependent only on time.
 Spin polarized $e^+e^-$ pair production via elliptical polarized laser fields has been reported by W\"ollert et al. in \cite{PhysRevD.91.125026Wollert}. \\
In this paper we study the pair creation mechanism for different state of beam polarization of the EM field  of two counterpropagating laser pulses. The characteristic parameter for the beam polarization is defined by the parameter of asymmetry $\mu$ \cite{narozhny2005relativistic} between e- and h-waves in the field expression at any arbitrary space time position. The main aim of revisiting this topic is to know how the pairs are distributed spatially and temporally for different values of $\mu$. It has been reported that while the counterpropagating beams of entirely e- and h-polarized $(\mu = \mp 1)$ effective for pair production, the beams with equal mixture of e- and h-polarization  $(\mu = 0)$ are the worst for pair production. In this paper we analyse this observation from the structure of underlying fields. Here we restrict our attention to these three cases of polarization state only, e.g., totally e- waves $(\mu = -1)$, h-waves$(\mu = 1)$, and equal mixture of e- and h-waves $(\mu = 0)$. While $\mu = 0$ case is not suitable for efficient pair production, it is found to be appropriate for generating shorter pulses of electrons and positrons.\\

This paper is organized as follow. In the Sect. \textbf{II} we discuss the basics understanding of the pair creation mechanism in the presence of EM field. The employed field model and the modification due to finite pulse duration is also discussed in this Section. The structure of EM fields, the field invariants and the fields in the Lorentz transformed frame is analysed for different values of $\mu$, with the reference to its possible role in pair production. In Sect. \textbf{III} we discuss the spatial distribution of EM fields, as function of the normalized longitudinal and transverse coordinates,  $\chi$ and  $\xi$ respectively (defined below) and also the azimuthal angle $\phi$. The fields are compared with those in the reference frame in which the electric and magnetic fields are parallel. The polarization dependence of the temporal distribution of the pairs for different values of $\mu$ is also presented in this section. We conclude in Sect. \textbf{IV}.
\section{Theoretical Background and Field Model}
The vacuum depletion probability in the presence of constant electric and magnetic field is given by the semiclassical theory \cite{berestetskii2012quantum},
\begin{equation}
|C_V |^2=|< 0|S_{if} |0>|^2 \propto \exp(-{2 Im\mathcal{L} \textnormal{VT}}/\hbar).
\end{equation}
 Here the initial (i) and final (f) vacuum states are taken as asymptotically far away in time $(t\rightarrow\mp\infty)$ and $\textnormal{VT}$ is the 4-focal volume. $S_{if}$ is the $S$- matrix element between the initial state 'i' and the final state 'f' state interacting with a strong uniform EM field. $Im\mathcal{L}$ is the imaginary part of the Heisenberg-Euler Lagrangian density for the interaction of EM field with the vacuum state of the spin $1/2$ charged particles \cite{Heisenberg}. The vacuum depletion gives rise to $e^-e^+$ particle production. The number of pairs created per unit volume per unit time is given by the Schwinger formula \cite{PhysRev.82.664} as
 \begin{equation}
 \label{w_e_average}
 w_{e^- e^+}=\frac{e^2 E_S^2}{4 \pi^2 \hbar^2 c}  \epsilon  \eta \coth(\frac{\pi \eta}{\epsilon})\exp(-\frac{\pi}{\epsilon}).
 \end{equation}
 As the EM field associated with a typical laser pulse having wavelength of the order of a micron and pulse duration of the order of 10 fs can be taken to be uniform in space and time over the compton length and time scales,  the total number of created pairs is calculated by the integration over whole space and time. We have \cite{PhysRev.82.664}-\cite{PhysRevLett.104.220404}
\begin{equation}
 \label{N_e_average}
 N_{e^- e^+}= \int dV\int dt w_{e^- e^+} = \frac{e^2 E_S^2}{4 \pi^2 \hbar^2 c} \int dV \int dt  \epsilon  \eta \coth(\frac{\pi \eta}{\epsilon})\exp(-\frac{\pi}{\epsilon}).
\end{equation}
Here $\epsilon = \mathcal{E}/E_S,$  $\eta = \mathcal{H}/E_S$, and $(\mathcal{E},\mathcal{H}) = \sqrt{(\mathcal{F}^2+\mathcal{G}^2)^{1/2} \pm \mathcal{F}}$ are the invariants that have the meaning of the electric and magnetic field strengths in the reference frame in which they are parallel to each other. $\mathcal{F} = \frac{1}{4}  F^{\mu \nu} F_{\mu \nu}=\frac{1}{2} (\textbf{E}^2-\textbf{H}^2 )$, $\mathcal{G} = \frac{1}{4} \epsilon ^{\mu \nu \rho \sigma} F_{\mu \nu} F_{\rho \sigma} = \textbf{E}\cdot\textbf{H}$ are Lorentz invariants of EM field. $F^{\mu\nu}$ is EM field tensor defined as $F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$ for EM four potential $A^{\mu}$ and $\mu,\nu$ are the Greek indices which run from $0,1,2,3$. $\epsilon ^{\mu \nu \rho \sigma}$ is totally antisymmetric Levi-Civita tensor with $\epsilon ^{0123} = 1$ \cite{Abramowitz:1974:HMF:1098650}. It is equal to $-1$ for non cyclic permutation of indices and is zero if any two indices are equal.  For a  plane wave both the Lorentz invariants $\mathcal{F}$ and $\mathcal{G}$ are zero and therefore pair production is not possible whatever maybe the intensity. To have the non-zero Lorentz invariants one can use focused EM fields. Depending on the values of $\epsilon$ and $\eta$ in the focal region, the expression of average particle creation can be approximated in a few special cases as follows:
 \begin{itemize}
 \item For $\eta\xrightarrow{} 0$
 \begin{equation}
 \lim\limits_{\eta\xrightarrow{}0 }\epsilon\eta\coth(\pi\eta/\epsilon) = \epsilon^2/\pi \nonumber
 \end{equation}
 \begin{equation}
 \label{N_eta0}
 N_{e^- e^+} \approx \frac{e^2 E_S^2}{4 \pi^3 \hbar^2 c} \int dV \int dt  \epsilon^2\exp(-\frac{\pi}{\epsilon})
 \end{equation}
 \cite{PhysRevLett.104.220404}.
 \item When $0<\eta<\epsilon$, then we have $\coth(\pi\eta/\epsilon) \approx (\epsilon/{\pi\eta}+\pi\eta/{3\epsilon})$, so the approximate form of  Eq. \ref{N_e_average} as,
 \begin{equation}
 \label{N_0<eta<epsilon}
 N_{e^- e^+}\approx\frac{e^2 E_S^2}{4 \pi^2 \hbar^2 c} \int dV \int dt  (\epsilon^2/\pi+\pi\eta^2/3) \exp(-\frac{\pi}{\epsilon}).
 \end{equation}
 \end{itemize}
 We consider the solution of the 3-dimensional field model based on the Maxwell's equations proposed by Narozhny and Fofanov \cite{Narozhny2000}. According to this model the focused EM field does not possess any definite state of polarization. However it can always be represented as a superposition of e-and h-wave. Here e (h)-wave is the totally transverse electric (magnetic) field with respect to the propagation direction \cite{Narozhny2000}. For a circularly polarized laser beam propagating in the z-direction and having its focal region at the origin one can write the expression for electric and magnetic fields at any arbitrary position and time as:
 \begin{equation}
  \begin{aligned}
  \textbf{E}(\textbf{r},t) &= \frac{(1-\mu)}{2}\textbf{E}^e+\frac{(1+\mu)}{2}\textbf{E}^h, \\
  \textbf{H}(\textbf{r},t) &= \frac{(1-\mu)}{2}\textbf{H}^e+\frac{(1+\mu)}{2}\textbf{H}^h.
  \end{aligned}
  \end{equation}
  Here $\textbf{E}^e$, and $\textbf{H}^e$ are the circularly e-polarized electric and magnetic fields given as \cite{Narozhny2000}:
\begin{multline}
\label{Electric_magnetic_field_e_single_beam}
  \textbf{E}^e = iE_0e^{-i\varphi}\{F_1(\textbf{e}_x\pm i\textbf{e}_y)-F_2e^{\pm 2i\phi}(\textbf{e}_x\mp i\textbf{e}_y)\},\\
  \textbf{H}^e = \pm E_0e^{-i\varphi}\{(1-i\Delta^2\frac{\partial}{\partial\chi})[F_1(\textbf{e}_x\pm i\textbf{e}_y)+F_2e^{\pm 2i\phi}(\textbf{e}_x\mp i\textbf{e}_y)]+2i\Delta e^{\pm i\phi}\frac{\partial F_1}{\partial\xi}\textbf{e}_z\}.
\end{multline}
  Here, $\omega$ is the central frequency of the laser pulse, $\lambda$ is the wavelength, $\Delta$ is the focusing or spatial inhomogeneity parameter; $x$, $y$, and $z$ are the spatial coordinates; and
 \begin{eqnarray}
 \varphi = \omega (t-z/c),  \xi = \rho/R,  \chi = z/L, \\
   \rho = \sqrt{x^2+y^2},  \exp(i\phi) = (x+iy)/\rho,\\
   \Delta = c/{\omega R} = \lambda/{2\pi R},  L = R/\Delta.
 \end{eqnarray}
  In Eqn.(\ref{Electric_magnetic_field_e_single_beam}), $\pm$ sign corresponds to right and left circular polarizations respectively. We will consider only the right circular polarization for all the expressions henceforth. It should be noted, however that the all results discussed in this paper and the conclusions thereof remain the same for the left circularly polarized case too.
 The expression for circularly polarized h-fields  can be calculated by the duality transformation of the EM field as \cite{Narozhny2000}:
 \begin{equation}
 \textbf{E}^h =  i\textbf{H}^e,      \textbf{H}^h = - i\textbf{E}^e.
 \end{equation}
 For the weakly focused EM field i.e., $\Delta \ll 1$ the form functions $F_1$ and $F_2$ have the form of a Gaussian beam \cite{Narozhny2000};
\begin{eqnarray}
F_1 = (1+2i\chi)^{-2} ( 1-\frac{\xi^2}{1+2i\chi} ) \exp(-\frac{\xi^2}{1+2i\chi}), \\
F_2 = -\xi^2(1+2i\chi)^{-3} \exp(-\frac{\xi^2}{1+2i\chi}).
\end{eqnarray}
The finite temporal pulse width of the laser beam is accounted by the transformations \cite{Narozhny2000}:
\begin{equation}
\exp(-i\varphi) \rightarrow if^{\prime}(\varphi),\\ \exp(-i\varphi)\Delta  \rightarrow if(\varphi)\Delta \nonumber
\end{equation}
where $f(\varphi) = g(\varphi/{\omega t})\exp(-i\varphi)$, $g(0) = 1$ and $g$ should decreases very fast at the periphery of the pulse for $|\varphi| \gg \omega \tau$. We take $g(t/\tau) = \exp(-4t^2/\tau^2)$ in the focal plane $z=0$ and take $\tau = 10 fs$.

 For two counterpropagating laser pulses having only circularly e-polarized components i.e. $\mu = -1$, we have the following expressions for the electric and magnetic fields
 \begin{multline}
 \label{mu = -1 E field}
 \textbf{E}^e = 2i E_0 ge^{-i\omega t}\frac{e^{-\frac{\xi^2}{{1+4\chi^2}}}}{(1+4\chi^2)}[\{\cos\left(\omega z/c-2\psi\right)-\frac{2\xi^2\sin{\phi}}{(1+4\chi^2)^{1/2}}\sin\left(\phi+\omega z/c-3\psi\right)\}\textbf{e}_x\\+i\{\cos\left(\omega z/c-2\psi\right)-\frac{2\xi^2\cos{\phi}}{(1+4\chi^2)^{1/2}}\cos(\phi+\omega z/c-3\psi)\}\textbf{e}_y],
 \end{multline}
 and
\begin{multline}
\label{mu = -1 H field}
\textbf{H}^e = 2i E_0 ge^{-i\omega t}\frac{e^{-\frac{\xi^2}{{1+4\chi^2}}}}{(1+4\chi^2)}[\{\sin\left(\omega z/c-2\psi\right)-\frac{2\xi^2\cos{\phi}}{(1+4\chi^2)^{1/2}}\sin\left(\phi+\omega z/c-3\psi\right)\}\textbf{e}_x\\+i\{\sin\left(\omega z/c-2\psi\right)-\frac{2\xi^2\sin{\phi}}{(1+4\chi^2)^{1/2}}\cos(\phi+\omega z/c-3\psi)\}\textbf{e}_y\\
-\frac{8\Delta\xi}{(1+4\chi^2)^{1/2}}(1-\frac{\xi^2}{2(1+4\chi^2)^{1/2}})\cos(\phi+\omega z/{c})\textbf{e}_z].
 \end{multline}
 Here we have neglected terms of the order of $\xi^4$ in (\ref{mu = -1 E field},\ref{mu = -1 H field}). Furthermore the term $\frac{2\chi\xi^2}{1+4\chi^2}$ has been omitted in the phase terms as it is negligible in comparison to the dominant term $\omega z/c$.
For $\mu = 1$ when the laser beams have only the circularly h-polarized components the electric and magnetic fields are the dual transformed of the e-polarized fields. The expressions of the electric and magnetic fields are:
\begin{multline}
\label{mu = 1 E field}
\textbf{E}^h = -2E_0 ge^{-i\omega t}\frac{e^{-\frac{\xi^2}{{1+4\chi^2}}}}{(1+4\chi^2)}[\{\sin\left(\omega z/c-2\psi\right)-\frac{2\xi^2\cos{\phi}}{(1+4\chi^2)^{1/2}}\sin\left(\phi+\omega z/c-3\psi\right)\}\textbf{e}_x\\+i\{\sin\left(\omega z/c-2\psi\right)-\frac{2\xi^2\sin{\phi}}{(1+4\chi^2)^{1/2}}\cos(\phi+\omega z/c-3\psi)\}\textbf{e}_y\\-\frac{8\Delta\xi}{(1+4\chi^2)^{1/2}}(1-\frac{\xi^2}{2(1+4\chi^2)^ {1/2}})\cos(\phi+\omega z/{c})\textbf{e}_z],
\end{multline}
 and
\begin{multline}
\label{mu = 1 H field}
\textbf{H}^h = 2E_0 ge^{-i\omega t}\frac{e^{-\frac{\xi^2}{{1+4\chi^2}}}}{(1+4\chi^2)}[\{\cos\left(\omega z/c-2\psi\right)-\frac{2\xi^2\sin{\phi}}{(1+4\chi^2)^{1/2}}\sin\left(\phi+\omega z/c-3\psi\right)\}\textbf{e}_x\\+i\{\cos\left(\omega z/c-2\psi\right)-\frac{2\xi^2\cos{\phi}}{(1+4\chi^2)^{1/2}}\cos(\phi+\omega z/c-3\psi)\}\textbf{e}_y].
\end{multline}
The electric and magnetic fields for $\mu = 0$ case are equal mixtures of e- and h-waves.
\begin{equation}
\label{mu =0 field}
\textbf{E} = \textbf{E}^e+\textbf{E}^h = \textbf{E}^e + i \textbf{H}^e,\\
\textbf{H} = \textbf{H}^e+\textbf{H}^h = \textbf{H}^e - i \textbf{E}^e.
\end{equation}
The explicit expression of the electric field
\begin{multline}
\label{Electric_field_mu0}
\textbf{E} = 2i E_0e^{-i\omega t}g \frac{e^{-\frac{\xi^2}{{1+4\chi^2}}}}{1+4\chi^2}\times\{[exp(i(\omega z/c-2\psi))-\frac{2i\xi^2}{(1+4\chi^2)^{1/2}}e^{-i\phi}\sin(\omega z/c-2\psi+\phi)]\textbf{e}_x\\+i[exp(i(\omega z/c-2\psi))-\\\frac{2i\xi^2}{(1+4\chi^2)^{1/2}}e^{-i\phi}\cos(\omega z/c-2\psi+\phi)]\textbf{e}_y-\frac{8\Delta\xi}{(1+4\chi^2)^{1/2}}(1-\frac{\xi^2}{2(1+4\chi^2){1/2}})\cos(\phi+\omega z/{c})\textbf{e}_z\},
\end{multline}
 and the magnetic field is $-\pi/2 $ out of phase with the electric field i.e., $\textbf{H} = exp(-i \pi/2)\textbf{E}$. Here $\exp(i\psi) = \frac{1+2i\chi}{r}$, $r=\sqrt{1+4\chi^2}$.\\
 Using Eqns.(\ref{mu = -1 E field}-\ref{mu = -1 H field}) we derive the expressions for the invariants for $\mu = -1$.

\begin{multline}
\label{F_mu-1}
\mathcal{F}(\mu = -1) = \frac{2E_0^2g^2e^{-\frac{2\xi^2}{{1+4\chi^2}}}}{(1+4\chi^2)^{2}}\times[\cos{2(\omega z/c-2\psi)}-\\\frac{2\xi^2}{(1+4\chi^2)^{1/2}}\{\cos(2\omega z/c-5\psi)+\cos{2\omega t}\cos(\psi-2\phi)\}+\mathcal{O}(\xi^4)],\\
\end{multline}
\begin{multline}
\label{G_mu-1}
\mathcal{G}(\mu = -1) = \frac{2E_0^2g^2e^{-\frac{2\xi^2}{1+4\chi^2}}}{(1+4\chi^2)^{2}}\times[\sin{2(\omega z/c-2\psi)}\\-\frac{2\xi^2}{(1+4\chi^2)^{1/2}}\{\sin(2\omega z/c-5\psi)+\cos{2\omega t}\sin(\psi-2\phi)\}+\mathcal{O}(\xi^4)].
\end{multline}\\
 The invariants for $\mu = 1$ can be calculated by the duality transformation of the EM field for $\mu = -1$.  We have
 \begin{equation}
 \begin{aligned}
 \mathcal{F} (\mu = 1) = -\mathcal{F} (\mu = -1),\\
  \mathcal{G} (\mu = 1) = \mathcal{G} (\mu = -1).
 \end{aligned}
 \end{equation}
The expressions of $\mathcal{F}$ and $\mathcal{G}$ for $\mu = 0$
\begin{multline}
\label{F_mu0}
\mathcal{F}(\mu = 0) = - \frac{8E_0^2g^2e^{-\frac{2\xi^2}{1+4\chi^2}}\xi^2}{(1+4\chi^2)^{5/2}}\left[\cos(2\omega t+2\phi-\psi)-\frac{\xi^2}{(1+4\chi^2)^{1/2}}\cos(2\omega t+2\phi)\right],
\end{multline}
 and
\begin{multline}
\label{G_mu0}
\mathcal{G}(\mu = 0) =  \frac{8E_0^2g^2e^{-\frac{2\xi^2}{1+4\chi^2}}\xi^2}{(1+4\chi^2)^{5/2}}\left[\sin(2\omega t+2\phi-\psi)-\frac{\xi^2}{(1+4\chi^2)^{1/2}}\sin(2\omega t+2\phi)\right].
\end{multline}
At this point it may be worthwhile to compare the expressions of invariants $\mathcal{F}$ and $\mathcal{G}$ for $\mu = 0$ with those for $\mu = \mp 1$. First, the amplitude part of $\mathcal{F}$ and $\mathcal{G}$ for $\mu = 0$ has a factor of $\xi^2$ which makes it negligibly small in the focal region where $\xi \ll 1$. Away from the focal region $\xi^2$ increases but the amplitude is exponentially suppressed by the gaussian profile factor $e^{-\frac{2\xi^2}{1+4\chi^2}}$. Therefore the amplitude of $\mathcal{F}$ and $\mathcal{G}$ for $\mu = 0$ are always much smaller to those for $\mu = \mp 1$ which do not have $\xi^2$ factor.
 Second, the phase part of invariants for $\mu = \mp 1$ shows oscillatory behaviour along the propagation direction with a length scale of the order $2\pi c/\omega$ which is quite expected feature associated with the standing wave formation of counterpropagating laser beams. And this type of interference, which gets carried over to the reduced field invariants $\epsilon$ and $\eta$ (see Eqns.(\ref{epsilon_mu_1},\ref{eta_mu_1})) is the root cause effective pair production by the counterpropagating laser beams. However, this interference is absent for $\mu = 0$. In other words, we have this interference for countepropagating e-polarized beams or h-polarized beams but it is washed out when the colliding beams are made up with the equal mixture of e-and h-polarizations. This intriguing observation can be explained by analysing the expressions of EM fields of colliding pulses. \\
 The real parts of the $x$-and $y$-components of the electric field $ReE^e_x$ and $ReE^e_y$ from the Eqn.(\ref{mu = -1 E field}) for $\mu = -1$
 \begin{multline}
 \label{ReE_x_mu_1}
 ReE^e_x = 2E_0\sin{\omega t}g \frac{e^{-\frac{\xi^2}{{1+4\chi^2}}}}{1+4\chi^2}[\cos(\omega z/c-2\psi)-\frac{2\xi^2}{(1+4\chi^2)^{1/2}}\sin{\phi}\sin(\omega z/c-3\psi+\phi)],
 \end{multline}
 and
 \begin{multline}
 \label{ReE_y_mu_1}
 ReE^e_y = -2E_0\cos{\omega t}g \frac{e^{-\frac{\xi^2}{{1+4\chi^2}}}}{1+4\chi^2}[\cos(\omega z/c-2\psi)-\frac{2\xi^2}{(1+4\chi^2)^{1/2}}\cos{\phi}\cos(\omega z/c-3\psi)].
 \end{multline}
 The expressions of the electric field components for $\mu = -1$ in Eqns.(\ref{ReE_x_mu_1}-\ref{ReE_y_mu_1}) show oscillatory behaviour in longitudinal direction as well as in time.
 The magnitude of the electric field
 \begin{multline}
 \label{ReE_mu_1}
 ReE = \sqrt{{ReE^e_x}^2+{ReE^e_y}^2}
 \approx \frac{2E_0 ge^{-\frac{\xi^2}{1+4\chi^2}}}{(1+4\chi^2)}|\cos(\omega z/c-2\psi)|\\\times[1-\frac{\xi^2}{\cos(\omega z/c-2\psi)(1+4\chi^2)^{1/2}}\{\cos(\omega z/c-3\psi)+\cos{2\omega t}\cos(3\psi-\omega z/c-2\phi)\}+\mathcal{O}(\xi^4)].
 \end{multline}
 We note that the oscillatory behaviour in time has vanished while it has survived in $z$.
Similarly the $x$-and $y$-components of the real part of the magnetic field show oscillatory behaviour in $z$ and $t$:
 \begin{multline}
 \label{ReH_x_mu_1}
 ReH^e_x = 2E_0\sin{\omega t}g \frac{e^{-\frac{\xi^2}{{1+4\chi^2}}}}{1+4\chi^2}[\sin(\omega z/c-2\psi)-\frac{2\xi^2}{(1+4\chi^2)^{1/2}}\cos{\phi}\sin(\omega z/c-3\psi+\phi)],
 \end{multline}
 \begin{multline}
 \label{ReH_y_mu_1}
 ReH^e_y = -2E_0\cos{\omega t}g \frac{e^{-\frac{\xi^2}{{1+4\chi^2}}}}{1+4\chi^2}[\sin(\omega z/c-2\psi)+\frac{2\xi^2}{(1+4\chi^2)^{1/2}}\sin{\phi}\cos(\omega z/c-3\psi+\phi)].
 \end{multline}
We neglect the $z$-component of the magnetic field as it is proportional with $\Delta (\ll 1)$ and this would only give a term proportional to $\Delta^2$ in the invariants.
 In this approximation the real part of the magnitude of the magnetic field
 \begin{multline}
 \label{ReH_mu_1}
 ReH = \sqrt{{ReH^e_x}^2+{ReH^e_y}^2}
 \approx \frac{2E_0 ge^{-\frac{\xi^2}{1+4\chi^2}}}{(1+4\chi^2)}|\sin(\omega z/c-2\psi)|\\\times[1-\frac{\xi^2}{\sin(\omega z/c-2\psi)(1+4\chi^2)^{1/2}}\{\sin(\omega z/c-3\psi)+\cos{2\omega t}\sin(3\psi-\omega z/c-2\phi)\}+\mathcal{O}(\xi^4)].
 \end{multline}
 The leading order term in the expressions of the electric and magnetic fields' magnitude in Eqns.(\ref{ReE_mu_1},\ref{ReH_mu_1}) are of the following form
 \begin{equation}
 \begin{aligned}
 ReE (\mu = -1)& \approx A |\cos{\theta}|,\\
 ReH (\mu = -1)& \approx A|\sin{\theta}|,
 \end{aligned}
 \end{equation}
 where $A$ is the slowly varying part of the amplitude and $\theta$ corresponds to argument of the fast varying part which basically gives rise to oscillation in the amplitude in $z$-direction,
 \begin{equation}
 \begin{aligned}
 A &= \frac{2E_0g\exp(-\frac{\xi^2}{{1+4\chi^2}})}{1+4\chi^2},\\
 \theta &= \frac{\omega z}{c}-2\varphi.\\\nonumber
 \end{aligned}
 \end{equation}
 One can write the electric and magnetic fields components in the leading order term as:
 \begin{equation}
 \begin{aligned}
 ReE^e_x &= A\sin{\omega t}\cos\theta,\\
 ReE^e_y &= -A\cos{\omega t}\cos\theta,\\
 ReH^e_x &= A\sin{\omega t}\sin\theta,\\
 ReH^e_y &= -A\cos{\omega t}\sin\theta.
 \end{aligned}
 \end{equation}
 It is easy to see that such electric and magnetic fields will give invariants
 \begin{equation}
 \begin{aligned}
 \mathcal{F} (\mu = -1) &\approx \frac{1}{2}A^2 cos{2\theta},\\
 \mathcal{G}(\mu = -1) &\approx \frac{1}{2}A^2\sin{2\theta}.
 \end{aligned}
 \end{equation}
 Thus oscillatory behaviour of the leading order term in fields with the same amplitude but with a phase shift of $\pi/2$ is responsible for the observed features in the expressions of invariants $\mathcal{F}$ and $\mathcal{G}$ for $\mu = -1$ discussed above.\\
For $\mu = 0$, the individual components of electric and magnetic fields show interference effects but the magnitude of the fields are independent of the oscillatory term along the propagation direction. In order to clarify this point we examine the real part of the electric and magnetic fields components and its magnitude. The $x$, $y$-components of the electric field in Eq (\ref{Electric_field_mu0}) can be written as
  \begin{multline}
  \label{ReE_x_mu0:eqn}
  ReE_x = 2E_0g \frac{e^{-\frac{\xi^2}{{1+4\chi^2}}}}{1+4\chi^2}[\sin(\omega t-\omega z/c+2\psi)+\frac{2\xi^2}{(1+4\chi^2)^{1/2}}\sin(\omega z/c-3\psi+\phi)\cos(\omega t+\phi)],
  \end{multline}
  and
  \begin{multline}
  \label{ReE_y_mu0:eqn}
  ReE_y = -2E_0g \frac{e^{-\frac{\xi^2}{{1+4\chi^2}}}}{1+4\chi^2}[\cos(\omega t-\omega z/c+2\psi)-\frac{2\xi^2}{(1+4\chi^2)^{1/2}}\cos(\omega z/c-3\psi+\phi)\cos(\omega t+\phi)].
  \end{multline}
  The components consist of interference terms in longitudinal axis and in time. Using these components we calculate the magnitude of the electric field.

  \begin{multline}
  \label{ReE_mu0:eqn}
  ReE = \sqrt{{ReE_x}^2+{ReE_y}^2}\\
   = 2E_0g \frac{e^{-\frac{\xi^2}{{1+4\chi^2}}}}{1+4\chi^2}[1-\frac{4\xi^2}{(1+4\chi^2)^{1/2}}\cos(\omega t+\phi)\cos(\omega t+\phi-\psi)+\frac{4\xi^4}{(1+4\chi^2)}\cos^2(\omega t+\phi)]^{1/2}.
  \end{multline}
 It is easy to see that the leading order term for the magnitude of the total electric field does not show the oscillatory behaviour of the electric field components along the z-axis. It is due to $\pi/2$ phase difference between the oscillations of the components.  The same conclusion holds good for the components and the magnitude of  the total magnetic field. For the sake of completeness  we write down  the $x$-and $y$-components of the magnetic field using Eq. (\ref{mu =0 field})
  \begin{multline}
  \label{ReH_x_mu0:eqn}
  ReH_x = 2E_0g \frac{e^{-\frac{\xi^2}{{1+4\chi^2}}}}{1+4\chi^2}[\cos(\omega t-\omega z/c+2\psi)-\frac{2\xi^2}{(1+4\chi^2)^{1/2}}\sin(\omega z/c-3\psi+\phi)\sin(\omega t+\phi)],
  \end{multline}
  and
  \begin{multline}
  \label{ReH_y_mu0:eqn}
  ReH_y = 2E_0g \frac{e^{-\frac{\xi^2}{{1+4\chi^2}}}}{1+4\chi^2}[\sin(\omega t-\omega z/c+2\psi)-\frac{2\xi^2}{(1+4\chi^2)^{1/2}}\cos(\omega z/c-3\psi+\phi)\sin(\omega t+\phi)].
  \end{multline}
Finally we calculate the magnitude of the real part of the magnetic field
  \begin{multline}
  \label{ReH_mu0:eqn}
  ReH = \sqrt{{ReH_x}^2+{ReH_y}^2}
   = 2E_0g \frac{e^{-\frac{\xi^2}{{1+4\chi^2}}}}{1+4\chi^2}[1-\frac{4\xi^2}{(1+4\chi^2)^{1/2}}\sin(\omega t+\phi)\sin(\omega t+\phi-\psi)\\+\frac{4\xi^4}{(1+4\chi^2)}\sin^2(\omega t+\phi)]^{1/2}.
  \end{multline}
The expressions of the real part of the electric and magnetic fields in Eqns.(\ref{ReE_mu0:eqn}-\ref{ReH_mu0:eqn}) show that their leading order terms  are the same. Hence in the expression for $\mathcal{F}$ which is basically the half of the difference between square of real part of the electric and magnetic fields, this term cancels out and the leading order term is proportional to $\xi^2$. Initially it increases with the increase in $\xi$ but eventually its magnitude falls off  because of the gaussian pulse profile factor. Eqns.(\ref{F_mu0}-\ref{G_mu0}) show the expression of invariants of EM fields with an equal mixture of e and h waves. These do not show oscillatory behaviour along the $z-$axis. However, the  oscillations are present behaviour present in $t$ and  $\phi$.\\
The expressions of the reduced field invariants for $\mu = -1, 1, 0$ are given as:
\begin{multline}
\label{epsilon_mu_1}
\epsilon (\mu = -1) \approx \frac{2E_0 ge^{-\frac{\xi^2}{1+4\chi^2}}}{(1+4\chi^2)}|\cos(\omega z/c-2\psi)|\times[1-\frac{\xi^2}{\cos(\omega z/c-2\psi)(1+4\chi^2)^{1/2}}\times\\\{\cos(\omega z/c-3\psi)+\cos{2\omega t}\cos(3\psi-\omega z/c-2\phi)\}+\mathcal{O}(\xi^4)]
\end{multline}
 and
\begin{multline}
\label{eta_mu_1}
\eta (\mu = -1) \approx \frac{2E_0 ge^{-\frac{\xi^2}{1+4\chi^2}}}{(1+4\chi^2)}|\sin(\omega z/c-2\psi)|\times[1-\frac{\xi^2}{\sin(\omega z/c-2\psi)(1+4\chi^2)^{1/2}}\times\\\{\sin(\omega z/c-3\psi)+\cos{2\omega t}\sin(3\psi-\omega z/c-2\phi)\}+\mathcal{O}(\xi^4)].
\end{multline}
The leading order term in the reduced fields $\epsilon$ and $\eta$ in the above Eqns.(\ref{epsilon_mu_1}-\ref{eta_mu_1}) can be approximated as,
\begin{equation}
\begin{aligned}
\epsilon & \approx A|\cos\theta|,\\
\eta & \approx A|\sin\theta|,
\end{aligned}
\end{equation}
where $A$ and $\theta$ have been defined earlier. These approximate expressions are identical to those for  the electric and magnetic fields for small values of $\chi$ and  $\xi$. In the next section we will comeback to this interesting observation and  discuss in details its possible ramification.
For h-waves the reduced electric and magnetic fields are calculated by the duality transformation. The expressions are
\begin{equation}
\begin{aligned}
\epsilon (\mu = 1) = \eta (\mu = -1),\\
\eta (\mu = 1) = \epsilon (\mu = -1).
\end{aligned}
\end{equation}
 The reduced fields for $\mu = 0$ are given as:
\begin{multline}
\label{eqn:epsilon_mu0}
\epsilon (\mu = 0) \approx \frac{4E_0 ge^{-\frac{\xi^2}{1+4\chi^2}}\xi}{(1+4\chi^2)^{5/4}}|\sin(\psi/2-\phi-\omega t)|\times[1-\frac{\xi^2}{4\sin^2(\psi/2-\phi-\omega t)(1+4\chi^2)^{1/2}}\{\cos(\psi-4\phi-4\omega t)\\-\cos{2(\omega t+\phi)}\}+\mathcal{O}(\xi^4)],
\end{multline}
 and
\begin{multline}
\label{eta_mu0}
\eta (\mu = 0) \approx \frac{4E_0 ge^{-\frac{\xi^2}{1+4\chi^2}}\xi}{(1+4\chi^2)^{5/4}}|\cos(\psi/2-\phi-\omega t)|\times[1-\frac{\xi^2}{4\cos^2(\psi/2-\phi-\omega t)(1+4\chi^2)^{1/2}}\{\cos(\psi-4\phi-4\omega t)\\+\cos{2(\omega t+\phi)}\}+\mathcal{O}(\xi^4)].
\end{multline}
 The above expressions for $\epsilon$ and $\eta$ are derived in  the small $\chi$, $\xi$ approximation in order to understand the physical origin of the pair production in terms of the structure of EM fields and the invariants  in the focal region.
It is clear that the qualitative features of the invariants $\mathcal{F}$ and $\mathcal{G}$ get translated into reduced field invariants $\epsilon$ and $\eta$. As before the amplitudes of $\epsilon$ and $\eta$ are the same in all the cases. While it is maximum for $\xi = 0$ for $\mu = \mp 1$ it identically vanishes for $\mu = 0$ case, and is much smaller for any other value of $\xi$ because of the presence of the factor $\xi$ in the leading order term for the amplitude in the latter case. For $\mu = \mp 1$ both $\epsilon$ and $\eta$ show oscillatory behaviour with phase difference of $\pi/2$ with spatial frequency $\approx 2\pi\omega/c$ in $z$-direction, the propagation direction. The origin of this oscillation, as discussed earlier, is the interference of the counterpropagating beams. This type of oscillation is absent in $\epsilon$ and $\eta$ for $\mu = 0$. However they show oscillatory behaviour in the temporal domain and with the azimuthal variable $\phi$.
We note here that in going from $\mu = -1$ to $1$, $\epsilon$ and $\eta$ get interchanged. For $\mu = -1$, $\epsilon$ shows a maximum for $\xi = 0$ and $\chi = 0$. Consequently the spatial distribution of $e^+e^-$ pairs would show a peak at the centre of the focal spot. However for $\mu = 1$ $\epsilon$ is maximum for $\xi = 0$, $\chi = 0$ and hence the spatial distribution of $e^+e^-$ pairs will show a dip right at the centre of the focal spot. We will return this point later.
Having discussed the expressions for the fields, invariants for various polarization state of counterpropagating pulses we present results in the next section.
\section{Results and Discussion}
As discussed in the previous section the structure of the electromagnetic fields and their relationship to the reduced invariant fields are quite sensitive to the polarization of the colliding pulses. We, therefore, first present the spatial variation of EM fields and the corresponding reduced fields $\epsilon$ and $\eta$ for $\mu = \mp 1$.

\subsection{Fields and particles for $\mu = \mp 1$ beams: }
We consider the EM field by the superposition of two counterpropagating laser pulses. For the field distribution in the focal plane we present the results from Eq.(\ref{mu = -1 E field}-\ref{mu =0 field}). Figure \ref{fig:E_H_epsilon_eta_xi_mu_1_chi0} shows the fields $|Re{\textbf{E}}|$, $|Re{\textbf{H}}|$, and invariants  $\epsilon$, and $\eta$ as a function of  $\xi$ for $\mu = -1$ at $z = 0$ plane and time $t = 0$. The electric field and $\epsilon$ show maximum at $\xi = 0$ and falls off in the peripheral region. The magnetic field and $\eta$ vanish for all values of  $\xi$ for $z = 0$-plane and at $t = 0$.
 Figure \ref{fig:E_H_epsilon_eta_xi_mu1_chi0164} presents the fields $|Re{\textbf{E}}|$, $|Re{\textbf{H}}|$, and invariants $\epsilon$, and $\eta$ as a function of  $\xi$ for $\mu = 1$ for $z = 0.0164L$ (because electric field has one of its maxima at this value of $z$, see below) and time $t = 0$.  The variation in  $|Re\textbf{E}|$ and $\epsilon$ with $\xi$ is same as that for $\mu = -1$ case. The variation in $|Re\textbf{H}|$ and $\eta$ is somewhat different. Both of them show a maximum in the peripheral region. However, their values are always much smaller than that of $|Re{\textbf{E}}|$ or $\epsilon$. \\
 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=75mm]{E_H_epsilon_eta_xi_mu_1_chi0.eps}
 \end{center}
 \begin{center}
 \caption{Variation of $|Re\textbf{E}|$, $|Re\textbf{H}|$, $\epsilon$, and $\eta$ with dimensionless transverse variable $\xi$ of the two counterpropagating focused laser beam for $\mu = -1$. Fields are normalized with $E_S$, $E_0 = 0.0565$, $\chi = 0$, $\phi = 0$, and $t=0$.}\label{fig:E_H_epsilon_eta_xi_mu_1_chi0}
 \end{center}
 \end{figure}
 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=75mm]{E_H_epsilon_eta_xi_mu1_chi0164.eps}
\end{center}
\begin{center}
\caption{Variation of $|Re\textbf{E}|$, $|Re\textbf{H}|$, $\epsilon$, and $\eta$ with dimensionless transverse variable $\xi$ of the two counterpropagating focused laser beam for $\mu = 1$. Fields are normalized with $E_S$, $E_0 = 0.0565$, $\chi = 0.0164$, $\phi = 0$ and $t=0$.}\label{fig:E_H_epsilon_eta_xi_mu1_chi0164}
\end{center}
\end{figure}
 Figure \ref{fig:E_H_epsilon_eta_chi_mu_1_2May2016} shows the distribution of the fields and the reduced invariant fields as a function of normalized longitudinal coordinate $\chi$ for $\mu = -1$. Here the  field distributions form standing wave patterns because of the superposition of two monochromatic EM waves. The decrease in the amplitude of oscillation is characterized by the form function $g$ and there are  multiple maxima although the central maximum is located at $\chi = 0$. Figure \ref{fig:E_H_epsilon_eta_chi_mu1_2May2016} shows the distribution of the fields and invariant fields with normalized longitudinal coordinate $\chi$ for $\mu = 1$. Here the fields and invariants show standing wave pattern similar to that in the case of $\mu = -1$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=75mm]{E_H_epsilon_eta_chi_mu_1_2May2016.eps}
\end{center}
\begin{center}
\caption{Variation of $|Re\textbf{E}|$, $|Re\textbf{H}|$, $\epsilon$, and $\eta$ with dimensionless longitudinal variable $\chi$ of the two counterpropagating focused laser beam for $\mu = -1$. Fields are normalized with $E_S$, $E_0 = 0.0565$, $\xi = 0$, and $t=0$.}\label{fig:E_H_epsilon_eta_chi_mu_1_2May2016}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=75mm]{E_H_epsilon_eta_chi_mu1_2May2016.eps}
\end{center}
\begin{center}
\caption{Variation of $|Re\textbf{E}|$, $|Re\textbf{H}|$, $\epsilon$, and $\eta$ with dimensionless longitudinal variable $\chi$ of the two counterpropagating focused laser beam for $\mu = 1$. Fields are normalized with $E_S$, $E_0 = 0.0565$, $\xi = 0$, and $t=0$.}\label{fig:E_H_epsilon_eta_chi_mu1_2May2016}
\end{center}
\end{figure}
The longitudinal extent of the focused field in both the cases is upto $\chi = \pm 0.2$ and is symmetrical about $\chi = 0$. In this region the field distribution shows oscillation with decreasing amplitude. The maxima of these oscillations are spaced by $\approx 0.03272L$ (Rayleigh lengths) along the propagation direction . For $\mu = -1$, the central lobe shows the maximum for the electric field and the minimum for magnetic field and vice versa for $\mu = 1$. \\
The distributions of the fields as shown in Figs.(\ref{fig:E_H_epsilon_eta_xi_mu_1_chi0},\ref{fig:E_H_epsilon_eta_xi_mu1_chi0164},\ref{fig:E_H_epsilon_eta_chi_mu_1_2May2016},\ref{fig:E_H_epsilon_eta_chi_mu1_2May2016}) reveal remarkable equality between $|Re\textbf{E}| (|Re\textbf{H}|)$, and $\epsilon (\eta)$. Infact, in Figs.(\ref{fig:E_H_epsilon_eta_chi_mu_1_2May2016}-\ref{fig:E_H_epsilon_eta_chi_mu1_2May2016}), they are just identical. Recalling that $\epsilon (\eta)$ has the meaning of a transformed electric field (magnetic field) in the Lorentz frame in which the electric and magnetic fields are parallel to each other. Such a frame is achieved for any non-orthogonal electric $\textbf{E}$ and magnetic $\textbf{H}$ fields by the Lorentz boost operation given by
\begin{equation}
\frac{\frac{\textbf{V}}{c}}{1+\frac{V^2}{c^2}} = \frac{\textbf{E}\times\textbf{H}}{|\textbf{E}|^2+|\textbf{H}|^2},
\end{equation}
see in Ref. {\cite{Landaufield}-\cite{griffiths1999introduction}}.\\
 The observation that in both cases ($\mu = \mp 1$) the fields in the lab frame and the reduced fields ($\epsilon$, and $\eta$) in transformed frame are identical or nearly identical suggests that the fields are parallel or nearly parallel in both the frame. This can be further understood by evaluating the cross product $\textbf{C}( = Re\textbf{E}\times Re\textbf{H})$ to calculate $\textbf{V}$. In Cartesian coordinate system
 \begin{multline}
 C_x = ReE^e_yReH^e_z-ReE^e_zReE^e_y\\
 = -\frac{8E_0^2g^2e^{-\frac{2\xi^2}{1+4\chi^2}}\Delta\xi\sin{2\omega t}}{(1+4\chi^2)^{4}}\times[\cos(\omega z/c-2\psi)-\frac{2\xi^2}{(1+4\chi^2)^{1/2}}\cos{\phi}\cos(\omega z/c+\phi-3\psi)]\\\times[-2(1+4\chi^2)^{3/2}\cos(\omega z/c+\phi-3\psi)+\xi^2(1+4\chi^2)\cos(\omega z/c+\phi-4\psi)],
 \end{multline}
 \begin{equation}
 V_x \approx -\frac{\Delta\xi}{(1+4\chi^2)^{2}}\sin{2\omega t}\cos(\omega z/c-2\varphi),
 \end{equation}
 \begin{multline}
 C_y = ReE^e_zReH^e_x-ReE^e_xReH^e_z\\
 = -\frac{8E_0^2g^2e^{-\frac{2\xi^2}{1+4\chi^2}}\Delta\xi(1-cos{2\omega t})}{(1+4\chi^2)^{4}}[\cos(\omega z/c-2\psi)-\frac{2\xi^2}{(1+4\chi^2)^{1/2}}\sin{\phi}\sin(\omega z/c+\phi-3\psi)]\\\times[-2(1+4\chi^2)^{3/2}\cos(\omega z/c+\phi-3\psi)+\xi^2(1+4\chi^2)\cos(\omega z/c+\phi-4\psi)],
 \end{multline}
 \begin{equation}
  V_y \approx -\frac{2\Delta\xi}{(1+4\chi^2)^{2}}\sin^2{\omega t}\cos(\omega z/c-2\varphi),
  \end{equation}
 and
 \begin{multline}
 C_z = ReE^e_xReH^e_y-ReE^e_yReH^e_x\\
 = -\frac{4E_0^2g^2e^{-\frac{2\xi^2}{1+4\chi^2}}\xi^2\sin{2\omega t}}{(1+4\chi^2)^{5/2}}[\sin(2\phi-\psi)-\frac{\xi^2}{(1+4\chi^2)^{1/2}}\sin{2(\omega z/c+\phi-3\psi)}],
 \end{multline}
 \begin{equation}
  V_z \approx -\frac{\xi^2}{2(1+4\chi^2)^{1/2}}\sin{2\omega t}\sin(2\phi-\varphi).
  \end{equation}
 As both $\xi$ and $\Delta\ll 1$, $\textbf{V}$ is negligibly small in the focal region and vanishes for $\xi = 0$. This explains the observation that the transformation from ($Re\textbf{E}$, $Re\textbf{H}$) to ($\epsilon$, $\eta$) is nearly identity transformation in the focal region and for the special case of $\xi = 0$ it is exactly identity transformation. The physical consequence of a very small value of $|\textbf{C}|$, in the focal region is that a very small amount of EM energy is flows out of the focal region and thereby resulting in an efficient pair production for two beam configuration for $\mu = \pm 1$. Furthermore, since $|\textbf{C}|$ is proportional to $\Delta$ a smaller value of $\Delta$ will lead to a larger number of pairs. This effect has been attributed to the increase in the focal volume in the literature \cite{bula}. However, the explanation given here is more direct and physical.\\
 The components of $Re\textbf{E}\times Re\textbf{H}$ are proportional to
 $\xi$. Therefore,  the electric and magnetic fields are not completely parallel as one
 moves to the regions in the focal volume where $\xi \ne 0$. This leads a
 small but finite amount of energy to flow out of the focal region due to
 the slight non-parallelism of the fields.
  In what follows we examine the effect of this on the possible mismatch
 between the electric and magnetic fields in both the frames. We show
 plots of  $(\left|Re\textbf{E}\right|,\epsilon)$ and $\left(|Re\textbf{H}|,
 \eta\right)$ in
 Figs.(\ref{fig:non_parallel_electric_mu_1},\ref{fig:non_parallel_magnetic_mu_1})
 respectively as a function of the scaled longitudinal variable $\chi$ for
 the values of $\xi=0.8$ and $t=0.4 fs$. It is clear from these plots that
 even going away from the focal region makes very little difference in the
 fields in  both the frames.

 It is then natural to examine if one can use the expressions for the magnitude of  electric and magnetic fields in the laboratory frame instead of those for  $\epsilon$ and $\eta$ for calculating number of pairs in Eq.(\ref{N_e_average}) for the counterpropagating laser beams with $\mu = \mp 1$.
  \begin{figure}[h]
  \begin{center}
  \includegraphics[width=75mm]{non_parallel_electric_mu_1.eps}
  \end{center}
  \begin{center}
  \caption{Variation of $|Re\textbf{E}|$, and $\epsilon$ with dimensionless longitudinal variable $\chi$ of the two counterpropagating focused laser beam for $\mu = -1$. Fields are normalized with $E_S$, $E_0 = 0.0565$,  $\xi = 0.8$, $\phi = 0$ and $t=0.4 fs$.}\label{fig:non_parallel_electric_mu_1}
  \end{center}
  \end{figure}
  \begin{figure}[h]
  \begin{center}
  \includegraphics[width=75mm]{non_parallel_magnetic_mu_1.eps}
  \end{center}
  \begin{center}
  \caption{Variation of $|Re\textbf{H}|$, and $\eta$ with dimensionless longitudinal variable $\chi$ of the two counterpropagating focused laser beam for $\mu = -1$. Fields are normalized with $E_S$, $E_0 = 0.0565$,  $\xi = 0.8$, $\phi = 0$ and $t=0.4 fs$.}\label{fig:non_parallel_magnetic_mu_1}
  \end{center}
  \end{figure}

  \begin{table}[ht]
  \caption{$N_{e^+e^-}$ for $\mu = -1$ using ($|Re\textbf{E}|$,$|Re\textbf{H}|$) and ($\epsilon$,$\eta$). Here $\Delta = 0.1$, $\tau = 10fs$, and $\lambda = 1\mu m$.}
  \centering
  \begin{tabular}{c c c }
  \hline\hline
  $I\times 10^{27}{W/cm^2}$ & $N_{e^+e^-}$($|Re\textbf{E}|$, $|Re\textbf{H}|$) & $N_{e^+e^-}$($\epsilon$, $\eta$)  \\ [0.5ex]

  \hline
  $0.2$ & $3.7157$ & $3.5269$  \\
  \hline
  $0.3$ & $2.1308 (4)$ & $2.0135(4)$  \\
  \hline
  $0.4$ & $4.1661(6)$ & $3.9253(6)$ \\
  \hline
  $0.5$ & $1.5907(8)$ & $1.4944(8)$  \\
  \hline
  $0.6$ & $2.4276(9)$ & $2.2782(9)$  \\
  \hline
  $0.7$ & $2.0694(10)$ & $1.9375(10)$  \\
  \hline
  $0.8$ & $1.1857(11)$ & $1.1091(11)$  \\
  \hline
  $0.9$ & $5.158(11)$ & $4.82(11)$  \\
  \hline
  $1$ & $1.7912(12)$ & $1.6723(12)$ \\ [1ex]
  \hline\hline
  \end{tabular}
  \footnote{The numbers in the brackets indicate in powers of $10$.}
  \label{table:N_mu_1_epsilon_ReE}
  \end{table}
 We see that the expressions of the EM fields in both the frame are same
 in the small $\xi$ approximation. So we use this field expressions and
 calculate the number of pairs in both cases which is tabulated in Table
 \ref{table:N_mu_1_epsilon_ReE}. The Table \ref{table:N_mu_1_epsilon_ReE}
 shows the numbers of pairs for $\mu = -1$, using fields
  ($|Re\textbf{E}|$, $|Re\textbf{H}|$) in the place of ($\epsilon$,
   $\eta$) in Eqn.(\ref{N_e_average}) in the 1st. column.  The second column shows the results using ($\epsilon$,
 $\eta$) in Eqn.(\ref{N_e_average}). It is seen that
 the number of pairs are almost same in column 1 and 2.  One immediate ramification of this observation is that  one can work in the laboratory frame for colliding pulses which circularly e- or h-polarized to study the pair production. This would offer enormous simplification for analytical calculation and thus may help in getting physical insight of the underlying process.  \\
 Having discussed the structure of the electromagnetic fields in the focal region and their relationship with the reduced field invariants, we now investigate the spatio-temporal distribution of the created pairs in the focal region. For convenience we define the differential particle distribution with respect to a particular space/time coordinate when $w_{e^-e^+}$ given by Eq. (\ref{w_e_average}) is integrated over all the other coordinates except for the coordinate under the consideration. This obviously gives the derivative of $N_{e^+e^-}$ with respect to that coordinate.  Such a differential particle distribution in spatio-temporal coordinates provides a measure  to know the space-time extension of the pair production in  the focal region. The variation of $dN_{e^+e^-}/{d\xi}$ as a function of   $\xi$ for $\mu = \pm 1$ is shown in Figs. (\ref{fig:dN_vs_xi_mu1_7May2016}-\ref{fig:dN_vs_xi_mu_1_7May2016}). In both the cases, it vanishes for $\xi=0$, starts increasing  with increase in  $\xi$ for small values of $\xi$, attains a maximum value and thereafter decays exponentially with further increase in $\xi$.  The extent  of $dN_{e^+e^-}/{d\xi}$ in the transverse direction $\xi$ can be quantified by the full-width-half-maxima (FWHM) of the respective curves. FWHM is $0.144R$ for $\mu = -1$ and $0.149R$ for $\mu = 1$, where $R$ is the focal radius defined earlier.
\begin{figure}[h]
\begin{center}
\includegraphics[width=75mm]{dN_vs_xi_mu_1_7May2016.eps}
\end{center}
\begin{center}
\caption{ Distribution of $dN_{e^+e^-}/{d\xi}$ as a function of  $\xi$ for  two counterpropagating focused laser beams with $\mu = -1$.  $E_0 = 0.0565E_S$, $\Delta = 0.1$, and $\tau=10fs$.}\label{fig:dN_vs_xi_mu_1_7May2016}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=75mm]{dN_vs_xi_mu1_7May2016.eps}
\end{center}
\begin{center}
\caption{Distribution of $dN_{e^+e^-}/{d\xi}$ as function of  $\xi$ for two counterpropagating focused laser beams with $\mu = 1$. $E_0 = 0.0565E_S$, $\Delta = 0.1$, and $\tau=10fs$.}\label{fig:dN_vs_xi_mu1_7May2016}
\end{center}
\end{figure}
The differential particle distribution $dN_{e^+e^-}/{d\chi}$ as function of $\xi$  for $\mu = \pm 1$ shows spiky behaviour in the subwavelength extension. For $\mu = -1$, as seen in the Fig.(\ref{fig:dN_vs_xi_mu_1_7May2016}) the distribution possesses a prominent  peak in  the central region. There are two small but finite bumps on either side of the central peak. In the Fig.(\ref{fig:dN_chi_mu1_9May2016})  $dN_{e^+e^-}/{d\chi}$ as a function of  $\chi$ shows two peaks located at $\chi =\mp 0.0164$ for $\mu = 1$. The effective longitudinal scale length at which maximum number of particles are created is of the order of $0.0048L \approx 0.076 \mu m$ for $\mu = -1$ and $0.0732\mu m$ for $\mu = 1$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=75mm]{dN_chi_mu_1_9May2016.eps}
\end{center}
\begin{center}
\caption{Distribution of $dN_{e^+e^-}/{d\chi}$ as function of $\chi$ for two counterpropagating focused laser beams having $\mu = -1$. $E_0 = 0.0565E_S$, $\Delta = 0.1$, and $\tau=10fs$.}\label{fig:dN_chi_mu_1_9May2016}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=75mm]{dN_chi_mu1_9May2016.eps}
\end{center}
\begin{center}
\caption{Distribution of $dN_{e^+e^-}/{d\chi}$ as function of $\chi$ for two counterpropagating focused laser beams having $\mu = 1$.  $E_0 = 0.0565E_S$, $\Delta = 0.1$, and $\tau=10fs$.}\label{fig:dN_chi_mu1_9May2016}
\end{center}
\end{figure}
 The differential particle distribution $dN_{e^+e^-}/{d\phi}$ as a function of  $\phi$ for $\mu = \mp 1$ is  shown in Figs. (\ref{fig:dN_vs_phi_mu_1_7May2016}-\ref{fig:dN_vs_phi_mu1_7May2016}). These figures  show  that pair production process does not have the azimuthal symmetry. It has a tendency to produce maximum number of pairs at $\phi = \pi/2$ and $3\pi/2$. FWHM for this distribution  is of the order of $0.2679 \pi$  for $\mu = -1$ and $0.2517 \pi$ for $\mu = 1$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=75mm]{dN_vs_phi_mu_1_7May2016.eps}
\end{center}
\begin{center}
\caption{Distribution of $dN_{e^+e^-}/{d\phi}$ as a function of $\phi$ for two counterpropagating focused laser beam with $\mu = -1$.  $E_0 = 0.0565E_S$, $\Delta = 0.1$, and $\tau=10fs$.}\label{fig:dN_vs_phi_mu_1_7May2016}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=75mm]{dN_vs_phi_mu1_7May2016.eps}
\end{center}
\begin{center}
\caption{Distribution of  $dN_{e^+e^-}/{d\phi}$ as a function of $\phi$ for two counterpropagating focused laser beam with $\mu = 1$. $E_0 = 0.0565E_S$, $\Delta = 0.1$, and $\tau=10fs$.}\label{fig:dN_vs_phi_mu1_7May2016}
\end{center}
\end{figure}
The distribution of $dN_{e^+e^-}/d{(t/\tau)}$ as a function of  $t/\tau$ for $\mu = \mp 1$, is shown in Figs.(\ref{fig:dN_vs_t_mu_1_8May2016}-\ref{fig:dN_vs_t_mu1_8May2016}) which present the differential particle production with time. One can see that the particles are produced over a much shorter time duration compared to the pulse duration of the laser pulses. It is possible to estimate the  bunch duration of electrons/positrons by  FWHM of the respective curves. FWHM is  of the order of $1.4 fs$ for $\mu = -1$ and $1.351 fs$ for $\mu = 1$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=75mm]{dN_vs_t_mu_1_8May2016.eps}
\end{center}
\begin{center}
\caption{Distribution of $dN_{e^+e^-}/d{(t/\tau)}$ as a function of  $t/\tau$ for  two counterpropagating focused laser beams with $\mu = -1$.  $E_0 = 0.0565E_S$, $\Delta = 0.1$, and $\tau=10fs$.}\label{fig:dN_vs_t_mu_1_8May2016}
\end{center}
\end{figure}
\begin{figure}[h]
 \begin{center}
 \includegraphics[width=75mm]{dN_vs_t_mu1_8May2016.eps}
 \end{center}
 \begin{center}
 \caption{Distribution of $dN_{e^+e^-}/d{(t/\tau)}$ as a function of  $t/\tau$ for  two counterpropagating focused laser beams with $\mu = -1$.  $E_0 = 0.0565E_S$, $\Delta = 0.1$, and $\tau=10fs$.}\label{fig:dN_vs_t_mu1_8May2016}
 \end{center}
 \end{figure}
Thus it is found that the pairs are produced over a very limited region of the focal region and over a very short time duration (compared to the pulse duration of the laser beam). The differential particle distribution as a function of  $\chi$ shows a spiky structure. For $\mu = 1$ it shows spike which are located at $\chi = \mp 0.0164$ where $\epsilon$ have peak values. The distribution of the pairs convey that central region is effective for $\mu = -1$ whereas for $\mu = 1$ the two side peak regions are important. The contribution of the other lobes in the longitudinal direction are not significant.\\
\subsection{Fields and particles for $\mu = 0$ beam:}
 Here we present the distribution of the fields in transverse and longitudinal spatial variables, $\xi$, and $\chi$ and consequently discuss the particle production mechanism.
 Figure \ref{fig:E_H_epsilon_eta_xi_mu0_chi0} shows the fields and invariants with $\xi$ for $\mu = 0$. The overall peak value of the field intensity is several order less in compare to the $\mu = \mp 1$ cases. The electric field is $\pi/2$ out of phase with the magnetic field which is also seen in the analytical expression in Eq.(\ref{mu =0 field}).
\begin{figure}[h]
 \begin{center}
 \includegraphics[width=75mm]{E_H_epsilon_eta_xi_mu0_chi0.eps}
 \end{center}
\begin{center}
\caption{Variation of $|Re\textbf{E}|$, $|Re\textbf{H}|$, $\epsilon$, and $\eta$ with dimensionless transverse variable $\xi$ of the two counterpropagating focused laser beam for $\mu =0$. Fields are normalized with $E_S$, $E_0 = 0.0565E_S$, $\chi = 0$, $\phi = 0$ and $t=0$.}\label{fig:E_H_epsilon_eta_xi_mu0_chi0}
\end{center}
\end{figure}
In the Fig.(\ref{fig:epsilon_xi_phi0}) we present the reduced electric field $\epsilon$ with $\xi$ for $\phi = 0$, $\pi/2$, $\pi/4$. It shows very strong dependence on $\phi$ which is basically confirmation with the analytical expression of $\epsilon$ given in Eq.(\ref{eqn:epsilon_mu0}).
\begin{figure}[h]
 \begin{center}
 \includegraphics[width=75mm]{epsilon_xi_phi0.eps}
 \end{center}
\begin{center}
\caption{Variation of $\epsilon$ with dimensionless transverse variable $\xi$ of the two counterpropagating focused laser beam of $\mu =0$ for $\phi = 0$, $\pi/2$, and $\pi/4$. Fields are normalized with $E_S$,$E_0 = 0.0565E_S$, $\chi = 0$, and $t=0$.}\label{fig:epsilon_xi_phi0}
\end{center}
\end{figure}
The distributions of the fields and invariants are shown in Figure \ref{fig:E_H_epsilon_eta_chi_mu0_xi01} with $\chi$ for the $\mu = 0$ beam configuration. Here the resultant field distributions do not have any interference pattern and peak value of the reduced field are several order less.
 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=75mm]{E_H_epsilon_eta_chi_mu0_xi01.eps}
 \end{center}
 \begin{center}
 \caption{Variation of $|Re\textbf{E}|$, $|Re\textbf{H}|$, $\epsilon$, and $\eta$ with dimensionless longitudinal variable $\chi$ of the two counterpropagating focused laser beam for $\mu = 0$. Fields are normalized with $E_S$, $E_0 = 0.0565E_S$, $\xi = 0.01$, and $t=0$.}\label{fig:E_H_epsilon_eta_chi_mu0_xi01}
 \end{center}
 \end{figure}
 Hence the distinguishing property of the field distribution is that the electric and magnetic fields  in  the two frames are quite different - both qualitatively and quantitatively . This non-parallelism of the fields in the lab frame can be visualised by the analytical expression of the cross product of $Re\textbf{E}$ and $Re\textbf{H}$. We calculate the $x$, $y$, and $z$-components of $Re\textbf{E}\times Re\textbf{H}$. However the $x$, $y$-components are negligible small because of the presence of the factor $\xi \Delta$ in their expressions. The most significant feature is contained  in the $z$-component of the cross-product $\textbf{C}$, which takes the form
 \begin{equation}
 C_z = \frac{4E_0^2g^2e^{-\frac{2\xi^2}{1+4\chi^2}}}{(1+4\chi^2)^{2}}[1-\frac{2\xi^2}{(1+4\chi^2)^{1/2}}\cos{\psi}].
 \end{equation}
 The above expression shows that the electric and magnetic fields are almost orthogonal to each other in the lab frame and the parallel portion of the fields goes as $\xi^2$.
 The reduced field invariants are, therefore,   for $\mu = 0$  are much less compared to  the fields in the lab frame. This feature  of the field tells us that there is finite EM field energy flowing out of the focal region. Hence this field configuration is not efficient for the pair production.  We have already  seen that in  the expressions of $\epsilon$ and $\eta$  the leading order terms are $\xi$ dependent. They are shown in Figs.(\ref{fig:E_H_epsilon_eta_xi_mu0_chi0}-\ref{fig:epsilon_xi_phi0}) as a function of $\xi$ for different values of $\phi$ and also as a function $\chi$. One can see that the reduced field invariants are very sensitive to the  azimuthal angle Fig.(\ref{fig:epsilon_xi_phi0}).\\
 The differential particle distribution a function of  $\chi$  shows  significantly smaller peaks  as shown in Fig. (\ref{fig:dN_chi_mu0_9May2016}). The central region shows a dip and there are two peaks on its either side. The extent of the effective region of pairs has increased in comparison  to those of  $\mu = \mp 1$ cases.  FWHM of each peak is $0.0139L$.
 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=75mm]{dN_chi_mu0_9May2016.eps}
 \end{center}
 \begin{center}
 \caption{Distribution of $dN_{e^+e^-}/{d\chi}$  as a function of  $\chi$ for two counterpropagating focused laser beams having $\mu = 0$. Here, integration has been performed over $\xi$, $\phi$,$t$ and $\chi$ is varied as a parameter. The $E_0 = 0.0565E_S$, $\Delta = 0.1$, and $\tau=10fs$.}\label{fig:dN_chi_mu0_9May2016}
 \end{center}
 \end{figure}
 Fig. (\ref{fig:dN_vs_xi_mu07May2016}) depicts the differential particle distribution as a function of  $\xi$. It shows the off-centred peak of the particle generation. The the transverse extent  of the particle distribution given by its  FWHM is $0.1269R$.
 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=75mm]{dN_vs_xi_mu07May2016.eps}
 \end{center}
 \begin{center}
 \caption{$dN_{e^+e^-}/{d\xi}$ as a function of  $\xi$ for two counterpropagating focused laser beam having $\mu =0$. Here, integration has been performed over $\xi$, $\phi$,$t$ and $\chi$ is varied as a parameter. The value of $E_0 = 0.0565E_S$, $\Delta = 0.1$, and $\tau=10fs$.}\label{fig:dN_vs_xi_mu07May2016}
 \end{center}
 \end{figure}
 \begin{figure}[h]
  \begin{center}
  \includegraphics[width=75mm]{dN_vs_phi_mu07May2016.eps}
  \end{center}
  \begin{center}
  \caption{Distribution of $dN_{e^+e^-}/{d\phi}$ as a function of  $\phi$ for two counterpropagating focused laser beams having $\mu =0$. Here, integration has been performed over $\xi$, $\chi$,  $t$ and $\phi$ is varied as a parameter. The value of $E_0 = 0.0565E_S$, $\Delta = 0.1$, and $\tau=10fs$.}\label{fig:dN_vs_phi_mu07May2016}
  \end{center}
  \end{figure}
  In the Fig.( \ref{fig:dN_vs_phi_mu07May2016}), we present the differential particle distribution as a function of  $\phi$ for $\mu = 0$. It shows a strong dependence on $\phi$ which is directly manifested by the reduced fields distribution $\epsilon$ and $\eta$. The peaks are located at $\phi = \pi/2$ and $3\pi/2$ having FWHM of the order of $0.2782$. An interesting feature is observes in Fig. (\ref{fig:dN_vs_t_mu08May2016}) which presents the differential particle distribution as a function of time. The distribution shows a very sharp peak of FWHM $449 as$. This implies that it is possible to generate ultra short duration  particle bunches using this configuration - much shorter than what can be obtained using laser pulses with $\mu = \mp 1$.
  \begin{figure}[h]
   \begin{center}
   \includegraphics[width=75mm]{dN_vs_t_mu08May2016.eps}
   \end{center}
   \begin{center}
   \caption{Distribution of $dN_{e^+e^-}/{t/\tau}$ as a function of  $t/\tau$ for two counterpropagating focused laser beams having $\mu =0$. Here, integration has been performed over $\xi$, $\phi$, $\chi$ and $t$ is varied as a parameter. The value of $E_0 = 0.0565E_S$, $\Delta = 0.1$, and $\tau=10fs$.}\label{fig:dN_vs_t_mu08May2016}
   \end{center}
   \end{figure}

\section{Conclusion}
We have discussed the particle production mechanism via Schwinger mechanism in the superposition of two counterpropagating focused laser beams for $\mu = \mp 1$, and $0$. The complete features of the pairs generations have been explained on the basis of the structure of the electromagnetic fields and their relationship with the invariants and the reduced field invariants. Analytical expressions of the resultant field distribution in both the frames are discussed. These analytical expressions are used to pinpoint why colliding beam configurations with  $\mu = \mp 1$ are particularly efficient for pair production and why that corresponding to  $\mu = 0$ gives much lower number of the pairs. It has been established that the configurations with  $\mu = \mp 1$ yields  electric and magnetic fields which are almost parallel to each other in the focal region. This minimizes the energy flowing out of the focal region and thereby producing maximum number of pairs. Just opposite situation arises for the configuration $\mu = 0$. In this case the resulting electric and magnetic fields are nearly orthogonal to each other and the most of electromagnetic energy flows out of the focal region thereby effecting less number of pairs. While  $\mu = 0 $ configuration is not efficient for pair production, it offers the possibility for generating ultra short bunches of electrons/positrons. The kinematic property of the particles or the momentum distribution \cite{Popov200283} is not discussed here.  There are interesting field models such as 'e -dipole' pulse \cite{PhysRevA.86.2012Gonoskov,PhysRevLett.111.060404Gonoskov} or tightly focused $\Delta (> 1)$ fields \cite{fedotov2009electron} which are quite promising for QED processes. It would be worthwhile extending the analyses discussed here to these models. Some of the outstanding issues will be addressed in future publications..
\section{Acknowledgements}
It is a pleasure to acknowledge helpful discussions with Dr. P. A. Naik and Dr. S. Krishnagopal.
\bibliographystyle{apsrev}

redpajama_set_name: RedPajamaArXiv


text:
Lupus & CTDs
FDA approves Idacio as eighth adalimumab biosimilar in U.S.
A biosimilar drug to the tumor necrosis factor inhibitor adalimumab, marketed as Idacio (adalimumab-aacf), has been approved by the Food and Drug Administration for use in the United States, according to a press release from manufacturer Fresenius Kabi.
U.S. biosimilar competition, use, and availability still lags behind European countries
Idacio is a citrate-free, low-concentration formulation of adalimumab and is now approved for use for all but three of the indications that currently apply to the reference adalimumab product (Humira): rheumatoid arthritis, polyarticular juvenile idiopathic arthritis, psoriatic arthritis in adults, ankylosing spondylitis, Crohn's disease in adults and children aged 6 years or older, ulcerative colitis in adults, and plaque psoriasis in adults. It does not apply to Humira's indications for hidradenitis suppurativa, uveitis, or ulcerative colitis in pediatric patients aged 5 years and older.
Wikimedia Commons/FitzColinGerald/Creative Commons License
Idacio is the eighth adalimumab biosimilar to be approved in the United States. Its approval was based on evidence of a similar profile of pharmacokinetics, safety, efficacy, and immunogenicity to Humira.
Idacio was first launched in 2019 and has been marketed in more than 37 countries worldwide, according to Fresenius Kabi. The U.S. launch is scheduled for July, and Idacio will be available as a self-administered prefilled syringe or prefilled pen.
A version of this article first appeared on Medscape.com.
Skinny-label biosimilars provide substantial savings to Medicare
MDedge Rheumatology
Commentary: COVID vaccines and combination therapy in RA, December 2022
Vaccination cuts long COVID risk for rheumatic disease patients
Randomized trial finds DMARD therapy for RA has a beneficial effect on vascular inflammation, CV risk
RA risk raised by work-related inhaled agents
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text:
using System.Collections.Generic;
using System.Linq;
using Grace.DependencyInjection;
using Grace.Tests.Classes.Simple;
using Xunit;

namespace Grace.Tests.DependencyInjection.Wrappers
{
    public class CompiledWrapperStrategyTests
    {
        [Fact]
        public void ExportWrapper_Non_Generic()
        {
            var container = new DependencyInjectionContainer();

            container.Configure(c =>
            {
                c.Export<MultipleService1>().As<IMultipleService>();
                c.ExportWrapper(typeof(ImportOneInstanceOfMultipleService))
                    .As(typeof(IImportOneInstanceOfMultipleService))
                    .WrappedType(typeof(IMultipleService));

            });

            var instance = container.Locate<IImportOneInstanceOfMultipleService>();

            Assert.NotNull(instance);
            Assert.IsType<MultipleService1>(instance.MultipleService);
        }

        [Fact]
        public void ExportWrapper_Non_Generic_Enumerable()
        {
            var container = new DependencyInjectionContainer();

            container.Configure(c =>
            {
                c.Export<MultipleService1>().As<IMultipleService>();
                c.Export<MultipleService2>().As<IMultipleService>();
                c.Export<MultipleService3>().As<IMultipleService>();
                c.Export<MultipleService4>().As<IMultipleService>();
                c.Export<MultipleService5>().As<IMultipleService>();
                c.ExportWrapper(typeof(ImportOneInstanceOfMultipleService))
                    .As(typeof(IImportOneInstanceOfMultipleService))
                    .WrappedType(typeof(IMultipleService));
            });

            var enumerable = container.Locate<IEnumerable<IImportOneInstanceOfMultipleService>>();

            Assert.NotNull(enumerable);

            var array = enumerable.ToArray();

            Assert.Equal(5, array.Length);
            Assert.IsType<MultipleService1>(array[0].MultipleService);
            Assert.IsType<MultipleService2>(array[1].MultipleService);
            Assert.IsType<MultipleService3>(array[2].MultipleService);
            Assert.IsType<MultipleService4>(array[3].MultipleService);
            Assert.IsType<MultipleService5>(array[4].MultipleService);
        }

        [Fact]
        public void ExportWrapper_Generic()
        {
            var container = new DependencyInjectionContainer();

            container.Configure(c =>
            {
                c.Export<MultipleService1>().As<IMultipleService>();
                c.ExportWrapper(typeof(DependentService<>)).As(typeof(IDependentService<>)).WrappedGenericArg(0);
            });

            var instance = container.Locate<IDependentService<IMultipleService>>();

            Assert.NotNull(instance);
            Assert.IsType<MultipleService1>(instance.Value);
        }

        [Fact]
        public void ExportWrapper_Generic_Enumerable()
        {
            var container = new DependencyInjectionContainer();

            container.Configure(c =>
            {
                c.Export<MultipleService1>().As<IMultipleService>();
                c.Export<MultipleService2>().As<IMultipleService>();
                c.Export<MultipleService3>().As<IMultipleService>();
                c.Export<MultipleService4>().As<IMultipleService>();
                c.Export<MultipleService5>().As<IMultipleService>();
                c.ExportWrapper(typeof(DependentService<>)).As(typeof(IDependentService<>)).WrappedGenericArg(0);
            });

            var instance = container.Locate<IEnumerable<IDependentService<IMultipleService>>>();

            Assert.NotNull(instance);

            var array = instance.ToArray();

            Assert.Equal(5, array.Length);
            Assert.IsType<MultipleService1>(array[0].Value);
            Assert.IsType<MultipleService2>(array[1].Value);
            Assert.IsType<MultipleService3>(array[2].Value);
            Assert.IsType<MultipleService4>(array[3].Value);
            Assert.IsType<MultipleService5>(array[4].Value);

        }

        public class DependentClassA
        {
            public DependentClassA(IDependentService<IBasicService> service)
            {
                Service = service;
            }

            public IDependentService<IBasicService> Service { get; }
        }

        public class DependentClassB
        {
            public DependentClassB(IDependentService<IBasicService> service)
            {
                Service = service;
            }

            public IDependentService<IBasicService> Service { get; }
        }

        [Fact]
        public void ExportWrapper_With_Conditions()
        {
            var container = new DependencyInjectionContainer();

            container.Configure(c =>
            {
                c.ExportWrapper(typeof(DependentService<>))
                    .As(typeof(IDependentService<>))
                    .When.InjectedInto(TypesThat.EndWith("A"));

                c.ExportWrapper(typeof(OtherDependentService<>))
                    .As(typeof(IDependentService<>))
                    .When.InjectedInto(TypesThat.EndWith("B"));

                c.Export<BasicService>().As<IBasicService>();
            });

            var instanceA = container.Locate<DependentClassA>();

            Assert.NotNull(instanceA);
            Assert.NotNull(instanceA.Service);
            Assert.NotNull(instanceA.Service.Value);
            Assert.IsType<DependentService<IBasicService>>(instanceA.Service);

            var instanceB = container.Locate<DependentClassB>();

            Assert.NotNull(instanceB);
            Assert.NotNull(instanceB.Service);
            Assert.NotNull(instanceB.Service.Value);
            Assert.IsType<OtherDependentService<IBasicService>>(instanceB.Service);
        }
    }
}